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Filter Bank Multi-Carrier Modulation

FREDERIC DE PORET

Master’s Degree ProjectStockholm, Sweden 2015

XR-EE-KT 2015:003

Acknowledgements

The work presented in this thesis have been realized with the radio access technologydepartment at Ericsson research and the Communication Theory Department of theElectrical Engineering School of the Royal Institute of Technology from September2014 to March 2015.

First of all I would like to thanks my supervisor, Ather Gattami, for his help andhis encouragements during the thesis work.

I would like to give a big thanks to Lars Rasmussen without whom I would nothave done this thesis.

I thank the Radio Access Technology Department of Ericsson Research at Kista fortheir welcome and all the good time I spend there and particularly Fredrik Lindqvist,Robert Baldemair, Goran Klang, Ning He who help me in my work

Hiou, Ali, Maxym, Naffis, Igor with whom I spend really nice time during the the-sis.

Finally, many thanks to my friends and my family who support me in my life andmy work.

i

Abstract

During the last years, multi-carrier modulations have raised a particular interest dueto their high spectral efficiency and the possible assumption of flat frequency fading.The most broadly used multi-carrier modulation is the CP-OFDM. It allows very sim-ple equalizations methods and MIMO transcoders which keep orthogonality betweencarrier waveforms. However, it does not allow any waveform flexibility and is not fullyspectrally efficient. Due to the low waveform flexibility its performance is quite lim-ited in scenarios like the frequency division multiple accesses. In order to improve itsperformance in this case, it is important to have a multi-carrier modulation with the pos-sibility to have a waveform well localized in time and frequency. Grey analysis showsthat the only way to get both full spectral efficiency and waveforms well localized inboth time and frequency domain is to give up the orthogonality in the complex field.Using filter bank multi-carrier (FBMC) with Offset-QAM (OQAM) is one combinationwhich achieves this task. In this thesis, we study this modulation, how it is possibleto efficiently modulate and demodulate it but also the transcoder (pre-coder, equalizeror both) that can be used when transmitting through multi-tap and MIMO channels.Another modulation, based on FBMC with OQAM, cyclic offset-QAM (COQAM)tries to make a tradeoff between spectral efficiency and simplicity of the equalizationand transcoding methods. In this thesis, FBMC based modulation schemes are testedthrough different scenario: unsynchronized multi-users, unsynchronized uplink, multi-taps channels, SIMO, MISO and MIMO channels. COQAM is tested with some ofthese scenario when it is considered as relevant to test it.

ii

Sammanfattning

Under senare ar har intresset for multi-carrier modulering okat pa grund av dess hogaspektrumeffektivitet under antagandet om slat frekvensfading. Den vanligaste multi-carrier moduleringen i praktiken ar CP-OFDM. Denna modulering tillater anvandningav enkla equalizeringsmetoder och MIMO transkodare vilket behaller ortogonalitetmellan vagformerna. Daremot tillater den inte nagon flexibilitet i val av vagform och arinte heller helt spektraleffektiv. Pa grund av inflexibilitet i val av vagform ar dess pre-standa begransad under till exempel frekvensmultiplexing (FDMA). For att forbattraprestandan hos CP-OFDM ar det viktigt att ha en modulering med mojlighet att envagform begransad i bade tid och frekvens. Grey analys visar att det enda sattet attuppna full spektrumeffektivitet, i kombination med vagformer som ar val begransadei bade tid och frekvens, ar att ge upp ortogonaliteten i komplexa faltet. Filter BankMulti-Carrier (FBMC) med Offset-QAM (OQAM) ar ett alternativ som uppnar detta.I den har avhandlingen studeras just denna modulation, hur det ar mojligt att gora ef-fektiv modulation och demodulation, men ocksa hur dess equalizer, pre-kodare ochtranskoder anvands vid multi-tap och MIMO-kanaler. En annan modulation, baseradpa FBMC med OQAM, ar cyclisk offset-QAM (COQAM). COQAM forsoker goraavvagning mellan spektral effektivitet och enkelhet av utjamnings- och transkodrarmetoder. I den har avhandlingen studeras bade FBMC-baserade modulationer ochCOQAM i en rad olika scenarios: osynkroniserade multianvandare, osynkroniseradupplank, flera vagar kanaler, SIMO, MISO och MIMO-kanaler.

iii

Contents

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Previous work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Project Purpose and Goal . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6 Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Filter Bank Multi-Carrier Modulation with Offset QAM 52.1 FBMC modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Practical implementation algorithms . . . . . . . . . . . . . . 72.1.2 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Application with Multi-Path and MIMO channels . . . . . . . . . . . 122.2.1 Multi-Path Channels . . . . . . . . . . . . . . . . . . . . . . 122.2.2 MIMO channel . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Cyclic Offset-QAM 363.1 The COQAM modulation . . . . . . . . . . . . . . . . . . . . . . . . 36

3.1.1 Practical implementation algorithm . . . . . . . . . . . . . . 373.1.2 Computational complexity . . . . . . . . . . . . . . . . . . . 40

3.2 Application with multi-path and MIMO channels . . . . . . . . . . . 413.2.1 Multi-path channel . . . . . . . . . . . . . . . . . . . . . . . 413.2.2 MIMO channel . . . . . . . . . . . . . . . . . . . . . . . . . 42

4 Simulation and results 454.1 Spectrum leakage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2 Unsynchronized Multi-user scenario . . . . . . . . . . . . . . . . . . 454.3 Performance of the equalizers when the transmitter is not synchronized

with the receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.4 Performance of the equalizers with multi-taps SISO channel . . . . . 484.5 Performance with SIMO channel . . . . . . . . . . . . . . . . . . . . 504.6 Performance with MISO channel . . . . . . . . . . . . . . . . . . . . 554.7 Performance with MIMO channel . . . . . . . . . . . . . . . . . . . 554.8 Summary tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.8.1 Notations in the tables . . . . . . . . . . . . . . . . . . . . . 564.8.2 Standard values of the parameters in the table . . . . . . . . . 564.8.3 Equalizers for FBMC, SISO channels . . . . . . . . . . . . . 604.8.4 Transcoder FBMC, MIMO channel . . . . . . . . . . . . . . 61

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5 Conclusion 625.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

v

List of Figures

2.1 General modulation scheme of the FBMC modulation . . . . . . . . . 62.2 General modulation scheme of the FBMC demodulation in the Hermi-

tian case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 General modulation scheme of the FBMC demodulation in the Eu-

clidean case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Implementation, K = 4 . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 Polyphase network modulation scheme . . . . . . . . . . . . . . . . . 92.6 The Hk filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.7 Global modulation scheme using the polyphase network implementation 102.8 Implementation of the receiver, K = 4 . . . . . . . . . . . . . . . . . 112.9 Polyphase network demodulation scheme . . . . . . . . . . . . . . . 122.10 Comparison of the complexity of the different modulation methods . . 132.11 Transmission scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 132.12 Equalization type 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.13 Equalization type 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.14 Equalization type 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.15 Transmission model . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.16 MIMO transmission model . . . . . . . . . . . . . . . . . . . . . . . 24

3.1 Two first modulation steps . . . . . . . . . . . . . . . . . . . . . . . 393.2 Process to add the cyclic prefix and the window . . . . . . . . . . . . 393.3 Computational complexity . . . . . . . . . . . . . . . . . . . . . . . 41

4.1 Power spectrum density using KM = 4096 . . . . . . . . . . . . . . 464.2 SIR as a function of the sub-channel for an unsynchronized multi-user

scenario, equal power between users . . . . . . . . . . . . . . . . . . 474.3 SIR as a function of the sub-channel for an unsynchronized multi-user

scenario, power of the interferer 104 times more powerful than the mainuser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.4 SINR as a function of the delay between the received signal and thedemodulator for 20dB SNR . . . . . . . . . . . . . . . . . . . . . . . 49

4.5 SINR as a function of the delay between the received signal and thedemodulator for 80dB SNR . . . . . . . . . . . . . . . . . . . . . . . 49

4.6 Capacity as a function of the SNR for an ”epa” channel with differentequalizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.7 SINR as a function of the SNR for an ”epa” channel with differentequalizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.8 Capacity as a function of the SNR for an ”etu” channel with differentequalizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

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4.9 SINR as a function of the SNR for an ”etu” channel with differentequalizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.10 SINR as a function of the sub-carrier for a particular ”etu” channel fordifferent equalizers with 20 dB SNR . . . . . . . . . . . . . . . . . . 53

4.11 SINR as a function of the sub-carrier for a particular ”etu” channel fordifferent equalizers with 80 dB SNR . . . . . . . . . . . . . . . . . . 53

4.12 Capacity as a function of the SNR for SIMO and SISO ”etu” channels 544.13 SINR as a function of the SNR for SIMO and SISO ”etu” channels . . 544.14 Capacity as a function β for a MISO channel (2× 1) for different SNR 564.15 Capacity as a function of the SNR for a MISO channel(2 × 1) and the

SISO channels taken independently . . . . . . . . . . . . . . . . . . 574.16 SINR as a function of the SNR for a MISO channel(2 × 1) and the

SISO channels taken independently . . . . . . . . . . . . . . . . . . 574.17 Capacity as a function of the SNR for MIMO channels with different

equalizers with 2 streams . . . . . . . . . . . . . . . . . . . . . . . . 584.18 SINR as a function of the SNR for MIMO channels with different

equalizers with 2 streams . . . . . . . . . . . . . . . . . . . . . . . . 584.19 Capacity as a function of the SNR for MIMO channels with different

equalizers with 4 streams . . . . . . . . . . . . . . . . . . . . . . . . 594.20 SINR as a function of the SNR for MIMO channels with different

equalizers with 4 streams . . . . . . . . . . . . . . . . . . . . . . . . 59

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Chapter 1

Introduction

1.1 BackgroundFor years, cellular networks have been in great expansion. Today, available bandwidthis a limiting factor. The performance of a radiowave system depends on the modulationused. It determines the symbol density, the power spectrum but also the robustness ofthe channel link. These properties influence the capacity of the transmission for eachuser. A well shaped power spectrum can also allow smaller guard band between userswithout too much interferences enhancement. Shannon and Nyquist have shown thetheoretical limit in therms of capacity of a channel with a limited bandwidth. However,this is a theoretical model which consider asymptotic systems and does not considerthe problem of computation complexity. Multi-carrier modulation schemes allow highspectrum efficiency with reasonable complexity for both modulation and equalization.However, according to the Balian-Low theorem [9], it is not possible to have the or-thogonality in the complex field, flexibility in the pulse shape and to transmit at theNyquist rate at the same time. The existing multi-carrier modulations make choicesbetween these properties. For example, CP-OFDM has the orthogonality in the com-plex field but do not transmit at the Nyquist rate and has no waveform flexibility. Inorder to improve the performances of a system in terms of data rate and frequency divi-sion multiple access, these two last properties are important. Filter bank multi-carrierwith OQAM use the orthogonality in the real field instead of the complex field andkeeps the two other properties.

1.2 Previous workTwo simple ways to generalize the CP-OFDM modulation in order to get more freedomin the waveform are the application of a window (WCP-OFDM) or of a filter (filteredCP-OFDM). Both technics can improve the spectral leakage of the pulse shape buthave a cost in terms of spectral efficiency. Moreover, due to the demodulation used forOFDM, these technics necessitate a filter at the receiver side to suppress the interfer-ence from unsynchronized users. Due to this filtering large guard bands are necessarybetween users. Another approach is the one of FBMC with QAM. In that case morefreedom in the waveform design is possible but it has a cost in terms of spectral effi-ciency. For example, if the waveform used is a root raised cosine with coefficient α,the spectral efficiency will be at most 1

1+α . It is then possible to reduce the guard band

1

between unsynchronized users to 1 sub-band. According to the Balian-Low theorem[9], the only way to have a pulse shape well localized in time and in frequency, seeAppendix Equation 5.3, and to transmit at the Nyquist rate is to reduce the constrainton the orthogonality. FBMC with Offset-QAM uses orthogonality in the real field in-stead of the complex field. It has been first introduced by R. W. Chang [6] and B. R.Saltzberg [14] during the mid-1960s. In 1974, an efficient way to modulate and demod-ulate FBMC has been developed by M. Bellanger [5]. More recently, OFDM modula-tions have been broadly studied. Many simple equalization and pre-coding approachesadapted to CP-OFDM appeared. These techniques result in perfect equalization in theparticular case of CP-OFDM. They can also be used with near perfect equalization withother multi-carrier modulation if the channel can be assumed flat frequency fading foreach sub-band. During the last few year, the necessity to improve the performances ofthe modulation used, mainly in terms of frequency division multiple access, appeared.Studies about FBMC, which appear as one of the solutions, started in order to see itscapabilities in different communication contexts. One of these studies is the PHY-DYAS project ([4], [3], [2]). Some reports from this project present some equalizationmethods for FBMC transmitted through multi-taps channels or MIMO channels. Theperformances of FBMC are always limited because of the intrinsic interference and thenon-perfect orthogonality. In order to suppress these problems new modulations suchas cyclic offset-QAM has been proposed in [13].

1.3 Project Purpose and GoalThe FBMC modulation aims to replace OFDM in some cases. In such context, it isimportant to estimate its performances and when it could improve OFDM or not. Whenit can improve the OFDM modulation, a question in terms of cost of this improvementis raised. The purpose of this thesis is to evaluate this improvement and this cost. Moreprecisely, it is to study FBMC and different equalization models and test them undersimple channel models relevant of a cellular network. Among them the unsynchronizedmulti-user scenario, the unsynchronized uplink scenario, more or less difficult multi-taps channels, MISO, SIMO and MIMO channels. A comparison is done betweenequalizer and precoding and between FBMC and OFDM. By this way, it is possible tocompare the performance and the cost of FBMC and OFDM. Another important fieldof comparison is the channel estimates and the computations necessary to obtain theequalizer.

1.4 OutlineThe rest of the thesis is organized as follow

Chapter 2 presents the FBMC modulation with OQAM, the practical implementa-tions of modulation and the demodulation as well as investigates different linear equal-ization and application to MIMO systems.

Chapter 3 presents a possible evolution of FBMC, the COQAM modulation and itssimple equalizations methods for both multi-taps and MIMO channels

Chapter 4 investigates the performance of both FBMC and COQAM modulations

2

and compare them with OFDM for different properties or scenario: Power spectrumdensity, unsynchronized multi-users, unsynchronized uplink, multi-taps channels, MIMOchannels. There is also a comparison of the different pre-coders and equalizers ofFBMC

Chapter 5 Summarizes the thesis, concludes the work and gives some ideas of pos-sible future work

The appendix presents the inner products (Euclidean and Hermitian one) overL2(C),the Balian-Low theorem, the derivation of the linear minimum mean square error filter,the maximum likelihood and minimum mean square error estimation in the case of asingle tap channel and, finally, the derivation of the orthogonality of the set of wave-forms used for FBMC with OQAM.

1.5 Notations

Table 1.1 Mathematical notationsE Denotes the expectation operatorX X denotes the estimate of XCn×m The space of complex matrices of size n×mOn,m Null matrix of size n×mIn Identity matrix of size n× n<(·) <(x) is the real part of x=(·) =(x) is the imaginary part of x≡ n ≡ m(k) mean that n−m is an integer multiple of kM∗ M∗ is the conjugate transpose matrix of MMᵀ Mᵀ is the transpose matrix of M〈·, ·〉H 〈a, b〉H is the Hermitian product between a and b〈·, ·〉E 〈a, b〉E is the Euclidean product between a and barg(·) arg(x) is the argument of x[[a, b]] [[a, b]] is the set of the integer higher or equal to a and lower or equal to bb·c bxc denotes the largest integer less than or equal to xiR x ∈ iR means real(x) = 0.∗ Linear convolution~ Circular convolutionCN (m,σ2) Complex normal distribution with mean m and variance σ2

δ(t) Distribution of Diracj j is the complex variable: j2 = −1

3

1.6 Acronyms

Table 1.2 AcronymsFBMC Filter bank multi-carrierOFDM Orthogonal frequency division multiplexingQAM Quadrature amplitude modulationOQAM Offset QAMCOQAM Cyclic offset QAMMSE Mean square errorMMSE Minimum mean square errorFFT Fast Fourier transformIFFT Inverse fast Fourier transformSNR Signal to noise ratioSIR Signal to interference ratioSINR Signal to interference-noise ratioSISO Single inputs single outputsSIMO Single input multiple outputsMISO Multiple inputs single inputsMIMO Multiple inputs multiple outputsPPN Polyphase Network MethodFF Refers to the modulation method using a filtering in the frequency domainSVD Singular values decompositionIIR Infinite impulse response filterFIR Finite impulse response filterNPR Near perfect reconstructionPR Perfect reconstruction

4

Chapter 2

Filter Bank Multi-CarrierModulation with Offset QAM

2.1 FBMC modulationThe FBMC modulation is a modulation which multiplexes several discrete time signalsinto one continuous signal. LetM be the number of discrete time signals, T the symboltime spacing, sln[i] the low rate signal for n ∈ [[1,M ]]. Each signal is modulated usinga certain pulse shape. Let hn(t) be the pulse shape for the discrete time signal n. Wedefine

sln(t) =∑i∈Z

sln[i]δ(t− iT )

sh(t) =

M∑n=1

sln(t) ∗ hn(t)

(2.1)

The block scheme of this modulation is presented in Figure 2.1. In order to demod-ulate the signal we should be able to reconstruct all the discrete time signals. Let sln[i]be the estimate of sln[i]. sln[i] is given by

sln[i] = 〈sh(t), hn(t− iT )〉

=

M∑m=1

〈slm(t) ∗ hm(t), hn(t− iT )〉

=

M∑m=1

∑i∈Z

sln[k]〈hm(t− kT ), hn(t− iT )〉

(2.2)

If the Hermitian product is used, sln[i], i ∈ Z is the result of the filtering of sh(t)using the matched filter of hn(t) followed by the sampling at the frequency 1

T . If itis the Euclidean product, we must take the real part after the matched filtering. Inorder to have sln[i] = sln[i], it is important that the different filters satisfy some prop-erties. First of all,∀n ∈ [[1,M ]], hn(t) should satisfy the Nyquist ISI criterion fora sampling frequency fs = 1

T in order to avoid inter-symbols interference. Then∀n,m ∈ [[1,M ]], n 6= m, i ∈ Z, 〈hm(t − iT ), hn(t)〉 = 0 in order to avoid inter-

5

Figure 2.1: General modulation scheme of the FBMC modulation

Figure 2.2: General modulation scheme of the FBMC demodulation in the Hermitiancase

Figure 2.3: General modulation scheme of the FBMC demodulation in the Euclideancase

6

carrier interference. Finally we must have 〈hn(t), hn(t)〉 = 1 for the normalization.We can consider here several categories in the FBMC modulation

• the inner product used is the Hermitian one and the criterions are perfectly satis-fied

• the inner product used is the Euclidean one and the criterions are perfectly satis-fied also called perfect reconstruction (PR)

• the inner product used is the Euclidean one and the criterions are almost satisfied(Usually the absolute value of the inner products are lower than 10−3) also callednear perfect reconstruction (NPR)

OFDM is one particular case of the first category of FBMC. However this categoryis particularly limiting considering the waveform design if we want to reach full spec-tral efficiency. We will only deal with OQAM-FBMC with NPR which is a subcategoryof the third category. This particular modulation is interesting because of the existenceof a computationally efficient implementation algorithm, it is spectrally efficient andallows some freedom in the choice of the filters. According to the Balian-Low theorem[9], it is a necessity to have only the Euclidean orthogonality and not the Hermitianone if we want both spectral efficiency and freedom in the design of the pulse shape inorder to have a good low pass and time limited filter. OQAM-FBMC with NPR, thebank of filters is generated from a filter, called the prototype filter. Let h, h(t), andH(f) be the filter, its time impulse, and frequency response. h must be half Nyquist(which means that h∗hmust satisfy the Nyquist ISI criterion) for a sampling frequencyfs = 1

T , be frequency limited within a bandwidth smaller than 2T , real, and symmetric.

Under these conditions, we can generate the bank of 2M filters

hI,n(t) = h(t)ejn( 2πT t+

π2 )

hQ,n(t) = jhI,n

(t− T

2

)= jh

(t− T

2

)ejn( 2π

T (t−T2 )+π2 )

∀n ∈ [[0,M − 1]]

(2.3)

All these filters are orthogonal in the Euclidean sense (see Appendix). We are in acase of perfect reconstruction. However, limited bandwidth means infinite time length.In order to avoid too much complexity and too much transition time, we need limitedtime filters. Let Th be the length of the prototype filter, we define the overlapping factor,noted K, K = Th

T . Due to this limitation in the time domain, some frequency leakageappears and the Nyquist ISI criterion is not anymore perfectly satisfied. Consequently,intrinsic interference appear. The bigger K, the less powerful are these interference.For example, the PhyDyas filter [4] as an intrinsic SIR of −65dB when K = 4.

2.1.1 Practical implementation algorithmsThe implementation is done in discrete time. The sampling time is Ts = T

M . Thereare several implementation algorithms. One of them consists of the calculation ofthe frequency representation of each block of symbols, the use of the IFFT algorithmand the overlap-sum method. Another algorithm uses the polyphase network method(PPN). That latter algorithm is the most efficient one but is not compatible with someequalization and pre-coding methods.

7

Figure 2.4: Implementation, K = 4

Filtering in the frequency domain

We want to calculate

sh(t) =

M∑n=1

∑i∈Z

slI,n[i]hI,n(t− iT ) + slQ,n[i]hQ,n(t− iT ) (2.4)

One solution is to calculate sI [i](t) =∑Mn=1 s

lI,n[i]hI,n(t − iT ) and sQ[i](t) =∑M

n=1 slQ,n[i]hQ,n(t − iT ) for each i and then sum up over i. The same process is

used to calculate sI [i](t) and sQ[i](t) for each value of i so it will only be describedfor sI [0](t). This signal is non-zero only during the duration of a symbol Th = KMTs. The idea of this algorithm is to modulate the signal in the frequency domain and thenapply the IFFT algorithm. The representation of sI [0](t) in the frequency domain is

SI [0](f) =

M∑n=1

slI,n[0]HI,n(f) (2.5)

As h has a bandwidth smaller than 2T , HI,n(f) has only 2K − 1 non-zero coef-

ficients. One of them is 1. By symmetry HI,n(f) has only K different coefficients.Consequently, this step requires only M(K − 1) real multiplications. Figure 2.4 il-lustrates these summations in the particular case of K = 4. One should notice thatHI,n(f) = ej

π2HI,n+1(f +K).

The polyphase network method

This method has been introduced by Maurice Bellanger [5] and reduce considerablythe computationnal cost of the modulation. The M filters used for the modulation havefor impulse responses

hI,n(t) = h(t)ejn( 2πM t+π

2 ), ∀n ∈ [[0,M − 1]] (2.6)

8

Figure 2.5: Polyphase network modulation scheme

In the Z-domain

HI,n(Z) =

KM−1∑t=0

h(t)ejn( 2πM t+π

2 )Z−t

=

KM−1∑t=0

h(t)ejn( 2πM t+π

2 )Z−t

=

M−1∑k=0

K−1∑t=0

h(tM + k)ejn( 2πM (tM+k)+π

2 )Z−(tM+k)

=

M−1∑k=0

Z−kK−1∑t=0

h(tM + k)ejn(2π(t+ kM )+π

2 )Z−tM

=

M−1∑k=0

(Z−kejn(2π k

M +π2 )

K−1∑t=0

h(tM + k)Z−tM

)

(2.7)

Let Hk(ZM ) be∑K−1t=0 h(tM + k)Z−tM . We get

HI,n(Z) =

M−1∑k=0

(Z−kejn(2π k

M )ejnπ2Hk(ZM )

)(2.8)

It is important to see that only the factor Z−kej(2π knM )ejnπ2 that depends on the fre-

quency. Moreover, ∀k ∈ [[0,M − 1]], the kth term of the summation contains exactlyall the terms of degree u, so that u ≡ k(M), in the variable Z. This particular decom-position shows that the modulation can be decomposed into three steps. The first oneis the pre-coding due to the OQAM modulation (for each frequency n, a multiplicationwith the coefficient ejn

π2 to get alternatively one real and one pure imaginary number).

The second is an IFFT of size M :∑M−1k=0 Z−kej(2π knM ) correspond to the modulation

of the current data at the frequency n over the discrete exponential n. Finally, to cal-culate the output at a delay u so that u ≡ k(M), we apply the filter Hk(ZM ). Thislast filtering can be seen as the filtering of the kth output of the IFFT with the filterHk(Z) before the parallel to serial step. This filter has only K non-zero coefficients.This process and the filters Hk(Z) are represented in Figures 2.5 and 2.6.

However, we have only modulated half of the symbols. The symbols modulatedusing the filters hQ,n have not been considered yet. In fact, the same implementation

9

Figure 2.6: The Hk filter

Figure 2.7: Global modulation scheme using the polyphase network implementation

can be used in that case, it is only important to include the multiplication with j in thepre-coding and a delay of M2 samples. The complete modulation scheme is representedin Figure 2.7

The receiver

Both practical implementations of the transmitter have their corresponding receiver im-plementation. They are built using the same principle as the transmitter. Moreover, thefilter h is real and symmetric. Consequently, its matched filter is also h. Figure 2.8shows the implementation of the decoding for the in-phase component when the filter-ing is done in the frequency domain. For the quadrature components, one should takethe imaginary part instead of the real part of the sum. Figure 2.9 shows the implementa-tion of the decoding for the in-phase component using the polyphase network method.The post-processing step suppresses the phase which has been introduced during thepre-processing and takes the real part of the result. For the quadrature components,one should delay the signal of −M2 samples and takes the imaginary part instead ofthe real part during the post-processing. The received signal has a total delay of KT .One should notice that for both modulations it can be interesting to keep both real and

10

Figure 2.8: Implementation of the receiver, K = 4

imaginary parts. In fact, the imaginary parts when decoding the in-phase component atthe time n correspond to the quadrature component at the time n− 1

2 . It is then easy toget the low rate signals at twice the original rate with a limited complexity cost.

2.1.2 ComplexityThe computation complexity is, here, calculated for of a block of M real symbols interms of real multiplications. No equalization is considered here and the complexity ofthe receiver is the same as the one of the transmitter. The computational complexitiesare calculated for either the receiver of the transmitter.

For the first modulation method

• filtering: (K − 1)×M

• IFFT: 2KM log2(KM)

For the second modulation method

• IFFT: 2M(log2(M)− 1)

• filtering: 4×K ×M

The overall computational complexity in terms of real multiplications for one mod-ulated complex symbol for either the transmitter or the receiver is

• Filtering in the frequency domain: 2K(1 + 2 log2(KM))− 2

11

Figure 2.9: Polyphase network demodulation scheme

• Polyphase network method: 4 log2(M) + 8K − 4

We can compare it to the computational complexity of OFDM: 2 log2(M). Oneshould notice that additional calculations are necessary when the channel is non-ideal.Figure 2.10 represents the evolution of the computational complexity per complex sym-bol in terms of real multiplication for OFDM, FBMC when K = 4 and FBMC forK = 8. We can see that the complexity of the first method is much higher and is muchmore sensible with K than the second one. For example, for K = 4 and M = 2048,the complexity of the first method is 4.86 times the one of OFDM. With the sameparameters, the complexity of the second method is 1.64 times the one of OFDM.

2.2 Application with Multi-Path and MIMO channelsThe interference due to near perfect reconstruction will be considered as a part of thenoise.

2.2.1 Multi-Path ChannelsWe consider a linear and time invariant channel g with additive white Gaussian noise.Let g(t) be the impulse response of this channel. We consider this channel as a timelimited channel with a maximum delay Tg . Let sht , shr and η be the signal sent by thetransmitter, the signal received by the receiver and the noise. The transmission modelwithout equalization is represented Figure 2.11.

shr (t) = g(t) ∗ sht (t) + η (2.9)

There are several ways to equalize the signal. The first possibility is to equal-ize after the demodulation. In that case, the processing including the modulation, thetransmission and the demodulation is a black box. After all these steps, there is inter-symbols interference and crosstalk between channels. The equalization step tries toestimate the original symbols using the outputs of the demodulation. The demodula-tion may process at a higher rate than the original symbol rate. This is illustrated inFigure 2.12.

The second possibility is to equalize shr . In other words, to find a filter w so thatsht (t) = w(t) ∗ shr (t) ≈ sht (t). This is illustrated in Figure 2.13.

12

100

101

102

103

104

0

50

100

150

200

250

300Complexity per complex symbol at the receiver side.

M

Num

ber

of r

eal m

ultip

licat

ion

per

com

plex

sym

bol

FBMC, method 1, K=4FBMC, method 2, K=4FBMC, method 1, K=8FBMC, method 2, K=8OFDM

Figure 2.10: Comparison of the complexity of the different modulation methods

Figure 2.11: Transmission scheme

13

The last possibility is to exchange, for each channel n, the matched filter h∗n(−t)used during the demodulation with a filter which is adapted to the channel. This isillustrated in Figure 2.14.

Equalization after demodulation

Effect of a multi-path channel over the sub-channels. We need to know the struc-ture of the ”black box” which represents the transmission: Structure of the noise, inter-symbols interference and crosstalk between channels. During the post processing, wesuppress the phase which has been introduced during the preprocessing and keep onlythe real or the imaginary part according to the channel we are demodulating (I or Q).As it has been noticed previously, it is possible to keep both real and imaginary partwhen we demodulate each channel. Then, the imaginary part resulting of the demod-ulation of the symbol from the I channel at the time i, correspond to the Q channel atthe time i − 1

2 . Respectively, the real part resulting of the demodulation of the sym-bol from the Q channel at the time i, correspond to the I channel at the time i + 1

2 .The result of the demodulation is then 2M real channels at twice the original rate. Foreach sub-carrier n, i ∈ 1

2Z, the first channel is <(〈sh(t), hI,n(t+ iT )〉H

), the sec-

ond one is =(〈sh(t), hI,n(t+ iT )〉H

). Let slI,n,r and slQ,n,r be these channels. We

will estimate the influence of the channel g over the different sub-channels. We canconsider n = 0 as it is eventually possible to make the transformation of the channelgn(t) = g(t)e2jπt nT and then treat the problem as if we were considering the basebandchannels. As the filter h is frequency limited and the channel is linear and time invari-ant, there is only crosstalk between adjacent channel. Consequently, slI,0 and slQ,0 arefunctions of slI,0,t, s

lQ,0,t, s

lI,1,t, s

lQ,1,t, s

lI,−1,t and slQ,−1,t. Our model is represented

by the scheme in Figure 2.2.1. The filter gI,n,0, gQ,n,0, gN,0 are the channel filterresulting of all the transmission steps. They can be represented as in Figure 2.15.

Let Lc, Lw and Ltot be the maximum of the lengths of the sub-channels, the lengthof the equalizer filters, and Ltot = (Lw+1)

2 + Lc. We define

• GI,n,0 and GQ,n,0, n ∈ −1, 0, 1, the convolution matrices of size (Lw + 1)×Ltot corresponding to the sub-channels gI,n→0 and gQ,n→0

• Γ0 the convolution matrix of size (Lw+1)×(KM+(Lw−1)M2 ) correspondingto gN→0 and η the noise vector (of length KM + (Lw − 1)M2 )

• SI,n,t and SQ,n,t, n ∈ −1, 0, 1 the vectors of length Ltot of the transmittedsignal (SI,n,t =

[sI,n(i− Lc) sI,n(i− Lc + 1) . . . sI,n(i+ Lw−1

2 )])

• SI,r and SQ,r the vectors of length (Lw + 1) of the received signals: SI,t =[sI,0,r(i) sI,0,r(i− 1

2 ) . . . sI,0,r(i+ Lw+12 )

].

The equation of the received signals can be written as

SI,r = <(GI,−1,0SI,−1,t +GQ,−1,0SQ,−1,t +GI,0,0SI,0,t +GQ,0,0SQ,0,t

+GI,1,0SI,1,t +GQ,1,0SQ,1,t + Γ0η)

SQ,r = =(GI,−1,0SI,−1,t +GQ,−1,0SQ,−1,t +GI,0,0SI,0,t +GQ,0,0SQ,0,t

+GI,1,0SI,1,t +GQ,1,0SQ,1,t + Γ0η)

(2.10)

14

Figure 2.12: Equalization type 1



15

Figure 2.15: Transmission model

1-tap equalizer. The goal of this equalizer is to be simple. We consider that foreach frequency the channel introduces only a phase and a gain. Let cn be the complexrepresenting this phase and this gain for the frequency n

sI,n,r(i) = <(cn)<(〈sht (t) + η(t), hn(t− iT )〉)−=(cn)=(〈sht (t) + η(t), hn(t− iT )〉)sQ,n,r(i) = =(cn)<(〈sht (t) + η(t), hn(t− iT )〉) + <(cn)=(〈sht (t) + η(t), hn(t− iT )〉)

(2.11)If i ∈ Z, then <(〈sh(t)+η(t), hi(t− iT )〉) ≈ sI,n,t(i) and =(〈sht (t), hn(t− iT )〉)

is a term of interference so

sI,n,t(i) =1

<(cn)2 + =(cn)2(<(cn)sI,n,r(i) + =(cn)sQ,n,r(i))

= <(

1

cn〈sht (t) + η(t), hn(t− iT )〉

) (2.12)

If n ∈ Z+ 12 ,=(〈sh(t)+η(t), hn(t−iT )〉) ≈ sQ,n,t(i) and<(〈sht (t)+η(t), hn(t−

iT )〉) is a term of interference so

sQ,n,t(i) =1

<(cn)2 + =(cn)2(−=(cn)sI,n,r(i) + <(cn)sQ,n,r(i))

= =(

1

cn〈sht (t) + η(t), hn(t− iT )〉

) (2.13)

MMSE equalizer Let ν be the delay of the equalizer. Here we want to estimatesI,0,t(i + ν) or sQ,0,t(i + ν) based on the vectors SI,r and SQ,r with the MMSE

16

equalizer. Let G0 be the matrix

G0 =[GI,−1,0 GQ,−1,0 GI,0,0 GQ,0,0 GI,1,0 GQ,1,0

](2.14)

The equation of the received signal can be written as

[SI,rSQ,r

]=

[<(G0)=(G0)

]SI,−1,t

SQ,−1,t

SI,0,tSQ,0,tSI,1,tSQ,1,t

+

[<(Γ0) −=(Γ0)=(Γ0) <(Γ0)

] [<(η)=(η)

](2.15)

According to the formula derived in Appendix, when we want to estimate sI,0,t(i+ν), the linear MMSE filter is

WI = e2Ltot+Lc+ν

[<(G0)=(G0)

]([<(G0)=(G0)

] [<(G0)=(G0)

]ᵀ+σ2η

σ2s

[<(Γ0) −=(Γ0)=(Γ0) <(Γ0)

] [<(Γ0) −=(Γ0)=(Γ0) <(Γ0)

]ᵀ)−1

(2.16)

For sQ,0,t(i+ ν), the linear MMSE filter is

WQ = e3Ltot+Lc+ν

[<(G0)=(G0)

]([<(G0)=(G0)

] [<(G0)=(G0)

]ᵀ+σ2η

σ2s

[<(Γ0) −=(Γ0)=(Γ0) <(Γ0)

] [<(Γ0) −=(Γ0)=(Γ0) <(Γ0)

]ᵀ)−1

(2.17)

Our estimates are

sI,0,t(i+ ν) = WI

[SI,rSQ,r

]sQ,0,t(i+ ν) = WQ

[SI,rSQ,r

] (2.18)

Multiband MMSE equalizer In this case, we want to estimate the transmitted sym-bols through the channels at the frequency i using the received symbols correspondingto the channels at the frequency i− 1, i and i+ 1. Let G−1, G0 and G1 be the matrices

G−1 =[GI,−2,−1 GQ,−2,−1 GI,−1,−1 GQ,−1,−1 GI,0,−1 GQ,0,−1 0 0 0 0

]G0 =

[0 0 GI,−1,0 GQ,−1,0 GI,0,0 GQ,0,0 GI,1,0 GQ,1,0 0 0

]G1 =

[0 0 0 0 GI,0,1 GQ,0,1 GI,1,1 GQ,1,1 GI,2,1 GQ,2,1

](2.19)

The received signals can be written as

17

SI,−1,r

SQ,−1,r

SI,0,rSQ,0,rSI,1,rSQ,1,r

=

<(G−1)=(G−1)<(G0)=(G0)<(G1)=(G1)

SI,−2,t

SQ,−2,t

SI,−1,t

SQ,−1,t

SI,0,tSQ,0,tSI,1,tSQ,1,tSI,2,tSQ,2,t

+

<Γ−1) −=(Γ−1)=(Γ−1) <(Γ−1)<(Γ0) −=(Γ0)=(Γ0) <(Γ0)<(Γ−1) −=(Γ−1)=(Γ−1) <(Γ−1)

[<(η)=(η)

]

(2.20)Let Sr, G, St, Γ, η be

Sr =

SI,−1,r

SQ,−1,r

SI,0,rSQ,0,rSI,1,rSQ,1,r

, G =

<(G−1)=(G−1)<(G0)=(G0)<(G1)=(G1)

, St =

SI,−2,t

SQ,−2,t

SI,−1,t

SQ,−1,t

SI,0,tSQ,0,tSI,1,tSQ,1,tSI,2,tSQ,2,t

,

Γ =

<(Γ−1) −=(Γ−1)=(Γ−1) <(Γ−1)<(Γ0) −=(Γ0)=(Γ0) <(Γ0)<(Γ−1) −=(Γ−1)=(Γ−1) <(Γ−1)

, η =

[<(η)=(η)

]

(2.21)

The previous equation can be written as

Sr = G St + Γη (2.22)

When we want to estimate sI,0,t(i+ ν), the multi-band MMSE filter is

WI = e4Ltot+Lc+νG

(G G

ᵀ+σ2η

σ2s

Γ Γᵀ

)−1

(2.23)

The estimate issI,0,t(i+ ν) = WISr (2.24)

When we want to estimate sQ,0,t(i+ ν), the multi-band MMSE filter is

WQ = e5Ltot+Lc+νG

(G G

ᵀ+σ2η

σ2s

Γ Γᵀ

)−1

(2.25)

The estimate issQ,0,t(i+ ν) = WQSr (2.26)

18

Equalization before demodulation

Equalizer based on the zero forcing. We are working at the same rate as the sam-pling rate. The zero forcing filter is an IIR filter and is usually not stable. We proposea filtering based on the zero forcing filter which can be implemented with an accept-able complexity without any stability problem. Let G be the convolution matrix ofthe channel, Sht the vector of the transmitted signal and Shr the vector of the receivedsignal.

Sht =

sht (−Tg)...

sht (0)sht (1)

...sht (N − 1)

, Shr =

shr (0)shr (1)

...shr (N − 1)

G =

gTg gTg−1 . . . g0 0 0 . . . 00 gTg . . . g1 g0 0 . . . 0...

. . . . . ....

0 . . . 0 gTg gTg−1 . . . g0 00 . . . 0 0 gTg . . . g1 g0

(2.27)

The received signals equation is

Shr = GSht + η (2.28)

In order to simplify the problem, we consider that we are in a case of a circularconvolution. This can be justified by the asymptotic equivalency of toeplitz matricesand circulant matrices (see Lemma 4.2, [10]). We can include part of the interferencein the noise. We define Sht , G and η

Sht =

sht (0)sht (1)

...sht (N − 1)

, G =

g0 0 . . . . . . 0 gTg gTg−1 . . . g1

g1 g0 0 . . . 0 0 gTg . . . g2

. . . . . . . . . . . . . . . . . . . . . . . . . . .0 . . . 0 gTg gTg−1 . . . g1 g0 00 . . . . . . 0 gTg gTg−1 . . . g1 g0

η = η +GSht − GSht

(2.29)The modified equation of the received signal is

Shr = GSht + η (2.30)

The matrix G is a circulant matrix. Let WN be the Fourier transform matrix of sizeN . We can write G = W−1

N DGWN with D the diagonal matrix defined as DG =

WN GW−1N . DG is diagonal because of G is a circulant matrix.

Sht = W−1N DGWN S

ht +WN η

D−1

GWNS

hr = WN S

ht +D−1

GWN η

(2.31)

19

D−1

GWNS

hr is an estimate of the Fourier transform of Sht . Let N = KM , the first

demodulation method uses the Fourier transform of Sht of size KM . Consequently, itis possible to apply this equalization during the demodulation using the first algorithm.The term η is in fact composed of two parts. The first one corresponds to the noiseand the intrinsic interference. The second part corresponds to the interference due tothe channel. It is important to study the influence of these interference. Let κ be theseinterference

κ = W−1KMD

−1

GWKM

(GSht − G× Sht

)

= W−1KMD

−1

GWKM

gTg gTg−1 . . . g1

0 gTg . . . g2

.... . . . . .

...0 . . . 0 gTgOKM−Tg,Tg

sht (−Tg)− sht (KM − 1− Tg)sht (1− Tg)− sht (KM − Tg)

...sht (−1)− sht (KM − 1)

(2.32)

We can consider sht (i − Tg) − sht (i + KM − 1 − Tg), i ∈ [0;Tg − 1] as a noise.This noise is white only in the case where the whole bandwidth is used and the powerrepartition is independent of the frequency. This might not be the case but the studyin the case of a white noise gives results which are general enough to be consideredin the non-white case. Considering the white noise case with σ2

s as variance for sht ,sht (i − Tg) − sht (i + KM − 1 − Tg) is zero-mean and has a variance of 2σ2

s . Theseinterference are filtered with the matched filter hn when we are demodulating. Let Gnbe

Gi =

hn(1)gTg hn(1)gTg−1 . . . hn(1)g1

0 hn(2)gTg . . . hn(2)g2

.... . . . . .

...0 . . . 0 hn(Tg)gTg

OKM−Tg,Tg

(2.33)

The interference κn for the sub-channel n are zero-mean and has a variance

Rκnκn =[

1 . . . 1]W−1KMD

−1

GWKMGn G

ᵀnW

−1KMD

−1

GWKM

1...1

(2.34)

When the channel length is short compared to the length of the symbol, the matrixGi is almost equal to zero. Consequently, the interference power is small when the sub-channel gain DG is not too low compared to the gain of the whole channel. However,when the sub-channel gain is small compared to the channel gain, the interference canbe powerful. One should see that, with this equalization, it is mainly the bad sub-channels which suffer from interference and not so much the good sub-channels.

MMSE. In the previous paragraph the filter used was based on the zero-forcing filter.It is possible to use an MMSE filter. Let w, Lp and Lf be the MMSE filter, it length inthe past and in the future (The time 0 correspond to the sample we want to estimate).

20

Here we write

Shr =

shr (k − TLp)...

shr (k + TLf )

, Sht =

sht (k − TLp − Tg)...

sht (k + TLf )

(2.35)

The equalization will be

sht (i) =

Lp∑k=−Lf

wkshr (i− k) (2.36)

The equation of the transmission is

Shr = GSht + η

sht (i) =[wTLp . . . wT−Lf

] shr (k − TLp)...

shr (k + TLf )

(2.37)

The minimum mean square criterion results in the filter[wTLp . . . wT−Lf

]= eTLp+1G (GRssG

∗ +Rηη)−1 (2.38)

Where eTLp+1 is the unit row vector with a 1 in position TLp + 1, Rss is the corre-lation matrix of transmitted signal and Rηη is the correlation matrix of the noise. If weassume a white noise, with zero mean and σ2

η as variance: Rηη = σ2ηITLp+Tg+TLf+1.

The value of Rss depends on the bandwidth which is used. In order to simplify theproblem it is possible to consider Rss = σ2

sITLp+Tg+TLf+1 with σ2s the power of

the signal. By this way, we obtain the MMSE filter. This filter must be long enoughin order to reduce the interference however this filtering may increase too much thecomputational complexity. In order to avoid such problem, it is possible to use filterin the frequency domain when we are applying the demodulation. However, duringthe demodulation we also apply the matched filter which is of length KM . The re-sulting filter (MMSE followed by the matched filter) will have a total length biggerthan KM . Consequently, we should either do a Fourier transform with a bigger lengthor keep the same length and consider new interference. This interference is the termswhich are consequence of the transformation of the linear convolution in a cyclic con-volution. Increasing the length of the Fourier transform might increase too much thecomplexity of the demodulation. Added to this interference, we should remember thatthis filter minimize the mean square error of the main signal and not the one of eachchannel taken independently. This has for consequence channel estimates which haveimportant bias. All these problems are solved with the next estimator.

Equalization during demodulation

Per sub-channel MMSE. In the previous paragraph, the MMSE filter was done inorder to minimize the mean square error between the transmit signal and the received-filtered signal. However this filtering does not consider the particular structure of thesignal and what is the final goal of the processing: the estimation of the symbols whichare transmitted through each sub-channel. It is possible to have an MMSE filter for

21

each sub-channel. Let Gn be the convolution matrix corresponding to the cascadeof the oversampling, the filtering using the pulse shape of the sub-channel n and thechannel impulse response.

shr (t− TLp)...

shr (t+ TLf )

= Gnsln(i) +G

sht (t− TLp − Tg)...

sht (t+ TLf )

−Gnsln(i) + η

sln(i) =[wTLp . . . wT−Lf

] shr (t− TLp)...

shr (t+ TLf )

(2.39)

sln is complex. From it, we can easily extract slI,n or slQ,n using the real or theimaginary part. The calculation of the MMSE gives[

wTLp . . . wT−Lf

]= Gn (GRssG

∗ +Rηη)−1 (2.40)

Here, the filter includes the matched filtering. Consequently, the filter w must be atleast as long as h. A good length for this filter is KM in order to avoid problems whenfiltering in the frequency domain during the demodulation. Let Rsnsn and σsn be thecorrelation matrix and the variance of the signal sln. The residual MSE, noted Ω, is

Ω = Rsnsn(1−Gᵀn(GRsG

ᵀRη)−1GnRsnsn)

= σ2sn(1−Gᵀ

n(σ2s

σ2sn

GGᵀ +σ2η

σ2sn

IKM )−1Gn)

SINR =1

(1−Gᵀn(

σ2s

σ2sn

GGᵀ +σ2η

σ2sn

IKM )−1Gn)

(2.41)

In the case where all the frequencies are used and an equal repartition of the powerover each channel σ2

η = σ2sn and

SINR =1

(1−Gᵀn(GGᵀ + 1

SNRIKM )−1Gn)(2.42)

The calculation can be done in real using the knowledge that the symbol is real. Itwill result two filters, one for the real part of the signal and one for the imaginary part.If we can expect an improvement of the results, it would increase a lot the demodulationcomplexity.

Complexity

The complexity is given in terms of real multiplications for the modulation of 1 com-plex symbol.

• 1-tap

– Demodulation: 4 log2(M) + 8K − 4 (PPN)

– Equalization: 4

– Total complexity: 4 log2(M) + 8K

22

• MMSE


– Equalization: 4× Lw– Total complexity: 4(log2(M) + 8K − 4 + Lw)

• Multiband MMSE


– Equalization: 12× Lw– Total complexity: 4(log2(M) + 8K − 4 + 3Lw)

• Zero Forcing Based Equalizer

– Demodulation: 2K(1 + 2 log2(KM))− 2 (FF)

– Equalization: 4× (2K − 1)

– Total complexity: 2K(5 + 2 log2(KM))− 6

• Per sub-carrier MMSE

– Demodulation: 2K(1 + 2 log2(KM))− 2 (FF)

– Equalization: 4× (2K − 1)

– Total complexity: 2K(5 + 2 log2(KM))− 6

2.2.2 MIMO channelLet Kt, Kr, gu,v be the number of transmitting antenna, number of receiving antennaand the impulse response of the channel from the transmitting antenna v to the receivingantenna u.

SIMO and MIMO without transmitter precoding

With a SIMO configuration, each receiving antenna receives one transmitted signalwhich passed through different channels and with different additive noises. It is possi-ble to use, for each received signal, an equalization method presented in the previoussection. Then, a simple diversity method can be used to obtain our estimate of thesymbols: maximum ratio combining, equal gain combining, selection combining, andso on. However, if the noise is independent from a received signal to another, the inter-ference is not. Consequently, even the maximum ratio combining will be suboptimal.The first approach can be improved by using all the received signals in the equaliza-tion method. This second approach can also be used with a MIMO configuration ifthere is no pre-coding. Let sI,n,r,u and sQ,n,r,u be the signals at the frequency band nreceived by the antenna u. We consider the same transmission model as the one pre-sented in Figure 2.2.1 for each transmitted and received signals. We redefine the filtersgI,n,m,u,v, gI,n,m,u,v and gN,m,u as shown in Figure 2.16.

• SI,m,r,u (respectively SQ,m,r,u) the vector of the received signal at the sub-channel I (respectively Q), at the frequency band m, received by antenna u.

• SI,n,t,v (respectively SQ,n,t,v) the vector of the sent signal at the sub-channel I(respectively Q), at the frequency band n, sent by antenna v.

23

Figure 2.16: MIMO transmission model

• GI,n,m,u,v (respectively GQ,n,m,u,v) the convolution matrix corresponding tothe filter gI,n,m,u,v (respectively gQ,n,m,u,v)

• Γm the convolution matrix corresponding to the filter gN,m

• ηu the noise received by antenna u.

The equations of the received signals are

SI,m,r,u =<

(n+1∑

m=n−1

Kt∑v=1

(GI,n,m,u,vSI,n,t,v +GQ,n,m,u,vSQ,n,t,v) + Γmηu

)

SQ,m,r,u ==

(n+1∑

m=n−1

Kt∑v=1

(GI,n,m,u,vSI,n,t,v +GQ,n,m,u,vSQ,n,t,v) + Γmηu

)∀u ∈ [[1,Kr]]

(2.43)

MMSE equalizer. As in the SISO case, we equalize the baseband sub-channel. Letν and v0 be the delay of the equalizer and the index of the transmitted signal we wantto equalize. We want to estimate sI,0,t,v0(i+ν) or sQ,0,t,v0(i+ν) based on the vectorsSI,0,r,u and SQ,0,r,u, u ∈ [[1,Kr]] with the MMSE equalizer. LetG0,u,v , G0,u, G0, Γ0,Sr,0,u, St,v , St be the matrices

G0,u,v =[GI,−1,0,u,v GQ,−1,0,u,v GI,0,0,u,v GQ,0,0,u,v GI,1,0,u,v GQ,1,0,u,v

]G0,u =

[G0,u,1 G0,u,2 . . . G0,u,Kt

]

G0 =

<(G0,1)=(G0,1)<(G0,2)

...<(G0,Kr )=(G0,Kr )

Γ0 =

[<(Γ0) −=(Γ0)=(Γ0) <(Γ0)

](2.44)

24

Sr,0,u =

[SI,0,r,uSQ,0,r,u

], St,v =

SI,−1,t,v

SQ,−1,t,v

SI,0,t,vSQ,0,t,vSI,1,t,vSQ,1,t,v

St =

St,1St,2

...St,Kt

(2.45)

Sr︷︸︸︷Sr,1Sr,2

...Sr,Kr

= G0St +

Γ0︷︸︸︷Γ0 0 . . . 0

0 Γ0. . .

......

. . . . . . 00 . . . 0 Γ0

η︷︸︸︷

<(η1)=(η1)<(η2)

...<(ηKr )=(ηKr )

(2.46)

Using Sr, Γ0 and η, the previous equation becomes

Sr = G0St + Γ0η (2.47)

Let Rηη be the correlation matrix of the vector η. The MMSE filters for the esti-mation of sI,0,t(i+ ν) is

wI = e2Lw+νG0

(G0G

ᵀ0 +

1

σ2s

Γ0RηηΓᵀ

0

)(2.48)

The MMSE filter for the estimation of sQ,0,t(i+ ν) is

wQ = e3Lw+νG0

(G0G

ᵀ0 +

1

σ2s

Γ0RηηΓᵀ

0

)(2.49)

The residual MSE (here e is either e2Lw+ν or e3Lw+ν) is

MSE = σ2Se

(IN −Gᵀ

0

(G0 G

ᵀ0 +

1

σ2S

Γ0RηηΓᵀ

0

)−1

G0

)eᵀ (2.50)

Multiband MMSE equalizer. This equalizer generalizes the multiband MMSE equal-izer that was done in the previous paragraph. We construct the matrix G as

25

Gᵀ−1,u,v

∆=

GᵀI,−2,−1,u,v

GᵀQ,−2,−1,u,v

GᵀI,−1,−1,u,v

GᵀQ,−1,−1,u,v

GᵀI,0,−1,u,v

GᵀQ,0,−1,u,v

0000

, Gᵀ

0,u,v∆=

00

GᵀI,−1,0,u,v

GᵀQ,−1,0,u,v

GᵀI,0,0,u,v

GᵀQ,0,0,u,v

GᵀI,1,0,u,v

GᵀQ,1,0,u,v

00

, Gᵀ

1,u,v∆=

0000

GᵀI,0,1,u,v

GᵀQ,0,1,u,v

GᵀI,1,1,u,v

GᵀQ,1,1,u,v

GᵀI,2,1,u,v

GᵀQ,2,1,u,v

Gu,v∆=

<(G−1,u,v)=(G−1,u,v)<(G0,u,v)=(G0,u,v)<(G1,u,v)=(G1,u,v)

G∆=

G1,1 G1,2 . . . G1,Kt

G2,1 G2,2 . . . G2,Kt...

.... . .

...GKr,1 GKr,2 . . . GKr,Kt

(2.51)

We construct the vector Sr as

Sr,u∆=

SI,−1,r,u

SQ,−1,r,u

SI,0,r,uSQ,0,r,uSI,1,r,uSQ,1,r,u

Sr∆=

Sr,1Sr,2

...Sr,Kr

(2.52)

We construct the vector St as

26

St,v∆=

SI,−2,t,v

SQ,−2,t,v

SI,−1,t,v

SQ,−1,t,v

SI,0,t,vSQ,0,t,vSI,1,t,vSQ,1,t,vSI,2,t,vSQ,2,t,v

St∆=

St,1St,2

...St,Kt

(2.53)

Let Γ be the matrix

Γ∆=

<(Γ−1) −=(Γ−1)=(Γ−1) <(Γ−1)<(Γ0) −=(Γ0)=(Γ0) <(Γ0)<(Γ−1) −=(Γ−1)=(Γ−1) <(Γ−1)

Γ =

Γ0 0 . . . 0

0 Γ0. . .

......

. . . . . . 00 . . . 0 Γ0

(2.54)

The equation of the received signal is then

Sr = GSt + Γη (2.55)

Let Rηη be the correlation matrix of the vector η. The MMSE filters for the esti-mation of sI,0,t(i+ ν) is

wI = e4Lw+νG

(GGᵀ +

1

σ2s

ΓRηηΓᵀ)

(2.56)

The MMSE filter for the estimation of sQ,0,t(i+ ν) is

wQ = e5Lw+νG

(GGᵀ +

1

σ2s

ΓRηηΓᵀ)

(2.57)

Per sub-carrier MMSE. In this paragraph, gu,v , u ∈ [[1,Kr]], v ∈ [[1,Kt]], arethe channel impulse responses of the channels from the transmitting antenna v to thereceiving antenna u. We assume that all these channels have a finite impulse responseand have the same length Lg . Gu,v , u ∈ [[1,Kr]], v ∈ [[1,Kt]], are the convolutionmatrices of size (KM) × (KM + Lg − 1). Let sht,v and shr,u be the signals sent

27

by transmitter antenna v and the signal received by receiver antenna u. We considerthat we want to equalize the complex symbol sln,1(i) sent by transmitter antenna 1 atfrequency n using the signals shr,u, u ∈ [[1,Kr]] at the times [[t, t+KM − 1]]. We alsoconsider the vector Gu,v0,n corresponding to the multiplication of Gu,v with the vectorof the pulse shape at the frequency n. We define Shr,u and Sht,v

Shr,u =

shr,u(t)...

shr,u(t+KM − 1)

Sht,v =

sht,v(t− Lg)...

sht,v(t+KM − 1)

(2.58)

We can write the equation of the transmission for each receiver as

Shr,u =

Kt∑v=1

Gu,vSht,v + ηu

= Gu,1,nsln,1(i) +

Kt∑v=1

Gu,vSht,v −Gu,1,nsln,1(i) + ηu

(2.59)

We need a model including all the received signals:

Shr︷︸︸︷Shr,1Shr,2

...Shr,Kr

=

G︷︸︸︷G1,1 G1,2 . . . G1,Kt

G2,1 G2,2 . . . G2,Kt...

.... . .

...GKr,1 GKr,2 . . . GKr,Kt

Sht︷︸︸︷Sht,1Sht,2

...Sht,Kt

+

eta︷︸︸︷η1

η2

...ηKr

(2.60)

Using the matrices Shr , G, Sht and η the equation can be written as

Shr = GSht + η (2.61)

The signal we want to extract isG1,1,n

G2,1,n

...GKr,1,n

sln,1(i) (2.62)

Let G1,n be the matrix

G1,n∆=

G1,1,n

G2,1,n

...GKr,1,n

(2.63)

Assuming equi-partition of the power, white noises uncorrelated from one anotherwith equal power and SNR the ratio between the power of the signal by the one of the

28

noise, the MMSE filter is

[w0 w1 . . . wKM

]= G1,n

(GG∗ +

1

SNRIKrKM

)−1

(2.64)

MISO

Even if the SIMO case is similar to the single antenna, when we are dealing withconfigurations with more transmitting antennas than receiving antenna the problem isdifferent. We should do a pre-coding in order to combine in the right way the differentsignals and improve the performance compare to a SISO configuration.

Regularized channel inversion pre-coding. This pre-coding is proposed by [12] butin another contex than FBMC. It is here adapted with the FBMC modulation. We wantto transmit, using the Q and the I channels at a particular frequency n, a symbol sothat at the receiver side we minimize the inter-symbol interference and the inter-carrierinterference. We will assume that we want to transmit one symbol sI,0,s(0) throughthe sub-channel I at frequency 0 at time 0. This result can be easily applied for allthe sub-channels and at all time. The result of the pre-coding is the complex signalst,0(i). We will precode this symbol, for each transmitting channel using the FIRfilters pI,v, pQ,v , v ∈ [1,Kt]. These filters can be non-causal. We want to design pvin order to minimize the mean square error at the receiver side. Let gn,v , v ∈ [[1,Kt]],n ∈ −1, 0, 1, be the channel filter corresponding to the transmission of a data at thefrequency 0 at the antenna v and received at the frequency n. gN,n(i), n ∈ −1, 0, 1be the filter corresponding to the reception of the noise at the frequency n. η be thenoise so that

sln,r(i) =

Kt∑v=1

gn,v(i) ∗ sl0,t,v(i) + gN,n(i) ∗ η(i)

slI,n,r(i) = <(sln,r(i))

slQ,n,r(i) = =(sln,r(i))

sl0,t,v(i) = slI,0,t,v(i) + jslQ,0,t,v(i)

i ∈ 1

2Z

(2.65)

All the filters pv , v ∈ [[1,Kt]], have the same length with a causal part of lengthLp+1 and a non-causal part of length Lf : its impulse response is pv(i) if i ∈ [[−Lf , Lp]]and 0 otherwise. Moreover, we assume that each channel gn,v have a causal part oflength Lg,p + 1 and a non-causal part of length Lg,f . Let

• PI,v and PQ,v be the convolution matrix of the filters pI,v and pQ,v of size (Lf +Lp + 1)× 1

• GI,n,v andGQ,n,v be the convolution matrix of the filter gn,v(i) and gn,v(i+ 12 ),

i ∈ Z of size (Lg,p + Lg,f + Lf + Lp + 1)× (Lf + Lp + 1)

• ΓI,n and ΓQ,n be the convolution matrix of the filter gN,n(i) and gN,n(i + 12 ),

i ∈ Z of size (Lg,p + Lg,f + Lf + Lp + 1)× (Lf + Lp + 1)

• SI,n,r, SQ,n,r be the received data at the frequency n, for the I and theQ channelin the time window [[−(Lf + Lg,f ), Lp + Lg,p]]

29

Using these notations we define

GI∆=

GI,−1,1 GI,−1,2 . . . GI,−1,Kt

GI,0,1 GI,0,2 . . . GI,0,KtGI,1,1 GI,1,2 . . . GI,1,Kt

GQ

∆=

GQ,−1,1 GQ,−1,2 . . . GQ,−1,Kt

GQ,0,1 GQ,0,2 . . . GQ,0,KtGQ,1,1 GQ,1,2 . . . GQ,1,Kt

ΓI

∆=

ΓI,−1

ΓI,0ΓI,1

ΓQ

∆=

ΓQ,−1

ΓQ,0ΓQ,1

(2.66)

Sr︷︸︸︷SI,−1,r

SI,0,rSI,1,rSQ,−1,r

SQ,0,rSQ,1,r

=

G︷︸︸︷[<(GI) −=(GI)=(GQ) <(GQ)

]

P︷︸︸︷

PI,1PI,2

...PI,KtPQ,1PQ,2

...PQ,Kt

sI,0,s(0)+

Γ︷︸︸︷[<(ΓI) −=(ΓI)=(ΓQ) <(ΓQ)

] η︷︸︸︷[<(η)=(η)

]

(2.67)Using Sr, G, P , Γ and η the equation become

Sr = G PsI,0,s(0) + Γ η (2.68)

Let eν be the unit column vector of length L ∆= 6(Lg,p + Lg,f + Lf + Lp + 1)

with a one in position ν ∆= (Lg,p + 2Lg,f + Lf + 2Lp + 2). A possible pre-coding is

the regularized channel inversion which always exists, avoids waste of power in orderto reduce the interference and reduces efficiently the interference

P = Gᵀ(G G

ᵀ+ βIL)−1 · eν (2.69)

The received signal is then

Sr = G Gᵀ(G G

ᵀ+ βIL)−1eν · sI,0,s(0) + Γ η

= eν · sI,0,s(0)− β(G Gᵀ

+ βIL)−1eν · sI,0,s(0) + Γ η

= eν · sI,0,s(0)− (1

βG G

ᵀ+ IL)−1eν · sI,0,s(0) + Γ η

(2.70)

We can see three terms in this expression

• the symbol eν · sI,0,s(0)

• the interference −β(G Gᵀ

+ βIL)−1eν · sI,0,s(0)

30

• the noise Γ η.

If G is full rank, it is possible to choose β = 0. In that case, the interference isequal to zero. However, this may result in a loss of power at the transmitter side if theeigenvalues of G G

ᵀare small. In the general case, the smaller β is, the smaller the

interference is. The decrease of the coefficient β has also for consequence the increaseof the ratio of the transmit power by the received power. Under conditions, it’s possibleto optimize β. If in the previous development, we have used βI as regularizationmatrix, it is possible to use other matrices. For example, if we want to reduce moreinter-carrier interference compared to inter-symbols interference, it’s possible to usethe regularization matrix D

β1 < β0

D =

β1IL6OL

6 ,L6OL

6 ,L6

OL6 ,L6

β0IL6OL

6 ,L6

OL6 ,L6OL

6 ,L6

β1IL6

OL2 ,L2

OL2 ,L2

β1IL6OL

6 ,L6OL

6 ,L6

OL6 ,L6

β0IL6OL

6 ,L6

OL6 ,L6OL

6 ,L6

β1IL6

(2.71)

This can be interesting when using an MMSE equalizer at the receiver as it reducesthe inter-symbols interference but not so much the inter-carrier interference.

Per sub-carrier least square error pre-coding and maximum ratio combining.This pre-coding requires a modulation using a filtering in the frequency domain andcan’t be used with the polyphase modulation method. This consists in the optimiza-tion, for each transmitter, of the modulation filter for a particular frequency so thatthe received signal will be as close as possible to the prototype filter at that frequency.Then, we can do an optimization of the received SINR using maximum ratio combin-ing at the transmitter side. For each transmitter v, we consider the convolution matrixGv , v ∈ [[1,Kt]] of size (KM + Lgv ) × KM of the channel. For each frequencyn ∈ [[0,M − 1]] we consider hn the pulse shape used for that frequency and Hn thevector of this pulse shape followed by Lgv zeroes. We want to find a pulse shape hn,vso that

hn ≈ gv ∗ hn,v (2.72)

Let Hn,v be the vector of the pulse shape. We want hn,v optimal according to theleast square.

Hn ≈ GvHn,v

Hv,n = (G∗vGv)−1G∗vHn

(2.73)

The resulting pulse shape hn,v is, usually, a low pass filter and can be used in themodulation process by multiplying its frequency response with the symbol we want tomodulate according to 2.1.1. As this filter depends on the frequency we are modulating,it is not possible to use the polyphase method. It is then possible to optimize the gainfor each transmitter at each frequency in order to improve the SINR under constrainton the transmission power.

31

MIMO with precoding

It is possible to use a MIMO system without pre-coding as shown in 2.2.2. It canalso be possible to pre-code in order to improve the transmission performance. Onepossibility is to have a similar approach as the one used with OFDM which uses thesingular value decomposition. First of all, we do the same approximation as the onedone in 2.2.1. We assume that all the channels have a length Let sht,v , shr,u be the signalsent by the antenna v and the received signal by the antenna u. We define

• the vectors of the transmitted signals Sht,v of size KM + Tg for all v in [[1,Kt]]

• the vectors of the received signals Shr,u of size KM for all u in [[1,Kr]]

• the vectors of the noise received by each antenna ηu of size KM for all u in[[1,Kr]]

• the convolutions matrices Gu,v , u ∈ [[1,Kr]] and v ∈ [[1,Kt]] of size (KM) ×(KM + Tg) corresponding to the channel gu,v .

For each receiving antenna, the equation of the transmission is

Shr,u =

Kt∑v=1

Gu,vSht,v + ηu (2.74)

As it has been done in 2.2.1, we define the circular convolution matrices Gu,v ofsize (KM) × (KM) the filter gu,v , the vector of the transmitted signals Sht,v of sizeKM and the modified noise vector ηu defined as

ηu =

Kt∑v=1

(Gu,vS

ht,v − Gu,vSht,v

)+ ηu (2.75)

Let W be the matrix of the discrete Fourier transform of size KM . The equationof the transmission can be written as

Shr,u =

Kt∑v=1

Gu,vSht,v + ηu

WShr,u =

Kt∑v=1

WGu,vW−1WSht,v +Wηu

(2.76)

For all u ∈ [[1,Kr]], v ∈ [[1,Kt]], Gu,v is a circulant matrix. Consequently,WGu,vW

−1 is a diagonal matrix. Let Du,v be these matrices. Then, the transmis-sion equation can be written as

WShr,1WShr,2

...WShr,Kr

=

D1,1 . . . D1,Kt...

. . ....

DKr,1 . . . DKr,Kt

WSht,1WSht,2

...WSht,Kt

+

Wη1

Wη2

...WηKt

(2.77)

32

The vectors WShr,u are the Fourier transforms of the vectors Shr,u. Let Shr,u,F bethese vector and shr,u,F (n), n ∈ [[1,KM ]] be the nth coefficient of the vector Shr,u,F .In the same way we define Sht,v,F the vector WSht,v and sht,v,F (m), m ∈ [[1,KM ]] bethe mth coefficient of this vector. Let du,v(n,m) be the coefficient of index (n,m) ofthe matrix Du,v . A consequence of the fact that Du,v is diagonal for all u and v is thepossible decomposition of this equality into KM different equalities

shr,1,F (n)

shr,2,F (n)...

shr,Kr,F (n)

=

D1,1(n) . . . D1,Kt(n)...

. . ....

DKr,1(n) . . . DKr,Kt(n)

sht,1,F (n)

sht,2,F (n)...

sht,Kt,F (n)

∀n ∈ [[1,KM ]]

(2.78)

Let M(n) be the matrix

M(n) =

D1,1(n) . . . D1,Kt(n)...

. . ....

DKr,1(n) . . . DKr,Kt(n)

(2.79)

For all n ∈ [[1,KM ]], it is possible to decompose M(n) into singular values. LetU(n), D(n) and V (n) be this decomposition so that M(n) = U(n)D(n)V ∗(n) withU and V Hermitian matrices (U−1 = U∗ and V −1 = V ∗). The structure of D is:

D =

λ1 0 . . . 0

0. . . . . .

......

. . . . . . 00 . . . 0 λKr

OKt−Kr,Kr

if Kr ≤ Kt

λ1 0 . . . 0

0. . . . . .

......

. . . . . . 00 . . . 0 λKt

OKr−Kt,Kt

if Kr > Kt

λ1 ≥ λ2 ≥ . . . ≥ λmin(Kr,Kt) ≥ 0

(2.80)

We assume that we want to transmit Ks streams through each frequency sub-channel. An intuitive pre-coding would be to calculate, for each time slot and foreach stream its frequency representation of size KM as it is done in 2.1.1. For eachfrequency n ∈ [[1,KM ]], pre-code from the stream frequency representation to thetransmit signal frequency representation using V (n). Then, to apply the IFFT algo-rithm and the overlap-sum as it is done in 2.1.1. However, one should remember thatwe are in the case of near perfect reconstruction which assumes that the pulse shapeused is a low pass filter at the transmitter and the receiver side. A condition in orderto stay in that case is that the resulting pulse shape of each stream at each transmitter

33

for each frequency remain a good low pass filter. It is not possible in general but, if thepre-coding is slow varying from one frequency to another in term of gain and phase,then the transformation due to the pre-coding will not affect too much the property oflow pass filter of the different sub-channels. In the case where the channels gu,v areflat frequency fading the gain of the pre-coding will not vary too much. The phase mayvary more due to the structure of the decomposition. It can be important to force thephase of each stream to be constant for at least one transmitter. This is possible if wedecompose each matrix V (n) into two matrices P (n) and Φ(n):

V (n) =

V1,1 . . . V1,Kt...

. . ....

VKt,1 . . . VKt,Kt

Φ(n) =

ej arg(V1,1) 0 . . . 0

0. . . . . .

......

. . . . . . 00 . . . 0 ej arg(VKt,Kt )

P (n) = Φ∗(n)V (n)

(2.81)

The pre-coding consists, for each frequency, for each time, in the multiplicationwith the matrix P (n) of the spectral representation of the streams at the frequencyn. We apply the IFFT algorithm to the obtained frequency representations and do anoverlap-sum algorithm.

At the receiver side, it is possible to take the frequency representation of each timesymbol and each received signal. Then we equalize per frequency by multiplying bythe matrix

E(n) =

ej arg(V1,1)

λ10 . . . 0

0. . . . . .

......

. . . . . . 0

0 . . . 0 ej arg(VKs,Ks)

λKs

[IKs OKr−Ks,Ks

]U∗

(2.82)Then we are in the frequency domain for each stream. It is possible to apply the

matched filter and demodulate.

Complexity (MIMO systems)

The computational complexity is expressed in terms of number of real multiplicationsfor the modulation of 1 complex symbol.

• MMSE

– Pre-coding: 0

– Modulation: 4 log2(M) + 8K − 4 (PPN)

– Total complexity, transmitter: 4 log2(M) + 8K − 4

– Demodulation: KrKt (4 log2(M) + 8K − 4) (PPN)

– Equalization: 4KrLw

34

– Total complexity, receiver: 4Kr(log2(M)+8K−4

Kt+ Lw)

• Multiband MMSE

– Pre-coding: 0



– Demodulation: KrKt (4 log2(M) + 8K − 4) (PPN)

– Equalization: 12KrLw

– Total complexity, receiver: 4Kr(log2(M)+8K−4

Kt+ 3Lw)

• Per sub-carrier MMSE

– Pre-coding: 0



– Demodulation: KrKt (2K(1 + 2 log2(KM))− 2) (FF)

– Equalization: 4Kr(2K − 1)

– Total complexity, receiver: Kr(2K(5+2 log2(KM))−2

Ks+ 4(2K − 1))

• Pre-coding

– Pre-coding: 4Kt(2K − 1)

– Modulation: KtKs

(2K(1 + 2 log2(KM))− 2) (FF)

– Total complexity, transmitter: Kt(2K(1+2 log2(KM))−2

Ks+ 4(2K − 1))

– Demodulation: KrKs (2K(1 + 2 log2(KM))− 2) (FF)

– Equalization: 4Kr(2K − 1)

– Total complexity, receiver: Kr(2K(5+2 log2(KM))−2

Ks+ 4(2K − 1))

35

Chapter 3

Cyclic Offset-QAM

FBMC modulation have some drawbacks due to intrinsic interferences, non-perfectequalization, complex equalizations methods and complex MIMO applications. Basedon FBMC modulation, the COQAM modulation (resp. CP-COQAM, WCP-COQAM)aims to provide the same equalization simplicity as that of OFDM, with perfect orthog-onality, by keeping the possibility of waveform design of FBMC.

3.1 The COQAM modulationCOQAM is a block based modulation scheme. Each block uses the FBMC modula-tion technique but in a finite time duration case using cyclic filtering instead of linearfiltering. The waveforms are, in the time domain, spread over the whole block. Inthe frequency domain, it is assumed that the prototype filter has a limited bandwidth.All the different blocks have the same structure and are multiplexed in time. Conse-quently, it is only interesting to understand how one block is modulated. We define Mand K such that 2M is the number of sub-carriers and 2K is the number of time sym-bols. Consequently there is a total of 4MK symbols which are modulated per block.OQAM modulation is used. Consequently, the symbols are real-valued and multipliedby either 1 or i. This is done in order to have an alternation between real (elementof R) and pure imaginary (element of iR) elements over frequency and time as it isillustrated in (3.1) where am(k) is the real symbol carried by the frequency m at thetemporal position k. This preprocessing is necessary in order to have orthogonality (inthe Euclidean vectorial space of the functions from R to C). This pre-coding is thesame as the one used for FBMC.

1× a1(1) i× a1(2) 1× a1(3) . . .i× a2(1) 1× a2(2) i× a2(3) . . .1× a3(1) i× a3(2) 1× a3(3) . . .

......

.... . .

(3.1)

We define φm,k so that we can write the resulting value as am(k)eiφm,k .

φm,k =

0 if m+ k is evenπ2 if m+ k is odd (3.2)

We define T and W such that T is the time duration of one block and W is thebandwidth of the signal. We define a prototype filter h(t) of length T whose shifted

36

versions hk,m(t) will carry the different samples am(k)eiφm,k . In the time domain, itis the circular shifting which is used so that hk(t) is defined as

hk(t) =

h(t+ k

2KT)

if t ∈[0, (1− k

2K )T]

h(t+ ( k

2K − 1)T)

if t ∈[(1− k

2K )T, T]

∀k ∈ [[0, 2K − 1]]

(3.3)

We can write hk,m(t) as a frequency shift of hk(t), i.e.

hk,m(t) = hk(t)e2πitmW2M

∀k ∈ [[0, 2K − 1]] m ∈ [[0, 2M − 1]](3.4)

The prototype filter should fulfill the Nyquist ISI criterion for a sampling rate fs =KT , its bandwidth should be less than W

M and its frequency response should be real andsymmetric. A prototype filter of an FBMC signal designed for an oversampling factorof value K can be used. These properties of the prototype filter provide Euclideanorthogonality to the waveforms eiφm,khk,m(t). As the am(k) coefficients are real,it is possible to demodulate them without interference between symbols in time andfrequency.

The modulated block is given by

s(t) =

2M−1∑m=0

2K−1∑k=0

am(k)eiφm,khk,m(t) (3.5)

A cyclic prefix can be added at the beginning of each block. If this cyclic prefixis at least as long as the channel impulse response, the receiver receives a block whichis cyclically convolved with the channel impulse response (once the cyclic prefix isremoved). Consequently the low complexity equalization is similar to that of CP-OFDM. It is also possible to add a window to reduce spectrum leakage.

3.1.1 Practical implementation algorithmTo simplify notations, cm(k) is used instead of am(k)eiφm,k . The implementation isdone in discrete time and frequency. The practical implementation of one block is asfollows.

s(t) =

2M−1∑m=0

2K−1∑k=0

cm(k)hm,k(t)

=

M−1∑l=0

(2K−1∑k=0

c2l(k)h2l,k(t) +

2K−1∑k=0

c2l+1(k)h2l+1,k(t)

)∀t ∈[[0, 2MK − 1]]

(3.6)

The idea of the practical implementation is:

• First, to calculate each sm(t) =∑2K−1k=0 cm(k)hm,k(t) independently in the

frequency domain.

• Then, to sum all the frequency contribution (sum over m ∈ [[0, 2M − 1]]) and tocalculate the signal of the block in the time domain.

• Finally, to complete the block with the cyclic prefix and the window parts.

37

First step LetCm(f) be the result of the FFT algorithm on the vector (cm(0), . . . , cm(2K − 1)).We want to calculate

Sm(f) =

2K−1∑k=0

cm(k)Hm,k(f)

=

2K−1∑k=0

cm(k)Hm,0(f)e−2πi(f+Km)kM

2KM

=Hm,0(f)

2K−1∑k=0

cm(k)e−2πi(f−Km)k

2K

=Hm,0(f)Cm(f −Km)

(3.7)

Hm,k(f) is equal to 0 if f /∈ [[Km−K + 1,Km+K − 1]]. It may be interestingto note that: Hm,0(f) = H0,0(f −Km) = H(f −Km)

Second step We want to calculate s(t) from the signals Sm(f). To do so, we willcalculate S(f) from Sm(f) and then use the IFFT algorithm to get s(t).

The sum can be processed directly however it can be interesting to remember thatSm(f) is non zero only for a few values of f . Sm(f) and Sm+2(f) are respectivelynon-zero for f ∈ [[Km−K+1,Km+K−1]] and f ∈ [[Km+K+1,Km+3K−1]].Consequently, there is no frequency where both of them are non-zero. In order to usethis property, it can be interesting to partition our sum into two sums: SI(f) which isthe sum of Sm(f) for m even and SQ(f) for m odd.

SI(f) = S2l(f)/l =

⌊f

2K− 1

2

⌋SQ(f) = S2l+1(f)/l =

⌊f

2K

⌋ (3.8)

Once we have calculated SI and SQ, we get

S(f) = SI(f) + SQ(f) (3.9)

After this calculation, the IFFT algorithm is used in order to get s(t). These twosteps are summarized in the scheme given in Figure 3.1

Third step Let Lw be the length of the transition time of each side of the windowand LCP be the length of the cyclic prefix. First, we copy the Lw first samples of theblock and add them to the end of the block. At the same time, we copy the Lw + LCPlast samples of the block and add them to the end of the block. We get an extendedblock of length L = LCP + 2Lw + Lblock. Then, we multiply this block with thewindow which is constituted of 3 parts: transition part of length Lw, a constant partof length LCP + Lblock and a transition time of length Lw. During the transmission,the transition parts of a block overlap with the transition parts of the previous and thefollowing blocks. This process is described Figure 3.2

Demodulation The demodulation can be done using the inverse processing as theone presented Figure 3.1. The result of this processing is complex symbols bm(k) =

38

Figure 3.1: Two first modulation steps

Figure 3.2: Process to add the cyclic prefix and the window

39

cm(k) + Im(k). Where Im(k) ∈ R if cm(k) ∈ iR or Im(k) ∈ iR if cm(k) ∈ R. Thenthe estimates of am(k) will be obtained by

am(k) = real(bm(k)e−iφm,k) (3.10)

3.1.2 Computational complexityComplexity at the transmitter side The computation complexity of a block of 4KMreal symbols in terms of complex multiplications is

• First step:

– FFT: K log2(2K)× 2M

– filtering: (2K − 1)× 2M

• Second step:

– IFFT: KM log2(2KM)

• Third step:

– Windowing: 2Lw

For the following discussions, we do not consider the complexity due to the win-dowing as it is neglegible compared to the other computations. The overall complexityfor the modulation of a block of length 2KM where 4KM real symbols are modulatedis KM(4 log2(K) + 8 + log2(M))−M .

Complexity at the receiver side The computation complexity in terms of complexmultiplications is

• First step:

– FFT: KM log2(2KM)

• Equalization (see 3.2.1)

– 2KM

• Second step:

– filtering: (2K − 1)× 2M

– IFFT: K log2(2K)× 2M

The overall computational complexity for the demodulation and the equalization(see 3.2.1) of a block of length 2KM where 4KM real symbols are modulated is:KM(4 log2(K) + 10 + log2(M))−M .

Figure 3.3 represent these complexities as a function of M for the transmitter andfor the receiver for COQAM with K = 4 and K = 8 and for an OFDM modulation.

40

100

101

102

103

104

105

0

5

10

15

20Complexity per complex symbol at the transmitter side.

MNum

ber

of c

ompl

ex m

ultip

licat

ion

per

com

plex

sym

bol

COQAM, K=4COQAM, K=8OFDM

100

101

102

103

104

105

0

5

10

15

20Complexity per complex symbol at the receiver side.

MNum

ber

of c

ompl

exm

ultip

licat

ion

per

com

plex

sym

bol

COQAM, K=4COQAM, K=8OFDM

Figure 3.3: Computational complexity

3.2 Application with multi-path and MIMO channelsThe main goal of this modulation method is to simplify the equalization steps compareto FBMC by having, inside each block, a circular convolution with the channel insteadof a linear convolution. This gives the possibility to equalize in the frequency domain.To transform linear convolutions in circular convolution a cyclic prefix is added to eachblock. Its length will be precise for the different applications.

3.2.1 Multi-path channelHere we consider a non ideal channel of length L+ 1. The cyclic prefix is consideredequal or longer than L. st(t), sr(t) and n(t) are respectively the transmitted signal, thereceived signal and the noise. t is the discrete time. The channel impulse response isrepresented by (c(0), . . . , c(L)). The channel is assumed as perfectly known.

The received signal is then:

sr(t) = st(t) ∗ c(t) + n(t) (3.11)

If we consider only one block, after removing the window and the cyclic prefix, thelinear convolution becomes a circular convolution, we get

sr(t) = st(t) ~ c(t) + n(t) (3.12)

In the frequency domain, we may write (3.12) as

Sr(f) = St(f)C(f) +N(f) (3.13)

41

Either maximum likelihood or minimum mean square error estimators can be used.In this case they are both linear. The maximum likelihood estimator is easier to obtainthan the MMSE as we only need to calculate the frequency response of the channel.In the following parts only this estimator is considered. The equalization is done withdiscrete frequency. It should be done by using a 2KM points FFT. It’s possible toinclude the equalization step in the demodulation once we are in the frequency domainbetween the step one and the step two.

3.2.2 MIMO channelIn this section we consider a MIMO channel with Kt transmitters and Kr receivers.bWe want to transmit through Ks streams (Ks ≤ min(Kt,Kr)). The cyclic prefix isconsidered equal or longer than the delay between the first tap of the fastest channeland last tap of the slowest channel. Its length is noted L + 1. It is then possible torepresent the channel impulse responses by ckr,kt(t) for t ∈ [[0, L]], kr ∈ [[1,Kr]], andkt ∈ [[1,Kt]]. Their corresponding frequency responses will be written as Ckr,kt(f)for f ∈ [[0, 2KM − 1]], kr ∈ [[1,Kr]], and kt ∈ [[1,Kt]]. The channels are assumed tobe perfectly known by the transmitter and the receiver.

Preprocessing and equalization

Like in the previous case we will focus in one particular block once the cyclic prefixhas been removed. It is possible to do all the derivations in the frequency domain.

Rkr (f) =

Kt∑kt=1

Ckr,kt(f)Tkt(f) +Nkr (f) (3.14)

It is possible to adopt the following matrix representation:

R =

R1

...RKr

, T =

T1

...TKt

, N =

N1

...NKr

,C =

C1,1 . . . C1,Kt...

. . ....

CKr,1 . . . CKr,Kt

∀f ∈ [[0,KM − 1]]

(3.15)

One should notice that, even if it is not explicitly written, all the coefficients of theentire matrix are functions of the frequency. The received signal is given by

R(f) = C(f)× T (f) +N(f)

∀f ∈ [[0,KM − 1]](3.16)

The streams are represented in the frequency domain by Sks(f) for f ∈ [[0, 2KM−1]] and ks ∈ [[1,Ks]]. The vector representation of the streams is denoted by

S(f) =

S1

...SKs

∀f ∈ [[0,KM − 1]]

(3.17)

42

Ideally we want to find a preprocessing matrix P (f) ∈ CKt×Ks and an equaliza-tion matrix E(f) ∈ CKs×Kr with the property that

IKs = E(f)× C(f)× P (f) ∀f ∈ [[0,KM − 1]] (3.18)

In order to find these matrices, we consider the decomposition in singular values ofC(f):

C(f) = U(f)×D(f)× V H(f) (3.19)

With U and V hermitian matrices (U−1 = UH and V −1 = V H ). The structure ofD is:

D =

λ1 0 . . . 0

0. . . . . .

......

. . . . . . 00 . . . 0 λKr

OKt−Kr,Kr

if Kr ≤ Kt

λ1 0 . . . 0

0. . . . . .

......

. . . . . . 00 . . . 0 λKt

OKr−Kt,Kt

if Kr > Kt

λ1 ≥ λ2 ≥ . . . ≥ λmin(Kr,Kt) ≥ 0

(3.20)

We choose:

P = V ×

IKs

OKs,Kt−Ks

E =

1λ1

0 . . . 0

0. . . . . .

......

. . . . . . 00 . . . 0 1

λks

× [ IKs OKr−Ks,Ks]× U−1

(3.21)

It is easy to see that:

E(f)× C(f)× P (f) = IKs (3.22)

Finally, if we transmit:T (f) = P (f)× S(f) (3.23)

43

We can easily equalize our received signal:

S(f) = E(f)×R(f)

= S(f) +

N1(f)λ1

...NKs (f)λKs

(3.24)

At the transmitter side, all the processing is done independently for each streamuntil we get the sum of SI and SQ. After this sum, we get the spectral representationof each stream of the current block. Then, we apply the preprocessing for the streamsfor each frequency, and we get the frequency representations of theKt signals we wantto transmit. Finally, we take the IFFT of each of these spectrums and get all the signalswe want to transmit.

At the receiver side, we receive one block through each antenna. First, we calculatethe spectrum of each received signals. Then we equalize in the frequency domain usingE(f). We get the spectrum of each received stream. Then we can continue the normalprocessing with each stream independently.

We can see the absence of interference between different symbols in the time, thefrequency and the space domains. However, one should be careful when choosingKs. If it is too high some of the used channels can show really low gain. Like in theOFDM case, it is possible to optimize frequency and spatial power allocation for thetransmission.

Complexity

At the transmitter side:

• For the FFTs: K log2(2K)× 2M ×Ks

• For the filtering: (2K − 1)× 2M ×Ks

• For the preprocessing (matrix multiplication for each frequency): 2KM ×Kt×Ks

• For the IFFT: KM log2(2KM)×Kt

At the receiver side:

• For the FFT: KM log2(2KM)×Kr

• For the equalization (matrix multiplication for each frequency): 2KM×Ks×Kr

• For the filtering: (2K − 1)× 2M ×Ks

• For the IFFTs: K log2(2K)× 2M ×Ks

44

Chapter 4

Simulation and results

4.1 Spectrum leakageFigure 4.1 represents the power spectrum density of a signal modulated using FBMC,CP-OFDM, WCP-OFDM (with a Bartlett window), CP-COQAM and WCP-COQAMwhen we use half of the bandwidth. We can see that the power spectrum density ofFBMC is far sharper than OFDM or windowed OFDM. It falls down to around−60dBfor the next band and then continues to decrease until almost−120dB. For OFDM, theout-of-band power spectrum still quite high even far from the guard band where the at-tenuation is around−40dB. The results are slightly better for COQAM than for OFDMbut the difference is not relevant. The window of WCP-OFDM and WCP-COQAM re-duces its out-of-band power spectrum density. However, for regular window length, itstills far higher than the out of band power spectrum density of FBMC.

4.2 Unsynchronized Multi-user scenarioOne parameter that can decrease the performance in terms of overall capacity of acellular system is the guard bands which are necessary between unsynchronized users.Due to the sharp spectrum of FBMC, it can be interesting to reduce as much as possiblethis guard band and use the newly available bandwidth. Figures 4.2 and 4.3 representthe SIR as a function of the sub-carrier when we demodulate a one user using thebandwidth ν ∈ [−0.3, 0.3] while another user, unsynchronized with the first one istransmitting over the rest of the bandwidth excepted the guard band of width 2 sub-carriers. The difference between the two figures is the fact that in Figure 4.2 the twousers transmit with similar power while in Figure 4.3 the interfering signal is 104 timesmore powerful than the one we are interested in. Both users use the same modulation,FBMC, OFDM, COQAM and so on. We can see that using the FBMC modulation, theSIR is not particularly affected by the interferer while, using the OFDM or COQAMmodulations, the performance are quite low and the windowing only improve slightlythe performance (≈ 3dB). Once the power of the interferer is huge compare to themain user, the performance of OFDM and COQAM continues to fall down while theSIR using FBMC remain almost the same excepted on the side of the bandwidth. Itcan be important to increase a little bit the guard band when the difference of powerbetween users is important. We can conclude that a guard band of 1 sub-band, betweenusers unsynchronized in time, and of 2 sub-bands, between two users unsynchronized

45

−0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

−60

−40

−20

0

Frequency

Mag

nitu

de (

dB)

Power spectrum density, with constant bloc length: KM=4096 (half of the bandwidth is used, for OFDM, K=1)

OFDMOFDM with bartlett window of length 2*50COQAM K=8 with bartlett window of length 2*50COQAM K=8COQAM K=4 (Phydyas prototype filter)COQAM K=4 (Phydyas prototype filter) with bartlett window of length 2*50FBMC (Phydyas prototype filter)

Figure 4.1: Power spectrum density using KM = 4096

in time and frequency, is enough when using the FBMC modulation. Using OFDMor COQAM, far larger guard band is necessary in order to filter the interfering signal.When using filtered OFDM, windowed OFDM, WCP-COQAM it is still necessary toapply a filter at the receiver side.

4.3 Performance of the equalizers when the transmitteris not synchronized with the receiver

Looking at the performance of FBMC modulation with the unsynchronized multi-userscenario, it can be interesting to be able to demodulate the signal without synchronizingthe receiver with the transmitter. A concrete example of this scenario is multiple mobileunits that transmit to the same base station using adjacent bandwidth. In that case, itcan be interesting to be able to demodulate all the signals with the same demodulator.This is possible in LTE, however, the base station has to synchronize the differentmobile units with its demodulator in order to be in a synchronized uplink case [7].Here consider that we have a simple additive white Gaussian noise channel with asimple delay between the received signal and the demodulator. Figure 4.4 shows theevolution of the SINR as a function of the delay when we have 20dB SNR. Figure4.5 shows the evolution of the SINR as a function of the delay when we have 80dBSNR. 1

T is the symbol rate for each sub-channel. We demodulate a symbol everyT2 . Consequently, for the delay higher than T

4 , it can be interesting to use the nextdemodulated symbol to estimate our current symbol. Then for the three equalizers:zero forcing equalizer, per sub-channel MMSE, 1 tap equalizer, the demodulation usedwas suboptimal for the delays between T

4 and T2 . The curves would be, in the optimal

case, symmetric centered around T4 . We can see that the 1-tap equalizer is not adapted

for such unsynchronized demodulation. For the other equalizers, the delay have alimited influence when the SNR is around 20dB. When the system is limited by theintrinsic interference (SNR = 80dB), we can see that the performance of the MMSEequalizer is quite limited in terms of SINR: when the delay is higher than 0.05T ,

46

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

−40

−20

0

20

40

60

SIR

dB

Frequency(ν)

Interference from an unsynchronized user on a main user in function of the frequency the Power ratio between the user and the interferer is 1

Guard band of 2 subbands

FBMCCOQAM without windowingCOQAM with Hanning window L

w=50

COQAM with Chebyshev window Lw

=50

OFDM without windowing

Figure 4.2: SIR as a function of the sub-channel for an unsynchronized multi-userscenario, equal power between users

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−100

−80

−60

−40

−20

0

20

40

60

SIR

dB

Interference from an unsynchronized user on a main user in function of the frequency the Power ratio between the user and the interferer is 10−4

Guard band of 2 subbands

Frequency(ν)

FBMCCOQAM without windowingCOQAM with Hanning window L

w=50

COQAM with Chebyshev window Lw

=50

OFDM without windowing

Figure 4.3: SIR as a function of the sub-channel for an unsynchronized multi-userscenario, power of the interferer 104 times more powerful than the main user

47

SIR ≈ SINR < 50dB. For the other equalizers, it seems that the influence of thedelay is quite limited and the unsynchronized uplink scenario seems to be possible.For OFDM and COQAM, this sort of scenario are not applicable as the channel plusthe delay must remain inside the cyclic prefix and consequently it is not possible tohave a random delay between the signal and the demodulator.

4.4 Performance of the equalizers with multi-taps SISOchannel

In order to evaluate the performance of the equalizers in the SISO case we simulate asystem considering a sampling frequency of 30.72MHz, 1024 sub-carriers in order tobe in conditions comparable to LTE (CP-OFDM with 30.72MHz sampling frequency,2048 sub-carriers) [7]. Consequently there is a spacing, for FBMC, of 30kHz betweensub-carrier and the main bandwidth of each sub-channel is 60kHz. We simulated us-ing extended typical urban model (”etu”) and the extended pedestrian A model (”epa”)standard channel [1]. These channels are multi-taps channels. The delay and the meanpower of each tap are defined by the model (see Appendix). All taps have classicalDoppler spectrum according to a speed of 5km/h and a carrier at 5Ghz. The resultshave been averaged over 200 channels realizations. Figure 4.6 and 4.7 represent theresults in terms of capacity and SINR as a function of the SNR for an ”epa” channel.Figure 4.8 and 4.9 represent the results in terms of capacity and SINR as a function ofthe SNR for an ”etu” channel. When we calculate the SINR we assume unbiased esti-mate, consequently for low SNR the capacity and the SINR are slightly over-evaluatedwhen we use MMSE equalizers. We will first compare the performance of the differ-ent equalizers for the two channels and then the general results of FBMC with OQAMwith OFDM and COQAM. In the case of the ”epa” channel, we can see that the per-formance of the different equalizers are quite similar excepted for the 1-tap equalizerwhich has quite limited performance. The curve of the SINR converge has expectedto a SIR almost equal to 65dB for all the equalizers excepted the multi-band MMSE:SIR ≈ 68dB and the 1-tap equalizer around SIR ≈ 38dB. The ”etu” channel is a harderchannel. As expected the SINR are lower than for the ”epa” channel. The SIR are alsomuch lower. However, one should remember that FBMC is a multi-carrier modulation.This ”etu” channel creates a lot of inter carrier interference and inter symbol interfer-ence for some sub-channels. It is not the case for all sub-channel. The mean SINR isusually limited by the bad channels. A way to evaluate the general performance of thesystem in general is to estimate the capacity. With the capacity we can see that the 1-tap equalizer still not performing really well. The other equalizers show results whichare quite similar to the ”epa” channel for regular SNR (between 0dB and 50dB). Forhigher SNR we can see that the performance are slightly worse than the ”epa” channel.It is then possible to see that the multi-band MMSE and the per-sub-channel MMSEpresents almost the same performance, then it is the zero-forcing based equalizer andfinally the MMSE equalizer. When we compare the results with the one of CP-OFDMand CP-COQAM, we can see that the SINR is similar in the case of OFDM than FBMCfor low and regular SNR. When the SNR gets higher the performance of OFDM getsbetter than to one of FBMC due to perfect orthogonality after equalization. However,one should remember that the energy bit is higher in the case of CP-OFDM than withFBMC. When comparing the capacity, the one of FBMC using a good equalizer is bet-ter than the one of OFDM except for very high SNR for both channels type. This is due

48

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

5

10

15

20

25Mean SINR after equalization in function of the delay for different equalization methods with regular SNR (20dB)

Delay (×T)

Mea

n S

INR

dB

MMSE multibandMMSEzero forcing based equalizerper subcarrier MMSE1 tap equalizer

Figure 4.4: SINR as a function of the delay between the received signal and the de-modulator for 20dB SNR

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

10

20

30

40

50

60

70

80Mean SINR after equalization in function of the delay for different equalization methods with very high SNR (80dB)

Delay (×T)

Mea

n S

INR

dB

MMSE multibandMMSEzero forcing based equalizerper subcarrier MMSE1 tap equalizer

Figure 4.5: SINR as a function of the delay between the received signal and the de-modulator for 80dB SNR

49

to the cyclic prefix that reduces the spectral efficiency. Finally, when the cyclic prefixis too short compare to the channel, we can see that the results of OFDM are similar tothe one of FBMC using the 1-tap equalizer. COQAM modulation has almost the samebehavior than OFDM. The main difference is the fact that COQAM modulation has abigger block length. This has for consequence a shorter cyclic prefix length relativelyto the block and a larger capacity. The performance of COQAM have been similar toOFDM along all the simulations and it seems that it is not particularly interesting touse COQAM instead of OFDM. Consequently it is not considered anymore in the nextsimulations.

Looking at the structure of the interference, we can see that with the MMSE equal-izer reduces mainly the inter-symbol interference but not so much the inter-carrier in-terference. For the zero forcing based equalizer and the per-subcarrier MMSE, it isthe inter-carrier interference which is well cancelled but not the inter-symbols inter-ference. Finally, the multi-band MMSE reduces inter-symbols interference as well asthe MMSE and reduces also part of the inter-carrier interference but not as much asthe equalizers placed before or during the demodulation. Figure 4.10 and 4.11 showthe SINR as a function of the sub-channel for an ”etu” channel with respectively 20dBand 80dB SNR. We can see that the performance of the different equalizers are quitesimilar when the SNR is 20dB. When the SNR is high compared to the intrinsic SIR,the SINR is almost equal to the SIR. Consequently, we can consider that the Figure4.11 represents the SIR per sub-carrier. The SIR for the MMSE equalizer is betterthan the zero forcing based equalizer and the per-subcarrier MMSE for the bad sub-carriers. However, it is the opposite for the good sub-carrier. The multiband MMSEkeeps the good performance of the MMSE when we have bad channels and increasesits performance when we have good channels.

4.5 Performance with SIMO channelHere, all the simulations the SIMO channels has been simulated using a set of SISO-”etu” or SISO-”epa” channels synchronized together. The results has been averagedover 100 channels. The simulations have been done with 256 sub-carriers of 60kHzbandwidth and 30kHz spacing between carrier. Figures 4.13 and 4.12 show the capacityand the mean SINR as a function of the SNR using the MMSE equalizer and the per-subcarrier MMSE. The results are compared with the ones when a single antenna isused with the MMSE and the multiband MMSE equalizer. In terms of capacity, we cansee that we have a gain of 4−5dB SNR by using two antenna instead of one. When thesystem becomes interference limited, the capacity is similar to the SISO case with themultiband MMSE. The difference of performance between the MMSE equalizer andthe per-subcarrier MMSE in the SIMO scenario is not particularly important. Whenlooking at the mean SINR as a function of the SNR, the performance of the SIMOscenario is far better that the one in the SISO scenario. This is the consequence oftwo things. First, we have two received signals with independent noise carrying thesame information. Consequently, it is possible to have a certain diversity gain. Then,by combining in an intelligent way the two signals, there is almost no more bad sub-channel. Consequently, there is far less sub-channels presenting low SINR. These badsub-channels reduce considerably the mean SINR.

50

0 10 20 30 40 50 60 70 800

5

10

15

20

25Capacity in function of the SNR for different equalization methods for an epa channel

SNRdB

Cap

acity

(B

its/s

/Hz)

MMSE multibandMMSEzero forcing based MMSEper subcarrier MMSE1 tap equalizerOFDM, L

CP=0.07M

OFDM, LCP

=0.25M

COQAM (LCP

=0.25M=0.0625KM)

Figure 4.6: Capacity as a function of the SNR for an ”epa” channel with differentequalizers

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80Mean SINR after equalization in function of the SNR for different equalization methods for an epa channel

SNRdB

Mea

n S

INR

dB

MMSE multibandMMSEzero forcing based MMSEper subcarrier MMSE1 tap equalizerOFDM, L

CP=0.07M

OFDM, LCP

=0.25M

COQAM (LCP

=0.25M=0.0625KM)

Figure 4.7: SINR as a function of the SNR for an ”epa” channel with different equal-izers

51

0 10 20 30 40 50 60 70 800

5

10

15

20

25Capacity in function of the SNR for different equalization methods for an etu channel

SNRdB

Cap

acity

(B

its/s

/Hz)

MMSE multibandMMSEzero forcing based equalizerper subcarrier MMSE1 tap equalizerOFDM (L

CP=0.07M < L

g)

OFDM (LCP

=0.15M = Lg)

OFDM (LCP

=0.25M > Lg)

COQAM (LCP

=0.25M=0.0625KM>Lg)

Figure 4.8: Capacity as a function of the SNR for an ”etu” channel with differentequalizers

0 10 20 30 40 50 60 70 80

0

10

20

30

40

50

60

70

80Mean SINR after equalization in function of the SNR for different equalization methods for an etu channel

SNRdB

Mea

n S

INR

dB

MMSE multibandMMSEzero forcing based MMSEper subcarrier MMSE1 tap equalizerOFDM (L

CP=0.07M<L

g)

OFDM (LCP

=0.15M=Lg)

OFDM (LCP

=0.25M > Lg)

COQAM (LCP

=0.25M=0.0625KM > Lg)

Figure 4.9: SINR as a function of the SNR for an ”etu” channel with different equaliz-ers

52

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−25

−20

−15

−10

−5

0

5

10

15

20

25

30

SIN

RdB

SINR per frequency for regular SNR (20dB) with different equalizers ("etu" channel)

Frequency(ν)

Zero−Forcing based equalizerMMSE equalizer1−tapPer subchannel MMSEMultiband−MMSECP−OFDM (L

CP=0.25M>L

g)

Figure 4.10: SINR as a function of the sub-carrier for a particular ”etu” channel fordifferent equalizers with 20 dB SNR

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

−40

−20

0

20

40

60

80

SIN

RdB

SINR per frequency for very high SNR (80dB) for different equalizer ("etu" channel)

Frequency(ν)

Zero forcing based equalizerMMSE equalizer1−tapPer subchannel MMSEMultiband MMSECP−OFDM (L

CP=0.25M>L

g)

Figure 4.11: SINR as a function of the sub-carrier for a particular ”etu” channel fordifferent equalizers with 80 dB SNR

53

0 10 20 30 40 50 60 70 800

5

10

15

20

25

Capacity as a function of the SNR for different equalization methods for SIMO channel(1×2 made with etu channels)

SNRdB

Cap

acity

(B

its/s

ampl

e)

MMSE Equalizerper subcarrier MMSE

Figure 4.12: Capacity as a function of the SNR for SIMO and SISO ”etu” channels

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

SINR as a function of the SNR for different equalization methods for SIMO channel(1×2 made with etu channels)

SNRdB

Mea

n S

INR

dB

MMSE Equalizerper subcarrier MMSE

Figure 4.13: SINR as a function of the SNR for SIMO and SISO ”etu” channels

54

4.6 Performance with MISO channelThe simulations have been done with 256 sub-carriers of 60kHz bandwidth and 30kHzspacing between carrier. For the MISO scenario, only the regularized channel inversionhas been tested. Using this pre-coding, theoretically no equalization is necessary. In areal context, a 1-tap equalizer should be used to avoid problem concerning the error inphase and gain as the transmitter cannot have a perfect knowledge of the channel. Firstwe will see the influence of the parameter β, then we will compare the performancesusing this method instead of only one transmitting antenna. This pre-coding has aparameter β. The performance of the equalizer depends mainly on this coefficient. Thesmaller β is, the less interference the received signal has but the more power is used atthe transmitter side. If β is too small, the system performance may decrease. Figure4.14 shows the capacity of the received signal as a function of beta for different valuesof SNR. One can see that, for low SNR, as expected, the capacity present a maximumas a function of the coefficient β. For example, for SNR = 0dB, it is possible toimprove the performance by 50% by optimizing the coefficient β. For relatively highSNR, the SINR collapse if β is too high, but is almost stable if β is too low. Theperformance of this pre-coding compare to the use of only one of the two channelsused alone is presented in the Figures 4.15 and 4.16. For this simulation, we have onlygenerated two ”etu” channels. We can see the performance of the system when bothof them are used at the same time, using the regularized inversion pre-coding, with theoptimal value of β in terms of capacity. The two other curves correspond to each ofthe channel taken independently using an MMSE equalizer at the receiver side. Onecan see that the performance, in terms of capacity, is slightly better in the MISO casethan when only one channel is used. One should remember that the SNR is the ratiobetween the power of the received signal and the power of the noise and not the powerof the transmit signal divided by the power of the noise. Consequently it is normalthat the gain from SISO to MISO is small. A more important gain could be seen if thecapacity was represented as a function of the ratio energy bit per N0. When estimatingthe mean SINR, the MISO performance is far better that the two SISO case. This is theconsequence of the possibility to use diversity. Consequently, the system avoids verylow SINR for some sub-channels which improve considerably the overall mean SINR.

4.7 Performance with MIMO channelIn this section, we will compare the different MIMO approach when we transmit 2 and4 streams. The scenarios using 2 streams and the one with 4 streams will be comparedindependently. Inside of each of these two sets of scenarios we will compare all at once.All the simulations the MIMO channels has been simulated using a set of SISO-”etu”or SISO-”epa” channels synchronized together. The results has been averaged over 50channels. The simulations have been done with 256 with the same sub-carriers band-width (60kHz) and the same spacing between sub-carrier (30kHz) as the single antennacase. These parameters have been chosen in order to be able to compare the results be-tween MIMO and SISO. Figures 4.17 and 4.18 represent the performance in terms ofcapacity and SINR of different MIMO systems using two streams. The different sce-narios used for these simulations are 2×2 ”epa” channels, 2×2 ”etu” channels and 2×3”etu” channels. For each of these scenarios, three different MIMO systems have beensimulated: the MMSE equalizer without pre-coding, the per sub-channel MMSE with-out pre-coding and the method using pre-coding. The mean SINR is not particularly

55

10−10

10−8

10−6

10−4

10−2

100

102

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5Capacity in function of the regularisation coefficient dor different SNR ("etu" channel)

β

Cap

acity

(bi

ts/s

/Hz)

SNR=0dBSNR=5dBSNR=10dBSNR=15dB

Figure 4.14: Capacity as a function β for a MISO channel (2× 1) for different SNR

relevant of the performance of the system. Consequently, we only comment the resultsin terms of capacity. Dealing with 2 × 2 channels, either ”etu” or ”epa”, the resultsare quite similar for all the system for SNR below 40dB. For higher SNR, one can seethat the performance using the pre-coding are better than the per sub-channel equalizerapproach which is better than the MMSE equalizer approach. Compare to the scenariousing 2 receiving antennas, the scenario using 3 receiving antenna improve the perfor-mance by ≈ 3dB for low SNR. The performances of the three equalizers are similar.For high SNR, it is the MMSE equalizer which is under-performing. The two otherspresent similar results, limited by the intrinsic interferences. In order to see the evo-lutions of the performance for more complicated scenario we have simulated MIMOchannels using 4 streams. Without pre-coding only the case with 4 × 8 antennas hasbeen simulated. With pre-coding 4× 8 and 8× 8 antennas has been tested. The resultsin terms of capacity and SINR are presented Figure 4.19 and 4.20. The differences ofperformance when using the ”etu” channels compare to the ”epa” channels are quitesmall. The three systems have almost the same performance. When 8 transmittingantennas are used with only 4 streams, which is only possible with the transcoder withprecoding, the system is performing better than with only 4 transmitting antenna.

4.8 Summary tables

4.8.1 Notations in the tables

4.8.2 Standard values of the parameters in the table• M = 1024 or 2048

• K = 4

• Lw ∈ [[3, 11]]. In the simulations we used Lw = 11 which gave us asymptoticresults. Reasonable values for Lw are 3 or 5.

56

0 10 20 30 40 50 60 70 800

5

10

15

20

25Capacity in function of the SNR for MISO and SISO "etu" channel

SNRdB

Cap

acity

(bi

ts/s

/Hz)

MISO with precodingSISO using channel 1 aloneSISO using channel 2 alone

Figure 4.15: Capacity as a function of the SNR for a MISO channel(2 × 1) and theSISO channels taken independently

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70SINR in function of the SNR for MISO and SISO "etu" channel

SNRdB

SIN

RdB

MISO with precodingSISO using channel 1 aloneSISO using channel 2 alone

Figure 4.16: SINR as a function of the SNR for a MISO channel(2× 1) and the SISOchannels taken independently

57

0 10 20 30 40 50 60 70 800

5

10

15

20

25

30

35

40

45

50Capacity in function of the SNR for different equalization methods for MIMO channels

SNRdB

Cap

acity

(B

its/s

/Hz)

MMSE Equalizer 2×2, epa channelsper subcarrier MMSE 2×2, epa channelsMIMO with precoding 2×2, epa channelsMMSE Equalizer 2×2, etu channelper subcarrier MMSE 2×2, etu channelMIMO with precoding 2×2, etu channelsMMSE Equalizer 2×3, etu channelper subcarrier MMSE 2×3, etu channelMIMO with precoding 2×3, etu channels

Figure 4.17: Capacity as a function of the SNR for MIMO channels with differentequalizers with 2 streams

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80Mean SINR after equalization in function of the SNR for different equalization methods for MIMO channels

SNRdB

Mea

n S

INR

dB

MMSE Equalizer 2×2, epa channelper subcarrier MMSE 2×2, epa channelMIMO with precoding 2×2, epa channelsMMSE Equalizer 2×2, etu channelper subcarrier MMSE 2×2, etu channelMIMO with precoding 2×2, etu channelsMMSE Equalizer 2×3, etu channelper subcarrier MMSE 2×3, etu channelMIMO with precoding 2×3, etu channels

Figure 4.18: SINR as a function of the SNR for MIMO channels with different equal-izers with 2 streams

58

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

100Capacity in function of the SNR for different equalization methods for MIMO channels

SNRdB

Cap

acity

(B

its/s

/Hz)

MMSE Equalizer 4×8, epa channelsper subcarrier MMSE 4×8, epa channelsMIMO with precoding 4×8, epa channelsMIMO with precoding 8×8, epa channelsMMSE Equalizer 4×8, etu channelper subcarrier MMSE 4×8, etu channelMIMO with precoding 4×8, etu channelsMIMO with precoding 8×8, etu channels

Figure 4.19: Capacity as a function of the SNR for MIMO channels with differentequalizers with 4 streams

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80Mean SINR after equalization in function of the SNR for different equalization methods for MIMO channels

SNRdB

Mea

n S

INR

dB

MMSE Equalizer 4×8, epa channelper subcarrier MMSE 4×8, epa channelMIMO with precoding 4×8, epa channelsMIMO with precoding 8×8, epa channelsMMSE Equalizer 4×8, etu channelper subcarrier MMSE 4×8, etu channelMIMO with precoding 4×8, etu channelsMIMO with precoding 8×8, etu channels

Figure 4.20: SINR as a function of the SNR for MIMO channels with different equal-izers with 4 streams

59

Table 4.1 Acronyms used in the summary tablesPPN Polyphase network methodFF Frequency filteringK Oversampling factorM Number of sub-carriersLc Length of the channelLw Length of the filterKt Number of transmitter antennasKr Number of receiver antennasKs Number of streamsSVD Singular values decomposition

• Ts ≈ 32.55ns (sampling time of LTE)

• Lc < 200

• Kr ∈ [[1, 8]]

• Kt ∈ [[1, 8]]

• Ks ∈ [[1, 8]]

4.8.3 Equalizers for FBMC, SISO channelsThe complexity are given it terms of real multiplication for the modulation of 1 com-plex symbol.

1-tap Equal-izer

MMSEEqualizer

MultibandMMSE

Zero ForcingBased Equal-izer

Per subcarrierMMSE

Demodulator PPN PPN PPN FF FFReceiver com-plexity

4 log2(M) +8K

4(log2(M) +8K−4+Lw)

4(log2(M) +8K − 4 +3Lw)

2K(5 +2 log2(KM))−6

2K(5 +2 log2(KM))−6

Performancechannel

Low Medium Very good Good Very good

Unsynchronizeduplink

Not accept-able

AcceptableSIR> 40dB

Very good Very good Very good

Estimate needed Channelfrequencyresponse (Mtaps)

All sub-channels andcross-talkchannels

All sub-channels andcross-talkchannels

Channelfrequency re-sponse (KMtaps)

Channelimpulseresponse

Computation ofthe estimator

M scalar in-versions

M inversionsof matrices ofsize 6(2K −1 + Lw)

M inversionsof matrices ofsize 10(2K −1 + Lw)

KM scalarinversions

Inversion of amatrix of sizeKM +Lc−1

Time over whichthe channelis consideredtime-invariant

KMTs (Lw − 1 +K)MTs

(Lw − 1 +K)MTs

KM KM

60

4.8.4 Transcoder FBMC, MIMO channelThe complexity are given it terms of real multiplication for the modulation of 1 com-plex symbol.

MMSE Equal-izer

Multiband-MMSE

Per-subcarier-MMSE

Precoding

Modulation PPN PPN PPN FFDemodulation PPN PPN FF FFComplexity transmit-ter

4 log2(M) +8K − 4

4 log2(M) +8K − 4

4 log2(M) +8K − 4

KtKs

(2K(1 +2 log2(KM))−2 + 4Ks(2K −1))

Complexity receiver 4KrKt (log2(M)+8K − 4 +KtLw)

4KrKt (log2(M)+8K − 4 +3KtLw)

KrKs

(2K(5 +2 log2(KM))−2 + 4Ks(2K −1))

KrKs

(2K(5 +2 log2(KM))−2 + 4Ks(2K −1))

Performance Good Not tested. Re-sults expected:better than oth-ers

Good Medium-good

Estimate needed All sub-channels andcrosstalk chan-nels (total of6KtKrM realchannels)

All sub-channels andcrosstalk chan-nels (total of6KtKrM realchannels)

KtKr chan-nels impulseresponse

KtKr channelsfrequency re-sponse (KMtaps)

Computation of theestimator

Inversion ofKtM matri-ces of size6Kt(2K − 1 +Lw)

Inversion ofKtM matri-ces of size10Kt(2K −1 + Lw)

Inversion ofone matrix ofsizeKt(KM+Lc − 1)

SVD of KMmatrices of sizeKr ×Kt

Constraints Kr ≥ Kt,Ks = Kt

Kr ≥ Kt,Ks = Kt

Kr ≥ Kt,Ks = Kt

No

61

Chapter 5

Conclusion

5.1 ConclusionIn this thesis, we have seen how well the FBMC modulation with OQAM keeps itspromises in terms of power spectrum and the possibility of unsynchronized multi-userswith very limited guard band. However, due to the non-perfect orthogonality betweensymbols, its performances are limited by the intrinsic interferences. When the channelsis not perfect, these interferences gets worse and simple equalizer, like the one usedwith CP-OFDM, cannot guaranty good performances for standard SNR ([0, 35]dB).In order to limit these interferences, more advanced equalizers are necessary. Then, achoice has to be done between performance, computation complexity and the estimatesabout the channel. With advanced equalizer, one great advantage of FBMC appear: thepossibility to demodulate the signal without synchronizing the received signal withthe demodulator with limited decrease of the system performance. This mean that itis possible to use frequency division multiple access with unsynchronized users, verysmall guard band and using only one demodulator for multiple users. Thanks to the fullspectrum efficiency of FBMC, the capacity of FBMC is higher than CP-OFDM whenthe channel is noise limited even if the SINR of FBMC is lower than OFDM. Thecyclic prefix of OFDM also increases the ratio Eb/N0 for similar SNR. There is notsuch energy loss for FBMC. When the channels gets more complex, like with MIMOchannels, FBMC equalizers gets more complex if we want to keep reasonable SIR. Thegap in terms of computation complexity between CP-OFDM and FBMC gets bigger.

COQAM modulation is a good way to solve the problems of non-perfect orthog-onality, non-perfect equalizations methods and computation complexity of FBMC. Itis possible to use very simple equalizations methods with very good results for bothSISO and MIMO channels. However, this improvement has drawbacks. This modula-tion has a similar power spectrum density than OFDM, necessitates large guard bandand filtering between unsynchronized users and does not allow unsynchronized uplink(the delay plus the channel should have a limited length). In fact one can see that CO-QAM is an evolution of OFDM by allowing multiplexing in the time domain and in thefrequency domain at the same time inside a block. It does not keep any of the strengthsof FBMC.

62

5.2 Future WorkWe have tested FBMC modulation using channels model. In order to see the realperformance of this new modulation, it will be interesting to test it with real channels.This is particularly important in the case of MIMO channels for which the channelsmodels were probably not general enough. In the meanwhile, it might be interestingto evaluate the influence of the non-perfect channel knowledge and the influence oftime varying channels with tracking. Another evolution on the channel model canbe the study of phase noise, I/Q impairment, saturation, Doppler effect and so on.Another interesting axis of future work can be the study of non-linear processing suchas interference cancellation methods or Tomlinson-Harashima pre-coding for bad orMIMO channels channels Finally, it might be interesting to go deeper in the study ofMIMO systems with pre-coding in order to improve the performance when the numberof antennas at the transmitter side increase.

63

Appendix

Inner products over L2(C)The vector space L2(C) consisting of the square-integrable functions from R to C canbe considered as a Hilbert space but also as a Euclidean space.

We define a Hermitian inner product over the Hilbert space as

〈f, g〉H =

∫Rf(t)g∗(t)dt

for f, g ∈ L2(C)

(5.1)

We also define an Euclidean inner product over the Euclidean space as

〈f, g〉E = <(〈f, g〉H)

for f, g ∈ L2(C)(5.2)

One should be careful about which vector space we are working with. First of all, inthe Euclidean space, the constellation must be real (usually ASK). In the Hilbert space,the constellation can be complex (PSK, QAM and so on). Another important thing isthe fact that the phase shift is, in the Euclidean space, more a problem of synchroniza-tion than a problem of equalization in contrast to the Hilbert vector space. The complexexponential is not anymore an eigenvector of every linear time invariant channel. How-ever, this decomposition gives the possibility to work over independent vector spacesof dimension 2. All these drawbacks are compensated by an orthogonality which ismuch easier to obtain and, consequently, we get much more freedom in the waveformdesign.

The Balian-Low theoremTheorem: Let g ∈ L2(R), G its Fourier transform and a, b two strictly positive realnumbers. If ∫

Rx2|g(x)|2dx <∞

and∫Rf2|G(f)|2df <∞

(5.3)

then, the family of Gabor functions (g, a, b) is not an orthogonal basis of the com-plex field L2(R).

64

Explanation and consequences: A family of Gabor functions is constructed using afunction, here g and two strictly positive real numbers, a and b. The family of Gaborfunction is then the set of all the functions: g(x − an)ejbmx, n ∈ Z, m ∈ Z. In otherwords, a correspond to the spacing in time between symbols for each sub-channel andb the frequency spacing between two sub-channel. If the family of Gabor functions(g, a, b) is a basis, it means that we transmit at the Nyquist rate. The orthogonalityof the family correspond to the orthogonality of the waveforms shifted in time andfrequency. The property, for a waveform, of being well localized in time and frequencyis the property of the Equation 5.3. Assuming the use of a Gabor family in order togenerate easily the waveforms and to be able to use the polyphase method, we have tochoose between:

• the spectral density (maximal only if the family is a base)

• the orthogonality in the complex field

• the property of being well localized in time and frequency (Equation 5.3)

Linear MMSE equalizationLet Υ and η be two random vectors of length L and Lη so that the correlation matricesof Υ and η are inner matrices: RΥΥ = σ2

ΥIL, Rηη = σ2ηILη . Let C and Cη be

matrices of length Lw × L and LwLη . We want to estimate the coefficient in positioni in the vector Υ, ti, using a vector R of length Lw defined as R = CΥ + Cηη. Wewant a linear estimator which minimizes the mean square error. The estimate is ti andthe condition of having a linear estimator means there exists a row vector W so thatti = WR. ei is the null row vector of length Lw with a 1 in position i. Let Ω be themean square error

Ω = E[|ti − ti|2

]= E

[|eiΥ−WR|2

]= eiE [ΥΥ∗] e∗i − 2WE [RΥ∗] e∗i +WE [RR]W ∗

(5.4)

The minimum is attained when the gradient is zero

dΩ

dW=d (eiE [ΥΥ∗] e∗i − 2WE [RΥ∗] e∗i +WE [RR]W ∗)

dW= −2E [RΥ∗] e∗i + 2E [RR]W ∗

E [RR∗]W ∗ = E [RΥ∗] e∗i

(5.5)

Now assume that E [RR∗] is invertible. Then

W = eiE [ΥR∗]E [RR∗]−1

= eiE [Υ(CΥ + Cηη)∗]E [(CΥ + Cηη)(CΥ + Cηη)∗]−1

= eiE [ΥΥ∗]C∗(CE [ΥΥ∗]C∗ + CηE [ηη∗]C∗η )−1

= eiσ2ΥC∗(σ2

ΥCC + σ2ηCηC

∗η )−1

= eiC∗(CC∗ +

σ2η

σ2Υ

CηC∗η )−1

(5.6)

In the case of a real vector space, the transpose conjugated matrix is replaced bythe transpose matrix.

65

The residual mean square error is

Ω = E[|ti − ti|2

]= E

[|eiΥ−WR|2

]= E

∣∣∣∣∣eiΥ− eiC∗(CC∗ +σ2η

σ2T

CηC∗η )−1 (CΥ + Cηη)

∣∣∣∣∣2

= σ2Υei(IL − C∗(CC∗ +

σ2η

σ2Υ

CηC∗η )−1C)e∗i

(5.7)

Equalization methods for single taps channelsThis section presents estimation methods of St given Sr = C×St+N. C is a constant.

Maximum likelihood Let N, St and Sr be the random variables representing thenoise, the transmitted signal and the received signal. N , St and Sr are their realizations.St takes its values in C and

N ∼ CN (0, σ2n)

Sr ∼ CN (CSt, σ2n)

The probability density function of Sr given St is

fSr|St=St(Sr) =1√

2πσ2e

(Sr−CSt)2

2σ2 (5.8)

σ is a constant so this density function reaches its maximum for Sr = CStThe estimate of St given Sr is: St = Sr

C

Minimum mean square error (MMSE) The MMSE equalizer (denoted Q) is theminimizing variable St = QSr of the following expression

minQ

E[‖St − St‖2

]= min

QE[‖Q(CSt +N)− St‖2

](5.9)

Let ∆ = E[‖Q(CSt +N)− St‖2

]Q is the solution of:

d∆

dQ= 0

0 = 2Q(C2σ2s + σ2

n)− 2Cσ2s

Cσ2s = Q(C2σ2

s + σ2n)

Q =Cσ2

S

C2σ2s + σ2

n

Q =1

C

1

1 +σ2n

C2σ2s

(5.10)

Let SNR be C2σ2s

σ2n

. The MMSE estimate is St = 1C

11+ 1

SNR

Sr.

66

Orthogonality of the filters hI,n and hQ,n

We want to verify the orthogonality between hI,n(t) and hI,m(t + iT ), hI,n(t) andhQ,m(t+ iT ) and hQ,n(t) and hQ,m(t+ iT ). We can distinguish 3 cases: |m−n| ≥ 2,|m− n| = 1 and n = m.

• |m − n| ≥ 2: As h is band limited with a bandwidth smaller than 2T , hI,n and

hQ,n have a spectrum equal to zero outside of the range [n−1T , n+1

T ]. Conse-quently, if |m− n| ≥ 2, hI,n and hQ,n are orthogonal with hI,m and hQ,m.

• |m − n| = 1: Without lost of generality we can assume m = n + 1. Let Σ be∫R hI,n(t)h∗I,m(t+ iT )dt

Σ =

∫ +∞

−∞jHI,n(f)H∗I,n(f − 1

T)e2jπiT(f− 1

T )df

=

∫ +∞

−∞jH(f)H(f − 1

T)e2jπiTfdf

=

∫ 12T

−∞jH(f)H(f − 1

T)e2jπiTfdf +

∫ +∞

12T

jH(f)H(f − 1

T)e2jπiTfdf

(5.11)In the second integral it is possible to change the variable using f2 = 1

T − f .Using the symmetry of H(f) and the fact that H(f) is real we get

Σ =

∫ 12T

−∞jH(f)H(f − 1

T)e2jπiTfdf +

∫ 12T

−∞jH(f2 −

1

T)H(f2)e−2jπiT (f2+ 1

T )df2

= j

∫ 12T

−∞2H(f)H(f − 1

T) cos(2πiTf)df

(5.12)

H(f) is real so Σ is a pure imaginary.We get 〈hI,n(t), hI,m(t + iT )〉E =<(Σ) = 0. With the same calculation we get the orthogonality between hQ,n(t)and hQ,m(t+ iT ). Let Σ2 be

∫R hI,n(t)hQ,m(t+ iT )dt

Σ2 =

∫ +∞

−∞H(f)H(f − 1

T)e−2jπ(i+ 1

2 )T(f− 1T )df

= −j∫ +∞

−∞H(f)H(f − 1

T)e−2jπ(i+ 1

2 )Tfdf

(5.13)

With the same calculation as the one does for Σ

Σ2 = −j∫ 1

2T

−∞2H(f)H(f − 1

T) cos(2π(i+

1

2)Tf)df (5.14)

Consequently 〈hI,n(t), hQ,m(t+ iT )〉E = <(Σ2) = 0

• m = n: As hI,n and hQ,n satisfy the Nyquist ISI criterion

∀n, i ∈ ZhI,n(t)⊥hI,n(t+ iT )

hQ,n(t)⊥hQ,n(t+ iT )

(5.15)

67

The orthogonality between hQ,n(t) and hI,n(t) is a consequence of hQ,n(t) =jhI,n(t+ T

2 )

〈hI,n(t), hQ,n(t+ iT )〉E = <(〈hI,n(t), jhI,n(t+ (i+

1

2)T )〉H

)= <

(−j〈hI,0(t), hI,0(t+ (i+

1

2)T )〉H

) (5.16)

As hI,0(t) is real, 〈hI,0(t), hI,0(t+(i+ 12 )T )〉H is real. Consequently 〈hI,n(t), hQ,n(t+

iT )〉E = 0

Delay and power attenuation of the taps in the ”epa” and ”etu” mod-els

Table 5.1 Extended Pedestrian A model (”epa”)Excess tap delay (ns) Relative power (dB)0 0.030 −1.070 −2.090 −3.0110 −8.0190 −17.2410 −20.8

Table 5.2 Extended Typical Urban model (”etu”)Excess tap delay (ns) Relative power (dB)0 −1.050 −1.0120 −1.0200 0.0230 0.0500 0.01600 −3.02300 −5.05000 −7.0

68

Bibliography

[1] 3GPP TS 36.104 version 8.2.0 release 8. Technical report.

[2] M. Bellanger et al. Equalization and demodulation in the receiver (single an-tenna). Technical report, PHYDYAS, 2008.

[3] M. Bellanger et al. MIMO channel matrix estimation and tracking. Technicalreport, PHYDYAS, 2009.

[4] M. Bellanger et al. PHYDYAS, FBMC physical layer: a primer. Technical report,PHYDYAS, 2010.

[5] M.G. Bellanger and J. Daguet. TDM-FDM transmultiplexer: Digital polyphaseand fft. IEEE Transactions on Communications, 22(9):1199–1205, Sep 1974.

[6] R. W. Chang. High-speed multichannel data transmission with bandlimited or-thogonal signals,. Bell Syst. Tech. J., December 1966.

[7] Stefan Parkvall, Erik Dahlman and Johan Skold. LTE/ LTE-Advanced for mobilebroadband. Elsevier Ltd., 2011.

[8] B. Farhang-Boroujeny. Ofdm versus filter bank multicarrier. Signal ProcessingMagazine, IEEE, 28(3):92–112, May 2011.

[9] H. G. Feichtinger and T. Strohmer. Gabor analysis and algorythms. Birkhauser,1998.

[10] Robert M. Grey. Toeplitz and circulant matrices: a review.

[11] B. Hirosaki. An orthogonally multiplexed qam system using the discrete fouriertransform. IEEE Transactions on Communications, 29(7):982–989, Jul 1981.

[12] F. Kaltenberger, M. Kountouris, L. Cardoso, R. Knopp, and D. Gesbert. Capacityof linear multi-user mimo precoding schemes with measured channel data. InSignal Processing Advances in Wireless Communications, 2008. SPAWC 2008.IEEE 9th Workshop on Communications, pages 580–584, July 2008.

[13] Hao Lin and Pierre Siohan. An advanced multi-carrier modulation for future radiosystem.

[14] B. Saltzberg. Performance of an efficient parallel data transmission system. IEEETransactions on Communication Technology, 15(6):805–811, December 1967.

69

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