Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Orthogonal Frequency Division Multiplexing
Stelios Stefanatos
Department of Physics,National & Kapodistrian University of Athens,
Athens, Greece
April 7, 2009
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Outline
1 Introduction
2 Frequency Division Multiplexing
3 Orthogonal Frequency Division Multiplexing
4 Signal Processing in OFDM
5 Summary
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
System Model (Complex Baseband Notation)
Information (transmitted) signal:s(t), t ∈ [0,T],S(f ) , (Fs)(f ), f ∈ [−W,W]
Energy Constraint: E{∫|s(t)|2dt} = E{
∫|S(f )|2df} ≤ P
LTI channel:c(t), t ∈ [0,Tc]C(f ) , (Fc)(f ), f ∈ [?, ?]
Received signal:y(t) , s(t) ∗ c(t) + w(t), t ∈ [0,T], (Tc << T; realistic?)Y(f ) = C(f )S(f ) + W(f ), f ∈ [−W,W]
Questions:information theoretic limitssignal structure for simple processing
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Capacity of the LTI Channel
Assumption: Tx and Rx have perfect Channel State Information(CSI)
System’s mutual information (in bits/sec/Hz):
I(y(t); s(t)) = 12W
∫ W−W log2
(1 + |C(f )|2P(f )
N0
)df
P(f ) , E{|S(f )|2} ≥ 0,∫ W−W P(f ) = P
Capacity achieved by:s(t) should be a white (circularly symmetric) Gaussian processPower P should be allocated in frequency in a waterfilling manner
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Waterfilling Power Allocation
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Channel compensation (equalization)
Single carrier linearly modulated signal:
s(t) =∑K−1
k=0 xkgT(t − kTs), t ∈ [0,T]
xk ∈ A, Ts ≈ 1/(2W) (Nyquist pulses)
Question: How do we choose A and gT(t) so that s(t)
1 is Gaussian(-like)?2 has a non–flat power spectral density
Generic Rx waveform processing (generation of sufficientstatistics)
Received signal y(t) goes through a low pass filter gR(t)Sampling at the symbol period (Nyquist sampling)
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Discrete-time signal processing
Discrete-time I/O relation:
y[k] , y(kTs) =∑Lh−1
l=0 h[l]xk−l + w[k], k = 0, . . . ,K − 1
h[k] = h(kTs), h(t) , gR(t) ∗ c(t) ∗ gT(t)K = bT/TscLh = bTh/Tsc : # of channel tapsSmaller Ts results in larger Lh (increasing ISI)!w[k] :AWGN
Matrix-vector relationship:
y = T (h)x + w
y = [y[0], y[1], . . . , y[K − 1]]T , h, x, w similarly definedT (h): K × K Toeplitz matrix with first column h (convolutionmatrix)
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Discrete-time signal processing
T (h) =
h[0]h[1] h[0]
h[1]. . .
......
. . .
h[Lh − 1]. . .
h[Lh − 1]. . .
h[Lh − 1] · · · h[1] h[0]
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Data Detection
Optimal decision rule:x = arg minx∈AK ||y− T (h)x||2
For {xk} i.i.d. from an M-ary constellation“Brute force” maximization complexity: O(MK)Dynamic programming (Viterbi) complexity: O(MLh−1)
Complexity unacceptable for high-rate/broadband applicationsExample: DSL channel has Lh ≈ O(100)
Low complexity channel compensation (equalization) is required
linear equalization (ZF, MMSE)non-linear equalization (DFE)
Non-optimal techniques (sub-optimal performance)
Need for a “channel-matched” signaling
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Frequency Division Multiplexing
Basic idea:
Divide the channel BW into small non− overlapping bandscentered at {fn}N−1
n=0 (say N bands)
Separate original high− rate data stream into N low− ratestreams (symbol period increases from Ts to NTs)
Transmit each low-rate signal within each of the frequency bands
If frequency band is “small” each stream faces a flat fading (onetap) channel H(fn)
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Frequency Division Multiplexing
Multiplexed signal:
s(t) =∑N−1
n=0∑
k xk[n]gT(t − kNTs)ej2πfn(t−kNTs)
k: time indexn: frequency index
set of sub− carriers: {gT(t − kNTs)ej2πfn(t−kNTs)} generatedfrom a prototype pulse gT(t) of small bandwidth
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Frequency Division Multiplexing
Advantages:waterfilling can be approximated by appropriate power loadingchannel effect can be easily compensated for by simple one tapequalization
Disadvantages:high Tx complexity (need for N parallel oscillators)need for frequency guard bands (reduced bandwidth efficiency)
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Orthogonal Frequency Division Multiplexing
Spectral efficiency requirement:zero guard bands (possibly allow for overlapping in frequency)desirable property: decoupled processing for each stream
Solution: Employ orthonormal sub-carriers∫gT(t − k1NTs)ej2πf1(t−k1NTs)g∗T(t − k2NTs)e−j2πf2(t−k1NTs))dt =
δ(f1 − f2)δ(k1 − k2)
the set of subcarriers {gT(t − kNTs)ej2πfn(t−kNTs)} is an(orthonormal) basis for its span
for maximum spectral efficiency we choose the span to be thespace of band- and time-limited functions (dimension of space?)
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Optimal Rx structure
Rx also requires a filter bank!!!
if a frequency selective channel is present a possible Rx structurewould divide each stream with the corresponding channel gain
problem: How to choose gT(t) and {fn}N−1n=0 ?
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Fourier Basis
one particularly simple choice of the orthonormal basis is basedon the rectangular pulse, i.e.,
gT(t) = 1/√
NTs, t ∈ [0,NTs], for all n
fn = n/(NTs), n = 0, . . . ,N − 1
the resulting basis is called the Fourier basis
in that case the transmitted signal (OFDM symbol) equals
s(t) = 1√NTs
∑N−1n=0 x[n]ej2πnt/(NTs), t ∈ [0,NTs]
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Time-frequency plane
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
OFDM Spectrum
nominal BW: 2W = 1/Ts
“sinc” shape of the sub-carriers spectrum introduces out-of-bandleakagePSD tends to a perfect rectangular for large number ofsub-carriers (increasing symbol duration)
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Discrete-time implementation
sampling the OFDM symbol at a rate 1/Ts results in thediscrete-time sequence
s[n] = s(nTs) = 1√NTs
∑N−1k=0 x[k]ej2πnk/N
important observation: this is the Inverse Discrete FourierTransform (IDFT) of the (symbol) sequence {x[k]}N−1
k=0
OFDM signal generator can be implemented as follows:
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Discrete-time implementation
(I)DFT transformation F : CN 7→ CN requires a complexity oforder O(N2)
In 1965 an efficient implementation of DFT was proposed: FastFourier Transform (FFT)
complexity scales as O(N log2 N)
IFFT-based implementation of OFDM has allowed for systemsemploying up to 8K sub-carriers (DVB-T)
FFT also utilized at the Rx also
sample the received signal at 1/Ts
perform an FFT on the N-length sequence
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
FFT vs DFT
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Channel effect
sampled received OFDM symbol (assuming no noise) can bewritten as:
y[n] =∑N−1
k=0 x[k]gk[n], n = 0, . . . ,N + Lh − 1
where gk[n] =∑Lh−1
l=0 h[l]ej2πk(n−l)/N
what happens for h[l] = aδ[l]?
The channel and the finite symbol duration introduce two effects:
received symbol now contained in [0,N + Lh − 1] (ISI)
received version of sub-carriers {gk[n]} no longer orthogonal
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Cyclic Prefix
Cyclic prefixed (CP) signal generation:x ∈ CN : vector of symbols, input to IFFTs = FHx ∈ CN : IFFT outputscp = [s[N − Lh], s[N − Lh + 1], s[N − 1], s]T = Tcps ∈ CN+Lh−1 :cyclic prefixed symbol (input to A/D)
sampled received signal (no noise): ycp = T (h)scp
drop CP:
y = Rcpycp
= RcpT (h)Tcps= C(h)s
C(h): N × N circulant matrix
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Cyclic Prefix
C(h) =
h[0] h[3] h[2] h[1]h[1] h[0] h[3] h[2]
h[1]. . . h[3]
......
. . .
h[Lh − 1]. . .
h[Lh − 1]. . .
h[Lh − 1] · · · h[1] h[0]
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Cyclic Prefix
The introduction of the CP has transformed the channel’s effectfrom linear convolution to circular convolution
Proposition: A circulant matrix is diagonalized by the Fourierbasis, i.e.,
C(h) = FHD(h)F
h , Fh (DFT of h; channel’s frequency response)DSP interpretation?Other examples of diagonalization?
received time-domain signal: y = FHD(h)x
performing an FFT on y results in N parallel (decoupled)channels: y = Fy = D(h)x
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Cyclic Prefix
End-to-end practical OFDM system:
OFDM system abstraction:
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Cyclic Prefix
Advantages of CP:eliminates ISI“diagonalizes channel”simple equalization (complexity scales as O(NM)) (why?)simple power (rate) allocationnegligible complexity
Disadvantages of CP:spectral efficiency and power loss: Lh/(N + Lh)increase ratio: N/Lh
DSL: “channel shortening” before FFT
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Channel Estimation (LTI case)
typical case for fixed wireless scenarios (DVB, WiFi)
channel is estimated at the beginning of transmission (frame) bytransmission of pilot OFDM symbols (preamble)
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Channel Estimation (LTI case)
Least Squares (LS) channel estimation
h[k] = y[k]/xp[k], k = 0, 1, . . . ,N − 1
typically a LS channel estimate is preformed for a number ofOFDM pilot symbols (usually two) and the average value isextracted
advantage: very simple implementation (e.g., set all pilotsymbols to 1)
Minimum Mean Square Error (MMSE) channel estimationchannel’s frequency response {h[k]} is correlated, i.e.,information of the channel response at frequency k providessome information about its values in other frequencies
more complex estimation procedure
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Channel Estimation (LTI case)
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Channel Estimation (LTV case)
case of slow mobility (e.g., walking speed)
channel assumed quasi− static during each OFDM symbolduration
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Channel Estimation (LTV case)
periodic pilot insertion is mandatory
channel is initially estimated at the “pilot slots” and subsequentlyinterpolated at the “data slots”
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Phase Impairments
Oscillator instabilities, non-perfect carrier synch., Doppler, e.t.c.,introduce a phase error on the received signal
Mathematical model (AWGN only):
y(t) = ejθ(t)s(t) + w(t)
θ(t) = 2πfFOt + φ(t)
frequency offset: fFO (constant)
phase noise: φ(t)typical model: dφ(t)/dt = n(t) =⇒ φ(t) =
∫ t0 n(t)dt (Wiener
process)n(t): AWGN process
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Phase Impairments
sample realization of φ(t)
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Phase Impairments
Consider FO only
Received OFDM signal (no channel):
y(t) =∑N−1
n=0 xnej2π(fFO+n/(NTs))t + w(t), t ∈ [0,NTs]
FFT output will be (with no FO compensation)
xn = ACPExn + ICIn + noisen
FO introduces two effects:Common Phase Error ACPE (same for all symbols)Inter Carrier interference (ICI)
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Phase Impairments
Demodulator output (64-QAM, SNR = 20dB, no FO)
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Phase Impairments
Demodulator output (FO = 0.1% of sub-carrier spacing)
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Phase Impairments
Demodulator output (FO = 0.5% of sub-carrier spacing)
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Phase Impairments
OFDM much more sensitive to FO that SC!!!“improved” pulse shape may be beneficial
compensation is mandatory
two methods:estimate the CPE in every OFDM symbol with pilots (ICI stillremains)estimate the value of fFO and (digitally) “de-rotate” the receivedsignal (ideally, no FO effect)
phase noise has exactly the same effect
compensation is more difficult since it is a random process
pilot aided CPE estimation only solution
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Summary
OFDM is the modulation of choice for transmitting high ratesover dispersive channels
major step for wider acceptance: use of FFT
simple channel equalization
increased sensitivity to phase impairments
not well suited for high mobility applications (due to ICI)
expected to be prevalent in future wireless standards
Stelios Stefanatos Orthogonal Frequency Division Multiplexing
Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary
Wireless standards based on OFDM
point-to-point transmissionDigital Audio Broadcasting (DAB)Digital Video Broadcasting (DVB)IEEE 802.11a, ETSI HiperLAN (WiFi applications, low mobility)
multiuser transmission (OFDMA)IEEE 802.16 (WiMAX)UMTS LTE
Stelios Stefanatos Orthogonal Frequency Division Multiplexing