CHAPTER 7 OFDM Xijun Wang

WEEKLY READING

1. Goldsmith, “Wireless Communications”, Chapters12

2. Tse, “Fundamentals of Wireless Communication”,Chapter 4

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I hear and I forget,I see and I remember, I do and I understand

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MULTICARRIER MODULATION

n Basic idea¨ Divide the transmitted bitstream into many different

substreams and send these over many different subchannels

¨ The corresponding subchannel bandwidth is much less than the total system bandwidth

¨ Each subchannel has a bandwidth less than the coherence bandwidth of the channel, so the subchannelsexperience relatively flat fading

n Implementation¨ Orthogonal frequency division multiplexing (OFDM) ¨ Vector coding

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ISI MITIGATION

n Consider a linearly-modulated system with data rate R and passband bandwidth B

n The coherence bandwidth for the channel is assumed to be Bc < B

n break this wideband system into N linearly-modulated subsystems in parallel

n each with subchannel bandwidth BN = B/N and data rate RN ≈ R/N

n For N sufficiently large, the subchannel bandwidth BN = B/N << Bc, and the symbol time TN ≈ 1/BN >> 1/Bc ≈ Tm

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MULTICARRIER TRANSMITTER

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MULTICARRIER RECEIVER

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DRAWBACK

n Near-ideal (and hence, expensive) low pass filters will be required to maintain the orthogonality of the subcarriers at the receiver.

n Requires n independent modulators and demodulators, which entails significant expense, size, and power consumption.

n This form of multicarrier modulation can be spectrally inefficient

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DRAWBACK

n each subchannel is modulated using raised cosine pulse shapes with rolloff factor β.

n The passband bandwidth of each subchannel is then BN = (1 + β)/TN.

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Let ε/TN denote the additional bandwidth required due to time-limiting of these pulse shapes.

The total required bandwidth for nonoverlapping subchannels is

OVERLAPPING SUBCHANNELS

n The total system bandwidth with overlapping subchannels is

n With N large, the impact of β and ε on the total system bandwidth is negligible

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GOOD, BUT HOW TO SEPARATE?

n The subcarriers must still be orthogonal so that they can be separated out by the demodulator in the receiver.

n a set of (approximately) orthogonal basis functions on the interval [0, TN]

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The minimum frequency separation required for subcarriers to remain orthogonal over the symbol interval [0, TN ] is 1/TN

MULTICARRIER RECEIVER

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MULTICARRIER RECEIVER

n If the effect of the channel h(t) and noise n(t) are neglected

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Since the subcarriers overlap, their orthogonality is compromised by timing and frequency offset.

DISCRETE IMPLEMENTATION

n Although multicarrier modulation was invented in the 1950’s, its requirement for separate modulators and de- modulators on each subchannel was far too complex for most system implementations at the time.

n However, the development of simple and cheap implementations of the discrete Fourier transform (DFT) and the inverse DFT (IDFT) twenty years later, combined with the realization that multicarrier modulation can be implemented with these algorithms, ignited its widespread use.

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DISCRETE FOURIER TRANSFORM

n Let x[n], 0 ≤ n ≤ N − 1, denote a discrete time sequence.

n The N-point DFT of x[n] is defined as

n When an input data stream x[n] is sent through a linear time-invariant discrete-time channel h[n], the output y[n] is

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DISCRETE FOURIER TRANSFORM

n The N-point circular convolution of x[n] and h[n] is defined as

¨ where[n−k]N denotes [n−k] modulo N.

n x[n−k]N is a periodic version of x[n−k] with period N.

n Circular convolution in time leads to multiplication in frequency

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CYCLIC PREFIX

n Consider a channel input sequence x[n] = x[0], . . . , x[N − 1] of length N

n a discrete-time channel with finite impulse response (FIR) h[n] = h[0], . . . , h[μ] of length μ + 1 = Tm/Ts, where Tm is the channel delay spread and Ts the sampling time associated with the discrete time sequence.

n This yields a new sequence

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CYCLIC PREFIX

n Suppose 𝑥"[n] is input to a discrete-time channel with impulse response h[n].

n The channel output y[n], 0 ≤ n ≤ N − 1 is

n Taking the DFT of the channel output in the absenseof noise then yields

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CYCLIC PREFIX

n for known h[n]

n The cyclic prefix¨ is not needed to recover x[n], 0 ≤ n ≤ N − 1 ¨ serves to eliminate ISI between the data blocks ¨ results in a data rate reduction of N/(μ + N) ¨ The transmit power associated with sending the cyclic

prefix is also wasted

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OFDM

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discrete frequency components

OFDM

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OFDM

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OFDM

The OFDM system effectively decomposes the wideband channel into a set of narrowband orthogonal subchannels with a different QAM symbol sent over each subchannel.

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MATRIX REPRESENTATION

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cyclic prefix

N × (N+𝜇)

The circular convolution operation u = h⊗x can be viewed as a linear transformation

MATRIX REPRESENTATION

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N × N

MATRIX REPRESENTATION

n DFT operation on x[n] can be represented by the matrix multiplication

n IDFT26

MATRIX REPRESENTATION

n Eigenvalue decomposition

¨ Λ is a diagonal matrix of eigenvalues of 𝐻%¨ MH is a unitary matrix whose rows comprise the

eigenvectors of 𝐻%.

n The rows of the DFT matrix Q are eigenvectors of 𝐻%, thus

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Complex exponentials are eigenfunctions for any linear system.

MATRIX REPRESENTATION

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By adding a cyclic prefix and using the IDFT/DFT, OFDM decomposes an ISI channel into N orthogonal subchannels and knowledge of the channel matrix H is not needed for this decomposition.

has the same noise autocorrelation matrix as ν

a natural rotation at the input and at the output to convert the channel to a set of non-interfering channels with no ISI

VECTOR CODING

n The singular value decomposition of H can be written as

¨ U is N × N unitary, V is (N + μ) × (N + μ) unitary, and Σis a diagonal matrix whose ith element σi is the ithsingular value of H.

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VECTOR CODING

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The precoding filter matrix must be known at the transmitter

VECTOR CODING

n Each element of X is effectively passed through a scalar channel without ISI, where the scalar gain of subchannel i is the ith singular value of H

n Insert a guardband or null prefix between the vector codeword (VC) symbols to eliminate ISI

31It is more efficient than OFDM in terms of energy.

PEAK TO AVERAGE POWER RATIO

n The PAR of a continuous-time signal is

n The PAR of a discrete-time signal is

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PEAK TO AVERAGE POWER RATIO

n A low PAR allows the transmit power amplifier to operate efficiently

n A high PAR forces the transmit power amplifier to have a large backoff in order to ensure linear amplification of the signal.

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A typical power amplifier response.

PEAK TO AVERAGE POWER RATIO

n A high PAR requires high resolution for the receiver A/D convertor, since the dynamic range of the signal is much larger for high PAR signals.

n PAR increases approximately linearly with the number of subcarriers.

n Although it is desirable to have N as large as possible in order to keep the overhead associated with the cyclic prefix down, a large PAR is an important penalty that must be paid for large N.

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FREQUENCY AND TIMING OFFSET

n Orthogonality is assured by the subcarrier separation ∆f = 1/TN.

n The frequency separation of the subcarriers is imperfect: ¨ Mismatched oscillators¨ Doppler frequency shifts¨ Timing synchronization errors

n Intercarrier interference (ICI)

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APPLICATIONS

n How to determine the parameters?¨ Guard interval TG

¨ Should be larger than delay spread TG > 2-4×Tm

¨ Data block length TN

¨ TN should be large to reduce the rate loss and powerloss

¨ TN should be small to reduce PAR and ICI¨ TN=5×TG

¨ The number of subcarriers¨ Determined by the bandwidth and subcarrier

separation

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APPLICATIONS

n Given¨ Bit rate 25Mbit/s ¨ Delay spread 200ns ¨ Bandwidth <18MHz

n Design¨ Guard interval TG = 4×Tm=0.8µs¨ TN=5×TG=4 µs¨ Symbol length = TG+TN=4.8 µs¨ subcarrier separation = 1/TN=250kHz¨ What is the required number of subcarrier?

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APPLICATIONS

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APPLICATIONS

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IEEE 802.11A WIRELESS LAN STANDARD

n Occupies 20 MHz of bandwidth in the 5 GHz unlicensed band

n Subcarrier¨ N = 64 subcarriers are generated, ¨ only 48 are actually used for data transmission, ¨ the outer 12 zeroed in order to reduce adjacent channel

interference, and 4 used as pilot symbols for channel estimation.

n Symbol¨ The cyclic prefix consists of μ = 16 samples, so the total

number of samples associated with each OFDM symbol, including both data samples and the cyclic prefix, is 80.

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IEEE 802.11A WIRELESS LAN STANDARD

n The transmitter gets periodic feedback from the receiver about the packet error rate, which it uses to pick an appropriate error correction code and modulation technique.

n The same code and modulation must be used for all the subcarriers at any given time. ¨ The error correction code is a convolutional code with

one of three possible coding rates: r = 1 , 2 , or 3 . ¨ The modulation types that can be used on the

subchannels are BPSK, QPSK, 16QAM, or 64QAM.

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IEEE 802.11A WIRELESS LAN STANDARD

n Since the bandwidth B (and sampling rate 1/Ts) is 20 MHz, and there are 64 subcarriers evenly spaced over that bandwidth, the subcarrier bandwidth is

n Since μ = 16 and 1/Ts = 20MHz, the maximum delay spread for which ISI is removed is

n The symbol time per subchannel is

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IEEE 802.11A WIRELESS LAN STANDARD

n The minimum data rate that can be transmitted is

n The maximum data rate that can be transmitted is

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For a 20 MHz LTE channel (one carrier), its effective transmission bandwidth is 18 MHz. One PRB is 180 kHz. Thus, a 20 MHz LTE channel has 100 PRBs, which can be allocated to multiple users (users 1 and 2) simultaneously. In contrast, all the band- width in a WiFi system can only be occupied by one user within each 4 msOFDM symbol.