EE456 Digital Communications Professor Ha Nguyen September 2015 - PowerPoint PPT Presentation

EE456 – Digital Communications Professor Ha Nguyen September 2015 EE456 – Digital Communications 1

Introduction to Orthogonal Frequency-Division Multiplexing (OFDM) EE456 – Digital Communications 2

Spectrum of M -FSK ∆ f ⋯ f f f f f f M − M 1 2 3 1 For FSK with N = 2 λ frequencies, only one of N frequencies is activated over one symbol duration of T s = λT b , where T b is the bit duration. What frequency that is activated over any symbol duration is determined by the mapping from λ information bits to the frequency value. FSK is not a spectral-efficient modulation scheme! Why not using all the carriers to carry information at the same time since they are orthogonal? This leads to OFDM (orthogonal frequency-division multiplexing) technique. EE456 – Digital Communications 3

OFDM (Orthogonal Frequency-Division Multiplexing) ∆ = 2 / T f 1 N T N H ( f ) f f f f f − − 0 1 2 N 2 N 1 N ≈ ⋅∆ = Bandwidth W N f T N In OFDM, the spectrum is divided into overlapping but orthogonal subcarriers. Each sub-carrier is independently modulated by QAM. The minimum subcarrier separation is 1 /T N , where T N is the OFDM symbol length. OFDM can be simply looked upon as a combination of amplitude , phase and frequency modulation techniques. EE456 – Digital Communications 4

Key Features of OFDM The data rate on each of the subchannels is much less than the total data rate, and the corresponding subchannel bandwidth is much less than the total system bandwidth. The number of subchannels can be chosen so that each subchannel has a bandwidth small enough so that the frequency response over each subchannel’s frequency range is approximately constant. This ensures that inter-symbol interference (ISI) on each subchannel is small. The subchannels in OFDM need not be contiguous, so a large continuous block of spectrum is not needed for high rate transmission. The modulation formats on different subchannels need not be the same. In fact one may adaptively choose different modulation schemes according to the instantaneous quality of the subchannels. The most attractive feature of OFDM is that its modulator and demodulator can be efficiently implemented with DSP. The main technical issues that impair performance of OFDM are frequency offset and timing jitter, which degrade the orthogonality of the subchannels. Having QAM signals transmitted simultaneously over all carriers makes the peak-to-average power ratio (PAPR) of the OFDM signal significantly higher than that of single-carrier QAM signal. This is a serious problem when nonlinear amplifiers are used. EE456 – Digital Communications 5

OFDM Viewed and Implemented as a Multiple QAM Systems Infor. bits r bps b π + φ j (2 f t ) e n n = t kT N Detected ∗ + w s t ( ) h t ( ) ( ) t infor. bits − π + ϕ j (2 f t ) e n n EE456 – Digital Communications 6

Review of DFT/IDFT Let x [ n ] , 0 ≤ n ≤ N − 1 , be a DT sequence. The N -point DFT of x [ n ] is N − 1 x [ n ]e − j 2 πni � DFT { x [ n ] } = X [ i ] � , 0 ≤ i ≤ N − 1 . N n =0 Given X [ i ] , 0 ≤ i ≤ N − 1 , the sequence x [ n ] can be recovered using the IDFT: N − 1 IDFT { X [ i ] } = x [ n ] � 1 X [ i ]e j 2 πni � , 0 ≤ n ≤ N − 1 . N N i =0 When N is a power of two, the DFT and IDFT can be efficiently performed using the fast Fourier transform (FFT) and inverse FFT (IFFT) algorithms. When the discrete-time sequence x [ n ] is passed through a discrete-time linear time-invariant system whose impulse response is h [ n ] , the output y [ n ] is the discrete-time convolution of the input and the channel impulse response: ∞ � y [ n ] = h [ n ] ∗ x [ n ] = x [ n ] ∗ h [ n ] = h [ k ] x [ n − k ] k = −∞ The N -point circular convolution of x [ n ] and h [ n ] , both with length N , is N − 1 � y [ n ] = h [ n ] ⊗ x [ n ] = x [ n ] ⊗ h [ n ] = h [ k ] x [( n − k ) mod N ] . k =0 Circular convolution in time leads to multiplication in frequency: Y [ i ] = DFT { y [ n ] = x [ n ] ⊗ h [ n ] } = X [ i ] H [ i ] , 0 ≤ i ≤ N − 1 EE456 – Digital Communications 7

Example of Circular Convolution and FFT/IFFT in Matlab EE456 – Digital Communications 8

Implementation of OFDM with DFT/IDFT x ɶ − µ x ɶ − µ + x N ɶ − ⋯ [ ], [ 1], , [ 1] (a) Transmitter X [0] M 0 -QAM ( ) x π f t [0] sin 2 0 modulator Add X x x t ɶ [1] [1] ( ) M 1 -QAM cyclic S/P Converter I p ( t ) IDFT prefix, r modulator s t bits/sec ( ) b or and ⋮ ⋮ x ɶ t ( ) IFFT parallel- Q p ( t ) to-serial X N − x N − [ 1] [ 1] converter M N -1 -QAM ( ) π f t cos 2 modulator 0 − µ − µ + − y y ⋯ y N [ ], [ 1], , [ 1] y Y [0] [0] ( ) M 0 -QAM π f t sin 2 0 = t nT demodulator s Remove Y y [1] [1] M 1 -QAM P/S Converter − p t prefix, ( ) FFT bits demodulator r t and ( ) t = nT or ⋮ ⋮ s serial-to- DFT parallel p − t ( ) converter Y N − y N − [ 1] [ 1] M N -1 -QAM ( ) demodulator π f t cos 2 0 (b) Receiver EE456 – Digital Communications 9

How Does the Cyclic Prefix Work in OFDM? − µ − µ + − − µ − µ + − ⋯ ɶ x x ɶ ⋯ x N ɶ ⋯ ⋯ y y ⋯ y N ⋯ , [ ], [ 1], , [ 1], , [ ], [ 1], , [ 1], Equivalent discrete-time channel { } µ h n [ ] n = 0 µ Append last symbols to the front ≈ x N − µ x N − µ + x N − x x x x N − µ − x N − µ x N − µ + x N − ⋯ ⋯⋯⋯ ⋯ [ ], [ 1], , [ 1] [0], [1], [2], , [ 1] [ ], [ 1], , [ 1] Cyclic prefix (CP) of length µ Original signal sequence of length N One has ˜ x [ n ] = x [ n mod N ] for − µ ≤ n ≤ N − 1 , which also means that ˜ x [ n − k ] = x [( n − k ) mod N ] for − µ ≤ n − k ≤ N − 1 . Given ˜ x [ n ] is the input of the channel (ignoring noise), the channel output between 0 ≤ n ≤ N − 1 can be computed as: µ µ � � y [ n ] = x [ n ] ∗ h [ n ] = ˜ h [ k ]˜ x [ n − k ] = h [ k ] x [( n − k ) mod N ] = x [ n ] ⊗ h [ n ] k =0 k =0 where the third equality follows from the fact that, for 0 ≤ k ≤ µ , ˜ x [ n − k ] = x [( n − k ) mod N ] for 0 ≤ n ≤ N − 1 . Taking into account AWGN, the DFT of the channel output yields Y [ i ] = DFT { y [ n ] = ( x [ n ] + w [ n ]) ⊗ h [ n ] } = X [ i ] H [ i ] + W [ i ] , 0 ≤ i ≤ N − 1 , EE456 – Digital Communications 10

µ Append last symbols to the front ≈ x N − µ x N − µ + ⋯ x N − x x x ⋯⋯⋯ x N − µ − x N − µ x N − µ + ⋯ x N − [ ], [ 1], , [ 1] [0], [1], [2], , [ 1] [ ], [ 1], , [ 1] Cyclic prefix (CP) of length µ Original signal sequence of length N x N − x N − x N − x [0], , [ 1] x [0], , [ 1] x [0], , [ 1] Data block CP Data block CP CP Data block y N − y N − y N − y [0], , [ 1] y [0], , [ 1] y [0], , [ 1] µ µ µ N N N An OFDM symbol is basically a super-symbol obtained by multiplexing many M -QAM symbols in a complicated manner. The length of a super-symbol ( T N ) becomes longer and hence more resistent to multipath effect. One can also use zero padding to create a guard interval between consecutive OFDM symbols, hence avoiding ISI. EE456 – Digital Communications 11

OFDM System with the Equivalent Discrete-Time Multipath Channel − µ − µ + − − µ − µ + − x [ ], [ x 1], , [ x N 1] y [ ], [ y 1], , [ y N 1] X [0] x [0] y [0] Y [0] y [1] X [1] x [1] Y [1] { } µ h n [ ] n = 0 X N − [ 1] x N − y N − Y N − [ 1] [ 1] [ 1] It was shown that the input of the IFFT block in the transmitter and the output of the FFT block in the receiver is related by Y [ i ] = X [ i ] H [ i ] + W [ i ] , 0 ≤ i ≤ N − 1 , where W [ i ] is Gaussian noise component. The use of CP and IFFT/FFT decomposes the wideband channel H ( f ) into a set of narrowband orthogonal subchannels, each with a different QAM symbol. The demodulator needs to know the channel gains H [ i ] to recover the original QAM symbols by dividing out these gains: X [ i ] = Y [ i ] /H [ i ] . This process is called frequency equalization . The benefits of adding a cyclic prefix come at a cost. Since µ symbols are added to the input data blocks, there is an overhead of µ/N and a resulting data-rate reduction of N/ ( µ + N ) . The transmit power associated with sending the cyclic prefix is also wasted since this prefix consists of redundant data. EE456 – Digital Communications 12

16-QAM vs. “Constellation” of Time Samples (plotted with 4 random OFDM symbols, i.e., 4 × N = 64 samples) 16-QAM of X [ i ], PAPR ≈ 2 . 55 dB x [ n ], N = 16, PAPR ≈ 7 . 96 dB 1 2 0.5 1 0 0 −1 −0.5 −2 −1 0 1 −0.5 0 0.5 EE456 – Digital Communications 13

“Constellation” of Time Samples (plotted with 4 random OFDM symbols) for N = 64 and N = 128 x [ n ], N = 64, PAPR ≈ 11 . 91 dB x [ n ], N = 128, PAPR ≈ 14 . 33 dB 1 1 0.5 0.5 0 0 −0.5 −0.5 −0.5 0 0.5 −0.5 0 0.5 The peak powers appear to be the same, but the average power of the right constellation is much smaller than that of the left one, giving rise to a higher PAPR. EE456 – Digital Communications 14