8.4 ORTHOGONAL FREQUENCY DIVISION MULTIPLEXING (OFDM)

Transcription

1 ORHOGONAL FREQUENCY DIVISION MULIPLEXING (OFDM) ORHOGONAL FREQUENCY DIVISION MULIPLEXING (OFDM) All of the modulation techniques discussed so far are single-carrier modulation techniques that employ a single RF carrier. Multi-carrier modulation techniques on the other hand transmit data symbols in parallel on multiple subcarriers. A block of N data symbols, each of duration s, is converted into a block ofn parallel data symbols, each of duration = N s. hen parallel data symbols modulaten subcarriers that are spaced uniformly in frequency f = 1/ Hz apart. o describe and work with OFDM waveforms, it is most convenient to use the complex envelope representation for bandpass signals. he OFDM complex envelope is given by s(t) = n b(t n,x n ), (8.61) where b(t,x n ) = 2E u (t) (8.62) n is the block index, k is the subcarrier index, N is the number of sub-carriers, and x n = {x n,0, x n,1,..., x n, } is the nth data symbol block. Note that the amplitude shaping pulse used on each subcarrier is the rectangular pulse g(t) = 2E u (t). he complex-valued data symbols x n,k = x I n,k + jxq n,k are usually chosen from a QAM or PSK signal constellation. he OFDM bandpass signal has the quadrature representation 2E s(t) = u (t n) n x I n,kcos ( ( 2π f c + k ) ( ( t ) x Qn,k sin 2π f c + k ) ) t (8.63) and the envelope-phase representation 2E ( ( s(t) = u (t n) x n,k cos 2π f c + k ) )t+φ n,k n (8.64) where φ n,k = an 1 (x Q n,k /xi n,k ). From the quadrature representation in (8.63) it is apparent thate represents the energy in the elemental bandpass waveform ( ( 2E s(t) = u (t)cos 2π f c + k ) ) t or ( ( 2E s(t) = u (t)sin 2π f c + k ) ) t From the envelope-phase representation in (8.64), consider the waveforms corresponding to two different subcarriers in thenth OFDM symbol ( ( 2E s n,k (t) = u (t) x n,k cos 2π f c + k ) )t+φ n,k ( ( 2E s n,l (t) = u (t) x n,l cos 2π f c + l ) )t+φ n,l

2 130 NON-BINARY BANDPASS MODULAION AND ERROR PROBABILIY Separating adjacent OFDM subcarriers in frequency by f = 1/ Hz ensures that the corresponding OFDM subchannels are mutually orthogonal regardless of the set random phases{φ n,k } that are imparted by the data modulation (see Problem??). A cyclic extension (or guard interval) is usually appended to the OFDM waveform in (8.61) and (8.62) to combat ISI as will be explained later. he cyclic extension can be in the form of a cyclic prefix, a cyclic suffix, or both. With a cyclic suffix, the OFDM complex envelope becomes { s(t), 0 t s g (t) =, (8.65) s(t ), t (1+α g ) whereα g is the length of the guard interval and s(t) is defined in (8.61) and (8.62). Note that the segments of the waveform s g (t) in the intervals [0,α g ) and [, + α g ) are identical. he OFDM complex envelope with a cyclic suffix can be rewritten in the form s g (t) = n b(t n g,x n ), (8.66) where b(t,x n ) = u (t) = u (t) +u αg(t ) +u αg(t ) x n,k e j 2πk(t ) = u g (t), (8.67) and g = (1+α g ) is the OFDM symbol period with the addition of the guard interval. Likewise, with a cyclic prefix, the OFDM complex envelope is given by (8.66) with { s(t+), αg t 0 s g (t) =, (8.68) s(t), 0 t and b(t,x n ) = u αg(t+α g ) x n,k e j 2πk(t+) = u αg(t+α g ) +u (t) +u (t) = u g (t+α g ) (8.69) Finally, the cyclic extension can be divided between a cyclic prefix and cyclic suffix. Let α g = α p +α s, where α p and α s are the length of the cyclic prefix and cyclic suffix, respectively. hen the OFDM complex envelope is given by (8.66) with s(t+), α p t 0 s g (t) = s(t), 0 t, (8.70) s(t ), t t α s

3 ORHOGONAL FREQUENCY DIVISION MULIPLEXING (OFDM) 131 and b(t,x n ) = u αp(t+α p ) x n,k e j 2πk(t+) +u αs(t ) x n,k e j 2πk(t ) = u αp(t+α p ) +u αs(t ) +u (t) +u (t) = u g (t+α p ) (8.71) DF-Based OFDM Baseband Modulator A key advantage of using OFDM is that the baseband modulator can be implemented by using an inverse discrete-time Fourier transform (IDF). Later, we will see that the baseband demodulator can be implemented by using a discrete-time Fourier transform (DF). In practice, the computationally efficient fast Fourier transform (FF) and inverse fast Fourier transform (IFF) is used to implement the DF and IDF, respectively. he FF and IFF are very common signal processing functions. Consider the OFDM complex envelope without guard interval, as defined by (8.61) and (8.62). During the intervaln t (n+1), the OFDM complex envelope has the form s(t) = Eu (t n) x n,k e j2πk(t n) = Eu (t n) x n,k e j2πkt Ns, n t (n+1), (8.72) where we have made the substitution = N s. Now suppose that the complex envelope in (8.72) is sampled at s second intervals to yield the sample sequence X n,m = s(m s ) = E x n,k e j2πkm N, m = 0, 1,..., N 1. (8.73) Note that there are N samples in each OFDM symbol, since = N s. Observe that the vector X n = {X n,m } is the IDF of the vector Ex n = E{x n,k }. Contrary to conventional notation used when working with transforms, with OFDM it is customary to use the lower case vector Ex n to represent frequency domain coefficients, while the upper case vectorx n is used to represent time domain coefficients. As mentioned earlier, a cyclic extension (or guard interval) is usually added to the OFDM waveform as described in (8.66) and (8.67) to combat ISI. When a cyclic suffix is used, the corresponding sample sequence is X g n,m = X n,(m)n (8.74)

4 132 NON-BINARY BANDPASS MODULAION AND ERROR PROBABILIY { x k } x 0 x 1 x x x N- 2 N- 1 IFF X X X X 0 1 N- 2 N- 1 { X n } insert guard { g X n } g X I n { } D/A ~ si ( t) X g X Q n { } D/A ~s Q ( t) Figure 8.9 Block diagram of IDF-based baseband OFDM modulator with guard interval insertion and digital-to-analog conversion. = E x n,k e j2πkm N, m = 0, 1,..., N +G 1, (8.75) where G is the length of the guard interval in samples, and (m) N is the remainder after dividing m by N. his gives the vector X g n = {Xg n,m }N+G 1, where the values in the first and last G coordinates of the vectorx g n are the same. Likewise, when a cyclic prefix is used, the corresponding sample sequence is X g n,m = X n,(m)n (8.76) = E x n,k e j2πkm N, m = G,..., 1, 0, 1,..., N 1. (8.77) his yields the vector X g n = {Xn,m} g m= G, where again the first and last G coordinates of the vectorx g n are the same. he sampling interval after insertion of the guard interval,, must be compressed in time such that g s (N +G) g s = N s he OFDM complex envelope can be generated by splitting the complex-valued output vector X m into its real and imaginary parts, R(X n ) and I(X n ), respectively. he sequences {R(X nm )} and {I(X nm )} are then input to a pair of balanced digital-to-analog converters (DACs) to generate the real and imaginary components s I (t) and s Q (t), respectively, of the complex envelope s(t), during the time interval n t (n + 1). As shown in Fig. 8.9, the OFDM baseband modulator consists of an IFF operation, followed by guard interval insertion and digital-to-analog conversion. It is instructive at this stage to realize that the waveform generated by using the IDF OFDM baseband modulator is not exactly the same as the waveform generated from the analog waveform definition of OFDM. Consider for example, the OFDM waveform without a cyclic guard in (8.61) and (8.62). he analog waveform definition uses the rectangular amplitude shaping pulse u (t) that is strictly time-limited to seconds. Later, we will see that the corresponding OFDM power spectrum has infinite bandwidth, and any finite sampling rate of the complex envelope will necessarily lead to aliasing and imperfect

5 ORHOGONAL FREQUENCY DIVISION MULIPLEXING (OFDM) 133 reconstruction. With the IDF OFDM baseband modulator, we apply the IDF/IFF outputs to a pair of balanced DACs as explained earlier. However, the ideal DAC is an ideal low-pass filter with cutoff frequency 1/(2 s ), with a corresponding noncausal impulse response given by h(t) = sinc(t/ s ). Since the ideal DAC is nonrealizable, a causal reconstruction filter is used instead. However, such a filter will necessarily generate a waveform that is not strictly bandlimited, thus leading to side lobes. In conclusion, the side lobes of the analog OFDM waveform are inherent in the waveform itself due to rectangular pulse shaping, whereas the side lobes with the IDF implementation arises from the non-ideal reconstruction filter, i.e., the DAC Error Probability for OFDM he OFDM baseband demodulator is usually implemented by using a fast Fourier transform (FF), as discussed in Section 8.4. Following the development in Section 8.4, suppose that the discrete-time sequence X g n = {Xn,m g }N+G 1 is passed through a balanced pair of digital-to-analog converters (DACs), as shown in Fig he real and imaginary components of the resulting complex envelope are applied to a quadrature modulator as shown in Fig.??, and the resulting bandpass OFDM waveform is transmitted over an AWGN channel. he receiver first uses a quadrature demodulator as shown in Fig. 6.3 to extract the received complex envelope r(t) = r I (t)+j r Q (t). Suppose that the quadrature components r I (t) and r Q (t) are each passed through an ideal anti-aliasing filter (ideal low-pass filter) having a cutoff frequency 1/(2s) g followed by an analog-to-digital converter (ADC) as shown in Fig his produces the received complex-valued sample sequence R g n = {Rg n,m }N+G 1, where R g n,m = Xg n,m +ñ n,m m = 0,1,...,N +G 1, (8.78) and the ñ n,m are complex-valued Gaussian noise samples. For an ideal anti-aliasing filter having a cutoff frequency1/(2 g s ), theñ n,m are independent zero-mean complex Gaussian random variables with varianceσ 2 = 1 2 E[ ñ n,m 2 ] = N o / g s, whereg s = N s/(n+g). Assuming a cyclic suffix as discussed in Sect , the receiver first removes the guard interval according to R n,m = R g n,g+(m G) N, 0 m N 1, (8.79) where(m) N is the residue ofmmodulon. his operation simply deletes the firstgvalues of Rn,m g in (8.88) and replaces them with the last G values of Rg n,m, resulting in a new length-n sequence R n,m in (8.79). Demodulation is then performed by computing the FF on the block R n = {R n,m } to yield the vector z n = {z n,k } of N decision variables z n,k = 1 N R n,m e j2πkm N = Ex n,k +ν n,k, k = 0,...,N 1, (8.80) where the noise terms are given by ν n,k = 1 N ñ n,m e j2πkm N, k = 0,...,N 1. (8.81)

6 134 NON-BINARY BANDPASS MODULAION AND ERROR PROBABILIY ~ r ( t ) I A/D { R } I,n R remove R guard 0 R 1 RN- 2 RN- 1 ~ r ( t ) Q A/D { } R Q,n FF z 0 z 1 N- 2 z z N- 1 { z } k serial metric computer µ( v m ) z Figure 8.10 Block diagram of OFDM receiver. It can be shown that the ν n,k are zero mean complex Gaussian random variables with covariance φ j,k = 1 2 E[ν n,jνn,k] = N o Ns g δ jk. (8.82) Hence, the z n,k are independent Gaussian random variables with mean Ex n,k and variancen o /Ns g. o be consistent with our earlier results for PSK and QAM signals, we can multiply thez n,k for convenience by the scalar Ns g. Such scaling gives z n,k = EN/(N +G)x n,k + ν n,k, (8.83) where the ν n,k are i.i.d. zero-mean Gaussian random variables with variance N o. Notice that EN/(N +G)x n,k is equal to the complex signal vector that is transmitted on the ith sub-carrier, where the term N/(N +G) represents the loss in effective symbol energy due to the insertion of the cyclic guard interval. For each of the z n,k, the receiver decides in favor of the complex-valued data symbol x n,k that minimizes the squared Euclidean distance µ( s n,k ) = z n,k EN/(N +G)x n,k 2, k = 0,...,N 1. (8.84) hus, for each OFDM block, N symbol decisions are made, one for each of the N subcarriers. It is apparent from (8.83) that the probability of symbol error is identical to that achieved with independent modulation on each of the sub-carriers. his is expected, because the sub-carriers are mutually orthogonal OFDM on ISI Channels Consider an OFDM system where the number of sub-carriers N is chosen to be large enough so that the channel transfer function (t, f) is essentially constant across sub-bands of width1/. hen (t,k). = (t,k f ), k f 1/(2) f k f +(1/(2), (8.85) and no equalization is necessary because the ISI is negligible. Viewing the problem another way, if the block length N is chosen so that N L, where L is the length of the discrete-time channel impulse response, then the ISI will only affect a small fraction of the

7 ORHOGONAL FREQUENCY DIVISION MULIPLEXING (OFDM) 135 symbol transmitted on each sub-carrier. Weinstein and Ebert [4] came up with an ingenious solution, whereby they inserted a guard interval in the form of a length-g cyclic prefix or cyclic suffix to each IDF output vector X n = {X n,m }. In fact, if the discrete-time channel impulse response has duration L G, a cyclic guard interval can completely remove the ISI in a very efficient fashion as we now describe. Suppose that the IDF output vector X n = {X n,m } suffix to yield the vectorx g n = {X g n,m} N+G 1, where is appended with a cyclic X g n,m = X n,(m) N (8.86) = E x n,k e j2πkm N, m = 0, 1,..., N +G 1, (8.87) G is the length of the guard interval in samples, and (m) N is the residue of m modulo N. o maintain the data rate R s = 1/ s, the DAC in the transmitter is clocked with rate Rs g = N+G N R s, due to the insertion of the cyclic guard interval. Consider a time-invariant ISI channel with impulse responseg(t). he combination of the ADC, waveform channel g(t), anti aliasing filter, and DAC yields an overall discretetime channel with sampled impulse response g = {g m } L, where L is the length of the discrete-time channel impulse response. he discrete-time linear convolution of the transmitted sequence {X g n} with the discrete-time channel produces the discrete-time received sequence{rn,m}, g where { m Rn,m g i=0 = g ix g n,m i + L i=m+1 g ix g n 1,N+G+m i +ñ n,m, 0 m < L L i=0 g ix g n,m i +ñ. n,m, L m N +G 1 (8.88) Notice that the first L samples of Rn,m g in OFDM block n are corrupted by ISI from the previous OFDM block n 1. o remove the ISI introduced by the channel, the first G received samples{rn,m} g G 1 are discarded and replaced with the last G received samples {Rn,m g }N+G 1 m=n, as shown in Fig If the length of the guard interval satisfies G L, then we obtain the received sequence R n,m = R g n,g+(m G) N L = g i X n,(m i)n +ñ n,(m i)n, 0 m N 1. (8.89) i=0 Note that the first term in (8.89) represents a circular convolution of the transmitted sequencex n = {X n,m } with the discrete-time channelg = {g m } L. As shown in Fig. 8.10, the OFDM baseband demodulator computes the DF (in practice the FF) of the length-n vectorr n after the guard interval has been removed. his yields the DF/FF output vector z n,i = 1 N 2πmi j R n,m e N = i Exn,i +ν n,i, 0 i N 1, (8.90)

8 136 NON-BINARY BANDPASS MODULAION AND ERROR PROBABILIY block block block n-1 n n+1 G ISI G ISI G ISI G G G Figure 8.11 Removal of ISI by using the cyclic suffix. where i = L 2πmi j g m e N (8.91) is the FF of the channel impulse response g = {g m } L, and the noise samples {ν n,i } are i.i.d with zero-mean and variancen o /(Ns g). Note that = { i} i=0 is the DF of the zero-padded sequence g = {g m } and is equal to the sampled frequency response of the channel. o be consistent with our earlier results, we can multiply the z n,i for convenience by the scalar Ns g. By following the same argument used in Sect , such scaling gives z n,i = EN/(N +G) i x n,i + ν n,i i = 0,...,N 1, (8.92) where the ν n,i are i.i.d. zero-mean Gaussian random variables with variancen o. Observe that each decision variable z n,i depends only on the corresponding data symbol x n,i and, therefore, the ISI has been completely removed. Once again, for each of the z n,i, the receiver decides in favor of the complex-valued data symbol x n,k that minimizes the squared Euclidean distance µ( s m ) = z n,i EN/(N +G) i x n,i 2. (8.93) hus, for each OFDM block, N symbol decisions must be made, one for each of the N sub-carriers.