Digital Baseband Systems. Reference: Digital Communications John G. Proakis
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1 Digital Baseband Systems Reference: Digital Communications John G. Proais Baseband Pulse Transmission Baseband digital signals - signals whose spectrum extend down to or near zero frequency. Model of the transmission lin Transmitter Channel Noise Receiver Input data G (f) T C(f) G (f) R To Threshold Detection Pulse generation and Shaping Consider a baseband binary PAM system i/p a(t) = a ( t T) o/p ( i.e. from the transmit filter into the channel) The receiver filter o/p s(t) = a g( t T) y(t) = a h( t T) + n(t) n(t) is filtered white noise. h(t) = g T (t) * c(t) * g R (t) in the frequency domain H(f) = G T (f). C(f). G R (f) The receive filter output y(t) is sampled at time t= it ( i integer) y(it) = a h[( i ) T] n( it) 75
2 = aih( 0) a h[( i ) T] n( it), i The first term a i h(0) represents the contribution of the i th transmitted pulse. The second term represents the residual effect of all other transmitted pulses on the decoding of the i th bit. The residual effect due to the occurrence of pulses before and after sampling instant it is called intersymbol interference (ISI). In the absence of noise and ISI y(it) = a i h(0) = a i if h(0) = 1 ( normalizing without loss of generality) i.e., under ideal conditions i th transmitted bit is decoded correctly. The presence of noise and ISI degrades the receiver performance and the objective is to minimize these effects and hence deliver the data to its destination with the smalleset error rate possible. When the signal to noise ratio is high (telephone system PSTN) the operation of the system is largely limited by ISI rather than noise. Thus first we neglect the effect of noise and focus attention on ISI. 76
3 Nyquist s Criterion for Distortionless Baseband Binary Transmission * Typically, the transfer function of the channel and the transmitted pulse shape are specified. Objective: To determine the transfer functions of the transmit and receive filters to reconstruct the original data sequence. * The ISI free condition is satisfied if i.e., if (i-) = n then 1,, h(it - T) = 0 i i (**) 1, n0 h(nt) = 0, n0 (***) where h(0) =1 by normalization. Thus ignoring noise term, y(it) = a i Therefore condition given in (**) ensures perfect reception in the absence of noise. Let s try to put this in the frequency domain perspective. If h(t) is sampled at a rate of 1/T, i.e., h(t) multiplied by a periodic impulse train with period T, gives us the signal i(t) = ( t nt) n h s (t) = h(t) i(t) = h(t) ( t nt) = h( nt) ( t nt) n n H s (f) = h( nt) e j 2 nt = 1 since h(0)=1, h (n0 T) = 0 n or 77
4 Hence the Fourier transform h s (t) = h(0) (t) = (t) H s (f) = F{(t)} = 1 Using frequency domain too we can obtain H s (f). The Fourier transform of a periodic impulse train is a periodic impulse train I(f) = 1 T n ( f nf s ) where f s = 1/T is the sampling rate (frequency) The Fourier transform of h s (t) is H s (f) = H(f) * I(f) = H(f) * 1 T = 1 T n Combining these results we have 1 T H( f nf s ) H( f nf s ) = 1 or (f s = 1/T) n n ( f nf s ) H( f n / T) = T n This is the frequency domain requirement for zero ISI. ---> Nyquist s Criterion for distortionless baseband transmission in the absence of noise. * Important to remember that H(f) refers to the overall system, incorporating the transmit filter, the channel, and the receive filter. Theorem (Nyquist) 78
5 The necessary and sufficient condition for h(t) to satisfy 1, n0 h(nt) = 0, n0 is that its Fourier transform H(f) satisfies H( f n / T) = T n * prove the sufficient condition. (?) Ideal Nyquist Channel Now suppose that the channel has a bandwidth of W. Then C(f) = 0 for f > W and hence H(f) = 0 for f > W. We have three cases. Let s define B(f) = H( f n / T) n Now the condition to be satisfied is B(f) = T for zero ISI. ( Insight: Since obviously B(f) is periodic it can be expanded as a Fourier series, i.e., B(f) = b e j2 nft and the condition says b 0 = T and b n = 0 for n 0. n n Also note that H s (f) = B(f)/T = 1/T b e j2 nft = h( nt) e j 2 nt This implies b n = T h(-nt) n n (This relation is used later.) n 79
6 1. When T < 1/2W or 1/T > 2W, B(f) consists of non-overlapping replicas of H(f) separated by 1/T as shown below. B(f) -1/T-W -1/T -1/T+W -W 0 W 1/T - W 1/T 1/T + W There is no choice for H(f) to ensure B(f) T in this case and hence there is no way that we can design a system with no ISI. 2. When T = 1/2W or 1/T=2W (the Nyquist rate), the replication of H(f) separated by 1/T are as shown below. B(f) -1/T+W = -W 0 W = 1/T - W It is clear that in this case there exists only one H(f) that results in B(f) = T, namely T, ( f W) H(f) = 0, ( otherwise) H(f) T -W 0 W 80
7 which corresponds to the pulse h(t) = sin( t / T ) t / T t sinc T This means that the smallest value of T for which transmission with zero ISI is possible is T=1/2W, and for this value h(t) has to be a sinc function. h(t) = sinc( t/t) t/t As we now, this is noncausal and physically nonrealizable. To be physically realizable a delayed version of it is used, i.e., sinc[(t - t0)/t] is used and t0 is chosen such that for t < 0, sinc[(t - t0)/t] 0. The sampling time should also be shifted to nt + t0. * Another difficulty is that rate of convergence to zero is slow. The tails of h(t) decay as 1/t, thus a small timing jitter at the receiver or mistiming error in sampling could produce an infinite series of ISI components. Another thing is that the amplitude characteristic of H(f) should be flat from -W to W and zero elsewhere. This is physically unrealizable because of the abrupt transitions at the band edges W. Therefore in practice trying to approach (sin x)/ x pulse at the receiver would usually be counter productive. 81
8 * Example of ideal Nyquist signalling - Notice zero ISI at sampling points Case No 3: When T > 1/2W, B(f) consists of overlapping replications of H(f) separated by 1/T as shown below. In this case there exist numerous choices for H(f) T. B(f) T -W -1/T+W 0 1/T - W W *The practical difficulties encountered with the ideal Nyquist channel can be overcome by reducing the signaling rate from the maximum value 2W to an adjustable value between W and 2W. A particular pulse spectrum, for the T > 1/2W case, that has desirable spectral properties and has been widely used in practice is the raised cosine spectrum. 82
9 The raised cosine frequency characteristic consists of a flat portion and a rolloff portion that has a sinusoidal form as follows, H( f ) T 1 0 f 2T T T cos f f T 2T 2T 0 1 f 2T is called the rolloff factor and taes the values in the range 0 1. The bandwidth occupied by the signal beyond the Nyquist frequency 1/2T is called the excess bandwidth and is usually expressed as a percentage of the Nyquist frequency( excess bw = 50% or 50% rolloff). The pulse having raised cosine spectrum h(t) is normalized s.t. h(0) = 1 cos( t / T) h( t) sin c( t / T) t / T The following figures illustrates the raised cosine spectral characteristics and the corresponding pulses for = 0,0.5 and 1.0. The tails of h(t) decay as 1/t 3 for > 0. Thus a mistiming error in sampling leads to a series of ISI terms that converges to a finite value. 2 83
10 1 0.8 T=0.25 h(t) =0 =1 T 2T 3T -0.2 = T=0.25 =0 =0.5 H(f) =
11 Note that for the pulse reduces to h(t) = sinc(t/t) and the symbol rate 1/T = 2W. When the symbol rate is reduced to 1/T = W. (i.e. rate reduces by W for this value of or we can say if we eep the original signaling rate (2W) to have no ISI we need a bandwidth of 2W which we considered to be fixed at W in the foregoing analysis) Due to smooth characteristics of the raised cosine spectrum it is possible to design practical filters for the transmitter and the receiver that approximate the overall desired frequency response. In the special case where the channel is ideal,i.e., C(f) = 1, f W we have H(f) = G T (f)g R (f) If the receiver filter G R (f) is matched to the transmitter filter G T (f) then I.e., ideally H(f) = G T (f) 2 G T (f) = H( f ) e j2 ft 0 and G R (f) = G* T (f) square root raised cosine filter where t0 is some delay required to ensure the physical reliability of the filter. Note also that an additional delay is necessary to ensure the physical realizability of the receiving filter as well. * Overall raised cosine spectral characteristic is split evenly between the transmitting filter and the receiving filter. Problem: Calculate the impulse response of the square root raised cosine filter Signals & Systems 85
12 Design of Bandlimited Signals with Controlled ISI - Partial-Response (PR) Signaling or Correlative-level Coding * To achieve zero ISI signaling rate has to be reduced from the Nyquist rate of 2W symbols/s to realize practical transmitting and receiving filters. * Here the objective is to maintain Nyquist rate of signaling allowing for a controlled amount of ISI. Condition for zero ISI B(f) = T [ h(nt) = 0 for n 0]. Now assume we have the following condition. h(nt) = 1 for n=0,1 and zero otherwise ( allowing one ISI term, how this is implemented is explained later) This is equivalent to i.e., b n = T for n = 0,-1 and zero otherwise, since B(f) = T + T e -j2ft b n = Th(-nT) As we have discussed before it is impossible to satisfy this condition for T < 1/2W. However for T=1/2W this is satisfied if H(f) = 1 ( 1 2W e j f / W ) f W 0 otherwise Hence h(t) = 1 2 W e j f / W f cos 2W f W 0 otherwise h(t) = sinc(2wt) + sinc[2(wt - 1/2)] This pulse is called duobinary signal pulse where duo implies doubling of the transmission capacity of a binary system. This particular form correlative-level coding is called class I partial response. 86
13 This can be represented in the following manner. a + c Ideal Nyquist Channel H (f) N T Delay Filter Consider the above binary system where a = 1. c = a + a -1 ; c { 0, 2} This will change the two level uncorrelated a sequence into a sequence of correlated three level pulses c. This correlation between the adjacent pulses may be viewed as introducing ISI into the transmitted signal in an artificial manner. However, the ISI introduced is under designer s control, which is the basis of correlative coding. Transfer response of the filter with delay H d (f) = 1 + e -j2ft This is combined with the ideal Nyquist filter H N (f) = T = 1/2W f < W Thus the overall response same as before. H(f) = T + T e -j2ft = 1 ( 1 2W e j f / W ) f W 0 otherwise 87
14 The spectrum is given below along with the pulse shape. Note that the spectrum decays to zero smoothly which means that physically realizable filters can be designed that approximate this spectrum very closely. * A symbol rate of 2W is achieved. Another special case that leads to physically realizable transmitting and receiving filters is specified by the samples h(nt) = 1 ( n 1) 1 ( n 1) 0 ( otherwise) The corresponding pulse and its spectrum is ( t T) ( t T) h(t) = sinc sinc T T H(f) = 1 2W e jf / W e jf / W j f ( ) sin W W f W 0 f W This pulse and its magnitude are illustrated in the figures below. It is called a modified duobinary signal pulse. The spectrum has a zero or a spectral null at f=0 (d.c.), maing it suitable for transmission over a channel that does not pass d.c. H(f) -W W f 88
15 Generalized form of Correlative-Level Coding (Partial-Response Signaling) One can obtain other filter characteristics by selecting different values for the sample h(nt) and more than two non zero samples. However as we select more samples getting rid of controlled ISI becomes more complicated. In general, the class of bandlimited signal pulses have the form 2 h(t) = h( nt)sinc2w t n 2W, T= 1/2W Corresponding spectra H(f) = 1 jnf / W h( nt) e f W 2W n 0 f W This can be realized in the manner given below. a ( t nt) n n T T T h 0 h 1 h N-1 Ideal filter 89
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