1 the Fourier Transform of Discrete Time Sequences

Sampling and Reconstruction Supplement 1 The Fourier Transform of Discrete Time Sequences Wehave found that x t, the analog signal sampled by a train of Dirac delta functions with s spacing T =1=f , had aFourier transform related to Ffxtg = X f by s 1 X X f mf 1 X f = f s s s m=1 This Fourier transform is p erio dic in f , with p erio d f , as is easily shown: s 1 X X f kf = f X f kf mf s s s s s m=1 1 X = f X f m + k f s s m=1 1 X X f lf ; l = m + k = f s s l =1 = X f : 2 s We de ned X f as b eing identical to X f . Since this function is p erio dic, it can be d s represented by aFourier series, which led us to the inverse DTFT Z f =2 s 1 j 2nT f x n = X f e df T d f f =2 s s Z 1=2T j 2nT f = T X f e df 3 d 1=2T This view tends to leave the reader with the impression that without the ideal impulsive sampling representation, we would be at a loss to deal with the DTFT. Actually, we can do nicely without resorting to impulsive sampling, if we like, for generating the DTFT. Consider the sequence x n, and supp ose that there is some continuous time signal xt, T whose instantaneous values at multiples of time T constitute the values of x n, as in Fig. T 1. The CT signal is equal to Z 1 j 2f t xt= X f e df 4 1 if X f = Ffxtg. But also, Z 1 j 2f nT X f e df; 5 x n=xnT = T 1 1 x(t) t -T 0 T 8T xT (n) n -1 0 1 8 Figure 1: The samples xnT =x n are shown here simply as the function xtevaluated T at the particular instants fnT g. X(f) m=0 m=1 m=-1 m=2 m=-2 f -3/(2T) -1/(2T) 1/(2T) 3/(2T) Figure 2: The inverse Fourier transform of X f can b e evaluated as the sum of integrals over these nite sub-intervals. The subsequent simpli cations are p ossible b ecause the complex exp onentials by which X f is multiplied in the inversion have exp onents j 2nT f , and are all therefore p erio dic on these intervals. where we've just chosen to evaluate the inverse FT only at the sample lo cations. Now, just for fun, we'll express the integral in 5 as the summation of integrals over nite intervals, each with length 1=T , as illustrated in Figure 2. Z m 1 1 + X T 2T j 2f nT X f e df x n = T 1 m m=1 T 2T Z 1 1 X m m 2T ~ j 2nT f m=T ~ ~ ~ = e df; for f = f + X f 1 T T m=1 2T Z 1 1 X m 2T j 2nT f j 2nm e e df 6 = X f 1 T m=1 2T But the second exp onential in 6 is equal to 1 for all integers n and m, and can b e discarded. This is what we get for restricting our consideration of the inverse FT to the discrete lo cations fnT g. Thus we have 1 Z 1 X m 2T j 2nT f x n = X f e df T 1 T m=1 2T 2 1 Z 1 X 1 m 2T j 2nT f = T e df X f 1 T T m=1 2T f Z s 1 X 1 2 j 2nT f f X f mf e df = s s f s f s m=1 2 f Z s 1 2 j 2nT f = X f e df 7 d f s f s 2 for any continuous time signal xt , X f which matches the samples x n. This is T exactly the inverse DTFT of 3 we found earlier, when the expression within the brackets was taken from the impulsively sampled signal. So while the impulse train sampling is a very useful to ol, it is not necessary to create x n and its DTFT. The impulsive representation T is p erhaps more imp ortant in analyzing the reconstruction of CT signals from samples. 2 Filters for Reconstructing Continuous Time Signals from Samples Often our system will, at some p oint, need to reconstruct from x n a CT signal x^t to T approximate xt. Our analytical description could b e a mo del for digital/analog conversion at the output of many digital systems such as CD players, digital telephones, etc. This transition from a DT signal to aCT signal will b e made by assigning to each sample x n T a CT delta function t nT , after which we can use familiar ltering analysis on the CT signal x t. s 2.1 Bandlimited Interp olation for Perfect Reconstruction of Nyquist Sampled Signals Xs (f) m=0 Ideal HR (f) m=-1fs X(f) m=1 f -fs 0 fs Figure 3: The sp ectrum X f of the sampled signal, x t, with xt bandlimited, and the s s Nyquist sampling rate satis ed. Simple extraction of the m = 0 sp ectral replication yields exactly x^t=xt. 3 For the moment, we'll assume that xt is bandlimited, X f = 0; jf j f < 1 8 h and T satis es the Nyquist sampling criterion of f > 2f , or T < 1=2f . Clearly, as s h h illustrated in Fig. 3, a CT signal with Fourier transform X f , or the m =0 replication of the CT sp ectrum in the DT transform, could b e reconstructed by p erfect lowpass ltering. This simple graphic argument suggests that a lowpass lter, with cuto frequency of f =2, should recover the bandlimited signal perfectly. This must corresp ond to convolution s of the impulsive samples in x t with some lter h t in the time domain. To see what s R sort of h t this pro duces, we lo ok at the inverse Fourier transform: R Z 1 j 2f t h t = H f e df R R 1 ! Z f =2 s 1 j 2f t e df = f f =2 s s jf t jf t s s e e = j 2f t s sinf t t s = = sinc : 9 f t T s A plot of this lter is given in Fig. 4, with T=0.1, and zero crossings marked. Bandlimited Sinc Interpolation Filter FT Magnitude of Bandlimited Interpolation Filter 0.1 1 0.08 0.8 0.6 0.06 0.4 0.04 0.2 0.02 0 0 -0.2 0 5 10 15 20 25 30 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Frequency (Hz) Figure 4: The bandlimited interp olation lter corresp onding to sample spacing T of 0.1. Note the extent of the \tails" of this highly non-causal lter. To use CT system and transform theory,wehave ascrib ed a Dirac delta function to each sample, and can apply the p erfect H f lter in terms of the time-domain op eration with R 4 1 F fH f g = h t. R R 1 X xt = x n t nT h t T R n=1 Z 1 1 X x n nT h t d = T R 1 n=1 1 X = x nh t nT : 10 T R n=1 In other words, this linear interp olation pro cess, whichisintended to give x de ned values between samples, amounts to adding a bunch of copies of h t, each scaled by the value of R a particular sample, and shifted by nT . Knowing that the H f in 9 repro duces exactly X f , and substituting the sinc for R h tinto 10, we have R 1 X t nT xt= x nsinc 11 T T n=1 This is very useful, esp ecially for intuitiveevaluations; it should o ccupy a place in your heart alongside the sampling theorem. In fact, 11 is typically included as part of the sampling theorem. In other words, 11 falls into the \don't you ever dare forget this" category. It gives us an interp olation lter for exact repro duction of xt from samples taken at at least Nyquist rate. In Fig. 5, an example of the approximation of a bandlimited signal by a truncated sinc expansion is illustrated. This example features a cosine wave of frequency 0.5, which is sampled well ab ove the Nyquist rate. If we were to provide the contributions of samples outside the window also, a p erfect reconstruction would b e p ossible. 2.2 Non-Ideal Non-Bandlimited Interp olation Lo oking at 11 as the means of computing the value of xt at, for example, t = t , we 0 see that to nd the interp olated value, we need the contributions of an in nite number of samples, including the very distant ones. This is b ecause the sinc function is noncausal, and in fact has very heavy \tails," falling o in magnitude only as 1=t. It is easy to see the e ects in Fig.

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