ENEE 420 FALL 2007 COMMUNICATIONS SYSTEMS SAMPLING

Transcription

1 c 2008 by Armand M. Maowsi 1 ENEE 420 FALL 2007 COMMUNICATIONS SYSTEMS SAMPLING In these notes we discuss the sampling process and properties of some of its mathematical description. This culminates in the celebrated Shannon-Nyquist Sampling Theorem. ectangular pulses With > 0, the rectangular pulse p : is defined by 1 if 0 t p (t) 0 otherwise. Fourier analysis of rectangular pulses (1) For each f 0 in, straightforward calculations show P (f) p (t)e j2πft dt 0 e j2πft dt e j2πf 1 j2πf sin(πf) e jπf πf while Therefore, (2) P (f), f 0. sin(πf) e jπf if f 0 πf P (f) if f 0.

2 c 2008 by Armand M. Maowsi 2 From now on, the parameters and T s are selected so that 0 < < T s, and we write f s 1 T s. Throughout, we use the notation to denote the summation 0,±1,... over all integers. Train of rectangular pulses The train of pulses associated with p is the signal c : given by c (t) : p (t T s ), t. The signal c being periodic with period T s, it admits a Fourier series representation, namely c (t) α ()e j2πfst, t with Fourier coefficients given by α () By virtue of (2) we find (3) (4) α () It is now plain that Natural sampling c (t) 1 Ts p (t)e j2πfst dt T s 0 1 T s P (f s ), 0, ±1, ±2,... 1 sin(πf s) T s πf s e jπfs if ±1, ±2,... T s if 0. α () e j2πfst, t. Let g : denote an information-bearing signal. Natural sampling gives rise to the signal g Nat, : defined by g Nat, (t) c (t)g(t) p (t T s ) g(t), t.

3 c 2008 by Armand M. Maowsi 3 Its Fourier transform is given by G Nat, (f) g Nat, (t)e j2πft dt c (t)g(t)e j2πft dt ) ( α ()e j2πfst g(t)e j2πft dt α () g(t)e j2πfst e j2πft dt α () g(t)e j2π(f fs)t dt (5) α ()G(f f s ), f. As a result, we also find G Nat, (f) g Nat, (t) e j2πft dt α () (6) G(f f s ), f. Ideal pulses It may seem natural to model an instantaneous (or ideal) pulse (at time t 0) as a mapping (or function) p : such that 0 if t 0 p(t) 1 if t 0. Unfortunately, such a definition is not a useful one due to the following fact: From the point of view of Fourier analysis, the function p is indistinguishable from the identically zero function. Instead, we model an ideal pulse by a Dirac function δ :. We draw attention to the fact that although we present the pulse δ as if it were a function, this is far from being the case! The terminology is an accepted one and we shall use it throughout.

4 c 2008 by Armand M. Maowsi 4 Formally, we can compute the Fourier transform of the Dirac function as (7) δ(t)e j2πft dt 1, f. Train of ideal pulses In analogy with the notion of train of natural pulses, we can associate with ideal pulses the corresponding notion of pulse train. We define such a train of ideal pulses as the mapping c δ : given by c δ (t) δ(t T s), t. Again caution is in order: While the train c δ of ideal pulses may have been presented as if it were a mapping, this is not so due to the (unresolved) conceptional difficulties mentionned earlier. Yet, despite the fact that such a train of ideal pulses has only been vaguely defined (if at all), this notion does serve a useful purpose, albeit a formal one, as will become apparent below. Again proceeding formally, we compute the Fourier transform of c δ as C δ (f) c δ (t)e j2πft dt ( ) δ(t T s) e j2πft dt δ(t T s )e j2πft dt (8) e j2πfts, f. Ideal sampling Let g : denote an information-bearing signal. Ideal sampling produces the signal g Ideal : given by g Ideal (t) c δ (t)g(t) δ(t T s)g(t) (9) g(t s)δ(t T s ), t.

5 c 2008 by Armand M. Maowsi 5 (10) Its Fourier transform is therefore given by G Ideal (f) g Ideal (t)e j2πft dt g(t s) δ(t T s )e j2πft dt g(t s)e j2πfts f. This expression turns out to be not too useful for our purposes, a state of affairs which prompts us to see a different approach for evaluating the Fourier transform G Ideal. Although the expressions to be given are formal expressions for the Fourier transform of an object which has not been fully defined, they will turn out to be useful for understanding the properties of the sampling process. From natural to ideal pulses For each > 0, the normalized rectangular pulse p : is defined by p (t) : p 1 if 0 t (t) 0 otherwise. Its Fourier transform is simply given by sin(πf) P (f) : P e (f) jπf if f 0 πf 1 if f 0. The convergence P P (f) (f) 0 0 1, f and the expression (7) for the Fourier transform of a Dirac function together suggest that the ideal pulse δ can be thought of as the it of the normalized pulse p as goes to zero. Conversely, the normalized pulse p with small can be interpreted as a good approximation for the ideal pulse δ. We symbolically summarize such a convergence as (11) p δ. 0

6 c 2008 by Armand M. Maowsi 6 We mae no attempt at giving a precise meaning to the convergence (11). In fact, a precise definition is certainly fraught with difficulties, some of which are already apparent from the pointwise convergence 0 if t 0 p 0 (t) if t 0. esolving these difficulties is beyond the scope of these notes. From natural to ideal sampling Let go to zero: Since it is plain that sin(πf s ) 0 πf s 1, 0 P α () (f s ) 1, 0, ±1, ±2, Therefore, formally we conclude that c (t) 0 0 α () ( ) α () 0 e j2πfst e j2πfst (12) 1 T s ej2πfst. (13) On the other hand, formally wielding (11) we find c (t) 0 0 p (t T s) 0 p (t T s ) δ(t T s) c δ (t), t.

7 c 2008 by Armand M. Maowsi 7 Combining we conclude that δ(t T s) 1 e j2πfst, t. T s This (formal) relation often appears in the literature, and is nown as Poisson s summation formula. From train of natural pulses to train of ideal pulses (14) Next, we see that In short, (15) G Nat, (f) G Ideal (f) 0 0 α () ( 0 α () G(f f s ) ) G(f f s ), f. G Ideal (f) 1 T s G(f f s), f. ecovering g from g Nat, Assume the signal g : to be band-ited with cut-off frequency W, i.e., (16) G(f) 0, f > W. The frequency f Nyq 2W plays a particular role and is nown as the Nyquist rate for g. Under the condition (16) the translates G(f f s ) and G(f lf s ) do not overlap if l whenever the condition (17) 2W < f s holds. More precisely, under (16) and (17), the translates G(f f s ) and G(f lf s ) with l cannot be simultaneously non-zero. As a result, in the expression for the Fourier transform of g Nat,, namely G Nat, (f) α () G(f f s ), f,

8 c 2008 by Armand M. Maowsi 8 at most one of the terms G(f f s ), 0, ±1, ±2,..., is ever non-zero for a given frequency f under the condition (17). With this in mind, consider a lowpass filter H with cutoff frequency W h, i.e., H(f) 0, f > W h. If we select W h so that (18) then W < W h < f s W, (19) H(f) G Nat, (f) α () H(f)G(f f s ) α (0)H(f)G(f), f since for all ±1,..., we have H(f)G(f f s ) 0, f. In particular, if we tae the lowpass filter H to be 1 if f W h H(f) 0 otherwise, then we obtain H(f) G Nat, (f) T s G(f), f. Thus, the lowpass information-bearing signal m : can be recovered fully from g Nat, by linear processing. The Shannon-Nyquist Sampling Theorem We now show that not only can g also be recovered from g Ideal, but that this signal can be reconstructed from the samples {g(t s ), 0, ±1,...}. Here as well we assume the signal g : to be band-ited with cut-off frequency W. Moreover, the condition (17) is enforced. From (15) we readily conclude that G Ideal (f) 1 G(f), f W T s so that G(f) T s G Ideal (f), f W

9 c 2008 by Armand M. Maowsi 9 eporting this fact into (10) we conclude that G(f) T s g(t s)e j2πfts, f W Since the signal g is band-ited with cut-off frequency W, this last relation already shows that the samples should be sufficient to reconstruct the original signal g! By Fourier inversion, we get g(t) G(f)e j2πft dt (20) W W W W G(f)e j2πft dt T s g(t s) T s (T s g(t s)e j2πfts ) e j2πft dt W W e j2πf(t Ts) dt g(t s) ej2π(t Ts)W e j2π(t Ts)W j2π(t T s ) so that g(t) T s g(t s) sin (2π(t T s)w) (21), t. π(t T s ) This expression is sometimes written in terms of the Nyquist rate for the signal g, namely g(t) T s g(t s) sin (π(t T s)f Nyq ) π(t T s ) (22) T s f Nyq g(t s) sinc ((t T s )f Nyq ), t where we have defined sin (πt) sinc(t), t. πt With f s f Nyq, we get T s f Nyq 1 and the last relation becomes g(t) g(t s) sinc ((t T s )f Nyq ) (23) g(t s) sinc (f Nyq t ), t.