16 4. Discrete Observations 4.0.1. Sampling. The above results show that a band-limited function can be reconstructed perfectly from an infinite set of (perfect) samples. Similarly, the Fourier transform of a time-limited function can be reconstructed perfectly from an infinite number of (perfect) samples (the Fourier Series frequencies). In observational practice, functions must be both band-limited (one cannot store an infinite number of Fourier coe!cients) and time-limited (one cannot store an infinite number of samples). Before exploring what this all means, let us vary the problem slightly. Suppose we have { (w) with Fourier transform ˆ{ (v) and we sample { (w) at uniform intervals pw without paying attention, initially, as to whether it is band-limited or not. What is the relationship between the Fourier transform of the sampled function and that of { (w)? That is, the above development does not tell us how to compute a Fourier transform from a set of samples. One could use (3=2) > interpolating before computing the Fourier integral. As it turns out, this is unnecessary. We need some way to connect the sampled function with the underlying continuous values. The  function proves to be the ideal representation. Eq. (2.13) produces a single sample at time wp = The quantity, X4{LLL (w)= { (w)  (w qw) > (4.1) q= 4 vanishes except at w = tw for any integer t= The value associated with {LLL (w) at that time is found b y integrating it in an infinitesimal interval % + tw w % + tw and one finds immediately that   {LLL (tw)= { (tw) = Note that all measurements are integrals ov er some time interval, no matter how Continued on next page...17 4. D ISC R E T E O B SE RVAT IO N S short (perhaps nanoseconds). Because the  function is infinitesimally broad in time, the briefest of measuremen t integrals is adequate to assign a value.1= Let us Fourier analyze {LLL (w) > and evaluate it in two separate ways: (I) Direct sum. Z X4ˆ{LLL (v)= 4 { (w)  (w pw) h 2lvw gw p= 4 4 X4= { (pw) h 2lvpw = (4.2) p= 4 (II) By convolution. à ! X4{LLL (v)=ˆˆ{ (v) F  (w pw) = (4.3) p=¡P¢ 4 What is F 4 4  (w pw) ? We have, by direct integration, p=Ã! X X4 4F  (w pw) = h 2lpvw (4.4) p= p= 4 4 What function is this? The right-hand-side of (4.4) is clearly a Fourier series for a function periodic with period 1@w> in v= Iassertthatthe periodic function is w (v) > and the reader should confirm that computing the Fourier series representation of w (v) in the v domain, with period 1@w is exactly (4.4). But such a periodic  function can also be written2 X4w  (v q@w) (4.5) q= 4 Thus (4.3) can be written X4{LLL (v)=ˆˆ{ (v) w  (v q@w) q= 4 Z X4ˆ0= 4 { (v 0) w  (v q@w v ) gv 0 q= 4 4 X³ ´4qˆ= w { v (4.6) w q= 4 We now have two apparently very dierent representations of the Fourier transform of a sampled function. (I) Asserts two important things. The Fourier transform can be computed as the naive dis-cretization of the complex exponentials (or cosines and sines if one prefers) times the sample values. Equally important, the result is a periodic function with period 1@w= (Figure 5). Form (II) tells us that 13functio ns a re meanin g fu l o nl y when integ rat e d . Lighth ill (1 95 8 ) is a good prim e r on hand lin g t h em. Much of th e b ook has been b oiled down to the advice that, if in doubt about the m eaning of an integral, “integrate by parts”. 2Bracewell (1978) gives a complete discussion of the behavior of these othewise pecu liar functions. Note that we are ignoring all question s of convergence, interchange of summ ation and integration etc. Everything can be justified by appropri a te limiting pr oces s es .18 1. F R E Q UENCY DO M A IN FO RM ULATIO N 2 1.8 1.6 1.4 1.2 REAL(xhat) 1 0.8 0.6 0.4 0.2 0 -3 -2 -1 0 1 2 3 s Figure 5. Real part of the periodic Fourier transform of a sampled function. The baseband is defined as 1@2w v 1@2w,(here w =1)> but any interval of width   1@w is equivalent. These intervals are marked with vertical lines. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s a REAL(xhat) -3 -2 -1 0 1 2 3 s Figure 6. vd is the position where all Fourier transform amplitudes from the Fourier transform values indicated by the dots (Eq. 4.6) will appear. The baseband is indicated b y the vertical lines and any non-zero Fourier transform values outside this region will alias into it. the value of the Fourier transform at a particular frequency v is not in general equal to ˆ{ (v) = Rather it is the sum of all values of ˆ{ (v) separated by frequency 1/w= (Figure 6). This second form is clearly periodic with period 1@w> consistent with (I). Because of the periodicity, we can confine ourselv es for discussion, to one interval of width 1@w= By con vention we take it symmetric about v =0> in the range 1@ (2w)  v 1@ (2w) which we call the 19 baseband. We can now address the question of when ˆ { (v)? The {LLL (v) in the baseband will be equal to ˆanswer follows immediately from (4=6): if, and only if, ˆ{ (v) vanishes for v  |1@2w| = That is, the Fourier transform of a sampled function will be the Fourier transform of the original contin uous function only if the original function is bandlimited and w is chosen to be small enough such that ˆ{ (|v| A 1@w)= 0= We also see that there is no purpose in computing ˆ{LLL (v) outside the baseband: the function is perfectly periodic. We could use the sampling theorem to interpolate our samples before Fourier transforming. But that would produce a function whic h vanished outside the baseband–and we would be no wiser. Suppose the original continuous function is { (w)= D sin (2v0w) = (4.7) It follows immediately from the definition of the  function that l ˆ{ (v)= { (v + v0)  (v v0)} = (4.8)2 If we choose w? 1@2v0,we obtain the  functions in the baseband at the correct frequency. We ignore the  functions outside the baseband because we know them to be spurious. But suppose we choose, either knowing what we are doing, or in ignorance, wA 1@2v0. Then (4.6) tells us that it will appear, spuriously, at