Chapter 6 Discrete Fourier Transform The Modern World According to some, the modern world began in 1965 when J. Cooley and J. Tukey published their account of an efficient method for numerical computation of the Fourier transform.1 According to some others, the method was known to Gauss in the mid 1800s; the idea that lies at the heart of the algorithm is clearly present in an unpublished paper that appeared posthumously in 1866. Take your pick. Whatever the history, the present and future demands are that we process continuous signals by discrete methods. Computers and digital processing systems can work with finite sums only. To turn the continuous into the discrete and finite requires that a signal be both time-limited and band-limited, something we know cannot be true, and that we take a finite number of samples, something we know cannot suffice. But it works. At least such approximations work to the extent that a large fraction of the world’s economy depends upon them, and that’s not a bad measure of success. Some would argue that one shouldn’t think in terms of “turning the continuous into the discrete”, but rather that measurements and data in the real world come to us in discrete form, and that’s how we should understand the world and work with it. Period. For some, “discrete” versus “continuous” rises to the level of a religious war, and this battle can be fought out in the different approaches to the discrete Fourier transform. I’m taking sides, at least initially, in favor of “from continuous to discrete” as a way of motivating the definition. For one thing, we have built up a lot of intuition and understanding for the Fourier transform, its properties, and its uses, and I hope some of that intuition can transfer as we now work with the discrete Fourier transform. My choice for now is to make the discrete look as similar to the continuous as possible. 6.1 From Continuous to Discrete Start with a signal f(t) and its Fourier transform Ff(s), both functions of a continuous variable. We want to: • Find a discrete version of f(t) that’s a reasonable approximation of f(t). • Find a discrete version of Ff(s) that’s a reasonable approximation of Ff(s). 1 Incidentally, Tukey is also credited with coining the term “bit” as an abbreviation for “binary digit” — how about that for immortality! 252 Chapter 6 Discrete Fourier Transform • Find a way that the discrete version of Ff(s) is related to the discrete version of f(t) that’s a reasonable approximation to the way Ff(s) is related to f(t). Good things to try for, but it’s not quite straightforward. Here’s the setup. We suppose that f(t) is zero outside of 0 ≤ t ≤ L. We also suppose that the Fourier transform Ff(s)is zero, or effectively zero (beyond our ability to measure, negligible energy, whatever) outside of 0 <s<2B. I’m taking the support of Ff to be the interval from 0 to 2B instead of −B to B because it will make the initial indexing of the sample points easier; this will not be an issue in the end. We’ll also take L and B to both be integers so we don’t have to round up or down in any of the considerations that follow; you can think of that as our first concession to the discrete. Thus we are regarding f(t) as both time-limited and band-limited, with the knowledge that this can only be approximately true. Remember, however, that we’re ultimately going to come up with a definition of a discrete Fourier transform that will make sense in and of itself regardless of these shaky initial assumptions. After the definition is written down we could erase all that came before it, or merely cast a brief glance backwards from the discrete to the continuous with a few comments on how the former approximates the latter. Many treatments of the discrete Fourier transform that start with the discrete and stay with the discrete do just that. We’re trying not to do that. According to the sampling theorem (misapplied here, but play along), we can reconstruct f(t) perfectly from its samples if we sample at the rate 2B samples per second. Since f(t) is defined on an interval of length L and the samples are 1/2B apart, that means that we want a total of L N = =2BL (note that N is therefore even) 1/2B evenly spaced samples, at the points 1 2 N − 1 t =0,t = ,t = ,..., t − = . 0 1 2B 2 2B N 1 2B To know the values f(tk) is to know f(t), reasonably well. Thus we state: • The discrete version of f(t) is the list of sampled values f(t0),f(t1),...,f(tN−1). Next, represent the discrete version of f(t) (the list of sampled values) “continuously” with the aid of a finite impulse train (a finite III-function) at the sample points: N−1 X δ(t − tn) n=0 that is, N−1 N−1 fdiscrete(t)=f(t) X δ(t − tn)= X f(tn)δ(t − tn) . n=0 n=0 This is what we have considered previously as the sampled form of f(t). The Fourier transform of fdiscrete is N−1 N−1 −2πistn Ffdiscrete(s)= X f(tn)Fδ(t − tn)= X f(tn)e . n=0 n=0 6.1 From Continuous to Discrete 253 This is close to what we want — it’s the continuous Fourier transform of the sampled form of f(t). Now let’s change perspective and look at things in the frequency domain. The function f(t) is limited to 0 ≤ t ≤ L, and this determines a sampling rate for reconstructing Ff(s) from its samples in the frequency domain. The sampling rate is 1/L.(Not 2/L: think about how you would derive the sampling formula when the function is nonzero over an interval from 0 to p/2 rather than −p/2to+p/2.) We sample Ff(s) over the interval from 0 to 2B in the frequency domain at points spaced 1/L apart. The number of sample points is 2B =2BL = N 1/L the same number of sample points as for f(t). The sample points for Ff(s) are of the form m/L, and there are N of them: 1 N − 1 s =0,s = ,...,s − = . 0 1 L N 1 L The discrete version of Ff(s) that we take is not Ff(s) evaluated at these sample points sm. Rather, it is Ffdiscrete(s) evaluated at the sample points. We base the approximation of Ff(s) on the discrete version of f(t). To ease the notation write F (s) for Ffdiscrete(s). Then: • The discrete version of Ff(s) is the list of values N−1 N−1 N−1 −2πis0tn −2πis1tn −2πisN−1tn F (s0)=X f(tn)e ,F(s1)= X f(tn)e ,...,F(sN−1)=X f(tn)e . n=0 n=0 n=0 By this definition, we now have a way of going from the discrete version of f(t) to the discrete version of Ff(s), namely, N−1 −2πismtn F (sm)= X f(tn)e . n=0 These sums, one for each m from m =0tom = N − 1, are supposed to be an approximation to the Fourier transform going from f(t)toFf(s). In what sense is this a discrete approximation to the Fourier transform? Here’s one way of looking at it. Since f(t) is timelimited to 0 ≤ t ≤ L, we have L Ff(s)=Z e−2πist f(t) dt . 0 Thus at the sample points sm, L Z −2πismt Ff(sm)= e f(t) dt . 0 and to know the values Ff(sm) is to know Ff(s) reasonably well. Now use the sample points tk for f(t) to write a Riemann sum approximation for the integral. The spacing ∆t of the points is 1/2B,so − − L N 1 1 N 1 1 Ff(s )=Z f(t)e−2πisnt dt ≈ X f(t )e−2πismtn ∆t = X f(t )e−2πismtn = F (s ) . m n 2B n 2B m 0 n=0 n=0 This is the final point: • Up to the factor 1/2B, the values F (sm) provide an approximation to the values Ff(sm). 254 Chapter 6 Discrete Fourier Transform Writing a Riemann sum as an approximation to the integral defining Ff(sm) essentially discretizes the integral, and this is an alternate way of getting to the expression for F (sn), up to the factor 2B.We short-circuited this route by working directly with Ffdiscrete(s). You may find the “up to the factor 1/2B” unfortunate in this part of the discussion, but it’s in the nature of the subject. In fact, back in Chapter 2 we encountered a similar kind of “up to the factor . ” phenomenon when we obtained the Fourier transform as a limit of the Fourier coefficients for a Fourier series. We are almost ready for a definition, but there’s one final comment to clear the way for that. Use the definition of the sample points n m t = ,s= n 2B m L to write N−1 N−1 N−1 −2πismtn −2πinm/2BL −2πinm/N F (sm)= X f(tn)e = X f(tn)e = X f(tn)e .