563489578934

Sec. 2–8 Discrete Fourier Transform 97
least N numbers must be stored, and furthermore, the average sampling rate must be at least
the Nyquist rate. That is, † f s Ú 2B
(2–175)
Thus, in this first type of application, the dimensionality theorem is used to calculate the number
of storage locations (amount of memory) required to represent a waveform.
The second type of application is the inverse type of problem. Here the dimensionality
theorem is used to estimate the bandwidth of waveforms. This application is discussed in detail
in Chapter 3, Sec. 3–4, where the dimensionality theorem is used to give a lower bound
for the bandwidth of digital signals.
2–8 DISCRETE FOURIER TRANSFORM ‡
With the convenience of personal computers and the availability of digital signal-processing
integrated circuits, the spectrum of a waveform can be easily approximated by using the discrete
Fourier transform (DFT). Here we show how the DFT can be used to compute samples
of the continuous Fourier transform (CFT), Eq. (2–26), and values for the complex Fourier series
coefficients of Eq. (2–94).
DEFINITION.
The discrete Fourier transform (DFT) is defined by
X(n) =
k=N-1
a x(k)e -j(2p/N)nk
k=0
(2–176)
where n = 0, 1, 2, Á , N - 1, and the inverse discrete Fourier transform (IDFT) is defined
by
n=N-1
X(n)e j(2p/N)nk
x(k) = 1 (2–177)
N a n=0
where k = 0, 1, 2, Á , N - 1.
Time and frequency do not appear explicitly, because Eqs. (2–176) and (2–177) are just definitions
implemented on a digital computer to compute N values for the DFT and IDFT, respectively. One
should be aware that other authors may use different (equally valid) definitions. For example, a
1/1N factor could be used on the right side of Eq. (2–176) if the 1/N factor of Eq. (2–177) is
replaced by 1/1N. This produces a different scale factor when the DFT is related to the CFT.
Also, the signs of the exponents in Eq. (2–176) and (2–177) could be exchanged. This would
reverse the spectral samples along the frequency axis. The fast Fourier transform (FFT) is a fast
algorithm for evaluating the DFT [Ziemer, Tranter, and Fannin, 1998].
† If the spectrum of the waveform being sampled has a line at f = +B, there is some ambiguity as to whether
the line is included within bandwidth B. The line is included by letting f s 7 2B (i.e., dropping the equality sign).
‡ The fast Fourier transform (FFT) algorithms are fast ways of computing the DFT. The number of complex
multiplications required for the DFT is N 2 , whereas the FFT (with N selected to be a power of 2) requires only (N/2)
log 2 N complex multiplications. Thus, the FFT provides an improvement factor of 2N/(log 2 N) compared with the
DFT, which gives an improvement of 113.8 for an N = 512 point FFT.