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The Fast Fourier Transform 197
N =2 ν , 0 ≤ ν ≤ 11. For these values of N, the radix-2 FFT algorithm is used.
For all other values, a mixed-radix FFT algorithm is employed. This shows that
the divide-and-combine strategy is very effective when N is highly composite.
For example, the execution time is 0.16 sec for N = 2048, 2.48 sec for N = 2047,
and 46.96 sec for N = 2039.
□
The MATLAB functions developed previously in this chapter should
now be modified by substituting the fft function in place of the dft
function. From the preceding example care must be taken to use a highly
composite N. Agood practice is to choose N = 2 ν unless a specific
situation demands otherwise.
5.6.4 FAST CONVOLUTIONS
The conv function in MATLAB is implemented using the filter function
(which is written in C) and is very efficient for smaller values of N (< 50).
For larger values of N it is possible to speed up the convolution using the
FFT algorithm. This approach uses the circular convolution to implement
the linear convolution, and the FFT to implement the circular convolution.
The resulting algorithm is called a fast convolution algorithm. In
addition, if we choose N =2 ν and implement the radix-2 FFT, then the
algorithm is called a high-speed convolution. Let x 1 (n) beaN 1 -point sequence
and x 2 (n)beaN 2 -point sequence; then for high-speed convolution
N is chosen to be
N =2 ⌈log 2 (N1+N2−1)⌉ (5.64)
where ⌈x⌉ is the smallest integer greater than x (also called a ceiling
function). The linear convolution x 1 (n) ∗ x 2 (n) can now be implemented
by two N-point FFTs, one N-point IFFT, and one N-point dot-product.
x 1 (n) ∗ x 2 (n) =IFFT [FFT [x 1 (n)] · FFT [x 2 (n)]] (5.65)
For large values of N, (5.65) is faster than the time-domain convolution,
as we see in the following example.
□ EXAMPLE 5.23 To demonstrate the effectiveness of the high-speed convolution, let us compare
the execution times of two approaches. Let x 1 (n) be an L-point uniformly
distributed random number between [0, 1], and let x 2 (n)beanL-point Gaussian
random sequence with mean 0 and variance 1. We will determine the average
execution times for 1 ≤ L ≤ 150, in which the average is computed over the
100 realizations of random sequences. (Please see the cautionary note given in
Example 5.22.)
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