Thus, the two-dimensional array representation for the input is

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After the five-point DFT of each row in the data array is computed. the resulting complex array is multiplied by the array of twiddle factors:

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The last step then involves computing the three-point DFT of each column. This produces the output array X ( k ) , which is

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The connections between the three- and five-point DFTs are shown in the following figure, along with the eight twiddle factors:

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(h) Using the prime factor algorithm with N I = 5 and N2 = 3, we set A = N 2 = 3 and B = N , = 5. Then, with C = N ~ ( ( N , ' ) ) N ,= 6 and D = N ~ ( ( N ; ' ) ) ~= 10, we have the following index mappings for n and k : ,

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CHAP. 71

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The two-dimensional array representation for the input is

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and for the output array we have

The interconnections between the five- and three-point DFTs are the same as in the mixed-radix algorithm. However, there are no twiddle factors, and the ordering of the input and output arrays is different. The 15-point prime fuctor algorithm is diagrammed in the figure below.

The savings with the prime factor algorithm over the mixed-radix FFT are the eight complex multiplies by the twiddle factors.

Supplementary Problems

Radix-2 FFT Algorithms

Let x ( n ) be a sequence of length 1024 that is to be convolved with a sequence h ( n ) of length L. For what values of L is it more efficient to perform the convolution directly than it is to perform the convolution by taking the inverse DFT of the product X (k)H ( k ) and evaluating the DFTs using a radix-2 FFT algorithm

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[CHAP. 7

Suppose that we have a 1025-point data sequence (1 more than N = 2"). Instead of discarding the final value, we zero pad the sequence to make it of length N = 2" so that we can use a radix-2 FFT algorithm. (a) How many multiplications and additions are required to compute the DFT using a radix-2 FFT algorithm (b) How many multiplications and additions would be required to compute a 1025-point DFT directly

FFT Algorithms for Composite N

7.19 7.20 7-21

In a radix-3 decimation-in-time FFT, how is the input sequence indexed How many complex multiplications are necessary in a radix-3 decimation-in-frequency FFT Consider the FFT algorithm given in Example 7.3.2. ( a ) How many multiplications and additions are required to compute a 12-point DFT (h) How many multiplications and additions are necessary if the 12-point DFT is computed directly

Prime Factor FFT

7.22 7.23

Find the index maps for a 99-point prime factor FFT with N I = I I and N2 = 9. How many complex multiplications are required for a 12-point prime factor FFT with N, = 4 and N3 = 3 if we do not count multiplications by + I and j

7.24 7.25

How many twiddle factors are there in a 99-point prime factor FFT with N I = I 1 and N2 = 9 How many complex multiplications are required for a 15-point prime factor FFT if we do not count multiplications by & I

Answers to Supplementary Problems

7.18 7.19 7.20 7.21

(a) 1 1.264. (b) 1,050,625. The index for x(n) is expressed in ternary form, and then the ternary digits are read in reverse order. The same as a decimation-in-time FFT, which is 2N log, N . (a) Each 4-point DFT requires no multiplies and 12 adds, and each 3-point DFT requires 6 multiplies and 6 adds. With 6 twiddle factors, there are 6 (4)(6) = 30 multiplies and (4)(6) (3)(12) = 60 adds. (b) 144 multiplies and 132 adds.

7.22 7.23 7.24 7.25

n = 9nI 24. None. 90.

+ 1 In2, and k = 45kl + %k2.

8

Implementation of Discrete-Time Systems

8.1 INTRODUCTION

Given a linear shift-invariant system with a rational system function H ( z ) , the input and output are related by a linear constant coefficient difference equation. For example, with a system function