Radix-4 Fft(Fast Fourier Transform) Algorithm

Radix-4 FFT Algorithm

When the seat out of information points northward inwards the DFT is a ability of 4 (i.e., N = 4^v), we can, of course, e'er usage a radix-2 algorithm for the computation. However, for this case, it is to a greater extent than efficient computationally to employ a radix-r FFT algorithm.
Let us laid about past times describing a radix-4 decimation-in-time FFT algorithm briefly. We split or decimate the N-point input sequence into 4 subsequences, x(4n), x(4n+1), x(4n+2), x(4n+3), n = 0, 1, ... , N/4-1.

Thus the four N/4-point DFTs F(l, q)obtained from the inwards a higher house equation are combined to yield the N-point DFT. The facial expression for combining the N/4-point DFTs defines a radix-4 decimation-in-time butterfly, which tin survive expressed inwards matrix cast as

The radix-4 butterfly is depicted inwards Figure TC.3.9a in addition to inwards a to a greater extent than compact cast inwards Figure TC.3.9b. Note that each butterfly involves iii complex multiplications, since W_N⁰ = 1, in addition to 12 complex additions.

Figure TC.3.9 Basic butterfly computation inwards a radix-4 FFT algorithm.

By performing the additions inwards 2 steps, it is possible to trim back the seat out of additions per butterfly from 12 to 8. This tin survive accomplished to expressing the matrix of the linear transformation mentioned previously every bit a production of 2 matrices every bit follows:

Figure TC.3.10 Sixteen-point radix-4 decimation-in-time algorithm alongside input inwards normal social club in addition to output inwards digit-reversed order

A 16-point, radix-4 decimation-in-frequency FFT algorithm is shown inwards Figure TC.3.11. Its input is inwards normal social club in addition to its output is inwards digit-reversed order. It has just the same computational complexity every bit the decimation-in-time radix-4 FFT algorithm.

Figure TC.3.11 Sixteen-point, radix-4 decimation-in-frequency algorithm alongside input inwards normal social club in addition to output inwards digit-reversed order.

For illustrative purposes, allow us re-derive the radix-4 decimation-in-frequency algorithm past times breaking the N-point DFT formula into 4 smaller DFTs. We have

From the Definition of the twiddle factors, nosotros have

The relation is non a N/4-point DFT because the twiddle ingredient depends on northward in addition to non on N/4. To convert it into an N/4-point DFT nosotros subdivide the DFT sequence into 4 N/4-point subsequences, X(4k),X(4k+1), X(4k+2), and X(4k+3), k = 0, 1, ..., N/4. Thus nosotros obtain the radix-4 decimation-in-frequency DFT as

where nosotros accept used the property W_N^4kn = W^kn_N/4. Note that the input to each N/4-point DFT is a linear combination of 4 signal samples scaled past times a twiddle factor. This physical care for is repeated v times, where v = log₄N.

Reference Book: Digital Signal Processing Fourth Edition past times John G. Proakis in addition to Dimitris G. Manolakis

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