Radix-4 FFT implementation - matlab

Question

The Octave radix-4 FFT code below works fine if I set power of 4 (xp) values case-by-case.
$ octave fft4.m
ans = 1.4198e-015
However, if I uncomment the loop code I get the following error
$ octave fft4.m
error: `stage' undefined near line 48 column 68
error: evaluating argument list element number 1
error: evaluating argument list element number 2
error: called from:
error: r4fftN at line 48, column 22
error: c:\Users\david\Documents\Visual Studio 2010\Projects\mv_fft\fft4.m at line 80, column 7
the "error" refers to a line the in fft function code which otherwise works correctly when xp is not set by a loop ... very strange.
function Z = radix4bfly(x,segment,stageFlag,W)
% For the last stage of a radix-4 FFT all the ABCD multiplers are 1.
% Use the stageFlag variable to indicate the last stage
% stageFlag = 0 indicates last FFT stage, set to 1 otherwise
% Initialize variables and scale to 1/4
a=x(1)*.25;
b=x(2)*.25;
c=x(3)*.25;
d=x(4)*.25;
% Radix-4 Algorithm
A=a+b+c+d;
B=(a-b+c-d)*W(2*segment*stageFlag + 1);
C=(a-b*j-c+d*j)*W(segment*stageFlag + 1);
D=(a+b*j-c-d*j)*W(3*segment*stageFlag + 1);
% assemble output
Z = [A B C D];
end % radix4bfly()
% radix-4 DIF FFT, input signal must be floating point, real or complex
%
function S = r4fftN(s)
% Initialize variables and signals: length of input signal is a power of 4
N = length(s);
M = log2(N)/2;
% Initialize variables for floating point sim
W=exp(-j*2*pi*(0:N-1)/N);
S = complex(zeros(1,N));
sTemp = complex(zeros(1,N));
% FFT algorithm
% Calculate butterflies for first M-1 stages
sTemp = s;
for stage = 0:M-2
for n=1:N/4
S((1:4)+(n-1)*4) = radix4bfly(sTemp(n:N/4:end), floor((n-1)/(4^stage)) *(4^stage), 1, W);
end
sTemp = S;
end
% Calculate butterflies for last stage
for n=1:N/4
S((1:4)+(n-1)*4) = radix4bfly(sTemp(n:N/4:end), floor((n-1)/(4^stage)) * (4^
stage), 0, W);
end
sTemp = S;
% Rescale the final output
S = S*N;
end % r4fftN(s)
% test FFT code
%
xp = 2;
% ERROR if I uncomment loop!
%for xp=1:8
N = 4^xp; % must be power of: 4 16 64 256 1024 4086 ....
x = 2*pi/N * (0:N-1);
x = cos(x);
Y_ref = fft(x);
Y = r4fftN(x);
Y = digitrevorder(Y,4);
%Y = bitrevorder(Y,4);
abs(sum(Y_ref-Y)) % compare fft4 to built-in fft
%end

Answer 1

The problem was the loop-bound for the exponent xp should start from 2 as the fft4 code assumes at least 2 stages of radix-4 butterflies
Sorry folks :(
-David

Answer 2

Please find below a fully worked Matlab implementation of a radix-4 Decimation In Frequency FFT algorithm. I have also provided an overall operations count in terms of complex matrix multiplications and additions. It can be indeed shown that each radix-4 butterfly involves 3 complex multiplications and 8 complex additions. Since there are log_4(N) = log_2(N) / 2 stages and each stage involves N / 4 butterflies, so the operations count is
complex multiplications = (3 / 8) * N * log2(N)
complex additions = N * log2(N)
Here is the code:
% --- Radix-2 Decimation In Frequency - Iterative approach
clear all
close all
clc
% --- N should be a power of 4
N = 1024;
% x = randn(1, N);
x = zeros(1, N);
x(1 : 10) = 1;
xoriginal = x;
xhat = zeros(1, N);
numStages = log2(N) / 2;
W = exp(-1i * 2 * pi * (0 : N - 1) / N);
omegaa = exp(-1i * 2 * pi / N);
mulCount = 0;
sumCount = 0;
M = N / 4;
for p = 1 : numStages;
for index = 0 : (N / (4^(p - 1))) : (N - 1);
for n = 0 : M - 1;
a = x(n + index + 1) + x(n + index + M + 1) + x(n + index + 2 * M + 1) + x(n + index + 3 * M + 1);
b = (x(n + index + 1) - x(n + index + M + 1) + x(n + index + 2 * M + 1) - x(n + index + 3 * M + 1)) .* omegaa^(2 * (4^(p - 1) * n));
c = (x(n + index + 1) - 1i * x(n + index + M + 1) - x(n + index + 2 * M + 1) + 1i * x(n + index + 3 * M + 1)) .* omegaa^(1 * (4^(p - 1) * n));
d = (x(n + index + 1) + 1i * x(n + index + M + 1) - x(n + index + 2 * M + 1) - 1i * x(n + index + 3 * M + 1)) .* omegaa^(3 * (4^(p - 1) * n));
x(n + 1 + index) = a;
x(n + M + 1 + index) = b;
x(n + 2 * M + 1 + index) = c;
x(n + 3 * M + 1 + index) = d;
mulCount = mulCount + 3;
sumCount = sumCount + 8;
end;
end;
M = M / 4;
end
xhat = bitrevorder(x);
tic
xhatcheck = fft(xoriginal);
timeFFTW = toc;
rms = 100 * sqrt(sum(sum(abs(xhat - xhatcheck).^2)) / sum(sum(abs(xhat).^2)));
fprintf('Theoretical multiplications count \t = %i; \t Actual multiplications count \t = %i\n', ...
(3 / 8) * N * log2(N), mulCount);
fprintf('Theoretical additions count \t\t = %i; \t Actual additions count \t\t = %i\n\n', ...
N * log2(N), sumCount);
fprintf('Root mean square with FFTW implementation = %.10e\n', rms);