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Decimation in Frequency 16point FFT/DFT MATLAB source code

This section of MATLAB source code covers Decimation in Frequency FFT or DFT matlab code. It compares the FFT output with matlab builtin FFT function to validate the code. This page covers 16 point Decimation in Frequency FFT/DFT with Bit reversed OUTPUT.

Most common and familiar FFTs are radix-2. However other radices viz. small numbers then 10 are sometimes used. For example, radix-4 is especially attractive because the twiddle factors are all 1,-1,j or -j, which can be applied without any multiplications at all.

What is FFT radix

Radix is the size of an FFT decomposition.

Twiddle factor

Twiddle factors are the coefficients used to combine results from a previous stage to inputs to the next stage.
W= exp(j*2*pi*n/N) where here N=16 point and n = 0 to 7.

Decimation in Time FFT

FFTs can be de-composed using DFTs of even and odd points , which is called Decimation in Time FFT.

Decimation in Frequency FFT

FFTs can be de-composed using a first half/second half approach, which is called Decimation in Frequency FFT.

Both the decimation in time and decimation in frequency can be implemented using the same method only butterfly structure is different as shown in the figure above.

decimation in time and frequency

%16 POINT FFT with Bit
%reversed OUTPUT....
%Decimation in Frequency
clear all;
close all;
clc;
%Input Coefficients
x0 = 1 + i*1;
x1 = 2 + i*1;
x2 = 1 - i*2;
x3 = 2 - i*1;
x4 = 2 + i*3;
x5 = 3 + i*2;
x6 = 1 + i*3;
x7 = 3 + i*1;
x8 = -3 + i*3;
x9 = 3 - i*3;
x10 = -1 - i*1;
x11 = -3 - i*3;
x12 = 3 - i*3;
x13 = -1 - i*1;
x14 = -3 - i*3;
x15 = 1 + i*1;
x=[x0 ;x1 ;x2 ;x3 ;x4 ;x5;
x6; x7; x8; x9; x10; x11;
x12 ;x13; x14 ;x15];

%Twiddle Factors
%3rd stage
tc0=1.0000;
tc1=0.9239 - 0.3827i;
tc2=0.7071 - 0.7071i;
tc3=0.3827 - 0.9239i;
tc4=0.0000 - 1.0000i;
tc5=-0.3827 - 0.9239i;
tc6=-0.7071 - 0.7071i;
tc7=-0.9239 - 0.3827i;

%Twiddle Factors
%2nd stage
tb0=1.0000;
tb1=0.7071 - 0.7071i;
tb2=0.0000 - 1.0000i;
tb3=-0.7071 - 0.7071i;

%1st stage
ta0=1.0000;
ta1=0.0000 - 1.0000i;

%%Stage 3 outputs
s3_0 = x0 + x8;
s3_1 = x1 + x9;
s3_2 = x2 + x10;
s3_3 = x3 + x11;
s3_4 = x4 + x12;
s3_5 = x5 + x13;
s3_6 = x6 + x14;
s3_7 = x7 + x15;
s3_8 = (x0 - x8)*tc0;
s3_9 = (x1 - x9)*tc1;
s3_10 = (x2 - x10)*tc2;
s3_11 = (x3 - x11)*tc3;
s3_12 = (x4 - x12)*tc4;
s3_13 = (x5 - x13)*tc5;
s3_14 = (x6 - x14)*tc6;
s3_15 = (x7 - x15)*tc7;

Stage3 =[s3_0;s3_1;s3_2;s3_3;s3_4;
s3_5;s3_6;s3_7;s3_8;s3_9;
s3_10;s3_11;s3_12;
s3_13;s3_14;s3_15];

%%Stage 2 outputs
s2_0 = s3_0 + s3_4;
s2_1 = s3_1 + s3_5;
s2_2 = s3_2 + s3_6;
s2_3 = s3_3 + s3_7;
s2_4 = (s3_0 - s3_4)*tb0;
s2_5 = (s3_1 - s3_5)*tb1;
s2_6 = (s3_2 - s3_6)*tb2;
s2_7 = (s3_3 - s3_7)*tb3;
s2_8 = s3_8 + s3_12;
s2_9 = s3_9 + s3_13;
s2_10 = s3_10 + s3_14;
s2_11 = s3_11 + s3_15;
s2_12 = (s3_8 - s3_12)*tb0;
s2_13 = (s3_9 - s3_13)*tb1;
s2_14 = (s3_10 - s3_14)*tb2;
s2_15 = (s3_11 - s3_15)*tb3;

Stage2 =[s2_0;s2_1;s2_2;s2_3;s2_4;
s2_5;s2_6;s2_7;s2_8;s2_9;
s2_10;s2_11;s2_12;
s2_13;s2_14;s2_15];
%%stage 1
s1_0 = s2_0 + s2_2;
s1_1 = s2_1 + s2_3;
s1_2 = (s2_0 - s2_2) * ta0;
s1_3 = (s2_1 - s2_3) * ta1;
s1_4 = s2_4 + s2_6;
s1_5 = s2_5 + s2_7;
s1_6 = (s2_4 - s2_6) * ta0;
s1_7 = (s2_5 - s2_7) * ta1;
s1_8 = s2_8 + s2_10;
s1_9 = s2_9 + s2_11;
s1_10 = (s2_8 - s2_10) * ta0;
s1_11 = (s2_9 - s2_11) * ta1;
s1_12 = s2_12 + s2_14;
s1_13 = s2_13 + s2_15;
s1_14 = (s2_12 - s2_14) * ta0;
s1_15 = (s2_13 - s2_15) * ta1;

%%stage 0
s0_0 = s1_0 + s1_1;
s0_1 = s1_0 - s1_1;
s0_2 = s1_2 + s1_3;
s0_3 = s1_2 - s1_3;
s0_4 = s1_4 + s1_5;
s0_5 = s1_4 - s1_5;
s0_6 = s1_6 + s1_7;
s0_7 = s1_6 - s1_7;
s0_8 = s1_8 + s1_9;
s0_9 = s1_8 - s1_9;
s0_10 = s1_10 + s1_11;
s0_11 = s1_10 - s1_11;
s0_12 = s1_12 + s1_13;
s0_13 = s1_12 - s1_13;
s0_14 = s1_14 + s1_15;
s0_15 = s1_14 - s1_15;

Y = [s0_0;s0_1;s0_2;s0_3;s0_4;
s0_5;s0_6;s0_7;s0_8;s0_9;
s0_10;s0_11;s0_12;s0_13;
s0_14;s0_15];
% i=0:15;
% de2bi(i);
% D=bi2de(ans,'left-msb');
% D1=D+1;
Y=[Y(1);Y(9);Y(5);Y(13);Y(3);
Y(11);Y(7);Y(15);Y(2);
Y(10);Y(6);Y(14);
Y(4);Y(12);Y(8);Y(16)];
figure;plot(abs(Y)); title('Our FFT result'); %plotting my FFT

MATLAB built-in FFT MATLAB function

Y_fft = fft(x,16);
figure; plot(abs(Y_fft)); title('MATLAB function FFT result'); %plotting matlab's built in FFT

16 point FFT MATLAB Output

16point FFT matlab output

16 point FFT matlab function output

16point matlab fft function

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