# Non-zero DFT components where zero is expected?

I am implementing DFT in Octave. Here's my code:

function [real_comp, imag_comp] = mydft (samples)
N = columns(samples);
for m = 1:N
real_accum = 0;
imag_accum = 0;
for n = 1:N
real_factor = cos((2 * pi * (n-1) * (m-1)) / N);
# imag_factor = -I * sin((2 * pi * (n-1) * (m-1)) / N);
imag_factor = sin((2 * pi * (n-1) * (m-1)) / N);
real_accum += real_factor * samples(n);
imag_accum += imag_factor * samples(n);
endfor
real_comp(m) = real_acuum;
imag_comp(m) = imag_accum;
endfor
endfunction


My goal is to compute the magnitude at each DFT bin and plot it. I am passing in a list of real-only samples and storing the real and imaginary components in different variables. Here's the sample vector that I am using:

samples = [0.3535, 0.3535, 0.6464, 1.0607, 0.3535, -1.0607, -1.3535, -0.3535]


The samples are carefully chosen to prevent spectral leakage at fs = 8KHz. The graph of the DFT magnitude is exactly what I expect. There's nothing wrong with that. But what I don't get is that some components in my 8 point DFT which are supposed be zero, are not exactly zero but rather a very small, close-to-zero value. The built-in fft() function, on the other hand, has zeroed out components where I expect.

For example, I print the elements of output vector produced by my DFT and octave's FFt using the code below:

fft_out = fft(samples);
[real, imag] = mydft(samples);
dft_out = real + (-I * imag);
for i=1:columns(samples)
disp(fft_out(i));
disp(dft_out(i));
printf("\n");
endfor


The point-to-point comparison is given in the table below:

bin_index fft_out dft_out
1 -1.0000e-04 -1.0000e-04
2 0.00000 - 3.99988i 4.7184e-16 - 3.9999e+00i
3 1.4141 + 1.4144i 1.4141 + 1.4144i
4 0.0000e+00 - 8.0820e-05i 1.3878e-16 - 8.0820e-05i
5 -1.0000e-04 -1.0000e-04 - 1.4350e-16i
6 0.0000e+00 + 8.0820e-05i 2.3037e-15 + 8.0820e-05i
7 1.4141 - 1.4144i 1.4141 - 1.4144i
8 0.00000 + 3.99988i -2.0817e-15 + 3.9999e+00i

Now, I wouldn't care too much about the tiny difference but if I compute the phase angles of fft_out and dft_out,

disp(arg(fft_out))
disp(arg(dft_out))


I get the sign flipped in some cases:

fft_out: 3.14159  -1.57080   0.78550  -1.57080   3.14159   1.57080  -0.78550   1.57080
dft_out:-3.14159  -1.57080   0.78550  -1.57080  -3.14159   1.57080  -0.78550   1.57080


So, is there something wrong with my DFT implementation? Is it just a rounding or precision issue?

• My my. That 'samples' input sequence certainly does look familiar! Jan 3 at 11:24

These are just rounding errors, and you only get a flipped sign for phase values of $$\pm\pi$$, which is not an issue because the phase value is ambiguous, i.e., adding or subtracting multiples of $$2\pi$$ doesn't change the value of the complex coefficient.