Amplitude and Phase of result of FFT in MATLAB - matlab

Question

I tried to extract amplitude & phase values from the fft function result in the Matlab. I implemented the script as below
clear;
sf = 100; %sampling frequency
si = 1/sf; dt=si; %time sampling interval
L = 10; %Length of signal
t = linspace(0,L,L/dt+1);%(0:L-1)*dt; %time vector
t(end)=[];
fr = 4 %frequency
data = cos(2*pi*fr*t);
df = sf/length(data3); %frequency increment
f = linspace(0,length(data3)/2,length(data3)/2)*df; %frequency
fft_result =fft(data)/length(data);
spec_fft = abs(fft_result); %amplitude
pha_fft = angle(fft_result); %phase
When I checked the results of amplitude and phase values, of course, they showed peak values at a specific frequency that I specified. But, other frequencies also have amplitude. Of course, their values are very very small, but because of this problem, the phase spectrum didn't show me a clear result. Why other frequencies also have amplitude values?
And I made a cosine function that is not shifted. So I think the phase value should show the zero value, but it wasn't. Why this problem occurred?

Answer 1

These values won't be exactly zero due to the floating point operations involved. You generated a 4 Hz cosine with amplitude of 1. Taking the single-sided amplitude and phase spectrum shows an amplitude of 1, and a phase of 0 radians at the 4 Hz bin:
clear;
sf = 100; %sampling frequency
si = 1/sf; dt=si; %time sampling interval
L = 10; %Length of signal
t = linspace(0,L,L/dt+1);%(0:L-1)*dt; %time vector
t(end)=[];
fr = 4; %frequency
data = cos(2*pi*fr*t);
df = sf/length(data); %frequency increment
N = length(data);
f = ((0:(N/2))/ N) * sf; %frequency
fft_result =fft(data)/N;
spec_fft = abs(fft_result); %amplitude
% single sided amplitude
amp_single_side = spec_fft(1:N/2+1);
amp_single_side(2:end-1) = 2*amp_single_side(2:end-1);
% single sided phase
phase_single_side = angle(fft_result(1:N/2+1));
four_hertz_bin = find(f == 4);
four_hertz_amp = amp_single_side(four_hertz_bin);
four_hertz_phase = phase_single_side(four_hertz_bin);
figure;
subplot(2,1,1);
plot(f, amp_single_side)
xlabel('Frequency');
ylabel('Amplitude');
hold on;
plot(f(four_hertz_bin), four_hertz_amp, 'ro');
subplot(2,1,2);
plot(f, phase_single_side);
xlabel('Frequency');
ylabel('Phase');
hold on;
plot(f(four_hertz_bin), four_hertz_phase, 'ro');