How to fft time signal and get given number of fft points and frequency resoution?

Hi!

I have a time domain signal x. I should calculate te spectrum of it, but I want to get the spectrum with a certain frequency resolution and certain number of fft points. Let's say for example, that the time signal is stored in a 1 by 10000 vector. It is sampled with fs = 1000 Hz. The frequency resolution is given by these two things. Still I want my data to hae a frequency resolution of 1Hz, and 5000 datapoints. How can I do that without deformind the signal? Is there maybe an other way than using fft and still get the spectrum with right amplitude and phase?

Answer 1

[Mathieu NOE]

hello again

your fft resolution depends on the sampling freq (Fs) and your fft length (nfft)

df = Fs / nfft

see example code below (there is more than just this point)

clc
clearvars

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% load signal
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%% dummy data
Fs = 10000;
samples = 20000;
t = (0:samples-1)'*1/Fs;
signal = sin(2*pi*50*t)+2*sin(2*pi*440*t)+1e-2*randn(samples,1); % two sine + some noise

%% decimate (if needed)
% NB : decim = 1 will do nothing (output = input)
decim = 5;
if decim>1
    signal = decimate(signal,decim);
    Fs = Fs/decim;
end
samples = length(signal);
t = (0:samples-1)*1/Fs;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% FFT parameters
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
NFFT = Fs;    % so delta f = 1 Hz
Overlap = 0.75;
w = hanning(NFFT);     % Hanning window / Use the HANN function to get a Hanning window which has the first and last zero-weighted samples.


%% notch filter section %%%%%%
% H(s) = (s^2 + 1) / (s^2 + s/Q + 1)

fc = 50;       % notch freq
wc = 2*pi*fc;
Q = 5;         % adjust Q factor for wider (low Q) / narrower (high Q) notch
                % at f = fc the filter has gain = 0

w0 = 2*pi*fc/Fs;
alpha = sin(w0)/(2*Q);

            b0 =   1;
            b1 =  -2*cos(w0);
            b2 =   1;
            a0 =   1 + alpha;
            a1 =  -2*cos(w0);
            a2 =   1 - alpha;
            
% analog notch (for info)
num1=[1/wc^2 0 1];
den1=[1/wc^2 1/(wc*Q) 1];

% digital notch (for info)
num1z=[b0 b1 b2];
den1z=[a0 a1 a2];

freq = (fc-1:0.01:fc+1);
[g1,p1] = bode(num1,den1,2*pi*freq);
g1db = 20*log10(g1+1e-6);

[gd1,pd1] = dbode(num1z,den1z,1/Fs,2*pi*freq);
gd1db = 20*log10(gd1+1e-6);

figure(1);
plot(freq,g1db,'b',freq,gd1db,'+r');
title(' Notch: H(s) = (s^2 + 1) / (s^2 + s/Q + 1)');
legend('analog','digital');
xlabel('Fréquence (Hz)');
ylabel(' dB')
 
% now let's filter the signal 
signal_filtered = filtfilt(num1z,den1z,signal);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% display : averaged FFT spectrum before / after notch filter
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

[freq,fft_spectrum] = myfft_peak(signal, Fs, NFFT, Overlap); %
sensor_spectrum_dB = 20*log10(fft_spectrum);% convert to dB scale (ref = 1)

% demo findpeaks
[pks,locs]= findpeaks(sensor_spectrum_dB,'SortStr','descend','NPeaks',2);

[freq,fft_spectrum_filtered] = myfft_peak(signal_filtered, Fs, NFFT, Overlap);
sensor_spectrum_filtered_dB = 20*log10(fft_spectrum_filtered);% convert to dB scale (ref = 1)

figure(2),plot(freq,sensor_spectrum_dB,'b',freq,sensor_spectrum_filtered_dB,'r');grid
title(['Averaged FFT Spectrum  / Fs = ' num2str(Fs) ' Hz / Delta f = ' num2str(freq(2)-freq(1)) ' Hz ']);
legend('before notch filter','after notch filter');
xlabel('Frequency (Hz)');ylabel(' dB')
text(freq(locs)+1,pks,num2str(freq(locs)))

function  [freq_vector,fft_spectrum] = myfft_peak(signal, Fs, nfft, Overlap)
% FFT peak spectrum of signal  (example sinus amplitude 1   = 0 dB after fft).
% Linear averaging
%   signal - input signal, 
%   Fs - Sampling frequency (Hz).
%   nfft - FFT window size
%   Overlap - buffer overlap % (between 0 and 0.95)
signal = signal(:);
samples = length(signal);
% fill signal with zeros if its length is lower than nfft
if samples<nfft
    s_tmp = zeros(nfft,1);
    s_tmp((1:samples)) = signal;
    signal = s_tmp;
    samples = nfft;
end

% window : hanning
window = hanning(nfft);
window = window(:);

%    compute fft with overlap 
 offset = fix((1-Overlap)*nfft);
 spectnum = 1+ fix((samples-nfft)/offset); % Number of windows
%     % for info is equivalent to : 
%     noverlap = Overlap*nfft;
%     spectnum = fix((samples-noverlap)/(nfft-noverlap));	% Number of windows

    % main loop
    fft_spectrum = 0;
    for i=1:spectnum
        start = (i-1)*offset;
        sw = signal((1+start):(start+nfft)).*window;
        fft_spectrum = fft_spectrum + (abs(fft(sw))*4/nfft);     % X=fft(x.*hanning(N))*4/N; % hanning only 
    end
    fft_spectrum = fft_spectrum/spectnum; % to do linear averaging scaling

% one sidded fft spectrum  % Select first half 
    if rem(nfft,2)    % nfft odd
        select = (1:(nfft+1)/2)';
    else
        select = (1:nfft/2+1)';
    end
fft_spectrum = fft_spectrum(select);
freq_vector = (select - 1)*Fs/nfft;
end