# Unit of Amplitude After FFT Analysis

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William Skidmore am 23 Mär. 2021
Beantwortet: Mathieu NOE am 23 Mär. 2021
I have a set of data for sound pressure (Pa) in the time domain that I have performed fft analysis on to convert into the frequency domain.
After carrying out fft on the data I have created a chart of frequency (Hz) vs amplitude.
Please can someone help me, what units will the amplitude be in? I thought the units would remain in Pa, however after carrying out fft on the sound pressure data, some of the magnitudes are returning very high values, therefore leading me to the conclusion that the amplitude cannot remain in Pa, but I do not know what else it could be measured in? Is the amplitue unitless and just a measure of the relative amplitude of each specific frequency?
Thank you.
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### Antworten (1)

Mathieu NOE am 23 Mär. 2021
hello
using fft requires some brainstorming about the scaling
thats why i also compare my fft output amplitude to a know sinus wave (for example)
to end up with permanent question about scaling, I now since long time use the same home made fft (with overlapping and averaging possibilities ) as in the example below
function [freq_vector,fft_spectrum] = myfft_peak(signal, Fs, nfft, Overlap)
this will give you a fft output amplitude of 1 if the sine wave is also 1 peak amplitude (whatever the others parameters)
NB : overlap is a percentage of nfft in my function
clc
clear all
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% dummy data
Fs = 10000;
samples = 25000;
t = (0:samples-1)'*1/Fs;
signal = sin(2*pi*1000*t)+sin(2*pi*2000*t)+sin(2*pi*3000*t)+1e-3*rand(samples,1);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% FFT parameters
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
NFFT = 1000; %
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 = 1000; % 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 = linspace(fc-1,fc+1,200);
[g1,p1] = bode(num1,den1,2*pi*freq);
g1db = 20*log10(g1);
[gd1,pd1] = dbode(num1z,den1z,1/Fs,2*pi*freq);
gd1db = 20*log10(gd1);
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
df = mean(diff(freq));
[pks,locs]= findpeaks(sensor_spectrum_dB,'SortStr','descend','NPeaks',3);
[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)+df,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
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