Why does the amplitude of a signal change when the sampling frequency is changed?

hi eveyone;

i have accelerometers data and i want to make signal processing on it.

Firstly; how can i find max frequeny of this signal? (it needs for Nquist criteria).

secondly; why does the amplitude (dB) change when i change the samplng frequency?

thirdly; i coldn't see noise signal of it, which filter is used for more smooth signal? otherwise should i use smooth filter?

Answer 1

[Mehmet Rizelioglu]

Dear NOE; Thank you very much because you reply my questions patiently. İf you don't boring, i want you to disturb with my questions after :)

[Mathieu NOE]

hello

no problem... if you still have questions, I don't mind spending some time answering them

[Mehmet Rizelioglu]

let's start from the beginning. I would know thatwhen a discrete signal is intended to be written in terms of continuous signals (cos, sin waves), these continous waves have to sampled as to Nyquist criteria, so my data's frequency abloyt 77 Hz and owing to Fs=Fmax/2, we have to get Fs=154 Hz for continuos signal.

[Mathieu NOE]

well

it's a bit confusion here ...

a continuus signal has by definition no sampling frequency - otherwise it's discrete.

to discretize a continuous signal, the sampling frequency must be at least twice as the highest frequency contained in the continuous signal

I would recommend you to read some publications on sampling , like this one :

Sampling (signal processing) - Wikipedia

Nyquist–Shannon sampling theorem - Wikipedia

Answer 2

[Mathieu NOE]

hello

try this code - there should not be any amplitude change if you record your data with a different frequency - as soon as you don't sample so low that the signal vanish after sampling

you can adapt the code to your own needs - here it's about FFT analysis

clc
clearvars
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% load signal
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%% data
[signal,Fs] = audioread('test_voice.wav');

[samples,channels] = size(signal);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% FFT parameters
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
NFFT = 1024;    % 
OVERLAP = 0.75;

% spectrogram dB scale
spectrogram_dB_scale = 60;  % dB range scale (means , the lowest displayed level is XX dB below the max level)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% options 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% if you are dealing with acoustics, you may wish to have A weighted
% spectrums 

% option_w = 0 : linear spectrum (no weighting dB (L) )
% option_w = 1 : A weighted spectrum (dB (A) )
option_w = 0;

%% decimate (if needed)
% NB : decim = 1 will do nothing (output = input)
decim = 4;
if decim>1
    for ck = 1:channels
    newsignal(:,ck) = decimate(signal(:,ck),decim);
    Fs = Fs/decim;
    end
   signal = newsignal;
end
samples = length(signal);
time = (0:samples-1)*1/Fs;

%%%%%% legend structure %%%%%%%%
for ck = 1:channels
    leg_str{ck} = ['Channel ' num2str(ck) ];
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% display 1 : time domain plot
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
figure(1),plot(time,signal);grid on
title(['Time plot  / Fs = ' num2str(Fs) ' Hz ']);
xlabel('Time (s)');ylabel('Amplitude');
legend(leg_str);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% display 2 : averaged FFT spectrum
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

[freq, sensor_spectrum] = myfft_peak(signal,Fs,NFFT,OVERLAP);

% convert to dB scale (ref = 1)
sensor_spectrum_dB = 20*log10(sensor_spectrum);

% apply A weigthing if needed
if option_w == 1
    pondA_dB = pondA_function(freq);
    sensor_spectrum_dB = sensor_spectrum_dB+pondA_dB;
    my_ylabel = ('Amplitude (dB (A))');
else
    my_ylabel = ('Amplitude (dB (L))');
end


figure(2),plot(freq,sensor_spectrum_dB);grid on
df = freq(2)-freq(1); % frequency resolution 
title(['Averaged FFT Spectrum  / Fs = ' num2str(Fs) ' Hz / Delta f = ' num2str(df,3) ' Hz ']);
xlabel('Frequency (Hz)');ylabel(my_ylabel);
legend(leg_str);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% display 3 : time / frequency analysis : spectrogram demo
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for ck = 1:channels
    [sg,fsg,tsg] = specgram(signal(:,ck),NFFT,Fs,hanning(NFFT),floor(NFFT*OVERLAP));  

    % FFT normalisation and conversion amplitude from linear to dB (peak)
    sg_dBpeak = 20*log10(abs(sg))+20*log10(2/length(fsg));     % NB : X=fft(x.*hanning(N))*4/N; % hanning only

    % apply A weigthing if needed
    if option_w == 1
        pondA_dB = pondA_function(fsg);
        sg_dBpeak = sg_dBpeak+(pondA_dB*ones(1,size(sg_dBpeak,2)));
        my_title = ('Spectrogram (dB (A))');
    else
        my_title = ('Spectrogram (dB (L))');
    end

    % saturation of the dB range : 
    % saturation_dB = 60;  % dB range scale (means , the lowest displayed level is XX dB below the max level)
    min_disp_dB = round(max(max(sg_dBpeak))) - spectrogram_dB_scale;
    sg_dBpeak(sg_dBpeak<min_disp_dB) = min_disp_dB;

    % plots spectrogram
    figure(2+ck);
    imagesc(tsg,fsg,sg_dBpeak);colormap('jet');
    axis('xy');colorbar('vert');grid on
    df = fsg(2)-fsg(1); % freq resolution 
    title([my_title ' / Fs = ' num2str(Fs) ' Hz / Delta f = ' num2str(df,3) ' Hz / Channel : ' num2str(ck)]);
    xlabel('Time (s)');ylabel('Frequency (Hz)');

end

function pondA_dB = pondA_function(f)
	% dB (A) weighting curve
	n = ((12200^2*f.^4)./((f.^2+20.6^2).*(f.^2+12200^2).*sqrt(f.^2+107.7^2).*sqrt(f.^2+737.9^2)));
	r = ((12200^2*1000.^4)./((1000.^2+20.6^2).*(1000.^2+12200^2).*sqrt(1000.^2+107.7^2).*sqrt(1000.^2+737.9^2))) * ones(size(f));
	pondA = n./r;
	pondA_dB = 20*log10(pondA(:));
end


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 percentage of overlap % (between 0 and 0.95)

[samples,channels] = size(signal);

% fill signal with zeros if its length is lower than nfft
if samples<nfft
    s_tmp = zeros(nfft,channels);
    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*ones(1,channels));
        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

[Mehmet Rizelioglu]

thanks for your answer, but i wonder why its decrease on using signal process toolbox

[Mathieu NOE]

can you share the code and data ?

tx

[Mehmet Rizelioglu]

i am sending my data. I am using signal analyzer toolbox in matlab

[Mathieu NOE]

hello again

so this is what I get , some lines are longer than others

which column of data are you interested in ?

-0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 9 40.222030 28.875528 0 0.00 19 18 41

-0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.01 0.04 0.00 0.00 0.00 0.00 0.00 0.01

0.02 0.01 0.00 0.00 0.00 0.00 0.00 0.00

-0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 9 40.222030 28.875528 0 0.00 19 18 41

0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00

-0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.01 0.00 0.00 0.00 0.00 0.00 0.01 0.00

etc............

[Mehmet Rizelioglu]

only first column

[Mathieu NOE]

and what is your sampling frequency ?

[Mehmet Rizelioglu]

actually, i dont know what it shold be. how can i calculate these values?

[Mathieu NOE]

well , you cannot compute it from the signal itself unless you know a priori its period (if it's a sine wave for example)

this is something that is dependent of the system you used for recording

your signal ahs no specific period , it's more a broadband signal with energy distributed accross the entire spectrum

this is a spectrogram (time / frequency analysis) , with Fs = 1 Hz by default

you can see there is energy from f = 0 to Nyquist frequency (Fs / 2 = 0.5 Hz here)

[Mehmet Rizelioglu]

what can you offer me? what should i do?

amount of data for per second is changable and accelerometer gets about 75-80 data per second.

in the left picture the discrete data could not be sampled but on the right the data is sampled as 2000 samples (sampling rate=2000)

[Mathieu NOE]

well , so you have the info ! it's your job to make sure you know at which rate the data is collected

also if you know the total time of recording, you can also compute the sampling rate by dividing the record duration by the amount of samples

with Fs = 80 , we can update the code and plots (they look the same but now we have the correct frequencies)

seems there is a maxima of energy around 4.4 Hz

clc

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

signal = importdata('18kmsaat.txt');
[samples,channels] = size(signal);

selected_channel = 1; % select your data channel here
signal = signal(:,selected_channel); 

Fs = 80; % to be checked 
dt = 1/Fs;

time = (0:samples-1)*dt;


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


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% FFT parameters
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
NFFT = 128;    % fft length
OVERLAP = 0.75; % between 0 and 1 max ;  buffer overlap = OVERLAP*NFFT

% spectrogram dB scale
spectrogram_dB_scale = 80;  % dB range scale (means , the lowest displayed level is XX dB below the max level)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% options 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% if you are dealing with acoustics, you may wish to have A weighted
% spectrums 

% option_w = 0 : linear spectrum (no weighting dB (L) )
% option_w = 1 : A weighted spectrum (dB (A) )
option_w = 0;


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% display 1 : averaged FFT spectrum
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

[freq, sensor_spectrum] = myfft_peak(signal,Fs,NFFT,OVERLAP);

% convert to dB scale (ref = 1)
sensor_spectrum_dB = 20*log10(sensor_spectrum);

% apply A weigthing if needed
if option_w == 1
    pondA_dB = pondA_function(freq);
    sensor_spectrum_dB = sensor_spectrum_dB+pondA_dB;
    my_ylabel = ('Amplitude (dB (A))');
else
    my_ylabel = ('Amplitude (dB (L))');
end


figure(1),plot(freq,sensor_spectrum_dB,'b');grid on
title(['Averaged FFT Spectrum  / Fs = ' num2str(0.1*round(10*Fs)) ' Hz / Delta f = ' num2str(0.1*round(10*(freq(2)-freq(1)))) ' Hz ']);
xlabel('Frequency (Hz)');ylabel(my_ylabel);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% display 2 : time / frequency analysis : spectrogram demo
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

[sg,fsg,tsg] = specgram(signal,NFFT,Fs,hanning(NFFT),floor(NFFT*OVERLAP));  

% FFT normalisation and conversion amplitude from linear to dB (peak)
sg_dBpeak = 20*log10(abs(sg))+20*log10(2/length(fsg));     % NB : X=fft(x.*hanning(N))*4/N; % hanning only

% apply A weigthing if needed
if option_w == 1
    pondA_dB = pondA_function(fsg);
    sg_dBpeak = sg_dBpeak+(pondA_dB*ones(1,size(sg_dBpeak,2)));
    my_title = ('Spectrogram (dB (A))');
else
    my_title = ('Spectrogram (dB (L))');
end

% saturation of the dB range : 
% saturation_dB = 60;  % dB range scale (means , the lowest displayed level is XX dB below the max level)
min_disp_dB = round(max(max(sg_dBpeak))) - spectrogram_dB_scale;
sg_dBpeak(sg_dBpeak<min_disp_dB) = min_disp_dB;

% plots spectrogram
figure(2);
imagesc(tsg,fsg,sg_dBpeak);colormap('jet');
axis('xy');colorbar('vert');grid
title([my_title ' / Fs = ' num2str(0.1*round(10*Fs)) ' Hz / Delta f = ' num2str(0.1*round(10*(fsg(2)-fsg(1)))) ' Hz ']);
xlabel('Time (s)');ylabel('Frequency (Hz)');




function pondA_dB = pondA_function(f)
	% dB (A) weighting curve
	n = ((12200^2*f.^4)./((f.^2+20.6^2).*(f.^2+12200^2).*sqrt(f.^2+107.7^2).*sqrt(f.^2+737.9^2)));
	r = ((12200^2*1000.^4)./((1000.^2+20.6^2).*(1000.^2+12200^2).*sqrt(1000.^2+107.7^2).*sqrt(1000.^2+737.9^2))) * ones(size(f));
	pondA = n./r;
	pondA_dB = 20*log10(pondA(:));
end


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 percentage of overlap % (between 0 and 0.95)

[samples,channels] = size(signal);

% fill signal with zeros if its length is lower than nfft
if samples<nfft
    s_tmp = zeros(nfft,channels);
    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*ones(1,channels));
        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

[Mehmet Rizelioglu]

for example; in aboove graphic, there is 920 sample and it has long been 12 second. this data's frequecy about 77 Hz. Is this value Fs (sampling frequency)? or Shouldn't I define it myself? as [Fs=Fmax/2]

[Mathieu NOE]

yes , here you have to enter 77 (Hz) yourself in the Signal Analyser App

[Mehmet Rizelioglu]

Why?, What happens if I enter more values? <77 Hz. Beacuse sampling frequency have to big doble than normal frequency.

so, Shouldn't 77 Hz be the normal frequency?

[Mathieu NOE]

I think you are talking sampling frequency and nyquist frequency (which is Fs/2)

Nyquist frequency - Wikipedia

yes the two frequencies are of course linked

the maximum frequency you can plot is the nyquist frequency (which is Fs/2)

I don't get what you call "normal" frequency ?

[Mehmet Rizelioglu]

I mean my data have 77 samples per second on avarage. Because 920 samples/ 12 second = 77 sample/second (Hz). But are not sampling frequency different from this? I know that we define it (sampling frequency) ourselves.

[Mathieu NOE]

what you describe (920 samples/ 12 second = 77 sample/second) is really the real sampling frequency.

there is no "other" sampling frequency

[Mehmet Rizelioglu]

So, what is max frequency of this data?

[Mathieu NOE]

as I answered above , you can only see signals with frequencies up to the Fs/2 limit

does not mean any signal will have energy at Fs/2

that's why we do spectral analysis to see the distribution of amplitude vs frequencies (from 0 to Fs/2)