Is it possible to define the "Fmax" on the fft function?

Hello everyone

I am currently developing an algorithm in matlab to analyze signals from a MEMS acelerometer. Before trying to acquire the signal and do that programming i must understand how do fft and filters in matlab work. I must say beforehand that this algorithm must be as close as to normal maintenance acquisition and signal processing systems are (or at least i intend it to be).

To do that i have defined the initial variables that in that equipment i define initially to start the acquisition of the signal and processing. Those variables are the Fmax, the maximum frequency i want to see in the spectrum and with that value it defines the sampling frequency of my ADC, by obeying to the shannon theorem, and the cutting frequency of the low-pass filter i also want. I also defined the sampling time, ts, and the number of lines i want to have as resolution. (numbers are just experimental, i would appreciate if anyone would help me on that too, because as i said i want it to be an algorithm of a system to acquire and process aceleration signals of machines).

clear
clc

%% Initial Variables
Fmax=100; %maximum frequency [Hz]
Fs=2.56*Fmax; %sampling frequency as the Shannon's Theorem  defines [Hz]
ts=1/Fs; %sampling time
linhas=1000; %lines number to work as resolution in the frequency spectrum

L=1000; %length of signal
t=(0:L-1)*ts; %time vector
% 

I have defined 2 sinusoidal functions with 2 different frequencies. Summed them and got my "real signal".

%% Construção do Sinal a analisar
f1=sin(2*pi*15*t); %Sinal sinusoidal da 1ª frequência sin(2*pi*[Hz])
f2=sin(2*pi*51*t); %Sinal sinusoidal da 2ª frequência sin(2*pi*[Hz])

%plot dos sinais sinusoidais simples
figure(1)
subplot(3,1,1)
plot(t,f1);
title('1º Sinal Sinusoidal');
xlabel('tempo (s)');
subplot(3,1,2)
plot(t,f2);
title('2º Sinal Sinusoidal');
xlabel('tempo (s)');

%plot do sinal a analisar
sinal=f1+f2; %Sinal constituído por 2 frequências
subplot(3,1,3)
plot(t,sinal);
title('Sinal Complexo');
xlabel('tempo (s)');

Next i did the spectral analysis of "sinal" just like this:

%% Spectral Analysis
NFFT=2^nextpow2(linhas);
RBW=Fs/2*linspace(0,1,NFFT/2+1);

F=fft(sinal,NFFT)/L;
figure(4)
subplot(1,1,1)
plot(RBW,2*abs(F(1:NFFT/2+1)));
title('Espetro de Frequências')
xlabel('Frequência(Hz)')
ylabel('Amplitude')

I know it misses the filter, but i'm adding complexity at slow rate. Like it is i obtain the correct frequencies and amplitude of the signal but i want the plot to adjust the x-axis to the Fmax i defined but i'm not seeing how, and if the resolution i want is working. Perhaps i'm asking too many questions but i want help as i'm no matlab or signal processing expertise.

[Star Strider]

The maximum frequency of the fft is going to be the Nyquist frequency, equal to half the sampling frequency. (You also need to be certain that the analog signal is regularly sampled, so that all the sampling intervals are the same. Digital signal processing techniques — including the fft — require this.) You can certainly plot fewer frequencies, such as the ‘Fmax’ you chose.

Ideally, you also need to use a lowpass Bessel filter, implemented in hardware, at the input of your ADC with a cutoff frequency equal to the Nyuquist frequency. This is an anti-aliasing filter, and will prevent the sampled signal from including aliased frequencies. (The Bessel design is phase-neutral, or linear-phase, so no phase distortion of the analog signal will occur in the process of being filtered.)

[Luís Tomé]

I know i can plot the frequency spectrum with the x-axis limited to my Fmax, but i was hoping if i could do it "automatically" in the fft function or another function that i might have forgotten to add, as the frequencies past Fmax are not of any interest to me and with that interval [0,Fmax] i could have a better resolution using the same number of lines of resolution if i used the whole spectrum.

If what i understood from my professor advisor is right, what i want is the resolution of the spectrum to adjust, considering the whole spectrum to be just that small [0,Fmax] interval, and if i'm using 1k lines of resolution and with Fmax defined to 100Hz, the RBW would be divided into 1k segments with each of the segments being just 0.1Hz. If my function has a frequency of lets say 50,12Hz with unitary amplitude, the way i'm watching is the spectrum would have more energy towards the 50,1 Hz than into 50,2Hz with 0,1Hz of resolution. And with more accurate resolution i could obtain more "thin" spikes in the spectrum...am i right...or thinking wrong, or not explaining the things right?

As for the filter i intended to "emulate it" in matlab as it is easier for me. But i have also another question, the Windows function is a filter right? In lessons my teacher said we would be mostly using the Hanning windows function because it has a good compromise between frequency and amplitude accuracy. So my question is: do i have to pass the signal through the low-pass filter first and then window it or i could have a hann window function that is still a low-pass filter?

The initial objective of this project was to design a low cost acquisition system and accurate at finding fault at machinery like unbalances and misalignments. But at things started to get harder for me to understand and program we decided to just do the algorith so another person could continue our project. Does the context of the project make anything clearer?

I appreciate the info and the help in advance.

[Star Strider]

First, you can choose ‘Fmax’ to be anything you want, providing it is not greater than the Nyquist frequency. The frequency resolution is limited by the sampling frequency. (You can zero-pad the time-domain signal to increase the frequency resolution, however this may not always be appropriate.)

Second, an anti-aliasing filter implemented after sampling is useless, the reason it must be implemented in hardware before the ADC.

‘... the Windows function is a filter right?’

If you mean by that a Hamming or Kaiser window, or some such, not strictly. The windows are used (usually with FIR filters) to mitigate the effects of the sampled signal being finite rather than infinite, as would the Fourier transform of an analytical signal. They are generally not required for IIR filters. (My favourite design strategy is to use the kaiserord function to design FIR filters.)

My pleasure.

[Luís Tomé]

So you're saying that an anti-aliasing filter for what i want, it's useless if implemented via software? In my signal processing cycle i have the acelerometer connected to arduino microcontroller then i would have the hardware implemented anti-alias filter and aftwerwards i would use matlab as the ADC, window applying and then the spectral analysis. Is this the correct way of doing it?

I appreciate the help, info and knowledge so far, you've been great

[Star Strider]

‘So you're saying that an anti-aliasing filter for what i want, it's useless if implemented via software?’

Yes. If the anti-aliasing filter is not implemented in hardware at the input of the ADC, the aliased frequencies will already be present in the sampled signal, and are then impossible to remove.

Once you have tha anti-aliased sampled signal, you can do whatever you want with it. There is always a good argument to be made for using the highest sampling frequency possible (1 kHz is standard) in order to provide the best time and frequency resolution.

My pleasure!