[Math] Rebuilding original signal from frequencies, amplitude, and phase obtained after doing an fft

fourier analysisfourier seriesMATLAB

Rebuilding original signal from frequencies, amplitude, and phase obtained after doing an fft.

Greetings

I'm trying to rebuild a signal from the frequency, amplitude, and phase obtained after I do an fft of a signal, but when I try and combine the fft data (frequency, amplitude, and phase) back to see if I get a similar signal, the pattern is a little off. I think it has to do with my formula which may be a little incorrect.

The formula I'm using to combine the data is:

amplitude*sin(2*pi*time*frequency+phase)+amplitude*cos(2*pi*time*frequency+phase);

Please note: At the moment I don't want to use IFFT due to the fact that I will be editing the amplitude and frequencies before the calculations/rebuilding of the signal are done

An image of the plot is below. The top one is the original signal and the bottom one is the signal created with the equation. If you want to know I'm using matlab but I think the problem is with the equation. enter image description here

Here's the function I use to get the frequency, amplitude and phase of the signal.

function [freq,amp,phase,sigfft,sigifft,sigphase]=rtplotfft(vp_sig_orig,Fs)
    vp_sig_orig=vp_sig_orig';
    vp_sig_len=length(vp_sig_orig); %get sample rate from vp fs_rate needs to be an even number?

    % Use next highest power of 2 greater than or equal to length(x) to calculate FFT.
    nfft= 2^(nextpow2(length(vp_sig_orig))); 
    % Take fft, padding with zeros so that length(fftx) is equal to nfft 
    fftx = fft(vp_sig_orig,nfft); 
    sigfft= fft(vp_sig_orig); 
    sigifft=ifft(sigfft);
    sigphase = unwrap(angle(sigfft')); %get phase of orginal signal
    % Calculate the number of unique points
    NumUniquePts = ceil((nfft+1)/2); 
    % FFT is symmetric, throw away second half 
    fftx = fftx(1:NumUniquePts); 
    % Take the magnitude of fft of x and scale the fft so that it is not a function of the length of x
    mx = abs(fftx)/length(vp_sig_orig); 

    % Take the square of the magnitude of fft of x. 
    %mx = mx.^2; rem'd out to get amplitude to work

    % Since we dropped half the FFT, we multiply mx by 2 to keep the same energy.
    % The DC component and Nyquist component, if it exists, are unique and should not be multiplied by 2.
    if rem(nfft, 2) % odd nfft excludes Nyquist point
    mx(2:end) = mx(2:end)*2;
    else
    mx(2:end -1) = mx(2:end -1)*2;
    end

    %yamp=(mx(1,:)/max(abs(mx(1,:)))*1); % keep at 1, dont use in function use in script file directly
    amp=mx;

    % This is an evenly spaced frequency vector with NumUniquePts points. 
    freq_vect = (0:NumUniquePts-1)*vp_sig_len/nfft; 
    freq=freq_vect';

    %get phase of new signal
    phase = unwrap(angle(fftx)); %get phase of orginal signal

tia

Best Answer

This is what the IFFT is for. Do your modifications to the spectrum, and then get them back into the form IFFT expects, and then do an IFFT. Don't bother trying to re-implement the IFFT when you don't have to. It will be slower and more error-prone.

So each point in an FFT is a complex number that contains information about the phase and amplitude at that frequency. You're extracting this information with these lines:

sigphase = unwrap(angle(sigfft')); %get phase of orginal signal
mx = abs(fftx)/length(vp_sig_orig);

You're getting the magnitude and phase of each complex number, converting from rectangular complex number to polar form.

After modifying the phases and magnitudes, to reverse them back into something suitable for the IFFT, you just need to undo these and convert each back into a rectangular complex number, using the formula:

$z = r(\cos \varphi + i\sin \varphi ).\, $

where z is the resulting complex number, r is the magnitude, and φ is the phase. You'll have to undo the scaling of the magnitudes (and undo any change from radians to degrees, etc.) You'd do this for each mag, phase pair in the two vectors, combining them back into a single vector of the original length, made up of complex numbers. Then feed this to the IFFT. If you hadn't made any changes, it should return the original signal.

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