Matlab - FFT/PSD Problem for Preemphasis

So my plan is the following (create "adaptive" pre/deemp. using Matlab 2015a):

• 1. Read an Audiodata ([y,fs]) and generate white Noise with a certain SNR ([n,fs])
• 2. Generate a Filter H which shapes the PSD(y) similiar to the PSD(n)
• 3. Generate an inverse Filter G=H^(-1) which reverts the effect of H.
• 4. Get the output signal

The code I was thinking to use for this is the following:

    [y,fs]=audioread('test.wav');
    snr=5;
    n=awgn(y,snr,'measured')-y;
    [pyy,f]=pwelch(y,[],[],length(y)*2-1,fs);
    [pnn,fn]=pwelch(n,[],[],length(y)*2-1,fs);

    H=sqrt(snr*pnn./pyy);
    G=1./H;
    outy=ifft(fft(y).*H.*G);
    outn=ifft(fft(n).*G);
    out=outy+outn;

    [pout,fout]=pwelch(outy,[],[],length(outy)*2-1,fs);
    [pnout,fnout]=pwelch(outn,[],[],length(outn)*2-1,fs);

There are two problems when using the code above:

• 1. The pnout should be shaped liked pout (which it isn't).
• 2. The outputsignals (outy,outn) are both complex and therefore it is not possible to play the outputfile.

If I try it another way:

[y,fs]=audioread('test.wav');
snr=5;
n=awgn(y,snr,'measured')-y;
N=length(y);
bin_vals=0:N-1;
fax_Hz= bin_vals*fs/N;
N_2=ceil(N/2);
pyy=(fft(y).*conj(fft(y)));
pnn=(fft(n).*conj(fft(n)));

H=sqrt(snr*pnn./pyy);
G=1./H
outy=ifft(fft(y).*H.*G);
outn=ifft(fft(n).*G);
out=outy+outn;

I don't come across the previous problems but in this case I don't use a real PSD but the squared absolute of the frequencydomain-values. (and as I understand it this would give me the power spectrum and not the power spectral density)

Any ideas why my first way is wrong or how to change my second way to get a real "proper" PSD and not just the PS?

Thanks! Klaus

Answer 1

[Youssef Khmou]

When using the reverse fft function, we take the real part, in one of the equations H.*G is not necessary since G=H.^-1, another method to generate noise is given in the following version, however it still needs scaling operation.

    [y,fs]=audioread('test.wav');
    snr=5;
    sy=std(y);
    sn=sy/snr;
    n=sn*randn(size(y));
    [pyy,f]=pwelch(y,[],[],length(y)*2-1,fs);
    [pnn,fn]=pwelch(n,[],[],length(y)*2-1,fs);
    H=sqrt(pnn./(pyy*snr));
    G=1./H;
    outy=(real(ifft(fft(y))));
    outn=(real(ifft(fft(n).*G)));
    out=outy+outn;
    [pout,fout]=pwelch(outy,[],[],length(outy)*2-1,fs);
    [pnout,fnout]=pwelch(outn,[],[],length(outn)*2-1,fs);
    figure; 
    plot(fnout,pout,fnout,pnout,'r');

[klaus ebert]

Thanks for your fast support! I thought about taking the real part as well but this gives me the peak of pnout at the wrong frequencies (e.g. with a monoton 1khz sound the peak of the outputnoise is at 2khz instead of 1khz)

Are there other probelms with my code or did I have a wrong thought at some point?

Answer 2

[Youssef Khmou]

I think the first question is not well interpreted, maybe it consists of generating noisy version of y instead of white noise, in this case, the result is sufficiently accurate :

    [y,fs]=audioread('test.wav');
    snr=5;
    n=awgn(y,snr,'measured');
    [pyy,f]=pwelch(y,[],[],length(y)*2-1,fs);
    [pnn,fn]=pwelch(n,[],[],length(y)*2-1,fs);
    H=sqrt(pnn./(pyy*snr));
    G=1./H;
    outy=real(ifft(fft(y).*H.*G));
    outn=real(ifft(fft(n).*G));
    out=outy+outn;
    [pout,fout]=pwelch(outy,[],[],length(outy)*2-1,fs);
    [pnout,fnout]=pwelch(outn,[],[],length(outn)*2-1,fs);
    figure; 
    semilogy(fout,pout,fnout,pnout,'r');grid on;
    xlabel('fr [Hz]');
    ylabel('magnitude ');

[klaus ebert]

If I use exactly this code for my 1khz testfile the result looks like the following. As you can see the peak is at the doubled frequency and I really don't know why/where this happens.

The result should be like the one with the power spectrum (way2) like this(with different y-axis scaling and the right psd of course):

Thanks again for investing so much time and trying to help me!

[Youssef Khmou]

from these tests, i think the problem is related to the method used to estimate the psd, especially the overlap issue and number of segments, let us try a last solution using standard psd function instead of pwelch, this requires a scaling of fft, try the following :

   N=2000;
    %[pyy,f]=pwelch(y,[],[],length(y)*2-1,fs);
    [pyy,f]=psd(y,N);
    %[pnn,fn]=pwelch(n,[],[],length(y)*2-1,fs);
    [pnn,fn]=psd(n,N);
    H=sqrt(pnn./(pyy*snr));
    G=1./H;
    outy=real(ifft(fft(y,N/2+1)));
    outn=real(ifft(fft(n,N/2+1).*G));
    out=outy+outn;
    %[pout,fout]=pwelch(outy,[],[],length(outy)*2-1,fs);
    [pout,fout]=psd(outy,N);
    %[pnout,fnout]=pwelch(outn,[],[],length(outn)*2-1,fs);
    [pnout,fnout]=psd(outn,N);

[klaus ebert]

Still the same problem as with pwelch. I am slowly really freaking out about the stretchfactor of 2 on the x-axis for the noise. Thank you for your intense support, do you maybe have any more ideas?

[Youssef Khmou]

Here is another solution using two sided psd rather than one sided (psd or pwelch) in this case, we use the absolute value of fft :

     % data
    clear;
    fs=1000;
    ts=1/fs;
    f=fs/4;
    t=0:ts:0.1-ts;
    y=sin(2*pi*f*t);

    % altered part
    snr=5;
    sy=std(y);
    sn=sy/snr;
    n=sn*randn(size(y));
    pyy=abs(fft(y));
    pnn=abs(fft(n));
    H=sqrt(pnn./(pyy*snr));
    G=1./H;
    outy=y;
    outn=real(ifft(fft(n,length(y)).*G));
    out=outy+outn;
    [pout,fout]=psd(outy);
    [pnout,fnout]=psd(outn);
    figure; 
    semilogy(fout,pout,fnout,pnout,'r');grid on;
    xlabel('fr [Hz]');
    ylabel('magnitude ');