why is periodogram(y,w) not periodogram(y.*w)

why is periodogram(y,w) not the same as periodogram(y.*w), where y is the signal and w the window. For a rectangular window it is the same but for a hanning window there is a difference that has something to do with the mean and variance of the window I guess. What exactly is this difference? And why is this difference there? Same holds for pwelch.

Answer 1

[Wayne King]

The difference is that the syntax periodogram(y,w) uses the window normalization constant explained here:

http://www.mathworks.com/help/signal/ug/spectral-analysis.html#f12-20709

Read about the modified periodogram.

While periodogram(y.*w) does not use that normalization because you are using a rectangular window

I would recommend using

periodogram(y,w)

However, if you look at the two periodograms, you'll see that one is simply a scaled version of the other and that scaling is due to window normalization you get with

periodogram(y,w)

[Wesley Ooms]

Thanks alot, That is a very good link

For other people who are interested: this is how to go from fft to periodogram (amplitude power spectrum) and from periodogram to pwelch:

    y = load('data.mat')                ;% samples
    t = load('time.mat')                ;% time   
    d = mean(diff(t))                   ;% delta time
    N = length(y)                       ;% number of samples
    a = 10                              ;% number of averages
    o = 1000                            ;% overlap
    l = floor((N+o*(a-1))/a)            ;% length of time vectors
    w = (1-cos(2*pi*(1:l)/(l+1)))       ;% window
    f = ceil(-N/2:N/2-1)/d/N            ;% frequency points
    P = pwelch(y,w,o,l)                 ;% ----->   pwelch    <-----
    n = floor(l/2)+1;p=eye(n,a);q=p     ;% initialize variables

    for k=1:a;
        z      = y((1:l)+(l-o)*(k-1))   ;
        p(:,k) = periodogram(z,w,l)     ;% -----> periodogram <-----
        Y      = fft(z'.*w)             ;% ----->     fft     <-----
        Y      = [Y(1)/sqrt(2) Y(2:n-1) Y(n)/sqrt(2-rem(l,2))];
        q(:,k) = Y.''.*Y/pi/(w*w')      ;
    end

    p = mean(p,2);      q = mean(q,2)   ;% average
    figure(1);clf                       ;% post processing
    subplot(311); plot(y);
    subplot(312); plot(P);hold all;plot(p);plot(q)
    subplot(313); plot(abs(P)-abs(q))
    linkaxes(get(figure(1),'children'),'x')