Windowing effect on spectral leakage and phase after FFT

Hi,
This is a great forum. I am learning a lot about DSP here. But I am
puzzled by the effect of windowing in the code below. (P.S., this is not a
homework problem. I am trying to learn how to do spectral analysis on
random signals). I am much obliged to your help.

1. Why did spectral leakage and gain loss occurred in when I windowed the
sinusoid? (I think I coded it correctly).

2. My time domain signal contains no noise. So why I don't get complete
zero magnitude at frequency other than 50 and 120Hz? (This is screwing up
my phase plot)

3. I need help in interpreting the phase plot. All I understand is that
phase is correct at 50Hz and 120 Hz. But if I have a random signal where I
have no a priori knowledge on which frequency is meaningful, what do I do?

format long;
Fs = 1000; % Sampling frequency
T = 1/Fs; % Sample time
L = 1000; % Length of signal
t = (0:L-1)*T; % Time vector
% Sum of a 50 Hz sinusoid and a 120 Hz sinusoid
x = 0.7*cos(2*pi*50*t+0.25*pi) + cos(2*pi*120*t+0.75*pi); 
windowHanning = window(@hann, length(x));
y = windowHanning.*x'; % multiply by Hanning window.
z = x'; % Not windowed version. 

NFFT=length(y);
Y = fft(y,NFFT)/L;
f = Fs/2*linspace(0,1,NFFT/2+1);

NFFTz=length(z);
Z = fft(z,NFFTz)/L;
fz = Fs/2*linspace(0,1,NFFTz/2+1);

% Plot single-sided amplitude spectrum.
figure(1); plot(f,2*abs(Y(1:NFFT/2+1)), fz, 2*abs(Z(1:NFFTz/2+1)), '-r');

legend('Windowed', 'not windowed');
title('Single-Sided Amplitude Spectrum of y(t)')
xlabel('Frequency (Hz)')
ylabel('|Y(f)|')

%Plot non-windowed phase spectrum.
phi=angle(Z);
figure(2); subplot(211); plot(f, phi(1:NFFT/2+1)/pi); %plot in radian.
title('Single-Sided Phase Spectrum of non-windowed orignal signal');
xlabel('Frequency (Hz)');
ylabel('Phase (rad)');
subplot(212);
phiy=angle(Y);
plot(f, phiy(1:NFFT/2+1)/pi); 
title('Single-Sided Phase Spectrum of windowed orignal signal');
xlabel('Frequency (Hz)');
ylabel('Phase (rad)');
abs(Y(51))*2%Magnitude at 50Hz.
abs(Y(121))*2%Magnitude at 120Hz.
angle(Y(51))/pi %Phase angle (rad) at 50Hz
angle(Y(121))/pi %Phase angle (rad) at120Hz 

shinchan75034 schrieb:
> Hi,
> This is a great forum. I am learning a lot about DSP here. But I am
> puzzled by the effect of windowing in the code below. (P.S., this is not a
> homework problem. I am trying to learn how to do spectral analysis on
> random signals). I am much obliged to your help.
>
> 1. Why did spectral leakage and gain loss occurred in when I windowed the
> sinusoid? (I think I coded it correctly).
>
> 2. My time domain signal contains no noise. So why I don't get complete
> zero magnitude at frequency other than 50 and 120Hz? (This is screwing up
> my phase plot)
>
> 3. I need help in interpreting the phase plot. All I understand is that
> phase is correct at 50Hz and 120 Hz. But if I have a random signal where I
> have no a priori knowledge on which frequency is meaningful, what do I do?
>
> format long;
> Fs = 1000; % Sampling frequency
> T = 1/Fs; % Sample time
> L = 1000; % Length of signal
> t = (0:L-1)*T; % Time vector
> % Sum of a 50 Hz sinusoid and a 120 Hz sinusoid
> x = 0.7*cos(2*pi*50*t+0.25*pi) + cos(2*pi*120*t+0.75*pi);
> windowHanning = window(@hann, length(x));
> y = windowHanning.*x'; % multiply by Hanning window.
> z = x'; % Not windowed version.
>
> NFFT=length(y);
> Y = fft(y,NFFT)/L;
> f = Fs/2*linspace(0,1,NFFT/2+1);
>
> NFFTz=length(z);
> Z = fft(z,NFFTz)/L;
> fz = Fs/2*linspace(0,1,NFFTz/2+1);
>
> % Plot single-sided amplitude spectrum.
> figure(1); plot(f,2*abs(Y(1:NFFT/2+1)), fz, 2*abs(Z(1:NFFTz/2+1)), '-r');
>
> legend('Windowed', 'not windowed');
> title('Single-Sided Amplitude Spectrum of y(t)')
> xlabel('Frequency (Hz)')
> ylabel('|Y(f)|')
>
> %Plot non-windowed phase spectrum.
> phi=angle(Z);
> figure(2); subplot(211); plot(f, phi(1:NFFT/2+1)/pi); %plot in radian.
> title('Single-Sided Phase Spectrum of non-windowed orignal signal');
> xlabel('Frequency (Hz)');
> ylabel('Phase (rad)');
> subplot(212);
> phiy=angle(Y);
> plot(f, phiy(1:NFFT/2+1)/pi);
> title('Single-Sided Phase Spectrum of windowed orignal signal');
> xlabel('Frequency (Hz)');
> ylabel('Phase (rad)');
> abs(Y(51))*2%Magnitude at 50Hz.
> abs(Y(121))*2%Magnitude at 120Hz.
> angle(Y(51))/pi %Phase angle (rad) at 50Hz
> angle(Y(121))/pi %Phase angle (rad) at120Hz

1-2: Get a pencil and paper and write down what windowing in the time 
domain is doing with your spectrum and you might understand why 
windowing does mess up the spectrum (hint: multiplication --> convolution).
3: When you do not have a reference frequency or your signal is purely 
random phase is not a good feature to describe the signal.

On 18 Nov, 18:24, "shinchan75034" <shinchan75...@gmail.com> wrote:
> Hi,
> This is a great forum. I am learning a lot about DSP here. But I am
> puzzled by the effect of windowing in the code below. (P.S., this is not a
> homework problem. I am trying to learn how to do spectral analysis on
> random signals). I am much obliged to your help.
>
> 1. Why did spectral leakage and gain loss occurred in when I windowed the
> sinusoid? (I think I coded it correctly).

Because that's what windows do. The purpose of applying windows
is to suppress side lobes in the spectrum. This is achieved
at the expense of spectral leakage etc.

> 2. My time domain signal contains no noise. So why I don't get complete
> zero magnitude at frequency other than 50 and 120Hz? (This is screwing up
> my phase plot)

Because that's what windows do. Again, windows are used
for specific purposes, which are achived at some expense.

> 3. I need help in interpreting the phase plot. All I understand is that
> phase is correct at 50Hz and 120 Hz. But if I have a random signal where I
> have no a priori knowledge on which frequency is meaningful, what do I do?

Disregard the phase plots for the spectrum coefficients
with 'small' magnitude.

Rune

>On 18 Nov, 18:24, "shinchan75034" <shinchan75...@gmail.com> wrote:
>> Hi,
>> This is a great forum. I am learning a lot about DSP here. But I am
>> puzzled by the effect of windowing in the code below. (P.S., this is
not a
>> homework problem. I am trying to learn how to do spectral analysis on
>> random signals). I am much obliged to your help.
>>
>> 1. Why did spectral leakage and gain loss occurred in when I windowed
the
>> sinusoid? (I think I coded it correctly).
>
>Because that's what windows do. The purpose of applying windows
>is to suppress side lobes in the spectrum. This is achieved
>at the expense of spectral leakage etc.
>
>> 2. My time domain signal contains no noise. So why I don't get
complete
>> zero magnitude at frequency other than 50 and 120Hz? (This is screwing
up
>> my phase plot)
>
>Because that's what windows do. Again, windows are used
>for specific purposes, which are achived at some expense.
>
>> 3. I need help in interpreting the phase plot. All I understand is
that
>> phase is correct at 50Hz and 120 Hz. But if I have a random signal
where I
>> have no a priori knowledge on which frequency is meaningful, what do I
do?
>
>Disregard the phase plots for the spectrum coefficients
>with 'small' magnitude.
>
>Rune
>

I should have multiply amplitude correction factor of 2 for Hanning
window. But I am still puzzled. I thought you apply window to reduce
spectral leakage by removing the discontinuity (See
www.ee.iitm.ac.in/~nitin/_media/ee462/fftwindows.pdf)?

On 18 Nov, 20:00, "shinchan75034" <shinchan75...@gmail.com> wrote:
> >On 18 Nov, 18:24, "shinchan75034" <shinchan75...@gmail.com> wrote:
> >> Hi,
> >> This is a great forum. I am learning a lot about DSP here. But I am
> >> puzzled by the effect of windowing in the code below. (P.S., this is
> not a
> >> homework problem. I am trying to learn how to do spectral analysis on
> >> random signals). I am much obliged to your help.
>
> >> 1. Why did spectral leakage and gain loss occurred in when I windowed
> the
> >> sinusoid? (I think I coded it correctly).
>
> >Because that's what windows do. The purpose of applying windows
> >is to suppress side lobes in the spectrum. This is achieved
> >at the expense of spectral leakage etc.
>
> >> 2. My time domain signal contains no noise. So why I don't get
> complete
> >> zero magnitude at frequency other than 50 and 120Hz? (This is screwing
> up
> >> my phase plot)
>
> >Because that's what windows do. Again, windows are used
> >for specific purposes, which are achived at some expense.
>
> >> 3. I need help in interpreting the phase plot. All I understand is
> that
> >> phase is correct at 50Hz and 120 Hz. But if I have a random signal
> where I
> >> have no a priori knowledge on which frequency is meaningful, what do I
> do?
>
> >Disregard the phase plots for the spectrum coefficients
> >with 'small' magnitude.
>
> >Rune
>
> I should have multiply amplitude correction factor of 2 for Hanning
> window. But I am still puzzled. I thought you apply window to reduce
> spectral leakage by removing the discontinuity (Seewww.ee.iitm.ac.in/~nitin/_media/ee462/fftwindows.pdf)?- Hide quoted text -
>
> - Show quoted text -

You apply a window to change the shape of the frequency spectrum. It
convolves the frequency spectrum of the signal with the window
'kernel' (whcih is the frequency spectrum of the window). That might
'reduce' leakage, or it might not - I don't find 'reducing leakage' a
highly informative way to describe what is going on.

Try reading harris on window functions (google f j harris window
functions). Or read my notes on this at:

http://www.bores.com/courses/intro/freq/3_window.htm

and

http://www.bores.com/courses/advanced/windows/index.htm

harris, and the second of my notes above, discuss the 'figures f
merit' for window functions - these include gain-related figures such
as Processing Loss, Coherent Gain, and Scalloping Loss as well as
leakage in terms of sidelobes.

The way I sometimes think of it, you choose  a window function whose
kernel best matches the shape that either you think the signal's
spectrum has, or that you want the signal's spectrum to have.

Chris
=======================
Chris Bore
BORES Signal Processing
www.bores.com