On the use of windows

[Fred Herzfeld <herzfeld@xxxxxxxxxxxx>]

Hello List,

Once again the use of Windows to remove the edge effects reared its ugly 
head and once again I must make it very clear that the use of any window 
is very dangerous.

One window that exemplifies most windows is the raised cosine function 
which can be written as:

W(j) = 0.5*(1-Cos(2*pi*(j-1)/N))  for j=0 to N-1	EQUATION (1)

where j is the bin number: N is the length of the FFT window in Bins

If W(j) is used by itself as the input to a FFT routine then of course 
Bin number zero will have the DC value of W(j) = 0.5 and since W(j) just 
exactly fits one cycle into the FFT the result in Bin number one will 
have an amplitude of 0.5 and a phase of 180 degrees.

Next consider a single sinusoid as input to the FFT using a rectangular 
window that is

W(j) = 1 for all j					EQUATION (2)
where the input signal is:

S(j) = 1.0 Cos(2*pi*5*(j-1)/N –phi/180*pi)		EQUATION (3)
where phi = 30 is the angle in degrees in this example.

The output of the FFT should then be all zeroes except for Bin 5 which 
should have the amplitude 1.0 and the phase in degrees of 30.

So far so good. But now add (actually multiply) the raised cosine window 
to the function in EQUATION (3). The output of the FFT will now be 
(because of the window):
All bins will be zero except for

Bin 5 will now have A=0.5 and phi=30 .  WOW! The function amplitude has 
decreased by half. And the adjacent bins Bin 4 and Bin 6 no longer have 
zero values ! Bin 4 and bin 5 will each have an amplitude of 0.5 and a 
phase angle of 120 degrees. So first the amplitude has decreased by a 
factor of 2 and the phase of these two components is now 120 degrees. 
The “spectrum” hardly looks like the original single sinusoid. Why does 
this happen with just the addition of the raised cosine window. The 
answere is very simple. We are in effect modulating the single sinusoid 
by the raised cosine. The fact that the raised cosine goes smoothly to 
zero at each end does not really matter. All that does is generate a DC 
component. The raised cosine is in effect a single sinusoid which falls 
exactly into the first Bin thus forming the product of two sinusoids. 
Think back about amplitude modulation. That IS what we have. We see the 
results as the two upper and lower) sidebands of the original signal in 
Bin 5. I call this the “Modulation Error” and as far as I know it has 
not  been dealt with in the literature. Therefore if the signal being 
transformed by the use of the FFT is a sum of sinusoids, the 
multiplication of this sum by ANY window will produce “Modulation 
Errors”!!

To further indicate how violent a window can act use  the  window  given 
 in  EQUATION (1) along with :

Bin 5: 1*Cos(2*pi*5*(J-1)/N – 30/180*pi)			EQUATIONS (4)
Bin 6: 2*Cos(2*pi*6*(J-1)/N – 30/180*pi)
Bin 7: 3*Cos(2*pi*7*(J-1)/N – 30/180*pi)
Bin 8: 4*Cos(2*pi*8*(J-1)/N – 30/180*pi)
Bin 3: -1*Cos(2*pi*35*(J-1)/N – 30/180*pi)

In Bin 5 I have the same function as in EQUATION (3) and the additional 
4 sinusoidal terms are as show in EQUATION (4). Without any window we 
then get zeroes in all Bins except:

Bin 3:  amplitude = 1;  phase = 210 degrees		EQUATIONS (5)		
Bin 5:  amplitude = 1;  phase =  30 degrees
Bin 6:  amplitude = 2;  phase =  30 degrees
Bin 7:  amplitude = 3;  phase =  30 degrees
Bin 8:  amplitude = 4;  phase =  30 degrees

Lastly if I now again add the raised cosine window I get:

Bin 2:  amplitude = 0.25  phase =  30 degrees		EQUATIONS (6)		
Bin 3:  amplitude = 0.50  phase = 210 degrees
Bin 8:  amplitude = 1.25  phase =  30 degrees
Bin 8:  amplitude = 1.00  phase = 210 degrees

Well, isn’t that interesting.  Without a window we have entries in 5 
Bins which agrees with the input to the FFT. With a raised cosine window 
there are entries in only 4 Bins and on top of that Bins 5,6,7 which 
have outputs without a window are now zero when the raised cosine is 
used.  The other entries Bins 2,3,4,8,9 are also incorrect.

It is now to be realized that with “Modulation Error” caused by any 
window each and every frequency component generates two sidebands which 
will affect the immediately adjacent (upper and lower) components if 
they exist and if they do not exist.

The exposition I have given above is the best reason I can give for not 
using a window function with the FFT, the STFFT, the Fourier Series etc.

I will be very please to receive your responses and will answer each one.

Fred Herzfeld

--
Fred Herzfeld, MIT '54
78 Glynn Marsh Drive #59
Brunswick, Ga.31525
USA