Subject: Re: On the use of windows From: "Maher, Rob" <rmaher@xxxxxxxx> Date: Thu, 6 Apr 2006 16:42:17 -0600Let's be careful here. I think the original request had to do with an FIR filter being implemented via FFT using the overlap-add or overlap-save procedure. I agree that no explicit window is needed in this case since the goal is just implementing the linear time-invariant filter. But it is a different story if the goal is to do spectral analysis or to do analysis-modify-synthesis processing. What you need to DO with the spectral data is the key here. The discrete Fourier transform (DFT) is of finite length, hence, there is ALWAYS a window--a rectangular window. FFT algorithms compute the DFT. Hence, the statement and spectral analysis example about "not using a window" actually means "use a rectangular window AND ensure that the waveform being analyzed is perfectly periodic AND that the period is exactly an integer divisor of the DFT length." Don't believe it? Well, change the frequency to 5.25 in your equation 3, keeping N the same, and see the spectral leakage in the FFT output. Since in most signal analysis applications it is not possible to choose an exact match between the waveform cycle and the DFT length, the data record truncation effects will be unavoidable in the practical case. The use of a smoothly tapered window is intended to reduce the spectral sideband levels, BUT this is done at the expense of "smearing" the frequency resolution. In other words, it is a tradeoff between resolving closely spaced spectral components vs. avoiding spectral leakage (sidelobes). Going for a narrow main lobe by using a rectangular window unavoidably dictates higher sidelobes. One old but good reference: fredric j. harris. On the use of windows for harmonic analysis with the discrete Fourier transform. Proceedings of the IEEE, 66(1):51--82, January 1978. Regards, Rob Maher -----Original Message----- From: AUDITORY Research in Auditory Perception [mailto:AUDITORY@xxxxxxxx On Behalf Of Fred Herzfeld Sent: Thursday, April 06, 2006 3:10 PM To: AUDITORY@xxxxxxxx Subject: On the use of windows 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