Assume a digital signal x,let us call its true spectrum X.then we apply a window w with x,we get y = x.*w. let us call y's true spectrum Y. Then my question is : Are X and Y the same? 1. If they,X and Y, are same, how to account for that they are different in time domain obviously? 2. If they,X and Y,are different,but we always analysis the spectrum of X by using windowing the original signal,avoiding the leakage of spectrum. Although it avoid the leakage,it lost the true things. Any comments would be appreciated. Regards HyeeWang

# About windowing question.

Started by ●April 1, 2009

Reply by ●April 2, 20092009-04-02

On Apr 1, 10:31�pm, HyeeWang <hyeew...@gmail.com> wrote:> Assume a digital signal x,let us call its true spectrum X.then we > apply a window w with x,we get y = x.*w. let us call y's true spectrum > Y. > > Then my question is : Are X and Y the same?not usually, Y is always the convolution of X against W. if W is a dirac impulse (oh, i mean a "kronecker delta", since you said are assuming "digital" or more-precisely "discrete-time" signals), then Y is the same as X, which means y is the same as x which means that w is 1. this is why 1 and the "impulse function" are necessarily a transform pair.> 1. If they,X and Y, are same, how to account for that they are > different in time domain obviously?premise is wrong. if X and Y are the same, they must be the same in the time domain.> 2. If they,X and Y,are different,but we always analysis the spectrum > of X by using windowing the original signal,avoiding the leakage of > spectrum. Although it avoid the leakage,it lost the true things. > > Any comments would be appreciated. >this is the consequence of looking at a finite amount of data (rather than the whole infinte set). if i present you with a true sine wave, it goes on forever and has two infinitely-skinny impulses as + and - the sinusoidal frequency on the spectrum. but if you look at only a finite section of that sine wave, you cannot possibly be looking at two infinitesimally-thin lines in the frequency domain. there will be spreading or leakage in the frequency domain for the sole and sufficient reason that you considered only a finite-sized segment in the time domain. r b-j

Reply by ●April 2, 20092009-04-02

On Apr 1, 7:31 pm, HyeeWang <hyeew...@gmail.com> wrote:>... > we always analysis the spectrum > of X by using windowing the original signal,avoiding the leakage of > spectrum.>Although it avoid the leakage,it lost the true things.What do you claim are the 'true things' of a finite discrete signal that are 'lost'?> > Any comments would be appreciated. > > Regards > HyeeWangThere is always leakage. With the rectangular window the good things are that the mainlobe response of a tone is narrow and the leakage from bin centered tones takes the value of zero on the output sample frequencies of the DFT. The bad thing is that the sidelobe peaks do not fall off very fast as you move away from the tone frequency. Some people want the leakage arranged differently and so pick other windows. Dale B. Dalrymple

Reply by ●April 2, 20092009-04-02

On Apr 1, 10:31�pm, HyeeWang <hyeew...@gmail.com> wrote:> Assume a digital signal x,let us call its true spectrum X.then we > apply a window w with x,we get y = x.*w. let us call y's true spectrum > Y. > > Then my question is : Are X and Y the same? > > 1. If they,X and Y, are same, how to account for that they are > different in time domain obviously? > 2. If they,X and Y,are different,but we always analysis the spectrum > of X by using windowing the original signal,avoiding the leakage of > spectrum. Although it avoid the leakage,it lost the true things. > > Any comments would be appreciated. > > Regards > HyeeWangYou should read the paper by Fred Harris on Windowing. There are also some good articles in the B&K technical manuals (they available for free online). In analyzing sampled systems - start by assuming that the signal is a sinusoid which falls exactly on a frequency domain bin i.e. an FFT bin - so I'm assuming sampling in frequency domain as well. The application of a window will result in a reduction in the peak of the fourier domain data. This is normally referred to "coherent loss" (or gain). Next if you change the frequency slightly then it no longer falls exactly on an FFT bin, now the signal spreads to near by adjacent FFT bins - this also causes a reduce in the peak of the fourier domain data. This is often referred to as scalloping loss or leakage. Things get more complicated in analyzing broadband signals. Cheers, David Cheers

Reply by ●April 2, 20092009-04-02

On Apr 1, 10:31�pm, HyeeWang <hyeew...@gmail.com> wrote:> Assume a digital signal x,let us call its true spectrum X.then we > apply a window w with x,we get y = x.*w. let us call y's true spectrum > Y. > > Then my question is : Are X and Y the same? > > 1. If they,X and Y, are same, how to account for that they are > different in time domain obviously? > 2. If they,X and Y,are different,but we always analysis the spectrum > of X by using windowing the original signal,avoiding the leakage of > spectrum. Although it avoid the leakage,it lost the true things. > > Any comments would be appreciated. > > Regards > HyeeWangHere is a DFT tutorial I put together. Part of the tutorial shows how windowing works: http://www.fourier-series.com/fourierseries2/DFT_tutorial.html Good Luck

Reply by ●April 2, 20092009-04-02

>Assume a digital signal x,let us call its true spectrum X.then we >apply a window w with x,we get y = x.*w. let us call y's true spectrum >Y. > >Then my question is : Are X and Y the same? > >1. If they,X and Y, are same, how to account for that they are >different in time domain obviously? >2. If they,X and Y,are different,but we always analysis the spectrum >of X by using windowing the original signal,avoiding the leakage of >spectrum. Although it avoid the leakage,it lost the true things. > >Any comments would be appreciated. > >Regards >HyeeWang >Here's a thought experiment for ya. What happens if you apply a hamming window (or any similar shaped window) to data with a pulse in it? How might the window affect the FFT output if the pulse is short in duration relative to your data length? Will it matter where in the data the pulse occured? I guess my point is you can lose "the true things" if you don't pay close attention to your window AND your input data. I also recommend checking out Fred Harris' paper on windowing. It's awesome.

Reply by ●April 2, 20092009-04-02

On Apr 1, 10:31�pm, HyeeWang <hyeew...@gmail.com> wrote:> Assume a digital signal x,let us call its true spectrum X.then we > apply a window w with x,we get y = x.*w. let us call y's true spectrum > Y. > > Then my question is : Are X and Y the same? > > 1. If they,X and Y, are same, how to account for that they are > different in time domain obviously? > 2. If they,X and Y,are different,but we always analysis the spectrum > of X by using windowing the original signal,avoiding the leakage of > spectrum. Although it avoid the leakage,it lost the true things. > > Any comments would be appreciated. > > Regards > HyeeWangthe thing you are missing is due to the discrete nature of an FFT the "true X" you talk about is pretty crappy to begin with, using a window makes it less crappy in some areas, in exchange for making it even more crappy in others.

Reply by ●April 2, 20092009-04-02

On 2 Apr, 04:31, HyeeWang <hyeew...@gmail.com> wrote:> Assume a digital signal x,let us call its true spectrum X.then we > apply a window w with x,we get y = x.*w. let us call y's true spectrum > Y. > > Then my question is : Are X and Y the same?The question as stated doesn't make much sense; the answer might be 'yes' or 'no' depending on the window function. To get any further, you first of all need to ask yourself where the window function comes from in the first place (not all window functions are applied explicitly), and why it is there. Whenever a window function *is* applied explicitly, it is for a reason: Either one wants to achieve some benefit compared to not using the window (this is the argument in FIR filter design), or one wants to avoid a problem caused by not using the window (like in spectrum estimation). As always, people do things for reasons other than "screwing with student's minds." Finding out *why* things are done in a certain way goes a long way to explain the effects of the same action. Rune

Reply by ●April 2, 20092009-04-02

On Apr 2, 10:34 am, "Impoliticus" <swis...@uiuc.edu> wrote:>... > I also recommend checking out Fred Harris' paper on windowing. It's > awesome.The harris paper is perhaps the best around on what to do and how with regard to windows and FFTs. For numerical correctness also check out Nuttall's paper for corrections for the two digit rounded three term Blackman and the optimized coefficient windows response and/or coefficients in the harris paper. On the use of windows for harmonic analysis with the discrete Fourier transform by: FJ Harris Proceedings of the IEEE, Vol. 66, No. 1. (1978), pp. 51-83. Nuttall, Albert H. "Some Windows with Very Good Sidelobe Behavior." IEEE Transactions on Acoustics, Speech, and Signal Processing. Vol. ASSP-29 (February 1981). pp. 84-91 Dale B. Dalrymple