Started by April 1, 2009
```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.

Regards
HyeeWang
```
```On Apr 1, 10:31&#2013266080;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
```
```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
> HyeeWang

There 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
```
```On Apr 1, 10:31&#2013266080;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
> HyeeWang

You 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
```
```On Apr 1, 10:31&#2013266080;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
> HyeeWang

Here 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
```
```>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.
>
>
>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

I also recommend checking out Fred Harris' paper on windowing.  It's
awesome.
```
```On Apr 1, 10:31&#2013266080;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
> HyeeWang

the 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.

```
```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
```
```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

```