14
votes
Accepted
Why would one use a Hann or Bartlett window?
In reviewing fred harris' Figures of Merit for various windows (Table 1 in this link) the Hamming is compared to the Hanning (Hann) at various values of $\alpha$ and from that it is clear that the ...
- 43.4k
12
votes
Accepted
Proof that the rectangular window has the best resolution
Wikipedia has a very nice article on window functions https://en.wikipedia.org/wiki/Window_function
The basic trade-offs in windows is main lobe width vs side lobe height and overall attenuation. The ...
- 37.8k
10
votes
FFT with asymmetric windowing?
I'll use the shorthand window for "window function".
With audio, any processing that creates something akin to pre-ringing or pre-echo will sound sloshy like a low bit-rate mp3. This happens when ...
- 12.9k
10
votes
Accepted
Is spectral leakage due to windowing 'different' for the DTFT and DFT?
The concept of spectral leakage is not dependent on the context, and it's the same thing for the DTFT as it is for the DFT. It's probably helpful to remember that the DFT of a finite length sequence ...
- 84.3k
9
votes
Accepted
Why do we use window in time domain rather than do FFT modify the spectrum and than inverse FFT
Windowing reduces spectral leakage.
Say you start out with a $\sin(y) = \cos(\omega_0 t)$. The period is obviously $2 \pi/ \omega_0$.
But if nobody told you that the period is $2 \pi/ \omega$ ...
- 106
9
votes
Accepted
Why are there so many windowing functions?
Aside from reduction of spectral leakage, there is a one major trade-off to be made when choosing a window function. Below you can see a figure with various parameters. Two of them are most important:
...
- 10.8k
9
votes
Accepted
Fourier transform artifacts
The cross pattern is typically a border effect, due to the periodicity induced by the standard implementation and hypotheses behind the Fast Fourier transform, when the image lacks periodicity from ...
- 31.1k
9
votes
Accepted
Why is each window/frame overlapping?
Why is each window/frame overlapping?
Windowing is a means to stationarize signals. Inside a small enough window, you can expect that the properties of the signal chunk do not vary too fast. And you ...
- 31.1k
8
votes
Accepted
python scipy fft on numpy hanning window smears peaks
Is this meant to happen?
Yes, that is absolutely a well known effect of using any window function.
Taking a look through the Wikipedia article on window functions, we find that the rectangular window ...
- 1,026
8
votes
Accepted
Effect of windowing on noise
UPDATE: My previous response did not answer the OP's question. The following addresses the question directly:
Bottom line: Prior to windowing in time, each sample in frequency is an IID Gaussian ...
- 43.4k
8
votes
Does windowing affect Parseval's theorem?
Parseval's theorem will hold, but take into account that your signal in the time domain will no longer be $x[n]$. Namely, if you have that
$$\sum_{n=0}^{N-1} \Big| x[n] \Big|^2 = \frac{1}{N} \sum_{...
- 4,930
8
votes
Accepted
What is the difference of windowing functions for FIR filtering?
So, from the discussion in the comments it's clear you know most you need to know.
The window method for FIR filter design is based on this idea:
We know the "ideal" frequency response $H(f)$ we ...
- 28k
7
votes
How to do continuous signal processing (i.e without windowing)?
Yes, it's possible to analyse sound the way ears do.
For example, you could compute the DFT of a signal continuously using several Goertzel filters.
$$
y_k[n] = e^{j2\pi k/N} y_k[n-1] + x[n]
$$
...
- 23.9k
7
votes
Why is each window/frame overlapping?
More overlap means you end up with more windows (of a given length) per second of audio. More windows (of a given length) requires more FFTs which requires more MACs or FLOPs which generally requires ...
- 34.5k
7
votes
Kaiser window approximation
just implement the Modified Bessel function. it's easy.
i always like my window definitions centered about zero, since pretty much all of them are even symmetry.
i'll do this in discrete-time, but ...
- 18.1k
6
votes
Accepted
What is the relation of the transition band's width and the filter order for the FIR windowing method
There are only heuristic formulas for estimating the filter order. For a Kaiser window (which is probably the most frequently used window for filter design) the required filter order can be estimated ...
- 84.3k
6
votes
Kaiser window approximation
You could try the exponential window:
$$w_n=\frac{ \exp \left[\alpha \sqrt{1-\left(\frac{n-M}{M}\right)^2}\right]}{\exp(\alpha)}$$
$$\alpha=-427.5*10^{-6} A_s^2+0.1808*A_s-3.516$$
or the hyperbolic ...
- 1,798
6
votes
Window period(overlap) and FFT
In addition to what others have already said, I'll try to answer it from a purely practical point of view (this is also a variant of the overlap-add technique).
If your FFT length is 2048, then an ...
- 911
5
votes
Accepted
Is it customary to correct for the gain of a window?
Yes, it is customary to correct for the gain of a window, except for some cases I refer to later. (If you are interested only in the relative amplitude, of course you do not need to correct for the ...
- 66
5
votes
Is the discrete Gaussian kernel an eigenfunction of the DFT?
This answer complements @CedronDawg's answer which introduced this family of eigenvectors. More specifically, this answer presents three algorithms and a hybrid algorithm for generating for a given ...
- 12.9k
5
votes
Is the discrete Gaussian kernel an eigenfunction of the DFT?
I have made a tremendous amount of progress on this issue in the last few weeks.
The Zeroing Sine Family of Window Functions
I have discovered a previously unrecognized class of window functions. ...
- 6,982
5
votes
Intuitively, why is windowing function a low-pass filter?
I do interpret the question "why is windowing function a low-pass filter?" in another direction: why can a (typical) window function be interpreted as the series of coefficients of a low-...
- 31.1k
5
votes
How to calculate resolution of DFT with Hamming/Hann window?
The frequency resolution of a DFT is indeed significantly affected by the choice of window. Two key parameters to consider when selecting a window are the frequency resolution and dynamic range. The ...
- 43.4k
5
votes
How is the energy of $x_1\cdot x_2$ related to the energies of $x_1$ and $x_2$?
Knowing the energies of $x_1$ and $x_2$ is not sufficient for determining the energy of $x_3=x_1x_2$. What you can do is determine an upper bound for the energy of $x_3$ given the energies of $x_1$ ...
- 84.3k
5
votes
Accepted
Zero padding - High amplitude
I'd like to apply zero padding to it, for better frequency bin resolution.
First of all, let's state it one more time that zero padding does not improve frequency resolution of DFT. It'll only ...
- 27.5k
5
votes
Does windowing affect Parseval's theorem?
While the Fourier transform, discrete or continuous, can be regarded as unitary transform i.e a naturally norm preserving change between orthonormal bases in a normed complex vector space, the ...
- 4,411
5
votes
Accepted
Seven term Blackman Harris window
In this document you can find the coefficients of a seven-term Blackman Harris window. Ignoring the bizarre notation, it seems like the window is defined by
$$w[n]=\sum_{k=0}^6a_k\cos\left(\frac{2\pi ...
- 84.3k
5
votes
Is the DCT prone to spectral leakage like the DFT?
A window function other than rectangular can be applied to suppress sidelobes also with the discrete cosine transform (DCT). Window functions are also sometimes used together with some flavors of DCT ...
- 12.9k
Only top scored, non community-wiki answers of a minimum length are eligible