8

Edited in response to revised question and additional comments by the OP. I disagree with @JasonR's assertion that filter ringing is due to Gibbs phenomenon. As described in the Wikipedia article linked to in Jason's answer, the Gibbs phenomenon is an observation about the asymptotic behavior of the truncated sum (first $n$ terms) of the Fourier series of ...


8

As Jason pointed out there a basic "uncertainty principle": everything that's very narrow in frequency is wide in time and vice versa. If you use minimum filters, there should be no pre-ringing, only post ringing. Pre-ringing only happens for linear phase filters. Pre-ringing is much more audible than post-ringing, so minimum filters tend to be the better ...


8

Your observations are an example of the Gibbs phenomenon. When you apply a filter with a very sharp transition band, you will observe oscillations in the filter output (or "ringing") near any sharp transitions in the input signal (e.g. boundaries of pulsed waveforms). The apparent "frequency" of the oscillations is dependent upon the bandwidth of the filter; ...


7

Sticking with linear systems, removing the ringing is nearly the same as adding back some of the spectral content that your really steep transition filters removed. Why use some crazy scheme to add back the stuff in the "softer" transitions that your hard-edged filters cut out? Just use a more reasonable total filter response in the first place. Going to ...


5

I would say that this is interesting. There has been a lot of work done regarding the study of Gibbs Phenomenon. You should check out the following document to get a better understanding of how it comes up in practical DSP applications: http://people.clarkson.edu/~ajerri/books/examples/Gibbs_Book.pdf The typical way to manage Gibbs Phenomenon is to use ...


5

The book "Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills", by P. Nahin, Princeton University Press, leads up to and contains an explanation of the Gibbs phenomena which might be suitable for someone with a good undergraduate university level math background.


4

I think the main issue is you are jumping ahead of yourself. You probably remember or read somewhere that the Fourier transform of a $rect$ function is a $sinc$ function. This is true; however, no where in this section does he mention Fourier transform! In fact, what he is doing is not Fourier transform. What he does in this section is to represent any ...


3

The Gibbs phenomenon occurs wherever a discontinuous function is approximated by a truncated (Fourier) series. For the design of systems, the Gibbs phenomenon occurs when approximating ideal brick wall filter responses (like low-pass, band-pass, etc.), but also for all-pass systems such as the Hilbert transformer because of its discontinuity at $\omega=0$: $...


3

A bandpass filter with steep transitions and a flat passband approaches a rectangular shape. A rectangle in one FT domain is a Sinc function in the other domain. This is true for a rectangular window in the time domain creating spectral "leakage" in the frequency domain. Or for a rectangular window in the frequency domain creating a spiral packet in the ...


2

You can always say that sin and cos have shape of circular, and never have shape of sharp edge no matter the frequency of the argument, so that is why you have to get infinity of them to draw a rectangular :)


2

Recursive sinusoids is the basic principle of FM synthesis (used in famous Yamaha DX7 etc.) : with such synthesis, oscillators (named "operators") can be added but also embedded like this : sin(sin(t+sin(...))+...)


2

Instead of nulling the 2 quadrants, multiply them by $i$. Then the real part will be one image and the imaginary part the other image. X = fftshift(fft2( x )); % x is the input (image above) X( 1:floor(size(X,1)/2), 1:floor(size(X,2)/2) ) = ... X( 1:floor(size(X,1)/2), 1:floor(size(X,2)/2) ) * 1i; X( ceil(size(X,1)/2)+1:end, ceil(size(X,2)/2)+...


1

I would expect that this is (at least roughly) the best way to do this. If you find that you need to soften the edges, you could use a cosine roll-off. I'm not sure about which bit to discard. I would look at the data and see where most of the energy lies - compare just taking the real part to just taking the magnitude (signed magnitude I guess), and see ...


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