You can best look at it in the frequency domain. If $x[n]$ is the input sequence and $h[n]$ is the filter's impulse response, then the result of the first filter pass is $$X(e^{j\omega})H(e^{j\omega})... View answer Accepted answer 41 votes OK, I'll try to answer your questions: Q1: the number of taps is not equal the to the filter order. In your example the filter length is 5, i.e. the filter extends over 5 input samples [x(n), x(n-1),... View answer Accepted answer 30 votes When choosing one of these 4 types of linear phase filters there are mainly 3 things to consider: constraints on the zeros of H(z) at z=1 and z=-1 integer/non-integer group delay phase shift (... View answer Accepted answer 25 votes Digital filter design is a very large and mature topic and - as you've mentioned in your question - there is a lot of material available. What I want to try here is to get you started and to make the ... View answer Accepted answer 22 votes I'll use the non-unitary Fourier transform (but this is not important, it's just a preference):$$X(\omega)=\int_{-\infty}^{\infty}x(t)e^{-i\omega t}dt\tag{1}x(t)=\frac{1}{2\pi}\int_{-\infty}^{\...

No, taking the Fourier transform twice is equivalent to time inversion (or inversion of whatever dimension you're in). You just get $x(-t)$ times a constant which depends on the type of scaling you ...

As correctly stated in Peter K.'s answer, this question is about aliasing. Since you can't sample at a rate that is sufficiently high to avoid aliasing - i.e., $f_s>50\textrm{ kHz}$ - you have to ...
If you imagine a Dirac delta impulse as the limit of a very narrow very high rectangular impulse with unit area centered at $t=0$, then it's clear that its derivative must be a positive impulse at $0^-... View answer Accepted answer 19 votes The real and imaginary parts of the Fourier transform of a signal$x(t)$are the Fourier transforms of the signal's even and odd parts, respectively: $$X_R(\omega)=\frac12[X(\omega)+X^*(\omega)]\... View answer 18 votes Jojek's answer is of course correct. I would just like to add some more information because much too often have I seen the terms "IIR" and "recursive" confused. The following implications always hold: ... View answer Accepted answer 18 votes The differences you see are due to numerical errors in floating point format. All operations needed to perform an FFT and an inverse FFT can only be done with finite precision and you've shown the ... View answer Accepted answer 17 votes I agree that the windowing filter design method is not one of the most important design methods anymore, and it might indeed be the case that it is overrepresented in traditional textbooks, probably ... View answer 16 votes I agree with pichenettes's answer but I would like to add that it is pretty common practice to use a simple inexpensive low-order analog anti-aliasing filter, and do the rest of the anti-aliasing ... View answer Accepted answer 15 votes The given single-pole IIR filter is also called exponentially weighted moving average (EWMA) filter, and it is defined by the following difference equation:$$y[n]=\alpha x[n]+(1-\alpha)y[n-1],\qquad ... View answer 15 votes The Fourier and the Laplace transform obviously have many things in common. However, there are cases where only one of them can be used, or where it's more convenient to use one or the other. First ... View answer Accepted answer 14 votes It's true that taking the Fourier transform will leave you without any (visible) information on time and vice versa, but of course you don't lose any information, you just represent it in a way such ... View answer 14 votes There are already some good answers, but I still feel like adding yet another explanation, because I consider this topic extremely important for the understanding of many aspects of digital signal ... View answer Accepted answer 14 votes I believe that the fallacy is to believe that a Dirac comb is localized in time. It isn't because it is a periodic function and as such it can only have frequency components at multiples of its ... View answer 14 votes I personally believe that the approach of studying something just in order to "feel ready" to study something else later on is not efficient and tends to overwhelm the student. (This of course only ... View answer Accepted answer 13 votes$X^*(j\omega)$is the complex conjugate of$X(j\omega)$. So if $$X(j\omega)=X_R(\omega)+jX_I(\omega)$$ then $$X^*(j\omega)=X_R(\omega)-jX_I(\omega)$$ and $$X^*(-j\omega)=X_R(-\omega)-jX_I(-\omega)$$ ... View answer Accepted answer 13 votes Of course they don't exist. But we can stop time and use systems that would be non-causal if we hadn't stopped time. Stop time? Yes, just store your data and work offline / non-realtime. Or work on ... View answer Accepted answer 13 votes I think there are actually 3 questions in your question: Q1: Can I derive the frequency response given the poles of a (linear time-invariant) system? Yes, you can, up to a constant. If$s_{\infty,i}$... View answer Accepted answer 13 votes There's a good overview article which appeared in 1996 in the IEEE Signal Processing Magazine: Splitting the unit delay: tools for fractional delay filter design. The nice thing about it is that there'... View answer 12 votes You need to define what you mean by "invertible". Do you mean invertible by a causal and stable system? If yes, then any system that is not minimum-phase is not invertible (because the inverse system ... View answer 12 votes According to your definition of autocorrelation, the autocorrelation is simply the covariance of the two random variables$Z(n)$and$Z(n+\tau)$. This function is also called autocovariance. As an ... View answer Accepted answer 12 votes It's true that zero-padding in the time domain corresponds to interpolation in the frequency domain. If you have a length$N$signal$x[n], its discrete Fourier transform (DFT) is given by $$X[k]=\... View answer Accepted answer 12 votes Matlab's fft functions are all based on FFTW (this is confirmed here), so I guess the obvious choice for you should be FFTW. FFTW is hardware-independent but it can take advantage of some hardware-... View answer 12 votes Let's consider a discrete-time state space model (the derivation for a coninuous-time system is completely analogous):$$\begin{align}\mathbf{q}[n+1]&=\mathbf{Aq}[n]+\mathbf{b}x[n]\\ y[n]&=\... View answer Accepted answer 12 votes You need Parseval's theorem. For the discrete-time Fourier transform (DTFT) you have the following relation: $$\sum_{n=-\infty}^{\infty}|x[n]|^2=\int_{-1/2}^{1/2}|X(f)|^2df\tag{1}$$ wheref$is ... View answer 12 votes Note that$1536 = 3\cdot 512\$. You can use a decimation in time method. See http://www.altera.com/literature/an/an480.pdf I'll add some extra information here in response to your comment to sansuiso'...