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# Tag Info

### Covariance vs Autocorrelation

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 ...
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### Understanding the mathematical proof for the alias frequencies in a sampled sine wave

The reason is that if it is true for any $m$, it is also true for $m=kn$. I will sketch the proof in another way. Call $f_s = 1/t_s$ sampling frequency where $t_s$ is sampling period, the two signal ...
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### how to evaluate derivative of convolution integral?

Note that option (b) is not correct, and that it is also not equal to what you came up with. Option (b) is just the multiplication of $x(t)$ and $y'(t)$, not the convolution. Your solution and option (...
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### Can $\delta(t+\infty)$ be a legitimate signal?

It's possible, in mathematics, to complete real numbers with "infinite" values, with sound topological properties; for instance non-standard analysis or the Extended real number line (discussion at ...
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### Hilbert transform pair proof

I agree that one of the easiest ways to compute the Hilbert transform in this case is to use the analytic signal. This is most easily obtained via the Fourier transform. Note that the Fourier ...
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### Equation for impulse train as sum of complex exponentials

The summation on the left side of your equation represents a time domain discrete-time periodic signal $x[n]$ whose period is N. $$x[n] = \sum_{k=-\infty}^{\infty} \delta[n-kN]$$ And the summation ...
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### Fourier Transform of Kernel Density Estimation - Convolution Theorem?

It's a bit contrived, but observe that by the "sifting property" of Dirac-delta function: $$K(x-X_j) = K(x) * \delta(x-X_j)$$ where $*$ denotes convolution and $\delta$ is the Dirac-delta function....
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### Find autocorrelation of exponential signal $a^nu[n]$

Let $x[n]=a^nu[n], |a|<1$. Autocorrelation is $$\phi_{xx}[n]=\sum_{m=-\infty}^{\infty}x[m]x[m-n]=\sum_{m=-\infty}^{\infty}a^mu[m]a^{m-n}u[m-n]$$ First assume that $n>0$. In this case, we have ...
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### Gradient of Total Variation (TV) Norm in Total Variation Denoising

I am by no means an expert on total variation, however I think you should check out this Wikipedia page. It doesn't directly answer your question, but I believe the lemma below illustrates the ...
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### Non-causal FIR. Is that possible?

Your mistake is in believing that FIR filters have all their poles at $z=0$. That's only true for causal FIR filters. In general, FIR filters can have poles at $z=0$ and at $z=\infty$. Examples: ...
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### Demodulating upper sideband (USB) signals

Apart from scale factors, your USB signal is $$f(t)=m(t)\cos(2\pi\nu_ct)-\hat{m}(t)\sin(2\pi\nu_ct)\tag{1}$$ (as you've correctly stated in your question). Now if you multiply with a (coherent) ...
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The integrand in the Hilbert transform formula is $h(t,u) = \frac{f(t)}{u-t}$. With a (non-dilated) cardinal sine, you get \frac{\sin(t)}{t(u-t)} = \frac{1}{u}\left( \frac{\sin(t)}{u-t}+\frac{\sin(t)...