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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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Where is the flaw in this derivation of the DTFT of the unit step sequence $u[n]$?

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8 votes
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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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Why is the signum function $=2u(t) - 1$?

The signum function is defined by $$\text{sgn}(t)=\begin{cases}-1,&t<0\\0,&t=0\\1,&t>0\end{cases}$$ Using the half-maximum convention, the unit step function is defined by $$u(t)=\...
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Proof of complex conjugate symmetry property of DFT

Hint: According to Euler's formula we have $$e^{-j2\pi k}=\cos(2\pi k)-j\sin(2\pi k)=\ldots$$
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6 votes

Where is the flaw in this derivation of the DTFT of the unit step sequence $u[n]$?

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5 votes
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How to prove that the peak of the autocorrelation function is at zero lag?

The Cauchy Schwarz inequality states that: $$ \left|\int_{-\infty}^{\infty}g_1(t)g_2(t) dt\right|^2 \leq \int_{-\infty}^{\infty}|g_1(t)|^2 dt \int_{-\infty}^{\infty}|g_2(t)|^2 dt $$ I'm going to ...
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Inverse Laplace transform Using Inversion Formula

In engineering practice, the complex inversion integral is hardly ever used. As an engineer, you will almost exclusively need to invert rational functions, and this can be done by partial fraction ...
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5 votes

Proof of complex conjugate symmetry property of DFT

Remember that $e^z$ has a very different meaning than $e^x$ (taking $z\in\mathbb{C}$ and $x\in\mathbb{R}$). If the exponent was real, then, as you state in your question: $$e^x = 1 \iff x=0$$ ...
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5 votes

Proof of complex conjugate symmetry property of DFT

Did you ever wonder about where $\pi $ came from? Watch out... Let us first draw this weird function complex exponential $e^{-2j\pi t}$ for several discrete values of $t\in[0,10]$ (the little blue ...
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Convolution of even functions is even

Just do a change of variables: \begin{align} z(t) &= x\star y \big|_{t}\\ &=\int_{\tau=-\infty}^{\tau = \infty} x(\tau)y(t-\tau) \,\mathrm d\tau &\scriptstyle{\text{Set}~\tau=-\lambda, \...
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struggling to understand why Fourier basis is orthogonal

Just use the formula for the geometric series (I use $l=h-k\neq mN$): $$\sum_{n=0}^{N-1}e^{-j\frac{2\pi}{N}nl}=\frac{1-e^{-j\frac{2\pi}{N}Nl}}{1-e^{-j\frac{2\pi}{N}l}}=\frac{1-e^{-j2\pi l}}{1-e^{-j\...
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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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4 votes
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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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4 votes
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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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4 votes

Laplace of step and integration are same?

This is because the impulse response of an integrator is $h(t)=u(t)$. The output which is the convolution with the impulse respoponse is $$y(t)=\int_{-\infty}^{\infty}x(\tau)h(t-\tau)d\tau$$ and with $...
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  • 4,105
4 votes
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Discrete-time Fourier Transform of the unit step sequence $u[n]$

Cedron Dawg posted an interesting initial point in this answer. It begins with these steps: $$ \begin{align} U(\omega) &= \sum\limits_{n=0}^{+\infty} e^{-j \omega n} \\ &= \lim_{ N \to \...
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  • 4,842
4 votes

Discrete-time Fourier Transform of the unit step sequence $u[n]$

I'll provide two relatively simple proofs that do not require any knowledge of distribution theory. For a proof that computes the DTFT by a limit process using results from distribution theory, see ...
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4 votes

Where is the flaw in this derivation of the DTFT of the unit step sequence $u[n]$?

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4 votes
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Proof of $E[y(n)]=E[x(n)] \, \sum h(k)$

The standard meaning of white noise includes an insistence (whether implicit or explicit) that the mean is $0$. Thus, what you want to prove is trivially true: since $$Y[n] = \sum_{k=-\infty}^\infty ...
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4 votes

Fourier components of $\cos(2\pi f_1t)$

HINT: Going from your last equation, $$\frac{\sqrt{T}}{2}\bigg(\frac{e^{j2\pi (f_1T-n)}-1}{j2\pi (Tf_1-n)} + \frac{e^{-j2\pi (f_1T+n)}-1}{-j2\pi (Tf_1+n)}\bigg)$$ This can be simplified further down ...
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Derivation of $ R_{N(t)}(\tau) $ from its $f_{N(t)}(\eta)$

As @MattL points out in a comment, a Gaussian pdf does not imply whiteness. Indeed, it can be argued that the assumption that the process is a continuous-time white noise process is contrary to the ...
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4 votes

Proving Fourier transform of cosine multiplied with another function

This is taken from Lapidoth's book on digital communications. It expands on Matt's comment about solving this problem in the time domain, and it may answer your last question. Let $y(t) = x(t)\cos(2\...
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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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3 votes
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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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3 votes

Hilbert transform pair proof

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)...
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