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Questions tagged [proof]

The tag has no usage guidance.

3
votes
2answers
120 views

Can $\delta(t+\infty)$ be a legitimate signal?

Mathematically speaking, when I try to use some signal to disprove a system is invertible, can I use the signal like $\delta(t+\infty)$ ($\delta$ representing the Dirac distribution)? For example, the ...
2
votes
1answer
165 views

Prove that convolution with a separable filter is equivalent to convolution on each 1D filter

I would like to prove that convolution of an image $I \in \mathcal{M}_{m_1 \times n_1}$ with respect to a separable 2D filter $F$, (i.e., $F = F_1 F_2$, where $F \in \mathcal{M}_{m_2\times n_2}(\...
0
votes
1answer
26 views

Equation Derivation: Help with intermediate steps

As per the attached image above, I don't know how the equation in the first line transformed to the form in second line. I tried solving it by multiplication but the result was different.
2
votes
3answers
900 views

Proof of linearity

I have this system: $$y[n] − 4y[n − 1] + 4y[n − 2] = 20x[n] + 10x[n − 1]$$ I have no idea how to prove if the system is linear because it depends on future outputs.
1
vote
2answers
80 views

Derivation of $ R_{N(t)}(\tau) $ from its $f_{N(t)}(\eta)$

How can we prove the auto-correlation function of white gaussian noise $\{ R_{N(t)}(\tau) \}$ is $\frac{N_0}{2} \delta(\tau)$ from its p.d.f in equation below? $$ f_{N(t)}(\eta)=\frac{1}{\sqrt{2 \pi \...
6
votes
2answers
1k views

Even and odd signal energy property

In Signals and Systems by A. V. Oppenheim, A. S. Willsky, S. Hamid Nawab, 2nd Edition, and Signals and Systems, Simon Haykins, Barry Van Veen, 2nd Edition there is a problem related to energy of real-...
1
vote
1answer
798 views

Proof of the convolution property of Fourier Series in continuous time

I am facing problem in understanding the proof of Convolution property of Fourier Series (FS) in continuous time CT; that is: $$\mathrm{FS} \big\{x_1(t)\star x_2(t)\big\}=T\sum_{n=-\infty}^{\infty}...
0
votes
1answer
68 views

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

I have the signal $s(t) = \cos(2\pi f_1t)$ and I am looking for its components vs the Fourier basis, over the interval $[0, T]$. The formula for computing the coefficients is $$ s_n = \int_{t_0}^{t_1} ...
0
votes
1answer
54 views

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

If $h(n)$ is the impulse response of the discrete LTI system, $x(n)$ is the white noise process with variance $\sigma_x^2$ and $y(n)$ is the output, prove that $$\mu_y = \mu_x \, \sum_{k=0}^\...
2
votes
2answers
839 views

how to evaluate derivative of convolution integral?

If the signal $x(t)$ has ordinary first derivative $\dot x(t)$, then $\dfrac{d}{dt}\big(x(t)\star y(t)\big)$ is: (a) $\dot x(t)y(t)$ (b) $x(t)\dot y(t)$ (c) $\dot x(t)\star y(t)$ (d) $\...
1
vote
1answer
266 views

DFT shift theorem proof

Learning DSP on my own time. Can't figure out the proof for DFT shift theorem which states the following: Given, $x[n]$ to be a periodic with period $N$, $\text{DFT}\{x[n]\} = X[k]$, then $$ DFT\{x[n-...
11
votes
9answers
382 views

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

This question is related to this other question of mine where I ask for derivations of the discrete-time Fourier transform (DTFT) of the unit step sequence $u[n]$. During my search for derivations I ...
9
votes
2answers
777 views

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

From text books we know that the DTFT of $u[n]$ is given by $$U(\omega)=\pi\delta(\omega)+\frac{1}{1-e^{-j\omega}},\qquad -\pi\le\omega <\pi\tag{1}$$ However, I haven't seen a DSP textbook that ...
2
votes
1answer
180 views

struggling to understand why Fourier basis is orthogonal

Studying DSP on my own time on Coursera. Was given a proof to why the Fourier basis is orthogonal, but I can't figure it out. Here is how it is proof goes. Consider the Fourier basis $$ \left\...
1
vote
0answers
40 views

Normalized LMS with a posteriori Error and Woodburry's Matrix Inversion

I was going through this paper and the author mentioned that we can prove the following using the Matrix Inversion Lemma (AKA Woodburry's Matrix Inversion Identity): Using matrix inversion lemma we ...
0
votes
1answer
314 views

DFT of time reversed signal

I was looking into proof and find something strange: The last part we obtain from DFT definition. $$X[k] = \sum^{N-1}_{n=0}x[n]W^{kn}_N, \quad\text{Where}\quad W^{kn}_N = e^{-j\frac{2\pi}{N}nk}$$ ...
1
vote
1answer
329 views

Is $y[n] = n x[n]$ an LTI system?

How can I test if $y[n] = n x[n]$ is an LTI system? And any other system for that matter? For example, how come $y[n] = \left( \frac{1}{2} \right)^n u[n]$, where $u[n]$ is the unit step, is an LTI ...
3
votes
2answers
134 views

Mathematical relationship between highpass and lowpass filtering

Let $g, h_{HP}, h_{LP}: \mathbb{R} \rightarrow \mathbb{R}$ and $G, H_{HP}, H_{LP}$ denote their continuous Fourier transforms under the Fourier operator $\mathcal{F}$. Let $*$ denote the continuous ...
1
vote
0answers
39 views

Is power density invariant to Fourier Transform? Does it hold through derivation?

Assumptions We have a finite discrete measurement. Introduction I had some troubles determining a power spectral densities needed for Wiener deconvolution. Looking through the continuous equations ...
1
vote
1answer
167 views

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

I'm struggling to get my head round the mathematical proof for the alias frequencies in a sampled sine wave. I understand that sampling a sine wave of frequency $f_0$ every $t_s$ seconds gives you: $...
-4
votes
4answers
76 views

Signal periodicity in discrete-time

Let $x[n]$ be a discrete-time signal, and let $$y_1[n]=x[2n]$$ You have to show that if: $x[n]$ is periodic, then $y_1[n]$ is periodic. $y_1[n]$ is periodic, then x[n] is periodic. So for the ...
0
votes
2answers
144 views

DT unit impulse function properties proof

How to prove the following properties of DT unit impulse function. If anyone got link to a proof please mention it. I have search on the web. But only found the properties, not a proper method of ...
1
vote
3answers
862 views

Convolution of even functions is even

Prove that the convolution of two even functions is an even function. I have my own proof which I have included as an answer, but it assumes a linear time-invariant system. I want to know if there is ...
1
vote
1answer
60 views

Laplace of step and integration are same?

Why do we have Laplace transform of a step function and integrator is same. \begin{align} \mathcal L\left[u(t)\right] &= \frac 1s\\ \mathcal L \left[ \int dt\right] &= \frac 1s \end{align}...
0
votes
2answers
1k views

Find autocorrelation of exponential signal $a^nu[n]$

I need to find the autocorrelation of the following discrete signal $$x[n]=a^nu[n] $$ So I tried finding the convolution of $x[n]$ and $x[-n]$. \begin{align} \phi_{xx}[n]&=\sum_{m=-\infty}^\...
1
vote
1answer
610 views

Downsampling: Mathematical derivation

The problem I am having is related to sample rate conversion and more precise to sample rate reduction. I have been working on the paper Interpolation and Decimation of Digital Signals Tutorial Review ...
10
votes
2answers
6k views

Covariance vs Autocorrelation

I'm trying to figure out if there is a direct relationship between these concepts. Strictly from the definitions, they appear to be different concepts in general. The more I think about it, however, ...
4
votes
3answers
3k views

Proof of complex conjugate symmetry property of DFT

According to the Proof : \begin{align} X_n &= \sum_{k=0}^{N-1}x_ke^{-j\frac{2\pi k n}{N}}\\ X_{N-n} &= \sum_{k=0}^{N-1}x_ke^{-j\frac{2\pi k (N-n)}{N}}\\ &=\sum_{k=0}^{N-1}x_k e^{-j 2\pi ...
3
votes
1answer
194 views

Conceptual problem : numberof symbols for nonuniform distribution using entropy : how to determine block size?

A sequence of data of length $N$ can be subdivided into equal sixed blocks each of length (size) $l$. For each block, $w$, we can calculate the entropy known as the block entropy. Considering, entropy ...
1
vote
1answer
82 views

Impulse property - needed mathematical proof

Prove: $$ \left. \int_{-\infty}^{\infty}\frac{d\delta(t)}{dt}\phi(t)dt=\frac{-d\phi(t)}{dt}\right|_{t=0} $$ $\delta(t)$ is impulse signal and other one is any general signal. Can anybody do the ...
3
votes
1answer
351 views

Fourier transform of Kernel Density estimate: convolution theorem?

I am reading this paper about density estimation (Appendix A), where the authors apply a Fourier transform to the estimated probability density (the $X_j$ are a sample of $N$ data points drawn ...
0
votes
1answer
68 views

$\mathcal Z$-transform of an equation [Exam question]: Verifying the solution

I'm studying for exams at the moment and I'm trying to reproduce a solution from my professor (I have the solutions). The following signal is given: The excercise says: Calculate the Fourier ...
0
votes
1answer
214 views

How to prove relationship between mutual information and differential entropy?

I need to prove that mutual information given by $$I(X;Y)=\int_{x,y}f(x,y) \log_2 \left( \frac{\left (f(x,y)\right)}{f(x) f(y)}\right) \, dx \, dy$$ is equivalent to $I(X;Y)=H(Y) - H(Y|X)$ I am ...
1
vote
1answer
871 views

Calculating cutoff frequency for Butterworth filter

I have a problem while calculating cutoff frequency, suppose we have these specs. Firstly, I calculated the order of the filter and got $N=5.8858$ and round it up to get $N=6$. Now I'm supposed to ...
0
votes
2answers
103 views

Conditions for symmetric and unimodal windows in both time and frequency domains

After a lecture on harmonic analysis and time/frequency methods, I reconsidered the Gaussian kernel, defined in continuous time. It is unimodal and symmetric, and its continuous Fourier transform is ...
1
vote
5answers
1k views

Equation for impulse train as sum of complex exponentials

Could someone please break down what's going on in this equation for me? I understand what the left side looks like, but not so much how the right side is the same thing. Impulse train: $$\sum_{m=-\...
0
votes
1answer
91 views

Help in analytical expression of BER for M-QAM [duplicate]

I am new in learning deriving BER expression. I am familiar for the case of BPSK in AWGN but cannot understand how the BER expression in Eq(12) in this paper, An Enhanced Spectral Efficiency Chaos-...
7
votes
1answer
305 views

How to derive $r(t) = c(t) \circledast \frac{1}{2} h_b(t, \tau)$?

Consider a linear time-variant channel. The transmitted signal is $x(t)$, the channel impulse response is $h(t, \tau)$, and the received signal is $y(t)$. Then $$ y(t) = \int_{-\infty}^\infty x(\tau) ...
1
vote
1answer
851 views

Power spectral density of a PAM signal

Say I have a PAM signal as $x(t) = \sum_n a_n h(t-nT)$ where $\{a_n\} \in \{\pm 1\}$ are equiprobable random binary bits and $h(t)$ is bandlimited to $[-1/(2T),1/(2T)]$. ($x(t)$ is random process and ...
-2
votes
1answer
3k views

Non-causal FIR. Is that possible?

I know that a FIR (Finite Impulse Response) filter has the same quantity of poles than zeros. And I believe all the poles are at $z=0$. And a FIR filter is always stable, so it's ROC has to include ...
0
votes
1answer
138 views

Generalized linear-phase filter

I know I can write the frequency response of a Generalized linear-phase system as: $$H(e^{j\omega}) = A(\omega) e^{-j\left(\alpha \omega - \beta\right)}$$ where $A(\omega)$ is real. I need to prove ...
1
vote
1answer
429 views

Derivation of Bessel filters

I was reading Winder's Analog and Digital Filter Design and the section on Bessel filter. I was hoping to see a complete derivation of the Bessel filter theory, but Winder's book gives only \begin{...
0
votes
1answer
88 views

Insertion loss equality - proof

From this paper (slide 11), the insertion loss (or attenuation) is defined as $$ L(\omega^2)=\frac{\lvert V_i(j\omega)\rvert^2}{\lvert V_o(j\omega)\rvert^2}=\frac{1}{\lvert H(j\omega)\rvert^2}=10 \...
2
votes
2answers
982 views

Gradient of Total Variation (TV) Norm in Total Variation Denoising

In this link, it says that the gradient is as follow The Gradient of the TV norm is $$ \mathrm{Grad}J(f)=\mathrm{div}\left(\frac{\nabla f}{\lVert\nabla f\rVert}\right). $$ From this other link, ...
0
votes
1answer
285 views

Solving for frequency deviation, $\Delta f $ (FM modulation)

I'm sorry if I posted this in the wrong section. I solved the question and got $50\textrm{kHz}$, but the solution says it is $100\textrm{kHz}$ without any proof. Im asking for someone to tell me ...
0
votes
1answer
284 views

Demodulating upper sideband (USB) signals

A modulating signal $m(t)= \sin(2 \pi \nu_m t)$ is transmitted via a carrier of frequency $\nu_c$ using upper sideband (USB) modulation. Show that the USB modulated signal can be demodulated using a ...
2
votes
1answer
932 views

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

Show that for a signal $f(\tau)$ with finite energy and energy autocorrelation function $\phi^e_{ff} (\tau),$$$|\phi_{ff}^e (\tau)| \leq \phi_{ff}^e (0), \ \ \forall \tau.$$ According to my textbook ...
0
votes
0answers
207 views

Power autocorrelation function calculation

Calculate the power autocorrelation function $\phi^p_{ff} (\tau)$ and power spectral density $\Phi^p_{ff} (\nu)$ for the signal$$f(t)=A \cos(2 \pi \nu_m t) \cos(2\pi v_c t).$$ Attempt: According to ...
0
votes
0answers
929 views

Autocorrelation function and energy spectral density

Find the energy autocorrelation function $\phi^e_{ff}(\tau)$ and the energy spectral density $\Phi^e_{ff}(\nu)$ of the signal $f(t) = e^{-\gamma |t|},$ where $\gamma>0$ is a real constant. Attempt:...
0
votes
2answers
58 views

Periodicity of transfer function of FIR filter proof (Parks and Burrus, Digital Filter Design)

In Digital Filter Design by Parks and Burrus, p. 19. The transfer function of an FIR filter is given by the $\mathcal Z$-transform of $h(n)$ as: $$H(z)=\sum_{n=0}^{N-1}h(n)z^{-n}$$ (where $h$ is ...