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Continuous Phase Modulation - Phase trajectory expansion

For continuous phase modulation (CPM), the circular phase trajectory is expressed as follows: $$\phi(t) = 2\pi h\int_0^t\sum_k \xi_k g(\tau - kT_s)d\tau + \Phi_0\tag{1.1}$$ Where: $h$ is the ...
Gilles's user avatar
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0 answers
53 views

Proof of Hilbert transform of a real function $x(t)$ is generally a complex function

I am looking for a formal proof of the result: The Hilbert Transform of a real signal $x(t)$ is generally a complex signal. Can this be proven and if so how? Thank you.
devsucksatcalculus's user avatar
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0 answers
57 views

Is this proof valid?

Consider a discrete-time sequence, $y[n]$, defined as: $$y[n] = \frac 12 x[n] + \frac 12 (-1)^n\: x[n]$$ where $x[n]$ is another discrete-time sequence. The DTFT of $y[n]$ defined as $Y(e^{j\omega}) = ...
Ahsan Yousaf's user avatar
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3 votes
2 answers
130 views

Is there a stable linear shift invariant system whose transfer function is $H(z) = z^*$

I couldn't find such a system but I have also not been able to prove otherwise. Firstly, I don't know exactly how to take the inverse Z-Transform of $z^*$. Secondly, I don't know the ROC associated ...
user avatar
0 votes
1 answer
63 views

proof of alias matlab sin wave and syntax for time array

I have been asked to prove the following; Show that a sinusoid of amplitude 10V and frequency 2kHz sampled at %fs = 10kHz is an alias of a 500Hz sampled signal. I have develped the code for the 500hz ...
Tam's user avatar
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3 votes
1 answer
224 views

Mathematically show the problem of Histogram Equalization

Wikipedia - Adaptive Histogram Equalization says about classic Histogram Equalization: This works well when the distribution of pixel values is similar throughout the image. However, when the image ...
hasanghaforian's user avatar
0 votes
2 answers
208 views

Proof that DFT is symmetric

I am working through the proof that the DFT is symmetric from Lyons - Understanding Digital Signal Processing I don't quite understand the use of the $N$ variable. My understanding is that in $X(N-m)$,...
Joseph's user avatar
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1 answer
119 views

Bandwidth of cosine of bandlimited signal

I have a signal $x(t)$ with bandwidth $B_x$, and I am taking its cosine to create $y(t) = cos(x(t))$. After checking the spectrum with FFT, it seems that $y(t)$ is also bandlimited. But, is there a ...
Olayo's user avatar
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2 votes
2 answers
3k views

Proving real and odd function has imaginary and odd Fourier Transform

Cheers, I am trying to prove that a real and odd function/signal has imaginary and odd Fourier Transform. Although it seems fairly easy, I can't find a way to achieve it, and searching online hasn't ...
average_discrete_math_enjoyer's user avatar
3 votes
2 answers
639 views

Proving Fourier transform of cosine multiplied with another function

Cheers, I have a impulse response that looks like this: $h(t) = h_1(t) \cdot \cos(8 \pi t)$ and I have to find its frequency response. In order to achieve this I am trying to use the fact that $$x(t)y(...
average_discrete_math_enjoyer's user avatar
2 votes
1 answer
312 views

Proving that the uncertainty can not increase during the update step of a Kalman filter - positive semidefiniteness

I am trying to prove mathematically that the update step in a Kalman filter can not result in a increase in uncertainty. I found the following proof which is based on the inversion lemma and the ...
MattSt's user avatar
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1 vote
2 answers
72 views

Value of $\sum\limits_{n=-\infty}^{\infty}(x*x)[n]$

If $x[n]=(0.5)^nu[n]$ and $y[n]=(x*x)[n]$ then what is the value of $\sum\limits_{n=-\infty}^{\infty}y[n]$ ? I calculated the $\mathcal{Z}$-transform of $x[n]$ and then applied the accumulation ...
edison's user avatar
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0 answers
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Let a LTI system be causal and stable with the transfer function being... show that

if the system is an IIR LTI causal and stable one, and the transfer function is \[H(z)=\sum_{n=0}^{\infty}h[n]z{^{n}}= \frac{G}{1 -\sum_{k=1}^{p}a_kz{^{-k}}}\] show that the cepstrum of this system ...
Nyquist-er's user avatar
3 votes
1 answer
654 views

Proof of Bedrosian's theorem

I was looking for a straightforward proof for Bedrosian's theorem which says the Hilbert transform of the baseband signal times the passband signal is the original baseband signal times the Hilbert ...
Elias's user avatar
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1 vote
2 answers
80 views

How do you reduce $H\left(e^{j\frac{\pi}{2}}\right)$ further according to a textbook solution

I want to know how I could get from the first line to the second. I've been trying to figure it out for a while with no luck. Thank you in advance! \begin{align} H\left(e^{j0.5\pi}\right) &= \frac{...
DukeOfDoors's user avatar
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2 answers
93 views

Convert complex valued sinusoid to real valued sinusoid

This is the homework problem: convert $x[n]=je^{j\pi n/8}-je^{-j\pi n/8}$ to a real valued sinusoid. I understand that $\sin\theta=\dfrac{e^{j\theta}-e^{-j\theta}}{2j}$ In the solution, the answers ...
Idr's user avatar
  • 143
1 vote
2 answers
780 views

Is this piecewise-defined system time-invariant?

\begin{equation} G_c \qquad y[n] = \begin{cases} u[n] & \text{if} & n<0\\ 2u[n] & \text{if} & n\ge 0 \end{cases} \end{equation} I found this question in ...
tonythestark's user avatar
1 vote
0 answers
58 views

How the power spectral density changes under frequency scaling?

Let the power spectral density PSD of a random process be $$\phi(w) = \sum_{k = \infty}^\infty r(k)e^{-i\omega k}$$ where $\omega \in [-\pi,\pi]$ and $r(k)$ is the correlation function. I have to show ...
Lorenzo Gutiérrez's user avatar
1 vote
1 answer
210 views

Prove a property using shift theorem and duality

I'm reading Lectures on the Fourier Transform and Its Applications and I'm going to prove shift theorem for the inverse Fourier transform using duality. According to the mentioned source, the duality ...
S.H.W's user avatar
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0 votes
1 answer
253 views

Trouble showing Time Invariance of recursive system

The system is described with the following recursive differences equation: $$y[n]-4y[n-1]+4y[n-2]=20x[n]+10x[n-1]$$ now lets say the input is delayed by k, then: $$y[n]-4y[n-1]+4y[n-2]=20x[n-k]+10x[...
bertington313's user avatar
0 votes
1 answer
169 views

About the proof of an equality related to the DFT [sampling the DTFT to obtain the DFT]

This wiki page about the DTFT says that the DFT can be obtained from the DTFT by sampling the latter in one cycle at $N$ points: When the DTFT is continuous, a common practice is to compute an ...
Hilbert's user avatar
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2 votes
1 answer
1k views

Is the decimation process time invariant?

I'm stuck at exercise 1.18 in Understanding Digital Signal Processing by Richard G. Lyons: There is an often-used process in DSP called decimation, and in that process we retain some samples of an $...
Sylvain Leroux's user avatar
1 vote
1 answer
860 views

Having difficulty checking for time invariance of discrete system

A system is given with the following equation: $$y(n) = 3y^2 (n-1) - nx(n) + 4x(n-1) - 2x(n+1)$$ I need to check for the linearity and time invariance of the system. By just looking at the equation I ...
bikalpa's user avatar
  • 211
0 votes
2 answers
2k views

Proof regarding the periodicity of a continuous-time sinusoid after sampling

Question A continuous-time sinusoid $x_a(t)$ with fundamental period $T_p = \frac{1}{F_0}$ is sampled at a rate $F_s = \frac 1 T$ to produce a discrete-time sinusoid $x(n) = x_a(nT)$. Show that $x(n)$...
bikalpa's user avatar
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0 votes
0 answers
125 views

Proof of weak stationary random process autocovariance always goes to zero?

Professor told me that if a random process is weak stationary, and it does not feature any periodic component, then its autocovariance always goes to zero. I can intuitively understand it, however, ...
Juà's user avatar
  • 21
-1 votes
1 answer
265 views

Proof that DFT does not require more than N points

I'm trying to show how the discrete Fourier transform (DFT) arises from the equation for the continuous-time Fourier Transform. I've run into an interesting caveat which I can't seem to find an ...
MarcinKonowalczyk's user avatar
2 votes
3 answers
4k views

Resolution of Discrete Fourier Transform is 1/T - Mathematical proof?

In many articles I see that the frequency resolution of the Discrete Fourier Transform (DFT) equals Fs/N where Fs is the sampling rate and N is the total number of samples. Fs/N is equivalent to 1/T ...
D.Cohen's user avatar
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3 votes
2 answers
152 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 ...
Danny's user avatar
  • 43
3 votes
1 answer
3k 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}(\...
Ridin's user avatar
  • 31
0 votes
1 answer
31 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.
Muhammad Aldakkak's user avatar
3 votes
3 answers
8k 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.
ronen's user avatar
  • 31
3 votes
2 answers
371 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 \...
Suresh's user avatar
  • 287
6 votes
2 answers
7k 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-...
Meet's user avatar
  • 95
3 votes
1 answer
4k 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}...
Suresh's user avatar
  • 287
0 votes
1 answer
119 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} ...
Enio's user avatar
  • 1
0 votes
1 answer
76 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}^\...
Suresh's user avatar
  • 287
2 votes
2 answers
9k 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) $\...
user avatar
3 votes
1 answer
2k 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-...
flashburn's user avatar
  • 195
12 votes
9 answers
1k 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 ...
Matt L.'s user avatar
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14 votes
2 answers
14k 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 ...
Matt L.'s user avatar
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2 votes
1 answer
407 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\...
flashburn's user avatar
  • 195
2 votes
0 answers
107 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 ...
Copernicus's user avatar
0 votes
1 answer
2k views

DFT of time reversed signal

I was looking into proof and find something strange: The last part we obtain from DFT definition. $$X[k] = \sum^{N-1}_{n=0}x[n]W^{kn}_N, \quad\text{Where}\quad W^{kn}_N = e^{-j\frac{2\pi}{N}nk}$$ ...
qqffx's user avatar
  • 57
1 vote
1 answer
5k 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 ...
anon's user avatar
  • 25
3 votes
2 answers
726 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 ...
ComFreek's user avatar
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1 vote
0 answers
54 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 ...
Lukáš Mrazík's user avatar
2 votes
2 answers
993 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: $...
IanR's user avatar
  • 123
-2 votes
4 answers
2k 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 ...
nor's user avatar
  • 13
0 votes
2 answers
630 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 ...
Jupin S's user avatar
2 votes
3 answers
5k 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 ...
Undertherainbow's user avatar