Questions tagged [signal-processing]

Questions on the mathematical aspects of signal processing. Please consider first if your question might be more suitable for http://dsp.stackexchange.com/

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24 views

Change of variables on a double summation to yield a single sum

I'm going through a proof in Monson's Statistical Digital Signal Processing and Modelling on page 98. They used the substitution $k=n-m$ in order to change the double summation into a single summation....
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Finding whether a system is linear and time-invariant given input x(t) and output y(t) in integral form.

I had this question which I could not figure out myself, if you could at least guide me in it, it would be good. The question says: The output y(t) of a system is related to its input x(t) by $$y(t)=\...
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1answer
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Given an affine space, will an orthogonal projection have 0 inner product?

I know that if we have a vector space $V$ and orthogonally project $x$ onto it to get $\hat x$, then we have $\langle v, x-\hat x \rangle$ for any $v \in V$. However, I do not know whether the same ...
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Control process with long deadtime

I am a programmer who lacks the mathematics side of things. I need help writing out a SOPDT equation and a Smith Predictor using simple math. I did not go to school for this and understand that there ...
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Absolute summability of summation of integer exponents of polynomial roots (BIBO stability)

Let $\lambda _{i} = g_{i}\angle\theta_{i}$ be the roots of a M degree polynomial $S= \sum _{n=0}^{\infty} \left ( \left | \sum _{r=1}^{M} \left ( K_{r} \lambda _{r}^{n}\right )\right | \right )$ , ...
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1answer
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How to integrate this complex interal in real and imaginary form in signal processing?

I have an engineering background. While reading signal processing, I came across this integral: $$\frac{1}{4}\int_{-\infty}^{\infty} X(u) Y^*(u) \text{d}u+\frac{1}{4}\int_{-\infty}^{\infty} X^*(v) Y(v)...
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Where is the sinc function zero? [closed]

Like in the question, where the $\text{sinc}(x)=0$? Thank you so much.
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24 views

Auto correlation of a cosine signal

I have to find the autocorrelation of $ x(t) = A cos ( 2\pi f_0 t) $ , and I know from theory that I should calculate $ lim_{x \rightarrow \infty } \frac{A^{2}}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} ...
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34 views

Why does the Z transform represent a delay?

I'm studying the Z-transform. I recently did by hand the Z transform of an discrete impulse delayed $\mathcal{z}\{\delta[n-k]\} = z^{-k}$ I get that this means that any signal can be represented as ...
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How to implement zero-padded convolution with FFT multiplication?

It is well known that we can implement circular convolution $$t\to(f*s)(t)$$ of two functions $$t\to f(t)\\t\to s(t)$$by utilizing the convolution theorem of Fourier Transforms: $$(f*s)(t) = \mathcal ...
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Convolution of x(t) and x(-t)

Consider the signal $x(t)=e^{-t}u(t)$ where $u(t)=\mathbb{1}(t\geq0)$, i.e. the Heaviside function. Find the signal $y(t)=x(t)*x(-t)$ My attempt: $y(t)=x(t)*x(-t)$ $=e^{-t}u(t)*e^{t}u(-t)$ $=e^{-...
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Why does Fourier Transform use a negative exponent in its formula?

I realize this question has been asked before but I don't understand the explanations. For example I have read this (https://en.wikipedia.org/wiki/Negative_frequency) as well as numerous other answers....
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band pass filter

I have a PPG signal contains frequencies that are related to the PPG signal itself. In order to extract the actual PPG signal, I need to implement a band-pass filter in the frequency interval (in Hz) [...
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Easy way to compute Fourier tranform of a box function

I have a big dilemma about Fourier transform... I know that: $\prod \frac{t}{T} \rightharpoonup T sinc (ft)$ but if I have something like $\prod (2t)$ or $\prod \frac{2t}{T}$ how can I compute the ...
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What is the Hilbert transform of $e^{-jwt}$?

What is the Hilbert transform of $e^{-jwt}$ ? I know how to do it via convolution , but i am looking for a less formal, more intuitive explanation such as the phase shift it imparts
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Amplitude or Power of a Certain Frequency over time

How do I measure the power or amplitude of a certain frequency or band of frequencies in a signal over time? I hear that Hilbert transform can help but, when I take the real part of the Hilbert ...
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1answer
50 views

Dirichlet conditions and the fundamental period

It's well known that Dirichlet conditions are: Over any period $x(t)$ must be absolutely integrable; that is, $$\int_T|x(t)|dt\lt\infty$$ In any finite interval of time, $x(t)$ is of bounded ...
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What does it mean to downconvert signals within a bandwidth, centered at another bandwidth?

So, conceptually, when a book talks about down-converting signals within a bandwith of say, $25$ MHz, with the bandwidth centered at say, $50$ MHz, down to baseband, what exactly does it mean?
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What do I need to know to implement Savitzky-Golay filter

I'm working on a project and I have data that needs to be smoothed out in a live environment. After researching and trying different kind of filters I believe the best filter for my problem is the ...
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1answer
44 views

Convolution property of Fourier transform

I have a signal $$ x(t) = \frac{1}{T} e^{- \frac{t}{T} } u(t) $$ and I know his Fourier transform $$ X(f) = \frac{1}{1+ i 2\pi f T } $$ and I have to find $$ z(t) = x(t) \circledast x(t) $$ using ...
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Linear Time Invariant Systems?

Give the linear time invariant system: $$\dot x = Ax + Bu, \quad x(0) = x_0$$ $$y = Cx$$ How would one solve for $x_0$, $y(t)$, or $u(t)$ given 2 of the 3 unknowns? Clarity from comments: Let's say $...
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Finding the magnitude and argument of complex f(z)

I have the complex f(z): $$f(z) = \frac{(2e^{j11z})}{(3+j5z)^7}$$ and I need to figure out the magnitude and the argument argf(z). So far for the magnitude I have: $$|f(z)| = \frac{\sqrt{(4(cos11z)^...
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What is the Fourier transform of this function? $ f(t)=\exp (-t / 2) \text { for } t>0,0 \text { for } t<0 $?

What is the Fourier transform of this function? $f(t)=\exp\left(\frac{-t}2\right) \text { for } t>0,0 \text { for } t<0$? I am having some problems simplifying the function.
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19 views

Discrete Fourier Transform Relation between two signals

I'm struggling into show the relation between $\hat g[k]$ and $\hat h[k]$ given $g[n]=(-1)^nh[n]$, where $h[n]$ is a discrete signal, let's say with $0\le n <N$. I made some computations that I ...
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9 views

Distribution theory and Shannon sampling theorem

Let $C$ denote the Dirac comb distribution and let $\mathcal{F}$ denote the Fourier transform for tempered distributions. Let $x$ be any function in the Schwartz class $\mathcal{S}$ with $X = \mathcal{...
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1answer
27 views

Why doesn't my expression for the fourier transform of a function match its numerical fourier transform?

I am seeing a frustrating discrepancy between my evaluation of $F(\omega) = \frac{1}{a + iw} $, and the fast fourier transform of $f(t) = e^{-at}u(t)$ (where $u(t)$ is the step function from 0 to 1) ...
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1answer
69 views

Inverting a digital filter

I want to understand a technique for calculating the inverse of a digital filter. Let A be a function from $R$ to $R$ such that $$ A(0) = 1;\quad A(1) = a_1;\quad A(2) = a_2; \quad\text{, otherwise } ...
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2answers
58 views

Alternative definition of Dirac function

Can someone explain why the formula below is a definition of the Dirac function? $$\int_I y(0) \delta(t)\, dt =\begin{cases} y(0) \text{ if 0}\in \text{I} \\\\ 0 \text{ otherwise} \end{cases}$$ ...
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1answer
29 views

Why does tridiagonal matrix reduce noise?

Let $$B = \begin{pmatrix} 1/3 & 1/3 & 0 & 0 & 0 & \dots & 0 & 0 \\ 1/3 & 1/3 & 1/3 & 0 & 0 & \dots & 0 & 0 \\ 0 & 1/3 & 1/3 & 1/3 &...
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0answers
90 views

Calculating convolution of two piecewise functions

Let $$x(t) =\begin{cases}2&-2\lt t \lt 0 \\ -2 &0\lt t \lt 1\\ 0& \text{otherwise}\end{cases}$$ and $$h(t) =\begin{cases}4(1-|t|)&|t|\le1 \\0& \text{otherwise}\end{cases}$$ ...
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1answer
86 views

PDF of difference in random variables via convolution

See the figure for reference: $f_i$ and $f_{i+1}$ are unobserved times of events, while $y_i$ and $y_{i+1}$ are observed times of subsequent events. We'd like to recover the distribution of $f_{i+1} -...
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1answer
26 views

Seeming discrepancy between two Fourier transforms

I am familiar with both of the following Fourier transformation identities: $$e^{jw_0t} \leftrightarrow 2\piδ(w-w_0)$$ and $$x(t)e^{jw_0t} \leftrightarrow X(w-w_0)$$ However I need to find out why ...
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0answers
22 views

Finding PDF and CDF of a signal with Rayleigh fading model

Considering a random signal $x(t)$ with $0$ mean and unit variance is transmitted via $n$ channels where $1\leq n$. The pdf and the amplitude of the $n$-th signal is faded with Rayleigh fading: $$f_{...
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1answer
9 views

Prove a necessary condition so that a complex discrete exponential function is periodic.

Consider the discrete complex function $f(n) = e^{iwn}$ with w is nonzero and n is in the set of integers. Assume f is periodic, which means there is an integer N st, $f(n+N) = f(n)$. From the book "...
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1answer
27 views

Summation of Sinc function approximated by sines and cosines

Suppose we can express the summation of a Sinc function as shown below: $$\sum_{n=-\infty}^{\infty}Sa(3\pi(t-n)) = a+bcos(\omega t)+csin(\omega t)$$ Then what is the relationship between those ...
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1answer
27 views

Converting a generated digital signal to time series in MATLAB

I'm given the periodic digital function $$ x(n) = e^{0.2}\cos\left(\frac{2\pi 500 n}{3000}\right) + \sqrt{3} \cos\left(\frac{2\pi 700 n}{3000}\right) $$ where 3000 is the sampling frequency. Initially,...
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19 views

Calculating signal energy $E_g$ of summation of triangular pulses.

I am trying to find the signal energy of this time-domain function: $$g(t)=2\triangle(\frac{t-0.004}{0.004})-4\triangle(\frac{t-0.005}{0.002})$$ The definition of signal energy I have is: $$E_g=\int{g^...
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1answer
47 views

is the concatenation of White noise process samples in a vector white also?

Given that $X_t \sim N\left( {0,{I_2}{\sigma ^2}} \right)$ is a bivariate white noise process, what are the characteristic of the new process ${Y_t} = \left[ {\begin{array}{*{20}{c}}{{X_t}}\\ \vdots \...
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44 views

Transfer Functions and Stability?

Given a transfer function: $$f = \frac{\sigma(\sigma+1)}{s - \sigma + 1}$$ where $\sigma \in \mathbb{R}$. I want to check the stability of the above function. Attempt: Pole(s) of the function: $s = \...
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1answer
13 views

How to find a generator matrix when codewords are given?

I do not understand how to determine a generator matrix. I have attached a question below. Can someone please explain to me how to do this question? The vectors $C_0=[10001]$, $C_1=[11010]$, and $C_2=...
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29 views

Property of real-valued Fourier transformation

When it is given that a signal $x(t)$ has a real-valued Fourier transformation $X(f)$ then is the signal necessarily real-valued? I have thought the following: We have that the real and imaginary ...
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1answer
50 views

Meaning of “generalization”? in math

What is the meaning of word "generalization"? Please kindly explain in simple words For example in wikipedia article of interpolation, in sub heading "polynomial interpolation " a line is written ...
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23 views

Filtering time series

I'm filtering the highest frequencies off of a time series signal, but something strange happens. What I am expecting is a trigonometric curve fitting the time series without taking into account all ...
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0answers
10 views

Calculatee impulse response for given system

i am given an exercise in which i need to find the impulse response for this given system: y[n] + 2y[n-1] = x[n] + 2x[n-3] This is the only information i have. ...
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1answer
15 views

Estimating a cost of a graph smoothness

For a given video scene, I have a task to evaluate a score for its stability. I have managed calculate their deltaX and deltaY ...
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0answers
11 views

Going from a Discrete time signal to a continue time signal given a sampling rate.

I am given a DT signal x[n] = sin((pi/100)*n^1.5) for n is an integer [0,1499] and a sampling rate Fs = 1000Hz. How can I obtain the original continue time signal which was sampled @ 1000 Hz to obtain ...
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2answers
103 views

Evaluating $\sum_{k=1}^{\infty}\left(\frac{\sin(tk)}{k}\right)^2$

Using Poisson Summation Formula, how do you evaluate the following infinite sum $\sum_{k=1}^{\infty}\left(\frac{\sin(tk)}{k}\right)^2$? The Poisson Summation Formula states that: $\sum_{k=-\infty}^{\...
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Can properties for (circular symmetric) complex random matrices automatically work for real random matrices?

I am dealing with a theorem which relates to circularly symmetric complex Gaussian random matrices (CSGRM). It seems quite tempting to assume that the theorem also extends to real-valued Gaussian ...
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18 views

DTFT of decaying sinusoid

I have been presented with the following question: Perform the discrete-time Fourier transform (DTFT) on $$x[n] = 5\cdot\!0.25^n \cdot\cos \left(\frac{2\pi}{3}n \right)\cdot u[n]$$ My workings ...
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0answers
8 views

Q: Levy process as a combination of wiener process poisson procerss and determenistic process

Is it true that any Levy process can be decomposed to a 3 processes: 1. Poisson process 2. Wiener process 3. determenistic process If so, what is the proof/intuitaion for that? thank you very much

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