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Determining if a function is linear, time invariant, both or not

I have the function $y(t)=t^2x(t-1)$ and I need to figure out if it is linear or not and time invariant or not. By the looks of it I guessed it to be not linear but the answer is linear but not time ...
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42 views

Difficult problem involving a percentage of the period of a sinusoid

Im having difficulty intuitively understanding how to solve this problem: $x(t) = A\cos(\omega t + \phi)$ $A > 0$ $\phi\in(−\pi,\pi]$. $x(t) ≥ 2.4$ for $18$% of each period takes $0.123$ ...
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21 views

Implementing Normalized Cross-Correlation using FFT - How to?

Is there any way to calculate the normalized cross correlation between 2 signals by using the FFT? (I managed to implement it already for standard cross correlation equation). Thanks in advance,
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Is constant system a Causal System?

Is y(t) = 1 a causal system? From the definition of causal systems , a causal system is a system where the output depends on past and current inputs. Here the system doesn't depend on any input. ...
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Is this system Causal?

The output system is: $x(t)$ -->(S)--> $y(t) = \int_{-\infty}^{t}x(\tau) d\tau$ Recall that the system is causal if the output at $t$ depends only on input before $t$, or if the impulse response ...
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42 views

building time signal after inverse FFT

I have managed to implement both forward and inverse FFT transforms in C#. And, i tested them by taking signal both ways and got the real part after iFFT equals the original signal. Now , I have ...
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15 views

Mathematical Expression for a Fourier Transform $s(T)$

$S(f)$ is the Fourier transform of a non-periodic signal, $s(t)$. $S(f)$ is given by: $S(f) = 1,$ for $−1/2 ≤ f ≤ 1/2$ and $0$ otherwise. What would be a mathematical expression for $s(t)$?
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non negative matrix factorisation [closed]

i am working on a project involving the use of non negative matrix (NMF) for the separation of a mixture of audio signals.can please give me the mathematical explanation of NMF so that i can ...
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18 views

When studying 2D gabor functions why is a gaussian called elliptical?

Consider $$G(x,y)=\frac{1}{2\pi\sigma\beta}e^{-\pi\left[\frac{(x-x_0)^2}{\sigma^2}+\frac{(y-y_0)^2}{\beta^2}\right]}e^{i[\xi_0x+\nu_0y]}.$$ This is the product of a complex plane wave and what this ...
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1answer
19 views

Specific question about downsampling in frequency domain

I'm confused about why Equation 4.74 can be expressed as Equation 4.76 after the summation index is expressed as in Equation 4.75. Could somebody please explain? Thanks! Downsampling DTFT
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29 views

What is the requirement for separable parameters in an LSQ fit?

I am trying to determine the amplitude of a sinus modulated sinus as accurate as possible. My sampling frequency is sufficently high. The entire model looks as follows: $$ ...
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1answer
45 views

Sampling the Sine Function

Consider the sampled sine function, $f(n)=\sin(\omega n)$, where $n$ is an integer. If $\omega_2 = 3\pi/2$, does there exist an $0 \leq \omega_1 \leq \pi$ such that $\sin(\omega_1 ...
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1answer
22 views

Why is autocorrelation used without normalization in signal processing field?

According to the wikipedia(Link), autocorrelation has two definition. Oh my god! In statistics, the definition of the autocorrelation between times $s$ and $t$ is like the following: $$\displaystyle ...
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1answer
25 views

Is interpolating well-sampled data (Nyquist-Shannon theorem) a cheat?

Suppose to sample a signal $s(t)$ with bandwidth $B$ with a sampling frequency $f_c$. Suppose also that the number of sample collected is $N$ (the duration of the signal acquisition is then $T = ...
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1answer
31 views

Orthogonality of Two Signals.

My last question's link: Reconciling different definitions of orthogonality However, I failed to understand why they are equivalent. If $f$ and $g$ are real, \begin{align} ...
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1answer
46 views

Reconciling different definitions of orthogonality

I want to establish about orthogonality in my mind. I knew the orthogonality of two functions $f$ and $g$ in interval T like the following: $$ \int_{<T>}f(t)g^*(t)~dt=0 \tag{1} $$ where $$ ...
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1answer
31 views

Application of Residue Theorem to inverse Fourier transform

I'm reading through a derivation in a book and am having trouble understanding a step. Here's a screenshot 3.46 is the equation in $(k,\omega)$ space. They're doing an inverse Fourier transform back ...
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1answer
35 views

Fast Fourier Transform as Matrix Factorization

I'm given a vector of length 4 and three matrices that correspond to a Fast Fourier Transform, I'm not exactly sure which one, but I guess it's supposed to be the Cooley-Tukey algorithm. Here is the ...
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How to create obtain aliased version of $f(t)$ by upsampling whenever $f(t)$ at every $t$ is available

Suppose there is original complex-valued $f(t)$ with $t$ ranging from $-\infty$ to $\infty$. It is possible obtain samples from original $f(t)$ at every $t$ with some negligible error. If one ...
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28 views

Eigenvalues for correlation matrix which have the form of an harmonic function

As a continuation to this question, I took the matrix $C_{2 \times 2}$ which is: $$C=\left[ \begin{array}{} a& ace^{-\frac{|\phi_1-\phi_2|}{2}}\\ ace^{-\frac{|\phi_1-\phi_2|}{2}} ...
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1answer
10 views

Unit of the second derivative of the power spectral density

To characterize a subtle oscillation embedded in a time varying voltage signal measured in microvolts, I took the second derivative of the PSD (which I computed as the fft of the autocorrelation) ...
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Is it possible to use regularization to minimize the (expected) number of non-zero digits in a number?

This question may be slightly related to this question on length of the representation of a number in a certain basis. Introduction / Background In image and video coding, particularly the ...
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7 views

How do I validate my ARMAX model?

Say I have some ouput $y_1, y_2, \ldots, y_N$ and inputs $x_1, x_2, \ldots, x_N$ which, by various time series methods, I've found to match an ARMAX(2,2,1) model. So I've found the estimations for ...
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Variance of $\hat{b} =\underset{b}{\mathrm{argmin}} \sum_{t=1}^N [y_t - bu_t]^2 $

We have $y_t = bu_t + e_t$ where $u_t$ is the input signal and I'm trying to find an expression for the variance of the estimate for b that is determined to be $$\hat{b} = ...
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2answers
114 views

How can I recover a sequence of numbers given a corrupted version of it?

I have an unknown sequence of real numbers $x_i$ and a known sequence of real numbers $y_i$; $y_i$ is a corrupted version of $x_i$, i.e., $$y_i=x_i+n_i$$ where $n_i$ is a random number distributed ...
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0answers
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Resampling a digital sound signal [migrated]

As a non maths-pro, I'm looking for some pointers. I am rewriting the audio core in my emulator to improve accuracy, and am getting a bit stuck on the specifics of the technique. I am trying to ...
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37 views

What is the output $y(t)$ when you have input $x(t) = \cos(2 \pi t) $ and frequency response response $h(t) = u(t) - u(t - 1/2)$?

The output $y(t)$ is the convolution of input $x(t)$ with impulse response $h(t)$: $$ y(t) = h(t) * x(t) $$ This is a linear, time invariant system. What is the output $y(t)$ in real form when you ...
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0answers
48 views

Solving a non-linear parametric equation

I am interested in solving a parametric equation where the unknown function is a function of time, and there is also an input. For example: $ y^{2}(t) + y(t) = \sin(t)$ I am coming from a signal ...
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19 views

Single Sideband LSB-SC Demodulation

The problem is how the phase φ effects the outcome when the input(message signal) is the DSB-SC LSB. It's : message: $m(t)=A_{m}cos(ω_{m}t)$ carrier: $c(t)=A_{c}cos(w_{c}t)$ I found that the LSB ...
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Can downsampling create energy at the Nyquist frequency?

I am a bit surprised by the following and would like to share it with you. I expect I am mistaken somewhere and will be happy to be corrected. I have searched StackExchange not only in Mathematics ...
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How to find out the Power of $x(t)$?

I am studying signals and system. I learned that \begin{align} P&=\lim_{L\to\infty} \frac 1{2L} \int_{-L}^{L} |x(t)|^2 dt\\ P&=\frac 1{T} \int_{<T>} |x(t)|^2 dt ~~~\mbox{, P could be ...
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1answer
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In signals processing why is the discrete sequence x[n] undefined (as opposed to 0) when n is not an integer?

In Oppenheim & Schafer's "Discrete Time Signals Processing" it's written that: ... it is important to recognize that x[n] is defined only for integer values of n. It is not correct to think of ...
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Why is this defined as $u[n]$?

In LTI systems there are two famous functions the unit step function and the unit impulse functions. And they are defined as follows. $$ u[n] = \begin{cases} 1, & \text{if $n$ $\ge$ 0} \\ 0, ...
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Is it possible to do effectively irrational-interval sampling of a continuous signal?

Suppose there is a real-valued $f(t)$, with $t$ being time. And one wishes to sample at interval of $\pi$, for example. Perfect irrational-interval sampling is not possible, but is there a way to do ...
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1answer
28 views

is there any good way to figure out number of fourier series frequencies of some signal?

Suppose you have $f(t)$, but you do not know the exact function and can only measure $f(t)$ at certain time. Assume $f(t)$ is complex-valued with $t$ being "time." One wishes to find out the number ...
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1answer
14 views

Relationship between short-time and large-frequency asymptotics in Fourier transform

I am trying to understand how the short-time behaviour of a function $f(t)$ influences the large-frequency asymptotics of its Fourier transform $g(\omega)=\mathcal{F}[f(t)](\omega)\equiv ...
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39 views

Signal processing and algebraic geometry

Signal processing is a pretty huge branch of what I would (maybe wrongly) call electrical engineering. I have heard here and there whispers of interesting connections between signal processing - in ...
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How to sketch frequency response obtained from H(z)?

How to sketch frequency response obtained from H(z)? I'm adding an example and its solution below. I did not understand some of the things for question 7.4, part D. Any help is appreciated! There ...
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1answer
23 views

Convolution Problem

while working on a signal processing problem i've reached to the following: So my aproach was: Am I doing something wrong? Is it valid Y(f)=[X(f) x H(f)]*W(f)=X(f) x [H(f)*W(f)] If you could ...
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30 views

Fast evaluation of an integral convolution with an “expanding kernel”

Suppose I have a 1-D integral convolution transform like this: $$ g(x) = \int_{-\infty}^{+\infty} dy\, f(y)\, K(x-y). \qquad (1) $$ Say the kernel $K(x)$ is a known analytic function, and say we have ...
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1answer
21 views

Find the inverse z transform of $H(z)=\frac{1}{8-6z^{-1}+z^{-2}}$

This question was on a homework assignment, and the solutions have been distributed but I'm having trouble reproducing the solutions. Given the initial conditions $y[-1]=y[-2]=0$ and the difference ...
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1answer
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The set of inputs x(n) to a system is described with a small superscript T - what is that?

I think it means TRANSPOSE, but I can't figure out the need to perform a transpose operation: Similarly, the set of weights that go with the inputs is written with a small T as well: There is ...
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1answer
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Graph of the Angle of a Fourier Transform

If I need to graph the magnitude and angle of a discrete Fourier transform which happens to be $X(e^{j\omega}) = 4\cos(4\omega)$, I know how to graph the magnitude, but how do you graph the angle? I ...
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1answer
39 views

A signal on a noisy channel is input to a filter

Question: A Wide-sense stationary (weakly stationary, (WSS)) random signal {X(t)}t∈R with Power spectral density(PSD) S_X(ω) is transmitted on a noisy channel where it is disturbed by an additive ...
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Inverse sum headache

I'm now extreamly tierd of not pulling off this equation. $$\sum_{i=1}^n (y_i-\alpha)^2= \frac{2n\sum_{i=1}^n (y_i - \alpha)}{\sum_{i=1}^n (\frac{1}{y_i - \alpha})}$$ Solve for $\alpha$, y is a ...
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1answer
30 views

Periodic product of sinusoids

(This is problem P-3.7 from the book 'Signal processing first') Let $x(t) = 2\cos(\omega_1t)\cos(\omega_2t) = \cos([\omega_1 + \omega2]t)+\cos([\omega_2 - \omega_1]t)$ where $0 < \omega_1 < ...
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1answer
32 views

Is this system time-invariant?

I think this system is not time-invariant, but I'm not really sure how to plug in a couple test cases to check. The system is: $x(t)$ -->(S)--> $y(t) = \int_{-\infty}^{3t}x(\tau) d\tau$ Without ...
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0answers
56 views

Source estimation for identification of anomalous events

I’m stuck on the following problem. There are two sources $S_A$ and $S_B$ at the ends of a channel. Both are made up of a white noise component $W_i$ plus an impulsive component $I_i$: $S_A = W_A + ...
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1answer
14 views

How to find period of a sum of periodic functions

I got this function: $$ x[n]=\sin(2*\pi*4/3*n) + \cos(2*\pi*5/2*n) $$ It is easy to see that period of the sin is 3/4 and the ...
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30 views

Deriving difference equation from a rational system function $H(z)$

If I have the system function $H(z)$ of a linear time-invariant system, how do I derive the difference equation relating its input $x(n)$ and output $y(n)$? The system function is given by $$H(z) = ...