Questions tagged [linear-systems]

A linear system operates on inputs with only linear operators so the response to a complex input can be analysed as the sum of the response to a set of simpler inputs. This mathematical property makes the analysis of linear systems much simpler than non-linear systems where this summation or superposition does not hold. Linear systems are generally further classified as time invariant, meaning that there characteristics do not change over time.

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13 views

Linearity of the system with Pearson correlation sliding window

I have a sampling system that consists of linear and non-linear components(analog filters, ADC, CPU, and so on). I made some HW changes and I want to verify that the linearity was preserved. My method:...
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85 views

Find moments of a discrete process using $\mathcal{Z}$-Transform method

For a linear process relating a variable $Y_i$ to random, independent variables $X_i$ using the equation: \begin{equation} Y_i = aY_{i-1} + (1-a)X_i \end{equation} which has the solution: \begin{...
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104 views

Are there any examples of "Causal" Continuous-Time Linear-Time-Invariant Systems (CT-LTI) with output given by finite-duration functions?

Are there any examples of "Causal" Continuous-Time Linear-Time-Invariant Systems (CT-LTI) with output given by finite-duration functions $y(t)$?? With output I am meaning that $y(t)$ is such ...
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52 views

How to find the impulse response from the following input/output relation

Suppose we have a system defined with input $x(t)$, and output $y(t)$, related by $$ y(t) = \sum_i \alpha_i(t)x(t-\tau_i(t)) \tag 1 $$ This is a linear system, (according to the textbook), so we can ...
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60 views

Is the filter $1/(1-s)$ anti-causal?

The filter with the response function $$ H(s) = \frac{1}{1 - s} $$ Produces a positive phase shift and a negative group delay for all frequencies Is it anti-causal? Is there a way to deduce such ...
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31 views

Time Invariance of Continuous Time System

I am trying to check whether the system given by the following input-output relation is LSI or not? $$y(t) = \cos(x(t)-x(0))$$ My work: The given relation: $$y(t) = \cos(x(t)-x(0))$$ If the input ...
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56 views

State-space model and/or TF for a $K$-th order linear system given the signal sequence

The discrete-time signal $x_n$ is a sum of $K$ discrete-time complex damped exponential: \begin{equation} x_n = \sum_{k=1}^{K} \underbrace{(a_k e^{j\phi_k})}_{c_k} {\underbrace{e^{\{(- \...
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62 views

Is this system linear or non linear? $y[n]=cos(\frac{5\pi}{8}*n+\frac{\pi}{4})$ [closed]

$y[n]=cos(\frac{5\pi}{8}*n+\frac{\pi}{4})$ If I let $(\frac{5\pi}{8}*n+\frac{\pi}{4})=x[n]$, I get the system as non linear. But one of my teacher says it is linear, whereas another teacher says it is ...
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170 views

Can we tell if a system is linear and time-invariant from its frequency response?

Given a system with a known frequency response in the S-domain. Is there a way to find whether the system is linear and time invariant? My current understanding is that we need to take the inverse ...
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35 views

What are the characteristics of a smartphone speaker?

What are the characteristics of a smartphone speaker? Whether it is linear or non-linear?
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1answer
30 views

FInd the amplitude function to a system

I'm studying a course in signal analysis and have come across an exercise where I find the analytical part a bit tricky. I am to find the amplitude function $|H( f )|$ for a system with the impulse ...
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54 views

Filtering a sum of cosines

The block diagram below represents a linear modulation system operating at the frequency of $1000 Hz$, $f_C = 1000 Hz$, transmitting the message $m(t) = 2\cos(400πt)$ At point B, i got the signal: $$...
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64 views

How to find a system without an input $x[n]$ is linear or non linear

The question is this: $$ y[n] = \cos\left(\frac{5\pi}{8}n + \frac{\pi}{4}\right)$$ This is what my teacher said when I asked him for help-: In any system, inputs are not given, then we have to assume ...
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100 views

Why aren't the integrator and the differentiator inverse systems?

I have a statement that leads to a paradox, but I'm incapable of finding the part where I'm wrong. The integrator system $$x(t) \mapsto y(t)=\int_{-\infty}^{t}{x(\tau) \, {\rm d} \tau}$$ is a linear ...
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How to update point spread function of blind deconbolution by conjugate gradient?

There is an unblurred image $g$ and a blurred image $x$. Their relationship is expressed by the following formula using $psf$(point spread fucntion, size is $5×5$ kernel). $g = x \otimes psf\tag 1$ ...
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70 views

What is the importance of obtaining a linear signal?

I am a computer scientist who has started doing some work in the electrical engineering space – in particular, photonics. While reading about interferometric systems, I have noticed that there seems ...
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1answer
43 views

Identify out signal from in signal and impulse response

Hi there I am studying a course in signal processing and systems. I was given an excecise which I am having a great deal of trouble solving. I am allowed to solved using matlab, but for top marks I ...
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58 views

Definition of frequency response

I studied various signal processing materials for a long time, and I have a question. Considering an LTI filter, one can define its frequency response by evaluating its transfer function $H(z)$ on the ...
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55 views

Mixing Kalman filter and least-squares

I'm not sure it is the right department. I try my chance I am wondering if there is a way to make a hybrid formulation of a least-square problem and a Kalman filter. Let me explain what I mean: The (...
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1answer
54 views

Zeros and poles from transfer function

I have a transfer function $$ H(z) = \frac{Y(z)}{X(z)} = 1 - 0.5z^{-1} \text{.}$$ I'm interested in zeros and poles. I know I need to adjust the function to $$ H(z) = \frac{\prod_i(z-n_i)}{\prod_i(z-...
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167 views

Difference equation to FIR filter coefficients

I have a difference equation $$y[n] = y[n-1] + 0.5x[n] - 0.5x[n-2] \text{.}$$ According to this answer, the filter should have finite impulse response. I just can't figure out how to get the FIR ...
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26 views

time invariance $ y (t) = x(\frac{t}{3})$

check time variance of this system $$y (t) = x(\frac{t}{3})$$ my solution $$y_1(t) = x_1(t/3)\\ y_2(t) = x_2(t)= x_1(\dfrac{t}{3}-t_o)$$ where $$ x_2(t) = x_1(t-t_o)\\ y_1(t - t_o) = x_1(\dfrac{t-...
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46 views

How to prove that this system is an invertible system or not?

How could i go throw proving that this system $y(t)=\int_{-\infty}^{t}e^{-(t-\tau)}x(\tau)d\tau$ is invertible system or not ?
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86 views

Linear system stability criteria

Suppose we have a closed-loop system $H(s)=\frac{A(s)}{1+A(s)f(s)}=\frac{A(s)}{1+T(s)}$. I've seen the stability of the system stated a couple ways: If $H(s)$ has any poles in the RHP, then it is ...
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1answer
62 views

When is a LTI IIR $H(z)$ system minimum phase

EDIT: I failed to mention that the system's inverse also needs to be causal and stable. I cannot wrap my mind around on how when a system and its inverse are both causal and stable and LTI IIR it is ...
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23 views

Please explain the following Linear Prediction estimate graph

I am reading this doc https://librosa.org/doc/latest/generated/librosa.lpc.html#librosa.lpc? I can't understand the graph for linear prediction coefficients. Please help explain it as I a new to this ...
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41 views

Signal separation

This is probably a blind signal separation problem of sorts, but it seems like it should be easier than I am finding it. Let’s say I have N time series, each of length [M x 1] that are a superposition ...
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1answer
156 views

Why is particular solution zero for an impulse excitation signal?

We were being taught the impulse response for a series RC Circuit- consisting of simply one resistance, one capacitor, and an impulse excitation all in series. I get that the homogeneous part of the ...
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2answers
203 views

Where to start with DSP?

I have a DSP exam coming up this summer and I got an abysmal mark on an assignment earlier this year. Obviously, the lecture slides are not enough for me to understand DSP, so I am wondering where can ...
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221 views

Impulse response for an LTI system

I'm new to signal processing and working my way through a textbook. There is an exercise where a causal LTI system is given that responds to a rectangular pulse. I have an exercise where a causal LTI ...
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115 views

Under what conditions is the convolution of an input signal with the system's impulse response periodic?

I'm currently solving the following convolution problem from Oppenheimer's book: In the solution, it was stated that "$x(t)$ periodic implies $y(t)$ is periodic" So I wondered if it's ...
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38 views

How do i know it is will be periodic

(system is LTI and Casual) 1)If a periodic signal is applied to the input of this system. Does output always have to be periodic ? 2)What conditions are required for this system to be linear? thanks ...
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84 views

Output of a stable LTI system (discrete)

Consider an LTI (linear and time invariant) system that is BIBO (bounded input bounded output) stable and is such that $x[n] = 0$ for all $n < 0$ (note: this is sometimes referred to as a relaxed ...
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63 views

Understanding LTI systems graphically

I'm currently working on this problem from Oppenheimer's book: Given $x_{1}(t)$ and $y_{1}(t)$ I should figure out $y_{2}(t)$ and $y_{3}(t)$ given that the system is LTI. My progress so far: I'm ...
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207 views

How do I check for linearity for the following piecewise-defined system?

The problem at hand: Where I'm currently stuck: I'm not entirely sure about how to move on from this point, I'm trying to find the superposition of the responses of the two individual signals so I ...
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50 views

Physically, what does the usage of two variables mean for convolution

My intuition of convolution is that it is just a way to depict multiplication of two signals where each signal is made up of various frequencies and phases. Since it isn't easy to find the value of $\...
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116 views

Scaling the input vs scaling the impulse response for an LTI system

Two different cases: We pass $x(t)$ to an LTI system with impulse response $h(2t)$ and get the output $y(t)$. We pass $x(2t)$ to an LTI system with impulse response $h(t)$ and get the output $z(t)$. ...
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30 views

Stable and causal system

How many stable and causal systems with the same magnitude response are there? I know this relates to an all pass system for two rational transfer functions but am not sure about the specifics of this
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37 views

Z-transform of an affine function

What is the transfer function of the system described by the following affine input ($x$)-output ($y$) relationship: $$ y[n] = \alpha x[n] + \beta. $$ Using the Z-transform we find: $$ Y[z] = \alpha X[...
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80 views

convolution of two exponential signals with imaginary numbers

I can solve problems without imaginary numbers, but when exponential contain imaginary numbers, I can't solve the problem. For example, $x(t)=e^{3jt}+e^{4jt}$, $y(t)=(e^{-3t}-e^{-4t})u(t)$ (where $j$ ...
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78 views

Unit impulse response of a cascade interconnection of three discrete-time systems

I am nearly at the end of finishing a problem in my textbook but I couldn't understand something in the answer; I did everything to the point I found the overall response of the system in terms of $...
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48 views

Calculate impulse response when output contributes in input

I have an exercise in which I need to find the impulse response for this given system: $$y(n)=\frac{1}{2}y(n−1)+x(n−1)+x(n)$$ As per my knowledge, I need to find the homogenous solution. My homogenous ...
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338 views

How do I get a faster system response?

I have this model in simulink (the graph is my output): The step input has amplitude 0.5 m/s, and it steps up after 0.1 seconds. The gain $K_p=5$. The saturation block is to keep the voltage between -...
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1answer
62 views

Unit impulse response $h[n]$ of a discrete time system with multiple characteristic roots 0

I am trying to obtain the unit impulse response of a system in the form: $$y[n+N]+...+a_{N-1}y[n+1]+a_{N}y[n]=b_0x[n+N]+...+b_{N-1}x[n+1]+b_{N}x[n]$$ $$Q[E]y[n]=P[E]x[n]$$ (where E is the unit advance ...
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1answer
63 views

Output of a linear time-invariant(LTI) system

I am very confused about one of the questions I received during an exam. How do I solve this question?
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72 views

Finding the impulse response given response to another signal

I was trying to solve this question : I respresented $x(t) = u(t+1)-u(t-1)$ writing the convolution as $[u(t+1)-u(t-1)]*h(t) = y(t)$ I then used the property of differentiation to convert from the ...
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1answer
79 views

Initial rest condition applied on $x(t)$ vs $h(t)$

Define the LTI system $\mathcal{H} : x\mapsto y$ Define the convolution for continuous-time system : $$ (x*h)(t)=\int_{-\infty}^{\infty}x(\tau)h(t-\tau)\;\text{d}\tau $$ The initial rest condition ...
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1answer
66 views

Impulse Response of real coefficient, LTI System

I'm trying to obtain the impulse response $h[n]$ of a system whose frequency response is $H(e^{j\omega})=R(\omega)e^{-25j\omega}$. I believed that $h[n]=h[n-25]$, would be the correct answer, however ...
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1answer
34 views

understanding echo cancellation model

so, here is a very simple model. The CCDE is given, but I was trying to derive it on my own and now I am stuck. first of all, the reverse y[n] when goes through the delayed system, it turns to y[n-M]....
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1answer
49 views

Applying Superposition Property : $x^{2}[n]$ vs $x[n^{2}]$

If we consider the mapping $\mathcal{H} : x[n]\mapsto y[n]$ and define the following output signal $y_{1}[n]:=\mathcal{H}\{x[n]\}:=x^{2}[n]$, then one can easily verify that such system is non-linear ...

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