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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Discretizing my continuous system in Matlab
I was hoping someone could help me with conducting my discretization in MATLAB.
I have a linear system with known data: sampling interval $T$, system matrix $\mathbf{A}$, input matrix $\mathbf{B}$, ...
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1answer
63 views
On the use of Permutation matrix to perform iFFT
I have a question about using the permutation matrix for performing $iFFT$ for such matrix and then reshape it row-wise and column-wise way.
Let's say that we have a random matrix $x$ whose size is (...
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3answers
40 views
System classification: unit-time delay
I'm reading a book on linear systems and I can't understand why the unit-time delay is a distributed system. This is the example given in the book:
I understand that the initial state of the system ...
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1answer
44 views
Autocorrelation of LTI system output
I'm having trouble showing the following relation:
The autocorrelation of a LTI system with impulse response $h[n]$, input $x[n]$ and output $y[n]=h[n]*x[n]$ is given as:
$r_{yy} = r_{hh}*r_{xx}$
...
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2answers
51 views
Filter response to sinusoidal input signal
This question is related to Does 46 dB gain of a filter for a frequency imply 200 times more amplitude? In the accepted answer it is mentioned that "For a sine wave input the amplitude of the steady ...
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1answer
21 views
DTFT of inverse of any function
In my book solution is given like this.
But i am solving like this , am i doing wrong??
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3answers
144 views
Estimating the Signal by Deconvolution with a Prior on the Filter Coefficients and the Signal Samples
Assume I have signal $y[n]$ which is a result of convolution between channel $h[n]$ and signal $x[n]$. which means:
$$y[n] = h[n] \ast x[n]$$ where $\ast$ is the convolution operation
The signal $...
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3answers
87 views
Whether $y(t) = x(t/3)$ is a memoryless system?
I am a newbie in signal and system, and feeling confused about the memoryless property, it says at the definition, if a system gives output only depends on present input, then its memoryless. So if at ...
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0answers
33 views
Trouble checking if a given difference equation corresponds to an LTI System
I'm having trouble on how to check if a given difference equation corresponds to an LTI system. For instance,
Does the following difference equation correspond to a LTI DT system ? Explain.
$$y[...
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3answers
209 views
Oscillations in a first-order discrete-time linear systems
According to my understanding, continuous first order systems do not exhibit oscillatory behavior because a first order system has only one energy storage element and thus oscillation is not possible. ...
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0answers
26 views
LTI system response to finite length pseudorandom noise [closed]
I am trying to understand how an LTI system responds to pseudorandom noise.
Periodic pseudorandom noise has a spectrum that is an impulse train. If it is an input to an LTI system the frequency ...
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1answer
43 views
What are the properties of continuous-time improper systems?
I am trying to better understand the properties of improper systems $H(s) = \frac{b(s)}{a(s)}$, for which the order of the numerator $b(s)$ is greater than the order of the denominator $a(s)$ (in the ...
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1answer
49 views
Proving linearity and time-invariance
This question was asked in one of our quiz. Question is
Determine if the system with the input-output relationship given by
$$y(t) = \int_{t-10}^{t} \cos(\tau)x(\tau)d\tau$$
is linear ...
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1answer
48 views
Time invariance and linearity of recursive system
I am confused about the definition of linearity and time invariance of recursive system
Given $$y[n] = y[n-1] - y[n-2] - x[n]$$
To test time invariance, we shift the input $x[n]$ for shifting and ...
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1answer
34 views
Difference equation with variable coefficients in MATLAB
Given the following difference equation
$$y[n]-\left(\dfrac{n}{n+1}\right)y[n-1] = n\cdot x[n]$$
How can we use MATLAB to solve it?
I know if the coefficients are constant we can simply use ...
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1answer
43 views
Determining if a system is Linear
Having $h[n]=u_0[n]+0.8u_0[n-1]+1.6u_0[n-2]$ and $x[n]=(u_1[n]-u_1[n-3])$
The goal is to determine if the system $y[n]=h[n]*x[n]$ is linear.
I know that I would have to test it for Homogenity and ...
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2answers
75 views
If the frequency response of a system is time-independent, is this system LTI?
Intuitively, I feel like a time-variant system would necessarily have a time-dependent frequency response, and vice-versa. So, is the time-independence of the frequency response necessary and ...
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2answers
61 views
Cross-correlation of maximum length sequence with noise
Background
I am using a maximum length sequence to find the impulse response of a linear system. I want to calculate the expected SNR. I know that my signal amplitude is scaled by the length of the ...
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2answers
65 views
Response of an ideal integrator to a cosine wave
It sounds like a very elementary question on system theory but I got quite confused about it, so hopefully you guys can enlighten me.
I'm considering an ideal analog integrator, i.e., a system with ...
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1answer
13 views
Frequency response of each component of a system given its global response
Given the following block diagram, find the frequency responses $H1(f)$ and $H2(f)$. The frequency response of the whole system has to be $H(f)=(\alpha_0+\alpha_1e^{-j2\pi T_1f}+\alpha_2e^{-j2\pi T_2f}...
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1answer
105 views
Frequency response of a system given its block diagram
Given the following block diagram, I am asked to find the frequency response $H(f)$ of the system.
This is what I have done: The output of the first block is $j2\pi fX(f)$, and after going through $...
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1answer
36 views
Impulse response of a system in z domain
The question 3-23 in the "Discrete-Time Signal Processing - Second Edition" is:
and the solution is:
I cannot understand the solution. In the second row of the answer when I multiply (-4) with ...
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2answers
38 views
Are there existing third system (linearity, non linear, !!!)?
I like to know your idea about is there any third system after linear, none linear classification.
Could we consider more universe dimension than 4 dimensions (time, XYZ) like 11 dimensions of ...
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1answer
103 views
LTI, causal, discrete time system output
Consider a discrete time LTI causal system $S: y = S(u)$, with its impulse response $h:{Z} \rightarrow R:h(n)=3^{n+1}{H(n)}$ with $H$ the Heaviside function. We know the values of input:
$$u(0) = 2$$...
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1answer
112 views
is y[n] = x[n] + n time invariant?
My steps were as follows:
$\ x_2[n] = x[n-k] $
$\ y[n-k] = x[n-k] + (n-k) $
and
$\ y_2[n] = x_2[n] + n = x[n-k]+(n-k)$
Does this mean that it is indeed time invariant?
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1answer
22 views
Does constant Q transform have linearity property in the transformed domain?
With FFT, I sometimes take advantage of the fact that I can pre-calculate a signals Fourier Transform, and then you can add the noise in the spectral domain:
$$ \mathscr{F} (x[n] + h[n]) = \mathscr{F}...
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2answers
84 views
LTI system output given input and frequency response
The question I'm trying to understand is as follows: A linear time-invariant continuous-time system has the frequency response function $$H(\omega)=\frac{1}{j\omega+1} $$
Compute the output response $...
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1answer
38 views
System memory, causality, stability
im new into systems and im supposed to solve if the system has memory, us causal, linear, stationery, BIBO stable...The problem is i have never had experience with this type of system where the actual ...
0
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1answer
63 views
Parallel LTI system
Imagine I know that a system comprises two LTI subsystems $H_a$ and $H_b$ connected in parallel. Also, suppose that I can measure the impulse response of the whole system, $H_a \| H_b$, and the ...
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0answers
52 views
Phase response of an LTI-system
I've got an LTI-system such as follows:
$$y'(t)+{2} y(t)={5} x(t-{3}), t>0,
\\ \\ y(0)=0 \ \text{ ja } x(t)=0, \ t<0.$$
From this i've already figured out the transfer function:
$$H(f)=\frac{5\...
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1answer
39 views
Is $y(t) = \cos(t) + x(t)$ a time-invariant system?
Is $y(t) = \cos(t) + x(t)$ a time-invariant system?
$y(t-k) = \cos(t-k) + x(t-k)$
But it isn't equal to $\cos(t) + x(t-k)$
So, would it be time-invariant?
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4answers
2k views
Are all LTI systems invertible? If not, what is a good counterexample?
I have been trying to figure this out for a while now. Everywhere I have looked I could easily find examples of invertible LTI systems, but I could not find any counterexamples. Can anybody shed some ...
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3answers
259 views
I somehow “proved” that given any LTI system, its transfer function has to be constant. What am I missing?
A transfer function is defined as the Laplace transform of the ratio of output to input.
Also, every LTI system has an eigenfunction. Given such eigenfunction as an input, the ratio of the output to ...
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1answer
38 views
Steady-State Output from Transfer Function
The progress I have made is as follows:
$\sin(t)$ is our signal therefore $\omega = 1 = 2\pi f$ and $f$ = $\frac{1}{2 \pi}$
Also, $f_s$ = 10Hz therefore T = $\frac{1}{f_s}$ = 0.1s
$H(z) = \frac{z}{...
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1answer
28 views
Show That a 2D Linear Transform $ T \left( \cdot \right) $ Is Homogeneous
By my understanding, a transform T is homogeneous if T[0] = 0.
Then to prove that a linear transformation is homogeneous we say that:
T[ax(n1, n2) + bx(n1, n2)] = aT[x(n1, n2)] + bT[x(n1, n2)]
What ...
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2answers
441 views
The spectrum of the sum of two periodic signals
Considering the continue, periodic signals $s_1(t)$ and $s_2(t)$, with the period $P_1$ and $P_2$ respectively.
Consider now a new signal $s$ the sum of the two pervious signals.
I come to know ...
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1answer
157 views
impulse response VS zero-input response
I am new in the field of systems and signals, and I have a rather basic for the majority of the group, question:
Can we find the impulse response function of homogeneous ODE, instead of its zero-...
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votes
4answers
807 views
Can every type of linear filter be modelled by a convolution?
I have an input time series going through a filter that creates another time series as output.
If I assume in first approximation that my filter is linear, does it necessarily mean that I can model ...
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2answers
250 views
Convolution of Signal with a Gaussian Filter / Kernel
Following up on Analytical Solution for the Convolution of Signal with a Box Filter, I am now trying to convolve a Gaussian filter with the sine signal by hand.
My method is to use the definition of ...
2
votes
1answer
258 views
Initial Rest Condition for LCCDE causal LTI systems
I am self studying Alan Opennheim's course Signals and Systems. I am a math major and have no background in EE.
I understand that for a LCCDE system to be linear its auxiliary conditions must be 0.
...
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2answers
168 views
Why can adaptive IIR filters result in unstable solutions?
For adaptive filtering, both finite and infinite impulse response (FIR/IIR) filters can be utilized. As an advantage of FIR filters in this context, guaranteed stability is often mentioned, while IIR ...
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1answer
150 views
transient and steady-state response of first order system
Considering this general 1st order transfer function
$$
H(z) = \frac{b_0 + b_1z^{-1}}{1-az^{-1}}
$$
How to find (analytically) the transient and steady-state responses?
With steady-state response I ...
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2answers
79 views
Are Chebyshev Type 1 filters very unstable at very high orders?
I am currently experimenting with a digital implementation of a Chebyshev Type 1 filter of mine, for audio purposes. I have written the code myself so I was wondering if the behavior I've noticed is ...
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0answers
46 views
Determining if a 2D system is LSI
After a long time away from school, I'm a bit rusty and struggling with this question:
Determine if the following discrete system is LSI: $y(m,n) = mn*x(m+n) + mn*x(m-n)$
So here's what I've done ...
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1answer
29 views
Invariance of $T[x[n]]=\sum_{k=n-1}^{n+2}x[k]$
I have to test whether the following system is invariant or not: $T[x[n]]=\sum_{k=n-1}^{n+2}x[k]$, so I want to verify that, if $y[n]=T[x[n]]$, then $y[n-N]=T[x[n-N]]$.
$$T[x[n-N]]=\sum_{k=n-1}^{n+2}x[...
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1answer
92 views
Analytical Solution for the Convolution of Signal with a Box Filter
I have an exercise in which I am trying to filter an input signal $y(x) = \sin(x)$. Ideally, I would like to apply a box filter to this signal.
Previously, I successfully convolved the input signal $...
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2answers
163 views
Why does knowing the impulse response allow you to determine the output for any LTI system?
Going over some notes on LTI systems and I’m quite confused about how knowing the impulse response of a system allows us to recover the systems’ response to any signal. In the textbook screenshot ...
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1answer
50 views
Representing a continuous LTI system as a discrete one
I am aware that there are different ways to represent a continuous time system in discrete domain (e.g. bilinear transform, impulse invariance transform).
But my problem is as follows: Given an ...
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2answers
95 views
Cross-correlation of filtered random processes
I have a wide-sense-stationary (WSS) process $\{x(t)\}$ and two linear filters with impulse functions $h_1$ and $h_2$.
Let $\delta(\omega)$ be the power spectrum of $\{x(t)\}$ and $$H_1:\omega\...
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1answer
49 views
Difference between the two forms of input-output relationships of an LTI system
$$y(t)=f(t)*h(t)\tag{1}$$
$$y(t)=H(s)e^{st}\tag{2}$$
$$H(s)=\int_{-\infty}^{\infty}h(t)e^{-st}dt\tag{3}$$
Let $f(t)$ in Eqn $(1)$ be $e^{st}$. In many worked out examples, I have found that the two ...
