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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questions
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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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2answers
64 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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1answer
36 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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0answers
37 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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1answer
50 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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3answers
139 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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2answers
44 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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1answer
70 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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24 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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2answers
31 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:
$$
Y[z] = \alpha X[...
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28 views
convolution of two exponential signals with imaginary numbers
I can solve problems without imaginary numbers, but when exponential contain imaginary numbers, I can't solve the problem.
For example, $x(t)=e^{3jt}+e^{4jt}$, $y(t)=(e^{-3t}-e^{-4t})u(t)$ (where $j$ ...
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1answer
21 views
Unit impulse response of a cascade interconnection of three discrete-time systems
I am nearly at the end of finishing a problem in my textbook but I couldn't understand something in the answer;
I did everything to the point I found the overall response of the system in terms of $...
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1answer
32 views
Calculate impulse response when output contributes in input
I have an exercise in which I need to find the impulse response for this given system:
$$y(n)=\frac{1}{2}y(n−1)+x(n−1)+x(n)$$
As per my knowledge, I need to find the homogenous solution. My homogenous ...
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1answer
106 views
How do I get a faster system response?
I have this model in simulink (the graph is my output):
The step input has amplitude 0.5 m/s, and it steps up after 0.1 seconds. The gain $K_p=5$.
The saturation block is to keep the voltage between -...
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1answer
36 views
Unit impulse response $h[n]$ of a discrete time system with multiple characteristic roots 0
I am trying to obtain the unit impulse response of a system in the form:
$$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]$$
(where E is the unit advance ...
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1answer
34 views
Output of a linear time-invariant(LTI) system
I am very confused about one of the questions I received during an exam. How do I solve this question?
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1answer
60 views
Finding the impulse response given response to another signal
I was trying to solve this question :
I respresented $x(t) = u(t+1)-u(t-1)$
writing the convolution as $[u(t+1)-u(t-1)]*h(t) = y(t)$
I then used the property of differentiation to convert from the ...
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1answer
49 views
Initial rest condition applied on $x(t)$ vs $h(t)$
Define the LTI system $\mathcal{H} : x\mapsto y$
Define the convolution for continuous-time system :
$$
(x*h)(t)=\int_{-\infty}^{\infty}x(\tau)h(t-\tau)\;\text{d}\tau
$$
The initial rest condition ...
1
vote
1answer
53 views
Impulse Response of real coefficient, LTI System
I'm trying to obtain the impulse response $h[n]$ of a system whose frequency response is $H(e^{j\omega})=R(\omega)e^{-25j\omega}$. I believed that $h[n]=h[n-25]$, would be the correct answer, however ...
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1answer
26 views
understanding echo cancellation model
so, here is a very simple model.
The CCDE is given, but I was trying to derive it on my own and now I am stuck.
first of all, the reverse y[n] when goes through the delayed system, it turns to y[n-M]....
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1answer
46 views
Applying Superposition Property : $x^{2}[n]$ vs $x[n^{2}]$
If we consider the mapping $\mathcal{H} : x[n]\mapsto y[n]$ and define the following output signal $y_{1}[n]:=\mathcal{H}\{x[n]\}:=x^{2}[n]$, then one can easily verify that such system is non-linear ...
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1answer
68 views
Kalman filter with multiple sensors
I have been reading more about EKF and I am a bit confused on how you handle predictions with multiple sensors.
Ex, I have IMU, GPS, Odom and Stereo Camera.
Each can be used to predict location, how ...
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2answers
68 views
From transfer function to differential equation
I have the below detailed solution (boxed in blue) that I don't understand completely:
I can reconstitute the differential equation from:
$$ (1+Ts) X(s) = K_v U(s) $$
$$ x(t) + T\dot x(t) = K_v u(t) $...
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2answers
48 views
How could I approach determining if this 2D system represented as a 2D summation formula is linear?
I have a given 2D system:
$$y(m,n) = \sum_{k_1=-\infty}^{m} \sum_{k_2=-\infty}^{n} x(k_1,k_2)$$
My usual approach to determining if a system is linear is to test if it is homogeneous and additive. ...
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3answers
471 views
How can convolution be a linear and invariant operation?
I'm having a slight breakdown right now with a seemingly simple question. Say I have a system that convolves an input function with itself to produce an output function:
$g(x) = f(x) ∗ f(x)$
I've ...
0
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0answers
33 views
Analyze the transient response of a system
Suppose I have a sytem with the following transfer function :
$$H(s) = \frac{N_H(s)}{D_H(s)}$$
I would like a general method which is not dependant on the order of the system to analyze what would be ...
0
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1answer
52 views
time linearity of LTI system
There is this following question.
Consider the transformation H {x}[n] = n x [n]. Does H define an LTI system?
What I understood from the question is:
x[n] -> H -> y[n]
or according to the ...
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1answer
76 views
Particular Solution to Difference Equation
I have a system given by
$$y[n] - \frac{1}{4} y[n-1] - \frac{1}{8} y[n-2] =3x[n] $$
I want to solve for $y[n]$ for $x[n]=(\frac{1}{2})^nu[n]$.
The complementary solution evaluates to $[k_1(\frac{1}{2})...
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1answer
28 views
Impulse Response for an Anti Causal Linear Shift Invariant System
I have a system given by
$$y[n] - \frac{1}{4} y[n-1] - \frac{1}{8} y[n-2] =3x[n] $$
I'm asked to determine the impulse response of both an anti-causal and a causal Linear Shift Invariant System ...
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1answer
59 views
What is the reason behind complex conjugate pairs in Linear Phase FIR filter analysis from the Pole Zero plot
When we say that a Generalized Linear Phase System must satisfy the pole zero plot with the condition that a complex zero not on the unit circle exist's in a pair of 4. Then I understand that I need ...
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36 views
Augment State Space Model with Derivative of State
Problem
How do you augment a state space model with the derivative of a state? I know how to augment a state space model with the integral of a state by doing the following. Given a linear system
$$ \...
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1answer
57 views
Determine if system is linear time variant
The system equation is given as:
$$y(n)=(n-1)x(n-1)+(n+1)x(n+1)$$
I solved that the system is time variant:
\begin{align}
y(n-k)&=(n-k-1)x(n-k-1)+(n-k+1)x(n-k+1)\\
H[x(n-k)]&=(n-1)x(n-k-1)+(n+...
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3answers
80 views
How can we show that $|H(e^{j\omega})|=|H(e^{-j\omega})|$?
Let $H(z)$ be the rational system function of an LTI system.
How can we show that $|H(e^{j\omega})|=|H(e^{-j\omega})|$?
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Matched Filter Impulse Response
At the receiver end of a digital communication system, the matched filter is designed with impulse response matched to the effective signal that is given as : $$x_{(z,1)}^{*}(T_p-t)-x_{(z,0)}^{*}(T_p-...
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2answers
100 views
Inverse system of sinc?
I'm doing some self-study for an important exam I'll have in late March and came across the following question:
So, using the convolution properties, if I want to find an identity system so that the ...
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1answer
37 views
What is the decay rate (time constant) of two One-Pole LPF's cascaded in series?
If you have a One-Pole LPF, and set the cutoff frequency to $\frac{1}{2πt}$ where $t$ is the time to $\frac{1}{e}$ amplitude, it will provide an exponential decay in response to a step function (eg. ...
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1answer
29 views
What is the relation between input and output PSDs given system transfer function $H(s)$
If I have the system transfer function $H(s)$ in the complex frequency domain, how would I relate the input/output power spectral densities?
I have come across the relation $P_{out}(f) = |H(f)|^2P_{in}...
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1answer
89 views
Why not use the same “standard” exponentials for both continuous and discrete time
In continuous time the standard exponential signal is usually defined as
$$
e^{st}, \quad\text{with}\quad s = \sigma+j \omega
$$
In discrete time the standard exponential signal is usually defined as
...
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2answers
22 views
Impulse response of forward difference cascaded with one sample delay
Below is the excerpt from Discrete Time Signal Processing by Alan Oppenheim.
I don't get how $(\delta[n+1] - \delta[n]) * \delta[n-1])$ becomes $\delta[n] - \delta[n-1]$. The convolution sum operator ...
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1answer
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If the impulse response of a causal LTIC system is $h(t) = \delta(t) + \sin(t)u(t)$, is it marginally stable or unstable (BIBO)?
If you take the $H(s) = \mathcal{L}[h(t)]$ the poles are on the imaginary axis so the system should be marginally stable, but is it?
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1answer
37 views
Linear systems: Square root of input product
Hi guys i'm studying signals and systems, and my professor ask us if this signal is linear or not
$$y(t) = \big[x(t − 1)x(t + 1)\big]^{\frac 12}$$
the fact that is in the form of $x\cdot x$ told me ...
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2answers
123 views
Determining Causality and Time-Invariance of a system
Consider the following system:
$$y(t-1)=\int_{-\infty}^\infty x(𝜏)u(𝜏-t) d𝜏 $$
where $u(t)$ is the unit step function, which is zero for $t<0$ and equals $1$ for $t>0$.
$(1)$ Is the system ...
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1answer
57 views
Why can you use the one-sided laplace transform to solve differential equation describing a causal LTI-system?
In an example, an equation describing a causal LTI-system is
$$
(D^2 + 5D + 6) y(t) = (D+1) x(t)
$$
where $y(t) = y_{zs}(t) + y_{zi}(t)$ and the initial conditions are $y(0^-) = 2, \dot{y}(0^-) = 1$.
$...
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2answers
35 views
Question about a transient response statement (Oppenheim-Schafer"s discrete-time signal processing book, 3rd ed)
This is the statement from the book:
Eq. (129) only ensures that $y_t[n]$ will be bounded for all values of $n$.
For example, this may include a case where $y_t[n]$ oscillates
and hence never ...
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2answers
36 views
Clarification on eigenfunction property from Oppenheim's Discrete-time signal processing, 3rd ed
Could anybody explain what is meant by the following statement on page 50 of Oppenheim and Schafer's book Discrete-Time Signal Processing (third edition)?
The eigenfunction property of complex ...
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2answers
77 views
Laplace Transform: zeros and corresponding impulse response $h(t)$
Poles and the impulse response
If our impulse response is in the form :
$$h(t) = e^{-\sigma_0 t}\cos(\omega_0 t) \, u(t)$$
(where $u(t)$ is the unit step function)
And its Laplace transform is :
$$H(s)...
2
votes
1answer
49 views
Why can't this system be adequately represented using a z-domain transfer function?
According to this question and answer the following system cannot be adequately captured by a z-transform transfer function.
$$y[n] = y[n-1] + F_{\psi}(y[n-1)) + F_{\phi}(x[n-1])$$
where $F_{\alpha}(z)...
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1answer
31 views
RoC and Stability of a Rectangular Signal
If we have a system with an impulse defined as: $$h(t)=u(t)-u(t-2)$$
Then the Laplace Transform of h(t) would be the transfer function: $$H(s)=\frac{1}{s}-\frac{e^{-2s}}{s}, \quad Re(s)>0$$
We also ...
1
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0answers
31 views
How to find parameters of Kalman filter using matrix information?
I'm trying to understand concepts on Kalman Filters. Consider the overdetermined system $Ax=y$;
$$\begin{bmatrix}1 \\ 1 \\ 1 \\1 \end{bmatrix} x = \begin{bmatrix} 3 \\ 5 \\ 4 \\ 8 \end{bmatrix}$$
Let $...
2
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3answers
165 views
Are all exponential functions eigensignals of LTI systems?
I know that complex exponential functions are eigensignals to LTI systems. Do these include real exponential functions? E.g. $2^t, e^t, ...$
Thanks for the help!
