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.
Questions tagged [linear-systems]
675 questions
8
votes
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Problem with the definition of Linearity
From the high school maths we know that y=mx+c is a linear equation. However, in DSP the linear system must satisfy Additivity properties which y=mx+c does not hold because of +c.
So, is the definition of Linearity different in DSP and Maths? if so…
Zahid Hasan
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7
votes
2 answers
Eigen Function of Linear Time Invariant (LTI) System
Whenever I try to find the derivation of eigen signal of a LTI system, they always start with assumption that the input signal is complex exponential. Is there any other way to prove that this is the case starting from the basic equation of LTI…
Interception
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7
votes
1 answer
How to determine if the system is invertible
Is there any systematic way to determine if the system is invertible?
My general approach is first trying to find the inverse system by using mathematical method; that is, solving for the output in terms of input. If this is difficult, my guess…
IgNite
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6
votes
1 answer
Causality, Linearity, and Time Invariance for Systems Described by Linear Constant Coefficient Differential Equations
I am currently using Signals and Systems by Alan Oppenheim as a reference to learn about LTI systems.
Before introducing systems represented by linear constant coefficient differential equations, the author first considers general systems of the…
user33568
5
votes
1 answer
Differential equations and LTI systems
I've seen in many textbooks on Signals and Systems that an LTI (Linear Time-Invatiant) system can be described as a constant-coefficient linear differential equation, such as
$$\sum_{k=1}^N a_k \frac{d^k}{dt^k}y(t) = \sum_{l=1}^M b_l…
GKH
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5
votes
1 answer
LTI system without constant coefficient differential equation
I have encountered a system where the output $y(t)$ and input $x(t)$ are related in the laplace domain as:
$$Y(s) = H(s)X(s) \tag1$$
which is typical. However, $H(s)$ is not a rational function of polynomials of $s$. Instead, $H(s)$ contains terms…
oblivious
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4
votes
1 answer
Why is the state space representation, $dx/dt = A x(t) + B u(t)$ linear?
One thing puzzles me about the state space representation and that is why is it claimed that such systems are linear and time-invariant but particularly why are they claimed to be linear? The right-hand side of $dx/dt = A x(t) + B u(t)$ does not…
rhody
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4
votes
3 answers
Does scaling property imply superposition?
For a system to be linear,it follow the principles of scaling and superposition.Does scaling imply superposition?If so why are two different conditions given for linearity?If not can u specify an example for which only one of them is satisfied.
skt9
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3
votes
3 answers
I somehow "proved" that given any LTI system, its transfer function has to be constant. What am I missing?
A transfer function is defined as the Laplace transform of the ratio of output to input.
Also, every LTI system has an eigenfunction. Given such eigenfunction as an input, the ratio of the output to input is a constant, which means the transfer…
Venuce
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3
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3 answers
Why $y[n] = x[-n]$ is not time-invariant?
I followed these steps, but the answer still says that this system is time-invariant
let: $x_2[n] = x[n-k]$
$$\begin{align}
y_2[n] &= x_2[-n] \\
&= x[-(n-k)] \\
&= x[k-n] \\
\end{align}$$
and
$$\begin{align}
y[n-k] &= x[(-(n-k)] \\
&=…
rmendes
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3
votes
1 answer
Is applying a linear operation to the input and output of a continuous linear operator OK?
The scientist and engineers guide to digital signal processing, Steven W. Smith, p. 134-135 states:
[...] imagine a linear system receiving an input signal, $x[n]$. and generating an output signal, $y[n]$. Now suppose the input signal is changed…
user45664
- 133
- 5
3
votes
1 answer
Using the linear operator to check for time invariance of a differential equation?
I have a differential equation $$\frac{d^2y(t)}{dt^2}+y(t) = \frac{dx(t)}{dt} + x^2(t)$$ and I need to see if this system with input $x(t)$ and output $y(t)$ is time invariant and linear. I tried to break up the equation by substituting a linear…
JobHunter69
- 191
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- 9
3
votes
2 answers
Causal systems - Paradox? Confusion of definition?
For a system such that $y[n]=\cos(x[n])$, I'd like to determine whether the system is causal or not.
Definition 1: A system is causal if and only if the output $y[n]$ is a function of a linear combination of inputs $x[n-k]$ such that $k \ge0 $.
We…
photon
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2
votes
1 answer
Is the system $y[n]=x[n]+2=T\{x[n]\}$ an LTI-System?
Is the system $y[n]=x[n]+2=T\{x[n]\}$ an LTI-System?
Linearity:
$ \alpha \cdot T\{x_1[n] \} + \beta \cdot T\{x_2[n] \} = T\{\alpha \cdot x_1[n]\ + \beta \cdot x_2[n] \} \\\alpha \cdot (x_1[n]+2) + \beta \cdot(x_2[n]+2)= \alpha \cdot (x_1[n]+2) +…
Manuel Natali
- 23
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2
votes
1 answer
Verifying Linear Time Invariance
I have a system of the form:
$$T(x(n))=x(n)+3x(n-2)-5x(n-3)x(2n)$$
I claim that
$$T(x(n-k))=x(n-k)+3x(n-k-2)-5x(n-k-3)x(2n-2k),$$
$$y(x(n-k)= x(n-k)+3x(n-k-2)-5x(n-k-3)x(2n-2k)$$
and
$$T(ax_1(n)+bx_2(n))=ax_1(n)+bx_2(n)+3\left(…
Paul
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