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.
504
questions
2
votes
1answer
34 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 ...
1
vote
0answers
34 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
$$ \...
0
votes
1answer
44 views
Determine if system is linear time variant
The system equation is given as:
$$y(n)=(n-1)x(n-1)+(n+1)x(n+1)$$
I solved that the system is time variant:
\begin{align}
y(n-k)&=(n-k-1)x(n-k-1)+(n-k+1)x(n-k+1)\\
H[x(n-k)]&=(n-1)x(n-k-1)+(n+...
0
votes
3answers
64 views
How can we show that $|H(e^{j\omega})|=|H(e^{-j\omega})|$?
Let $H(z)$ be the rational system function of an LTI system.
How can we show that $|H(e^{j\omega})|=|H(e^{-j\omega})|$?
0
votes
0answers
30 views
Matched Filter Impulse Response
At the receiver end of a digital communication system, the matched filter is designed with impulse response matched to the effective signal that is given as : $$x_{(z,1)}^{*}(T_p-t)-x_{(z,0)}^{*}(T_p-...
2
votes
2answers
73 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 ...
0
votes
0answers
32 views
Calculate system function using pole and zero locations? [closed]
The pole zero of a digital system is given as zeros at $z=+j,z=-j,z=1,z=-1$ and 4 poles at $z=0$. As far my understanding this is a FIR filter. But I am not getting how to solve the system function ...
0
votes
1answer
30 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 $\frac{1}{2Ļt}$ where $t$ is the time to $\frac{1}{e}$ amplitude, it will provide an exponential decay in response to a step function (eg. ...
0
votes
1answer
23 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}...
5
votes
1answer
74 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
...
0
votes
2answers
19 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 ...
0
votes
1answer
41 views
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?
0
votes
1answer
34 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
$$y(t) = \big[x(t ā 1)x(t + 1)\big]^{\frac 12}$$
the fact that is in the form of $x\cdot x$ told me ...
1
vote
2answers
118 views
Determining Causality and Time-Invariance of a system
Consider the following system:
$$y(t-1)=\int_{-\infty}^\infty x(š)u(š-t) dš $$
where $u(t)$ is the unit step function, which is zero for $t<0$ and equals $1$ for $t>0$.
$(1)$ Is the system ...
1
vote
1answer
50 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 $y(0^-) = 2, \dot{y}(0^-) = 1$.
$...
1
vote
2answers
34 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 ...
1
vote
2answers
34 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 ...
2
votes
2answers
64 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 :
$$H(s)...
2
votes
1answer
45 views
Why can't this system be adequately represented using a z-domain transfer function?
According to this question and answer the following system cannot be adequately captured by a z-transform transfer function.
$$y[n] = y[n-1] + F_{\psi}(y[n-1)) + F_{\phi}(x[n-1])$$
where $F_{\alpha}(z)...
1
vote
1answer
24 views
RoC and Stability of a Rectangular Signal
If we have a system with an impulse defined as: $$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$$
We also ...
2
votes
0answers
29 views
How to find parameters of Kalman filter using matrix information?
I'm trying to understand concepts on Kalman Filters. Consider the overdetermined system $Ax=y$;
$$\begin{bmatrix}1 \\ 1 \\ 1 \\1 \end{bmatrix} x = \begin{bmatrix} 3 \\ 5 \\ 4 \\ 8 \end{bmatrix}$$
Let $...
2
votes
3answers
147 views
Are all exponential functions eigensignals of LTI systems?
I know that complex exponential functions are eigensignals to LTI systems. Do these include real exponential functions? E.g. $2^t, e^t, ...$
Thanks for the help!
2
votes
2answers
203 views
How can initial conditions be taken into account to calculate a system's terminal value using the final value theorem or some other technique
I wish to calculate the Final Value of the following system given initial conditions not at rest, and assuming the X will not change from its initial condition value.
The transfer function is
$$H(z) = ...
2
votes
1answer
48 views
Finding impulse response $h(t)$ by impulse matching; problem understanding Example 2.3 in “Linear systems and signals”, Lathi 3rd edition
Before the example it has been stated that in a system described by
$$
Q(D)y(t) = P(D)x(t), \quad (1.) \iff \\
(D^N + a_1D^{N-1} + ... + a_{N-1}D + a_N)y(t) = (b_{N-M}D^M + b_{N-M+1}D^{M-1} + ...+b_{N-...
1
vote
2answers
74 views
Why this system is linear?
Hi guys i'm studying signals and systems, and my professor told us that
$$y(t) = \int\limits_{ t+T }^{t-T/2} {x(a+T/2)}\mathrm{d} a$$
is a linear system.
But a primitive of $x$ isn't $ x^2$ ? How it'...
1
vote
1answer
114 views
Stability of a system in time-domain
Consider a system for which the input $x(t)$ and output $y(t)$ are related by the differential equation $$\frac{d^2y(t)}{dt^2} + \frac{3}{2}\frac{dy(t)}{dt} - y(t) = x(t) \tag{1}$$Determine $h(t)$ for ...
3
votes
1answer
49 views
Discrete time Final Value Theorem applied to feedback system
I wish to calculate the Final Value of systems in which a high pass filter of the output feeds back into the input.
A simple example would be:
where is a 1st order high pass filter with transfer ...
1
vote
3answers
74 views
Linear approximation of a system described by a logarithm - how?
The system described by a logarithm of any base, let's call it $y(t) = \ln(t)$, is non-linear.
Is it possible, by any means (transforms, operators etc.), and for small values of $t$ (for example, $t&...
1
vote
1answer
33 views
Disjoint ROCs of input and system function
I am trying to understand how the output behaves when the input and the system function do not have a common region of convergence (ROC) for an LSI system.
Consider an LSI system with $x[n]$, $h[n]$, $...
0
votes
1answer
25 views
Does fading memory mean impulse response with finite support?
Given a continuous-time impulse response $h(t)$, bandlimited to $B$.
The discrete-time $h[n]=h(n/(2B))$ is then a unique and perfect representation of $h(t)$ and a discrete-time system $h[n]$ is then ...
0
votes
0answers
53 views
Richardson-Lucy iterative deconvolution is producing erroneous deconvoluted images
I am working on producing deconvoluted image from degraded image where the process of degradation is known to be linear and space invariant. I have used Richardson-Lucy iterative deconvolution method ...
4
votes
2answers
54 views
Moving from deterministic signals to stochastic signals in s-domain (Power Spectral Density)
Assume we have the following system (coming from control systems theory, hence in s-domain)
$ Y(s) = H_A (s) \cdot A(s) - H_B (s) \cdot B(s) $
I now wish to consider $a(t)$ and $b(t)$ as white noise ...
2
votes
2answers
97 views
Output of a stable LTI system
Let $\mathcal{L}$ be a stable LTI system. Is it true that if input has finite energy then output also has finite energy? I'm not sure about that. We know that $$\int_{-\infty}^{+\infty}|h(t)|dt\lt\...
2
votes
1answer
69 views
Find the length of the impulse response for the given output and input
Homework Question:
Consider a signal $x[n]=\alpha e^{j \omega_{0} n}+\beta e^{j \omega_{1} n}+\gamma e^{j \omega_{2} n} .$ What is the length of impulse
response $h[n]$ of a system (non-trivial) such ...
0
votes
0answers
24 views
Discrete Time Systems with cosine()
I am trying to see if
y[n] = [cos(Ļn)]x[n]
is casual, stable, linear and shift-invariant.
I came up with the solution that it is not stable since it is not "...
0
votes
1answer
27 views
Testing for Linearity and Shift-Invariance
I started school a week and a half ago and we have gotten already a big chunk of material and I wanted to check if what I am currently doing is correct. Here below are the problems to work out that ...
-1
votes
1answer
60 views
Why isn't option C correct answer for the given question?
W
Please answer this question I am getting option c as my answer by putting input frequency of 3 radians in frequency response of system.
1
vote
1answer
56 views
Fourier transform of time division
I know that Fourier transform of $t^n f(t)= i^n \frac{d}{d\omega^n} F(\omega)$.
But does this work when $n<0$?
Is there any direct relation to compute the Fourier transform of $\frac{f(t)}{t}$?
1
vote
1answer
74 views
Confusion in initial condition of differential equation using Laplace transform transform
I'm confused in solving linear constant coefficients differential equations (LCCDEs) by Laplace transform if initial conditions are given at time
just before $t=0$
just after $t=0$
exactly at $t=0$
...
1
vote
2answers
50 views
Stability of passive LTI system
Is a passive LTI system will always be a stable(bibo and lyapunov) system?
Or in other words
Passivity of a LTI system implies stability?
If above statements are true then there must be some ...
4
votes
2answers
153 views
Derivation of the LMMSE (Linear Minimum Mean Squared Error) Estimate and the MMSE Under Gaussian Prior
I am learning estimation theory through Steven M. Kay - Fundamentals of Statistical Signal Processing, Volume 1: Estimation Theory. In the ...
1
vote
2answers
79 views
Output of marginally stable systems
In' b.p lathi's book of systems and signal it is written that
"marginally stable systems have one important application in the oscillator, which is a system that generates a signal on its own ...
2
votes
1answer
67 views
Determining a system's causality using its impulse response
I have the following input-output relation for a system:
$$y(t) = Odd Part Of [x(t)]$$
My question is: Is the system causal?
What my approach has been:
I expressed $y(t)$ alternatively as:
$$y(t) = \...
1
vote
1answer
39 views
Transients in marginally stable system
I know that if a system is asymptotically stable then after infinite time its output contains only characteristic of input signal and it is also known as steady state and there are no transient ...
0
votes
1answer
174 views
Time invariance of a summation sequence
The following sequence is given which is supposed to be time-variant:
$$y[n] = \sum_{k=n_0}^n x[k]$$
I'm having difficulties proving the time-variance or finding a counterexample for it being time-...
0
votes
2answers
91 views
How to compute transfer function $G(s) = \exp \left( - \sqrt{s} \right)$ in Matlab / Simulink?
How to compute transfer function
$$G(s) = \exp \left( - \sqrt{s} \right)$$
in Matlab / Simulink?
I am trying to calculate a PID controller for this function. This function describes heat transfer via ...
1
vote
0answers
121 views
Time-Invariant (non)autonomous Systems
Regarding potential distinctions between autonomous, non-autonomous, time-invariant and time-varying systems, I have found out opinions supporting that:
autonomous systems are time-invariant and non-...
0
votes
1answer
36 views
Compute output given input, transfer function and initial conditions
The problem statement is
Consider a causal LTI system whose transfer function $H(s)$ is given as $$H(s)=\frac{s+2}{(s+3)(s+4)}$$ Compute the output $y(t)$ for an input $x(t)=e^{-2t}u(t)$ when $y(0)=1$...
1
vote
1answer
45 views
Behaviour of integrator at steady state
I wanted to calculate response of integrator of sinusoidal input at steady state via these two methods as mention in image but these two methods give two different answers at steady state, so where ...
12
votes
5answers
1k views
Are there any real world applications for complex-valued signals or impulse responses?
I was just curious...
$$x[n] {\longrightarrow} \boxed{h[n]} {\longrightarrow} y[n]$$
I've never seen a real world filter where the coefficient of $h[n]$ were complex, or where $x[n]$ was a complex ...
