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.
2
votes
1answer
30 views
How to detect algebraic loop in a system
I came across a system (screen below) that looks like it should result in a algebraic loop, but when writing the equations using the Z-transform, it obviously does not (calculation below too). But I ...
0
votes
1answer
19 views
Why does the Y component in YCbCr represent luma/intensity?
As far as I know, the RGB to YCbCr conversion is just a linear transformation, yet the Y channel successfully captures the intensity value of each pixel. How?
In other words, where do the following ...
1
vote
1answer
33 views
confused in difference between calculations of convolution and multiplication?
How are two signals multiplied? And how is it different from convolving two signals??
I have read the above question on dsp stack exchange and i also tried it in my matlab , code is as follow:
...
0
votes
1answer
22 views
Is it possible to use k-means algorithm with just one vector
Suppose I have a vector $X = (x_1, x_2 , . . . ,x_n)$, $x_i$ is the maximum of $X$ and $x_k$ is the minimum. Is it possible to use k-means algorithm to cluster the values in vector $X$ into two ...
1
vote
1answer
62 views
Output of an LTI system given its transfer function and input
Given the transfer function $$T(s) = \frac{100}{1 + \frac{s}{10^{6}}}$$ and the input $$v_i(t) = 0.1 \sin(100t)$$ find the output, $v_o(t)$.
My approach was to use $v_o(t) = \mathcal{L^{-1}}\left\{T(...
1
vote
2answers
49 views
time invariance concept?
First of all ,please let me know?Is cosine time invariant or time variant?
If latter is the case then if a time varying input like cos is given to a time invariant system,how will it behave?
I have a ...
1
vote
1answer
59 views
Cost function for LTI system identification
I am currently reading and trying to understand a paper (Kulkarni and Colburn, 2004) that utilizes system identification methods to approximate head-related transfer functions.
The general approach ...
1
vote
0answers
18 views
Region of convergence of transfer function
I posted this question Mathematics SE and got no answer so I have posted it here.
I learned in my signal processing class that an LTI system can be defined using a linear constant coefficient ...
0
votes
0answers
8 views
Comparison Between Mahoney Filter and Kalman Filter in Euler Angles (Yaw, Pitch, Roll) Estimation
What are the advantages of Mahoney Filter over Kalman Filter for Euler Angle (Yaw, Pitch, Roll) estimation?
Could anyone compare them for Angular State Estimation?
Resources
Mahoney Filter ...
0
votes
1answer
33 views
Why is the ROC of Laplace transform independent of imaginary part of s?
An integral is defined as converging if it yields a finite value upon application of limits of integration. It is divergent otherwise.
Now sticking to the mathematical notation of Laplace transform, ...
0
votes
1answer
20 views
Equivalence of difference equation with system of difference equations
I have this system and tried to write down its difference equation as follows
$$ y[n] = b_0v[n] + b_1v[n-1]\\
v[n] = x[n] - a_1v[n-1] $$
I was asked to prove that the system can be ...
0
votes
1answer
36 views
Consider the integrator and check for time invariance
I'm trying to check the following system for for time-invariance.
$$ y(t) = \int_0^t x(\lambda) d\lambda$$
Please explain me why it is time-invariant.
1
vote
1answer
25 views
confusion in time invariance?
I am trying to learn shift invariance(time invariance) and i came across a function/system
$$y(t)=3x(t)+2\cos(\pi t/3)$$
I am confused.If it was alone 3x,it should be definitely time invariant(i ...
2
votes
2answers
43 views
Contradiction while using the convolution sum for a non-LTI system
In a recent quiz, we were given the following problem:
The cascaded LTI systems $\mathcal{T}_1$ and $\mathcal{T}_2$ respectively have impulse responses $h_1[n]=\delta[n+3]$ and $h_2[n]=\delta[5−n]$....
1
vote
2answers
59 views
Confusion in linearity?
I am studying linear systems from the book "Signals and Systems Laboratory with MATLAB". I have performed an example on MATLAB myself in this regard.
Code is as follow:
...
1
vote
0answers
14 views
Range of updated/new time axis after upsampling? [closed]
I am reading the book "Signals and systems laboratory with MATLAB" by Alex Palamides and I am reading system properties on pg 154 of the book but I am confused about an example shown in attached photo....
1
vote
2answers
73 views
Why can convolution only be applied to compute the output of a linear filter?
We apparently cannot compute the output of a bilateral filter (BF) using convolution (with the image) because the BF is a non-linear filter. In general, why can convolution only be applied to compute ...
0
votes
2answers
25 views
System Response Terminology
If I have a system specified by
$$P(D)y(t)=Q(D)x(t)$$
and I specify initial conditions $y(0^-)=a, \ y'(0^-)=b,\ x(0^-)=c$ does the term $x(0^-)=c$ correspond to the zero state response or zero ...
1
vote
1answer
28 views
confused in practical example of shift invariance?
I am reading book of w smith ,"The scientist and engineer guide to digital signal processing" 2 edition
I am trying to understand topic of shift invariance from theory given on pg 92
I have attached ...
0
votes
1answer
41 views
How does one describe the “frequency response” of a mixer?
Suppose one uses a vector network analyzer to measure the $S_{21}$ parameter of an RF mixer over a frequency range. Would it be appropriate to call this the “frequency response” of the mixer? I find ...
0
votes
1answer
49 views
What does this notation $g(x) = S\{\delta(t) \}(x)$ mean?
What does this notation mean $g(x) = S\{\delta(t) \}(x)$ mean, where $S$ is a "system", $\delta(t)$ is the impulse function and $g(x)$ the output function of the system? I am really not familiar with ...
2
votes
3answers
77 views
Causal unstable system turn into stable anticausal?
I would appreciate it very much if someone would be able to provide some clarity, help or comment on this problem.
I have been reading several papers on time series identification such as
https://www....
3
votes
1answer
65 views
Higher-order moment of output of LTI system
Assume a very simple LTI system. Assume $x$ is white Gaussian i.i.d. with variance $\sigma^2$.
The output variance is straightforward to obtain. For example, for a continuous-time system:
$$\mbox{...
3
votes
1answer
73 views
Can all systems be described by differential/difference equations?
All LCCODEs (linear, constant-coefficient ordinary differential equation) with initial rest conditions describe causal LTI systems (is this a correct statement?)
Is the converse true, i.e. can all ...
3
votes
2answers
85 views
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 ...
1
vote
1answer
47 views
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) + \...
2
votes
1answer
41 views
Causal LTI system having exponential input
I know that for an LTI system having complex exponential input,
i.e, $x(t)=\exp(j w_o t)$ & $h(t) \to $ LTI System ;
then, its output { $y(t) \} =M \exp(j w_o t + \phi)$ , where $M= |H(j w)|_{|w= ...
1
vote
0answers
40 views
Moving average and linearization of two piecewise linear systems
I have 2 oversampling ADC's running parallelly, each to process data in a specific range of the input as shown below:
Each ADC can process only half cycle range of a sine wave. Each ADC adds its own ...
3
votes
2answers
133 views
Can $\delta(t+\infty)$ be a legitimate signal?
Mathematically speaking, when I try to use some signal to disprove a system is invertible, can I use the signal like $\delta(t+\infty)$ ($\delta$ representing the Dirac distribution)? For example, the ...
1
vote
0answers
61 views
How to obtain the steady state response of the sampled periodic signal data?
Below the green plot Vin is 5Hz sine voltage signal from a function generator output sampled at 12kHz sampling rate:
The blue plot Vout is the processed/filtered data by using the following code in ...
0
votes
2answers
44 views
Cannot figure out the convolution operation in this problem
A linear system S has the relationship $y[n]=\sum_k {x[k]*g[n-2k]}$, k ranging from $-\infty$ to $\infty$, between its input $x[n]$ and output $y[n]$, $g[n]=u[n]-u[n-4]$.
Determine $y[n]$ when $x[n]=...
0
votes
0answers
35 views
Linear difference equation and method of Least Squares
I'm reading the book "Fault-Diagnosis Systems" by Isermann in the par. 9.2.1a.
The author explains how to estimate the parameter of a linear difference equation using Least Squares.
We start with a ...
0
votes
0answers
26 views
Co-prediction of signals
Is there any plausible method to estimate or predict one signal, on the basis of known value of another signal, provided the two signals bear a strong correlation?
For instance, the audio amplitude ...
0
votes
1answer
89 views
ROC of inverse system function
If the region of convergence (ROC) for system function $H(z)$ is $R_h$, what is the ROC of the inverse function $G(z)=\frac{1}{H(z)}$?
$$$$
3
votes
3answers
245 views
Why does decimation make a system time variant?
On Wikipedia I read this : "The Discrete Wavelet Transform, often used in modern signal processing, is time variant because it makes use of the decimation operation."
Why does decimation makes system ...
-1
votes
2answers
63 views
How does an LTI system produce multiple values from a unit impulse
I understand that in a discreet system, an impulse is 1 at the origin point and 0 everywhere else. I've seen many examples showing the impulse response of an LTI system to be many points, have many ...
0
votes
1answer
60 views
Explain the formula of convolution and convolution theorem
As stated in the title, I have two questions
Why convolution is defined as $(f*g)(x) = \int_{-\infty}^\infty f(t) g(x - t) dt$ instead of just $\int_{-\infty}^\infty f(t) g(x + t) dt$ ? Why we need ...
0
votes
2answers
43 views
expected value of two LTI output signals multiplied (cross correlation)
I have an input signal x (assumed to be iid Gaussian with $\mu=0$, $\sigma^2$) which is fed into two linear systems:
$y_1 = h_1 * x$
$y_2 = h_2 * x$
Now I would like to calculate $\mathbb{E}[y_1 y_2]...
0
votes
1answer
43 views
Derivation of Toeplitz Matrix
I'm having a difficult time understanding why the matrix for LTI systems is a Toeplitz matrix. I can see why $h_{n,m} = h_{n' + q,m' + q}$ given that $n' = n - q$ and $m' = m - q$, and $$\sum_{m'= -\...
0
votes
1answer
40 views
Acoustic Echo Cancellation
I have the following diagram for acoustic echo cancellation
I'm having a hard time figuring out what's present on the top and bottom lines in terms of reflected echos, original signal etc. Here's ...
5
votes
3answers
783 views
Check if the system is linear
The system:
$$ T(x[n]) = ax[n] + bx[n-3] $$
For me it seems that the system is linear:
$$
\begin{align}
T(\alpha_1x_1[n]+\alpha_2x_2[n]) & = a(\alpha_1x_1[n]+\alpha_2x_2[n]) + b(\alpha_1x_1[n-...
1
vote
0answers
223 views
Linear Predictive Coding example in MATLAB
I have some data that is highly correlated and I wanted to see if I could try and encode it using linear predictive coding (LPC). Here is how I've been understanding the process:
Encoding
Generate ...
1
vote
1answer
23 views
Linearity with Difference Period in Fourier Transform
I know that a system is linear if it satisfies
$$\mathscr{F}\{ a\,x(t)+b\,y(t) \} = a\,X(\omega)+b\,Y(\omega)$$
for Fourier transform, $X(\omega)\triangleq\mathscr{F}\{x(t)\}$
But what if $x(t)$ ...
0
votes
1answer
77 views
Figuring magnitude and phase response
I've got a linear time-invariant system $$y[n]=\frac{8}{9}y[n-1]+x[n]$$
which I transformed into a transfer function $$Y(z)=\frac{8}{9}Y(z)*z^{-1}+X(z) =>\frac{Y(z)}{X(z)}=\frac{1}{1-\frac{8}{9}*z^{...
1
vote
0answers
52 views
Understanding linear predictive coding in MATLAB
I want to test my understanding of linear prediction by running it on some test data in MATLAB. The way I understand it is if I have some data that is correlated, I can encode the signal with linear ...
1
vote
1answer
57 views
How do you determine the properties of a differential equation?
If I was given a differential equation of the form $$y'(t) + a(t)y(t) = x(t)$$ how would I be able to decide its linearity, time-invariance, and causality?
0
votes
1answer
48 views
How to find the coefficients of the following differential equation
An arbitrary signal $v(t)$ pass through the following system,
$w'(t) + 5 w(t) = v'''(t) + 320v''(t) + 40 v' (t) + 40v(t)$
How to determine the coefficients of the following differential equation, ...
-1
votes
1answer
224 views
Add echo as described by difference equation to audio signal (in MATLAB) [closed]
A simple linear system is echo. It can be described by equation
$$
y[n] = x[n]+ k\,x[n−d],
$$ where $n$ represents sample index, $k$ represents an attenuation coefficient, and $d$ represents time lag....
1
vote
0answers
22 views
Finding the component vectors for a grid of points
I have a 2d 'grid' of xy points (2000 of them).
I want to find the average vector between points
I have drawn the vectors on the graph, how do I find these vectors mathematically? I want to find the ...
6
votes
7answers
471 views
Intuition behind commutativity of convolution in LTI systems
Why is convolution commutative, as it seems to treat two signals in a different way in an LTI system?
If you imagine $y[n] = x[n] \star h[n]$ with $x[n]$ being an input signal and $h[n]$ being the ...
