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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7 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
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0answers
31 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
125 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 ...
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0answers
20 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 ...
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2answers
39 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]=...
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0answers
33 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 ...
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0answers
22 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 ...
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1answer
41 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 $G(z)=\frac{1}{H(z)}$?
$$$$
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3answers
219 views
Why does decimation make a system time variant?
On Wikipedia I read this : "The Discrete Wavelet Transform, often used in modern signal processing, is time variant because it makes use of the decimation operation."
Why does decimation makes system ...
-1
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2answers
51 views
How does an LTI system produce multiple values from a unit impulse
I understand that in a discreet system, an impulse is 1 at the origin point and 0 everywhere else. I've seen many examples showing the impulse response of an LTI system to be many points, have many ...
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1answer
42 views
Explain the formula of convolution and convolution theorem
As stated in the title, I have two questions
Why convolution is defined as $(f*g)(x) = \int_{-\infty}^\infty f(t) g(x - t) dt$ instead of just $\int_{-\infty}^\infty f(t) g(x + t) dt$ ? Why we need ...
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1answer
28 views
expected value of two LTI output signals multiplied (cross correlation)
I have an input signal x (assumed to be iid Gaussian with $\mu=0$, $\sigma^2$) which is fed into two linear systems:
$y_1 = h_1 * x$
$y_2 = h_2 * x$
Now I would like to calculate $\mathbb{E}[y_1 y_2]...
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1answer
33 views
Derivation of Toeplitz Matrix
I'm having a difficult time understanding why the matrix for LTI systems is a Toeplitz matrix. I can see why $h_{n,m} = h_{n' + q,m' + q}$ given that $n' = n - q$ and $m' = m - q$, and $$\sum_{m'= -\...
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1answer
31 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 ...
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4answers
715 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:
$$
\begin{align}
T(\alpha_1x_1[n]+\alpha_2x_2[n]) & = a(\alpha_1x_1[n]+\alpha_2x_2[n]) + b(\alpha_1x_1[n-...
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0answers
99 views
Linear Predictive Coding example in MATLAB
I have some data that is highly correlated and I wanted to see if I could try and encode it using linear predictive coding (LPC). Here is how I've been understanding the process:
Encoding
Generate ...
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1answer
21 views
Linearity with Difference Period in Fourier Transform
I know that a system is linear if it satisfies
$$\mathscr{F}\{ a\,x(t)+b\,y(t) \} = a\,X(\omega)+b\,Y(\omega)$$
for Fourier transform, $X(\omega)\triangleq\mathscr{F}\{x(t)\}$
But what if $x(t)$ ...
0
votes
1answer
70 views
Figuring magnitude and phase response
I've got a linear time-invariant system $$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
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0answers
38 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
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1answer
46 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?
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1answer
46 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, ...
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votes
1answer
132 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
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0answers
18 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
423 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 ...
1
vote
1answer
37 views
MIT exercise 6.003 HW2 - Concept of system initially at rest
I am following the MIT open course you can find here. My question is about one of the exercises given as homework in the latter and more specifically I think I am missing something on the concept of "...
3
votes
0answers
60 views
How to implement the RLS for matrices
I need to implement the RLS algorithm but it's for matrices instead for vectors, I have made the below code, but still something wrong is not working well,
EDIT:
The code should be done as below,
...
1
vote
2answers
55 views
Check whether a given equation is linear
$ a = (x, y) \in \mathbb{Z}^2 $ is given as a pixel. My equation in which $g$ is image(matrix) is defined as,
$f(x, y) = 56g(x,y)+93g(x−1,y)+92g(x+1, y)−57g(x, y−1)+555g(x, y+1) $
How can we know ...
0
votes
1answer
60 views
System that has derivative of input is non causal
Consider a system $y(t) = \dot{x}(t)$ where $y$ is the output and $x$ is the input. Given an initial condition $x_0$ and two inputs $x_1$ and $x_2$ such that $$x_1(t)=x_2(t) , 0 \le t < t_0$$ the ...
2
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1answer
87 views
Block Diagram for a difference equation
I have this little doubt regarding how to draw a block diagram representation of a difference equation. Let us implement $ y(n) = ay(n-3) + by(n-2)+cy(n-1) + x(n)$ in block diagram where $a, b, c$ are ...
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1answer
41 views
Signals and Systems - LTI - Transforms - Impulse Response
I have $x(t)-> LTI -> y(t)$ where $LTI=h(t),H(jw)$.
and
$H(jw)=ab/((a+jw)(b+jw))$ where a and B are real numbers.
I am wanting to find the impulse response $h(t)$ as well as the input/...
0
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1answer
36 views
3-tap FIR filter: simple expression for $H(e^{j\omega})$ using trigonometric identities
We have a linear time-invariant system described by the input-output relation
$$y[n] = x[n] + 2x[n - 1] + x[n - 2]$$
Below is my approach to analyze this system.
The impulse response of this system ...
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1answer
102 views
How to find the impulse response of this system with complex roots?
I've been working trying to analyze a causal stable system. Hopefully a numeric example such as the one I am working on, and the problems I face during it could be useful to others.
I have this ...
4
votes
3answers
143 views
How to find the difference equation directly from Direct Form II signal flow graph
I am trying to solve for the difference equation of the following signal flow graph:
I am aware that Direct Form II can be converted to Direct Form I, which finding the difference equation directly ...
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1answer
80 views
Linearity of the given system
I am given the following system and I am checking the additive property:
$$y(t)=x(e^t)$$
where $y(t)$ is the output and $x(t)$ is the input given to the system.
Now this is what I did so far:
\...
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1answer
32 views
Why is the impulse response function of this system 0?
Suppose I have an system $ y(t) = t^{2}x(t)$.
The impulse response of this system would be: $h(t) = t^{2} \delta(t)$.
Since $\delta(t) = 0$ for $t \neq 0 , h(t) = 0$ for $t \neq 0$.
And at $t=0, h(...
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votes
1answer
50 views
classify the system if it's linear , non-linear , time variants or invariants [closed]
classify the system if it's linear , non-linear , time variants or invariants
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2answers
83 views
Conversion from stationarity to non-stationarity
Is there any way to convert a non-stationary signal to a stationary one, perform operations on it meant for a stationary signal and then convert it back to the non-stationary one?
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2answers
573 views
What does it mean for a function to have frequencies?
In a lecture, my professor mentioned that $\cos$ has two frequencies. I see that using the inverse Euler's formula we can express $\cos$ as a some linear combination of complex exponentials, each with ...
2
votes
3answers
125 views
Question on Wiener Filtering
I have read that a Wiener filter is a filter used to produce an estimate of a desired or target random process by linear time-invariant (LTI) filtering of an observed noisy process.
Now, my doubt ...
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1answer
57 views
Linear combination of DT unit impulse
Came across this example in class but I'm not sure how the expression $x(n)$ was derived.
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1answer
40 views
Transfer function of open loop system
Suppose i have the system equation
$Y(s) = G(s)X(s)+ 3T(s)$
Then what is the transfer function of the system?
I know that the transfer function is $Y(s)/X(s)$, but i can't get that expression.
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1answer
59 views
Damped Harmonic Oscillation as an LTI
The goal is to create an LTI filter which is exactly, or approximates, damping of harmonic modes.
The equation of course is:
$$\frac{d^2 x}{dt^2} + 2 \xi \omega \frac{dx}{dt}+\omega^2x=0$$
This can ...
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3answers
96 views
Discrete state space model - Why are we calculating $x[k+1]$ instead of $\dot{\textbf{x}}(t)$?
A continuous state space model is defined as follows.
$$
\dot{\textbf{x}}(t)=\textbf{A}\textbf{x}(t)+\textbf{B}\textbf{u}(t) \\
\textbf{y}(t)=\textbf{C}\textbf{x}(t)+\textbf{D}\textbf{u}(t)
$$
If we ...
-1
votes
1answer
95 views
Calculate the Output of Linear Time Invariant System Given it Impulse Response [closed]
A filter is defined as $ h \left[ n \right] = \delta \left[ n \right] - \delta \left[ n - 1 \right] $.
Given a signal $ h \left[ n \right] $ defined as:
$$ x \left [ n \right ] = \begin{cases}
1 &...
0
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2answers
71 views
Discrete filter $y[n] = \frac{1}{3} x[n] + \frac{1}{3} x[n-1] + \frac{1}{3} x[n-2]$
Consider the filter which equation can be represented by $y[n] = \frac{1}{3}x[n] + \frac{1}{3}x[n-1] + \frac{1}{3}x[n-2]$, in $x[n]$ and $y[n]$ are sequence of input and output of the system ...
1
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1answer
314 views
Determine if $ y[n] = ny[n-1] + x[n]$ is linear time invariant and BIBO stable
Check if the following system is linear time invariant and BIBO stable..
$$
y[n] = ny[n-1] + x[n]
$$
for $n\ge 0$. We are also given that the system is at rest (i.e. $y[−1] = 0$).
I know that to ...
2
votes
2answers
105 views
Frequency response of marginally stable LTI systems
The frequency response of a system is defined as:
$$\int_0^\infty{h(t)e^{-j\omega t}dt}$$
where $h(t)$ is the impulse response. But in marginally stable systems, $h(t)$ does not decay so the integral ...
3
votes
2answers
331 views
Are discrete systems defined by LCCDE always LTI?
Suppose a discrete-time system is defined by linear constant-coefficient difference equation
$$\sum_{k=0}^{N} a_k y[n-k] = \sum_{k=0}^{M} b_k x[n-k]$$
where at least two different coefficients $a_i,...
2
votes
3answers
952 views
Proof of linearity
I have this system:
$$y[n] − 4y[n − 1] + 4y[n − 2] = 20x[n] + 10x[n − 1]$$
I have no idea how to prove if the system is linear because it depends on future outputs.
0
votes
0answers
20 views
Relationship between signal amplitude and input output correlation
I have a linear system with input $ x(t) $ and output $ y(t)$ given by $$ y(t) = \int_0^\infty K(t')x(t-t')dt', $$ where $K(t)$ is a known kernel, with some parameters.
The functional form of $K(t)$ ...
