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 ...
0
votes
1answer
21 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 ...
4
votes
4answers
662 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:
$$
\begin{align}
T(\alpha_1x_1[n]+\alpha_2x_2[n]) & = a(\alpha_1x_1[n]+\alpha_2x_2[n]) + b(\alpha_1x_1[n-...
1
vote
0answers
40 views
Linear Predictive Coding example in MATLAB
I have some data that is highly correlated and I wanted to see if I could try and encode it using linear predictive coding (LPC). Here is how I've been understanding the process:
Encoding
Generate ...
1
vote
1answer
21 views
Linearity with Difference Period in Fourier Transform
I know that a system is linear if it satisfies
$$\mathscr{F}\{ a\,x(t)+b\,y(t) \} = a\,X(\omega)+b\,Y(\omega)$$
for Fourier transform, $X(\omega)\triangleq\mathscr{F}\{x(t)\}$
But what if $x(t)$ ...
0
votes
1answer
61 views
Figuring magnitude and phase response
I've got a linear time-invariant system $$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
30 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
43 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
42 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
98 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....
0
votes
0answers
60 views
Recursive signal output using linear algebra and initial conditions
I am messing around with signal processing, and I figured out the equation of a delay feedback loop, but I want to write it out in linear algebra. The problem is that I'm not quite sure how to format ...
1
vote
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 ...
0
votes
0answers
12 views
Modeling an affine system with a linear model
I have a system that's reasonably linear and I want to model as a linear system. The systems can be represented as FIR filter with following decimation. Suppose the input to the system is $x[n]$ and ...
0
votes
0answers
27 views
Likelihood / Error of a Zero Input Sequence
How can I measure how well a known system describes an unknown sequence?
In my work I need to evaluate how likely a particular observation sequence is given any parameterization of an LTI system. The ...
6
votes
7answers
387 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
29 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
55 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
50 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
33 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
votes
1answer
46 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 ...
0
votes
1answer
40 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
votes
0answers
16 views
Fitting filter coefficients within a narrow band of frequencies, and using them as multi-rate
Intro:
I have a vibrating system that I am trying to model. I've input a sine sweep up to 1200Hz and measured the device's input and output into MATLAB. I am interested in finding the inverse filter ...
0
votes
1answer
31 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 ...
0
votes
1answer
76 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
76 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 ...
0
votes
1answer
75 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:
\...
0
votes
0answers
43 views
Depending on how a system function is written, will it always have the same zeros and poles?
There are many ways to write a system function. In terms of direct forms, cascade, parallel, transposed. Also the system function depends on if you write the poles and zeros in terms of $1+az^{-1}$ or ...
0
votes
1answer
29 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(...
0
votes
0answers
17 views
Correlation between elements of a video
Should there be an expected correlation between the audio signal and the frame pixel data, considering the frames of a video?
If so, what is the reason for such correlation? Pondering over the topic ...
-2
votes
1answer
38 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
1
vote
2answers
75 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?
0
votes
0answers
17 views
Wiener Filter: spacing of the data
I read in an article that for the discrete version of Wiener filter as proposed by Levinson, the data can be arbitrarily spaced. What is implied by this? I believe that the spacing does matter to ...
0
votes
2answers
571 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
83 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 ...
1
vote
1answer
33 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.
0
votes
1answer
39 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.
0
votes
1answer
53 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 ...
0
votes
3answers
95 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
65 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:
$$ x \left [ n \right ] = \begin{cases}
1 &...
0
votes
2answers
60 views
Discrete filter $y[n] = \frac{1}{3} x[n] + \frac{1}{3} x[n-1] + \frac{1}{3} x[n-2]$
Consider the filter which equation can be represented by $y[n] = \frac{1}{3}x[n] + \frac{1}{3}x[n-1] + \frac{1}{3}x[n-2]$, in $x[n]$ and $y[n]$ are sequence of input and output of the system ...
1
vote
1answer
122 views
Determine if $ y[n] = ny[n-1] + x[n]$ is linear time invariant and BIBO stable
Check if the following system is linear time invariant and BIBO stable..
$$
y[n] = ny[n-1] + x[n]
$$
for $n\ge 0$. We are also given that the system is at rest (i.e. $y[−1] = 0$).
I know that to ...
2
votes
2answers
93 views
Frequency response of marginally stable LTI systems
The frequency response of a system is defined as:
$$\int_0^\infty{h(t)e^{-j\omega t}dt}$$
where $h(t)$ is the impulse response. But in marginally stable systems, $h(t)$ does not decay so the integral ...
3
votes
2answers
237 views
Are discrete systems defined by LCCDE always LTI?
Suppose a discrete-time system is defined by linear constant-coefficient difference equation
$$\sum_{k=0}^{N} a_k y[n-k] = \sum_{k=0}^{M} b_k x[n-k]$$
where at least two different coefficients $a_i,...
2
votes
3answers
331 views
Proof of linearity
I have this system:
$$y[n] − 4y[n − 1] + 4y[n − 2] = 20x[n] + 10x[n − 1]$$
I have no idea how to prove if the system is linear because it depends on future outputs.
0
votes
0answers
17 views
Relationship between signal amplitude and input output correlation
I have a linear system with input $ x(t) $ and output $ y(t)$ given by $$ y(t) = \int_0^\infty K(t')x(t-t')dt', $$ where $K(t)$ is a known kernel, with some parameters.
The functional form of $K(t)$ ...
0
votes
1answer
45 views
Explicit a succession using z inverse transform
Is it possible to explicit $y(n)$ of this mathematical succession in recursive form using z inverse transform?:
$ y(0) = 1 \\ y(n+1) = 2y(n) + 3 $
I can't write ...
0
votes
3answers
96 views
Least Squares with blocks/updates
I have a continuous-time system that I want to fit via least squares. I just send $N$ digital samples $x[n]$ through the system and receive (via analog signal chain, ADC etc) $N$ digital samples $y[n]$...
1
vote
1answer
85 views
Is this two input discrete system linear? [closed]
Given two inputs $\: x_1[n]\: x_2[n]\:$Is the system $\:y[n]=x_1[n]\times x_2[n]\:$ linear ?
My Approach:
$(x_1\times x_2)[n]=S_1[n]\rightarrow Y_1[n]$
$(x_3\times x_4)[n]=S_2[n]\rightarrow Y_2[n]$
$...
0
votes
3answers
197 views
Distinguishing FIR and IIR from difference equation
Find the transfer function of the difference equation $$y_n = x_n + 1.2y_{n-1}$$
I fail to understand how one can distinguish between an FIR filter and an IIR filter by looking at the equation given ...
0
votes
0answers
45 views
Relationship of the cross correlation and the amplitude of the signal
I would like to characterise the maximum amplitude $|x(t)|$ (or similar quantity $|x(t)|$ or $|x(t)|^2$) of the following linear system
$$ x(t) = \int_0^\infty K(t')S(t-t')dt',$$
where $ S(t) $ is ...
-1
votes
1answer
367 views
How to conclude LTI, causality and BIBO stability of a system represented by a differential equation?
I have started to learn about systems represented by differential equations in Oppenheim's Signals & Systems, and I got really confused about it. I am trying to understand how I can show that a ...
