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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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29 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 ...
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2answers
120 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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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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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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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
38 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
218 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 ...
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50 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
41 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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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
714 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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95 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 $x(t)$ ...
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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^{...
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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 ...
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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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1answer
128 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....
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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 ...
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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 ...
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1answer
35 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 "...
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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, ...
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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 ...
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1answer
57 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 ...
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1answer
85 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/...
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1answer
35 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
99 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 ...
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3answers
132 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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1answer
49 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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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 ...
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3answers
121 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
58 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 ...
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1answer
89 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 &...
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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 ...
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1answer
298 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 ...
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2answers
103 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 ...
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2answers
316 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,...
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3answers
895 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.
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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 $K(t)$ ...
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1answer
47 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 ...