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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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47 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 ...
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15 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 ...
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4 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 ...
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29 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, ...
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16 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 ...
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34 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.
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24 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 ...
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42 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 $h_2[n]=\delta[5−n]$....
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36 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: ...
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12 views

Range of updated/new time axis after upsampling?

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....
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59 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 ...
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23 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 ...
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25 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 ...
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34 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 ...
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44 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 ...
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If an otherwise linear system is scaled/normalized by one of the inputs, does the system become nonlinear?

$\dot{\boldsymbol{x}}(t)=\boldsymbol{Ax}(t)+\boldsymbol{Bu}(t)$ $\boldsymbol{y}(t)=\boldsymbol{Cx}(t)+\boldsymbol{Du}(t)$ but say that $y(t)=\frac{Y(t)}{u_1(t)}$, where $u_1(t)$ is one of the system ...
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3answers
69 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....
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1answer
62 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{...
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1answer
69 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 ...
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2answers
84 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 ...
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1answer
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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) + \...
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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= ...
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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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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 ...
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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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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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25 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
69 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
235 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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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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55 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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31 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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38 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
38 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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756 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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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
22 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
75 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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45 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
55 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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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, ...
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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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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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459 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
48 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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76 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
60 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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114 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
169 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 ...