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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1answer
125 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
278 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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2answers
74 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
91 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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2answers
50 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
48 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
52 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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3answers
827 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
301 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
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 $x(t)$ ...
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1answer
81 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
59 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
65 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
53 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
262 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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0answers
25 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
496 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
66 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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0answers
111 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
67 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
177 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
307 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
44 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
62 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
199 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
327 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
93 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
39 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
80 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
108 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
580 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
195 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
96 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
42 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
74 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
100 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
216 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
72 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
836 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
137 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
451 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
2k 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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1answer
52 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 ...
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3answers
102 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 $y[n]$...
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1answer
92 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: $(x_1\times x_2)[n]=S_1[n]\rightarrow Y_1[n]$ $(x_3\times x_4)[n]=S_2[n]\rightarrow Y_2[n]$ $...
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3answers
589 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 ...
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1answer
869 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 ...
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2answers
97 views

What Is the Derivative of the Function $ s \left( t \right) = \left( 1 - {e}^{\frac{-t}{RC}} \right) u \left( t \right) $?

What is the derivative of the following function $$ s \left( t \right) = \left( 1 - {e}^{\frac{-t}{RC}} \right) u \left( t \right) $$ with respect to $t$, where $u(t)$ is a unit step ...
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1answer
160 views

Eigenfunction property for LTI sinusoidal and the sinusoidal steady-state response

All LTI systems possess the eigenfunction property for complex exponential inputs. That is (restricting our attention to periodic complex exponentials), if $e^{j\omega_k t}$ is an input to the LTI ...
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2answers
202 views

Time-invariant and Time-varying Systems

Determine whether the following system is time-invariant or not: $y(t)=x(t)\sin 10\pi t$ Solution: Given: $y(t)=x(t)\sin 10\pi t$ $y(t)=T[x(t)]=x(t)\sin 10\pi t$ The output due to input delayed ...