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
74 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
111 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
1k 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
684 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
26 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
97 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
95 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
85 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
60 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
381 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
36 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
593 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
105 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
129 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
104 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
295 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
572 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
47 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
96 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
379 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
625 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
107 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
42 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
110 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
136 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
585 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
263 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
193 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
44 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
99 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
150 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
509 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
74 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 ...
2
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1answer
2k 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
179 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 ...
4
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2answers
716 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
4k 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
58 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
117 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
95 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
972 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
1k 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
102 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 ...
2
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1answer
271 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
287 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 ...
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1answer
350 views

Signals and systems book with linear algebra approach

I am currently taking a course on discrete-time signals and systems. I am using B. P. Lathi's Signal Processing and Linear Systems, which I don't like at all, as it doesn't draw any paralells to ...
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3answers
4k views

Why $y[n] = x[-n]$ is not time-invariant?

I followed these steps, but the answer still says that this system is time-invariant let: $x_2[n] = x[n-k]$ $$\begin{align} y_2[n] &= x_2[-n] \\ &= x[-(n-k)] \\ &= x[k-n] \\ \end{align}...
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1answer
87 views

Is it possible to replace an integrator system with an equivalent differentiator?

I have a system whose input-output relation is as follows $$y(t)=x(t)+\int_{-\infty }^{t} x(\tau) \,\mathrm d \tau$$ Can I create an equivalent system by using differentiators rather than ...
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1answer
60 views

Finding the output of a system where the input is a sum of complex exponentials

So, I have to find $H\{ x(t)\})$ (which is an LTI system), where $$x(t) = \sum_{k=0}^{\infty} a_ke^{ \ jw_kt}$$ and where the impulse response of the system is given by: $$h(t) = \frac{\delta(t+\tau)-\...
5
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1answer
1k views

Does “improper” imply that a system cannot be stable and causal?

This answer and the comments in it made me wonder whether the following statement is true: If a transfer function is improper, then that system cannot be causal and stable at the same time. I had ...

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