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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32 views

If the frequency response of a system is time-independent, is this system LTI?

Intuitively, I feel like a time-variant system would necessarily have a time-dependent frequency response, and vice-versa. So, is the time-independence of the frequency response necessary and ...
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50 views

Cross-correlation of maximum length sequence with noise

Background I am using a maximum length sequence to find the impulse response of a linear system. I want to calculate the expected SNR. I know that my signal amplitude is scaled by the length of the ...
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Response of an ideal integrator to a cosine wave

It sounds like a very elementary question on system theory but I got quite confused about it, so hopefully you guys can enlighten me. I'm considering an ideal analog integrator, i.e., a system with ...
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12 views

Frequency response of each component of a system given its global response

Given the following block diagram, find the frequency responses $H1(f)$ and $H2(f)$. The frequency response of the whole system has to be $H(f)=(\alpha_0+\alpha_1e^{-j2\pi T_1f}+\alpha_2e^{-j2\pi T_2f}...
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1answer
87 views

Frequency response of a system given its block diagram

Given the following block diagram, I am asked to find the frequency response $H(f)$ of the system. This is what I have done: The output of the first block is $j2\pi fX(f)$, and after going through $...
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1answer
35 views

Impulse response of a system in z domain

The question 3-23 in the "Discrete-Time Signal Processing - Second Edition" is: and the solution is: I cannot understand the solution. In the second row of the answer when I multiply (-4) with ...
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Are there existing third system (linearity, non linear, !!!)?

I like to know your idea about is there any third system after linear, none linear classification. Could we consider more universe dimension than 4 dimensions (time, XYZ) like 11 dimensions of  ...
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1answer
92 views

LTI, causal, discrete time system output

Consider a discrete time LTI causal system $S: y = S(u)$, with its impulse response $h:{Z} \rightarrow R:h(n)=3^{n+1}{H(n)}$ with $H$ the Heaviside function. We know the values of input: $$u(0) = 2$$...
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107 views

is y[n] = x[n] + n time invariant?

My steps were as follows: $\ x_2[n] = x[n-k] $ $\ y[n-k] = x[n-k] + (n-k) $ and $\ y_2[n] = x_2[n] + n = x[n-k]+(n-k)$ Does this mean that it is indeed time invariant?
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22 views

Does constant Q transform have linearity property in the transformed domain?

With FFT, I sometimes take advantage of the fact that I can pre-calculate a signals Fourier Transform, and then you can add the noise in the spectral domain: $$ \mathscr{F} (x[n] + h[n]) = \mathscr{F}...
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56 views

LTI system output given input and frequency response

The question I'm trying to understand is as follows: A linear time-invariant continuous-time system has the frequency response function $$H(\omega)=\frac{1}{j\omega+1} $$ Compute the output response $...
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31 views

System memory, causality, stability

im new into systems and im supposed to solve if the system has memory, us causal, linear, stationery, BIBO stable...The problem is i have never had experience with this type of system where the actual ...
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43 views

Parallel LTI system

Imagine I know that a system comprises two LTI subsystems $H_a$ and $H_b$ connected in parallel. Also, suppose that I can measure the impulse response of the whole system, $H_a \| H_b$, and the ...
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52 views

Phase response of an LTI-system

I've got an LTI-system such as follows: $$y'(t)+{2} y(t)={5} x(t-{3}), t>0, \\ \\ y(0)=0 \ \text{ ja } x(t)=0, \ t<0.$$ From this i've already figured out the transfer function: $$H(f)=\frac{5\...
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35 views

Is $y(t) = \cos(t) + x(t)$ a time-invariant system?

Is $y(t) = \cos(t) + x(t)$ a time-invariant system? $y(t-k) = \cos(t-k) + x(t-k)$ But it isn't equal to $\cos(t) + x(t-k)$ So, would it be time-invariant?
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Are all LTI systems invertible? If not, what is a good counterexample?

I have been trying to figure this out for a while now. Everywhere I have looked I could easily find examples of invertible LTI systems, but I could not find any counterexamples. Can anybody shed some ...
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237 views

I somehow “proved” that given any LTI system, its transfer function has to be constant. What am I missing?

A transfer function is defined as the Laplace transform of the ratio of output to input. Also, every LTI system has an eigenfunction. Given such eigenfunction as an input, the ratio of the output to ...
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1answer
33 views

Steady-State Output from Transfer Function

The progress I have made is as follows: $\sin(t)$ is our signal therefore $\omega = 1 = 2\pi f$ and $f$ = $\frac{1}{2 \pi}$ Also, $f_s$ = 10Hz therefore T = $\frac{1}{f_s}$ = 0.1s $H(z) = \frac{z}{...
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27 views

Show That a 2D Linear Transform $ T \left( \cdot \right) $ Is Homogeneous

By my understanding, a transform T is homogeneous if T[0] = 0. Then to prove that a linear transformation is homogeneous we say that: T[ax(n1, n2) + bx(n1, n2)] = aT[x(n1, n2)] + bT[x(n1, n2)] What ...
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417 views

The spectrum of the sum of two periodic signals

Considering the continue, periodic signals $s_1(t)$ and $s_2(t)$, with the period $P_1$ and $P_2$ respectively. Consider now a new signal $s$ the sum of the two pervious signals. I come to know ...
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82 views

impulse response VS zero-input response

I am new in the field of systems and signals, and I have a rather basic for the majority of the group, question: Can we find the impulse response function of homogeneous ODE, instead of its zero-...
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755 views

Can every type of linear filter be modelled by a convolution?

I have an input time series going through a filter that creates another time series as output. If I assume in first approximation that my filter is linear, does it necessarily mean that I can model ...
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141 views

Convolution of Signal with a Gaussian Filter / Kernel

Following up on Analytical Solution for the Convolution of Signal with a Box Filter, I am now trying to convolve a Gaussian filter with the sine signal by hand. My method is to use the definition of ...
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1answer
112 views

Initial Rest Condition for LCCDE causal LTI systems

I am self studying Alan Opennheim's course Signals and Systems. I am a math major and have no background in EE. I understand that for a LCCDE system to be linear its auxiliary conditions must be 0. ...
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140 views

Why can adaptive IIR filters result in unstable solutions?

For adaptive filtering, both finite and infinite impulse response (FIR/IIR) filters can be utilized. As an advantage of FIR filters in this context, guaranteed stability is often mentioned, while IIR ...
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1answer
113 views

transient and steady-state response of first order system

Considering this general 1st order transfer function $$ H(z) = \frac{b_0 + b_1z^{-1}}{1-az^{-1}} $$ How to find (analytically) the transient and steady-state responses? With steady-state response I ...
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2answers
61 views

Are Chebyshev Type 1 filters very unstable at very high orders?

I am currently experimenting with a digital implementation of a Chebyshev Type 1 filter of mine, for audio purposes. I have written the code myself so I was wondering if the behavior I've noticed is ...
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33 views

Determining if a 2D system is LSI

After a long time away from school, I'm a bit rusty and struggling with this question: Determine if the following discrete system is LSI: $y(m,n) = mn*x(m+n) + mn*x(m-n)$ So here's what I've done ...
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1answer
29 views

Invariance of $T[x[n]]=\sum_{k=n-1}^{n+2}x[k]$

I have to test whether the following system is invariant or not: $T[x[n]]=\sum_{k=n-1}^{n+2}x[k]$, so I want to verify that, if $y[n]=T[x[n]]$, then $y[n-N]=T[x[n-N]]$. $$T[x[n-N]]=\sum_{k=n-1}^{n+2}x[...
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82 views

Analytical Solution for the Convolution of Signal with a Box Filter

I have an exercise in which I am trying to filter an input signal $y(x) = \sin(x)$. Ideally, I would like to apply a box filter to this signal. Previously, I successfully convolved the input signal $...
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118 views

Why does knowing the impulse response allow you to determine the output for any LTI system?

Going over some notes on LTI systems and I’m quite confused about how knowing the impulse response of a system allows us to recover the systems’ response to any signal. In the textbook screenshot ...
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1answer
46 views

Representing a continuous LTI system as a discrete one

I am aware that there are different ways to represent a continuous time system in discrete domain (e.g. bilinear transform, impulse invariance transform). But my problem is as follows: Given an ...
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2answers
74 views

Cross-correlation of filtered random processes

I have a wide-sense-stationary (WSS) process $\{x(t)\}$ and two linear filters with impulse functions $h_1$ and $h_2$. Let $\delta(\omega)$ be the power spectrum of $\{x(t)\}$ and $$H_1:\omega\...
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1answer
34 views

Difference between the two forms of input-output relationships of an LTI system

$$y(t)=f(t)*h(t)\tag{1}$$ $$y(t)=H(s)e^{st}\tag{2}$$ $$H(s)=\int_{-\infty}^{\infty}h(t)e^{-st}dt\tag{3}$$ Let $f(t)$ in Eqn $(1)$ be $e^{st}$. In many worked out examples, I have found that the two ...
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1answer
40 views

how do I derive the system equation for a simple delay with feedback?

I am a software engineer, and just learning digital signal processing formally, though I've hacked around before a fair amount. I'm implementing a delay audio VST and I'm trying to wrap my head ...
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33 views

Controllability's dependence on B matrix

Assuming the LTI system: $\dot{x}=Ax+Bu\\$ , where $x\in R^n$ I understand that the $B$ matrix is usually given, and it is up to us to derive the optimal $u$ given an initial state of the system ...
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15 views

External/Internal Description of a System

When an LTI system is NOT completely controllable/observable, does the linear differential equation governing the system give the internal or the external description of a system?
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210 views

Unit Impulse : Bounded or Unbounded?

As the title suggests, I have a confusion here. In a Systems text I am going through, it mentions of the Unit Impulse as an unbounded signal. Yes, the unit impulse "height" is unbounded, but the "...
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4answers
110 views

Determining LTI system response to a scale change

We know that for an LTI system, if $y(t)$ is the output for $x(t)$ then the response for $x(t-2)$ will be $y(t-2)$ and so on. But my question is what will be the system response for the input $x(-2t)$...
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78 views

What is the difference between a controller and a compensator?

I recently revisited some texts on control theory. In some textbooks the term compensator is used, seemingly interchangeably with controller (see e.g. The Control Handbook by Levine, chapter 14 on ...
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4answers
202 views

LTI system and initial conditions

I suspect this is rather obvious once explained; I must be misunderstanding something. From reading various answers here (such as this one), I understand that an LTI system must have zero output for ...
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33 views

Does phase margin exists only for linear systems?

If possible, how to determine the phase margin for nonlinear systems?
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76 views

Why the following discrete type system,variable accumulator, is time invariant?

$y[n]=T{x[n]}=\displaystyle\sum_{k=n-n_{0}}^{n+n_{0}} x[k]$ it is some sort of moving summer which computes $n^{\text{th}}$ output sample by adding all samples lying within length $n_{0}$ around some ...
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65 views

Using Kalman filter vs Extended Kalman filter for differential drive robot with IMU

I have an IMU that provides me with a heading that is pretty accurate and accurate encoders on the wheels of my differential drive robot which provides me with pretty accurate velocity but has ...
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23 views

Inversion problem for big dataset

Suppose my goal is to solve for $x$ in $Ax = b$ using CG method and $A$ is Toeplitz. The problem is that $b$ is extremely large in size and I can't just read in the whole vector $b$ and then solve for ...
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61 views

System identification with limited bandwidth

Given a FIR filter $h[n]$. Its action can described as: $$ \mathbf{y} = \mathbf{H} \mathbf{x} \\ \mathbf{y} = \mathbf{X} \mathbf{h} $$ where $\mathbf{H}$ and $\mathbf{X}$ is a Toeplitz matrix. If $h$...
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2answers
116 views

What is the difference between y[n] and y(k) (using square brackets)?

Can I modify y[n] = αy[n-1] + x[n] the same way I would do it with y(k) --> Y(z)? I need H(z), but I don't know what is the difference between using (k), and (n). I often encountered [k] notation, ...
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1answer
54 views

A question about Fourier transformation

Hello, this is my first time actually asking in stackexchange. I am a computer engineering student and currently i am doing a linear system course (i don't really know how this is equivalent in ...
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477 views

How to Prove a System Is Invertible?

what i know is that for a system to be invertibel it should be one-one , but I am confused that if i am given a transfer function of a LTI system how can I prove or verify if it is invertible. ...
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1answer
36 views

Reconstructed output mismatch for LTI system

I have a system with measured input (u) and output (y). I assume that this is an linear time-invariant (LTI) system and I want ...