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.

Filter by
Sorted by
Tagged with
75
votes
4answers
200k views

What is meant by a system's “impulse response” and “frequency response?”

Can anyone state the difference between frequency response and impulse response in simple English?
6
votes
1answer
3k views

Do Causal Discrete-time systems have proper transfer functions?

In the case of continuous-time systems, if the system is causal, its Laplace transfer function is strictly proper (the degree of the numerator is less than the degree of the denominator). Is this ...
9
votes
3answers
8k views

What is the difference between natural response and zero input response?

I am new to DSP and was going through different responses of a system subjected to an input. My understanding of zero input response is: it is the response/output of the system when the input signal ...
5
votes
1answer
4k views

Initial conditions for the LTI systems described as a difference equations

Why do we need the initial conditions to be zero for the LTI systems described as a difference equations? First question is why do we need it for linearity? I can't think of any example of the non ...
12
votes
4answers
3k views

Are complex exponentials the only eigenfunctions of LTI systems?

Is there an example of an eigenfunction of a linear time invariant (LTI) system that is not a complex exponential? Justin Romberg's Eigenfunctions of LTI Systems says such eigenfuctions do exist, but ...
2
votes
3answers
491 views

Is this system linear?

I have this system: $$ y[n] = -\frac{1}{2} x[n+2] - y[n+1] $$ I have no idea how to prove if the system is linear or not, because, it depends on future outputs... Thanks for the help
1
vote
2answers
304 views

Output of a discrete-time LTI system different form than input?

This question is related to this one. I'm going through old exams for a 2nd year systems and transforms course, and came across this question. I'm posting this question just in case my other question ...
10
votes
3answers
1k views

Is there a way to obtain the impulse response of a discrete system by just knowing it's response to the discrete unit step function?

In continuous time it was possible; $$ u(t){\longrightarrow} \boxed{\quad\textrm{system}\quad} {\longrightarrow} y(t)\implies \delta(t)=\frac{du(t)}{dt}{\longrightarrow}\boxed{\quad\textrm{system}\...
8
votes
3answers
3k views

Is Ideal LPF BIBO unstable?

In one of other discussions : How to find frequency response, stability, and causality of a linear system? I found a comment which was quite strong and definitely caught my attention. An ideal ...
3
votes
1answer
2k views

Convolution of $\sin(\omega t)$ and $\cos(\omega t)$?

If $x(t)=\sin(\frac{\pi t}{4})$ and $y(t)=\cos(\frac{\pi t}{4})$ then i need to find the Convolution $$z(t)=x(t) \circledast y(t)$$ So convolution will be $$\begin{align} z(t) &=\int_{-\infty}^{...
3
votes
2answers
653 views

A system that perfoms Fourier Transform operation - is it an LTI system?

If a system takes input as the time domain signal and outputs the frequency domain signal, is such a system an LTI system? For if the input time domain signal can be represented as a linear ...
7
votes
4answers
449 views

Does instability make an otherwise LTI system nonlinear (or time-variant)?

I am spinning this question off from the question from johnny. Matt L. and I have had directly opposite conclusions to johnny's question. I want to decouple the question from issues of causality and ...
5
votes
2answers
6k views

Given the input and the output, how to determine the impulse response?

I would like to find the impulse response, $h[n]$, of an LTI system given the input $$x[n] = [1,-3,2]$$ and the output $$y[n] = [1,-1,-4,4]$$ I know that $y[t]=x[t]*h[t]$, but I am having hard ...
2
votes
3answers
2k views

Why unit impulse function is used to find impulse response of an LTI system?

Hello i am working in digital image restoration field, recently i have studied concept of convolution, i studied that to find the impulse response/point-spread function of an LTI system, an unit ...
-2
votes
1answer
170 views

LTI system output

I compute the output of a LTI system, can someone tell me if my answer is right..? and help me with my others questions? The impulse response is: $h(n) = \left(\frac{1}{2}\right)^nu(n)$ , entry is $x(...
2
votes
6answers
3k views

Why do we always characterize a LTI system by its impulse response?

Why do we always characterize a LTI system by its impulse response and not by another response, like the step response? What does the impulse response have that is so special?
1
vote
0answers
154 views

Basic “filtering” in digital communications

I am completely stuck on solving some problems for my digital communications class. Junior level undergrad class, so fairly basic. I have been searching around all day for related articles etc but can'...
0
votes
3answers
715 views

Help in understanding the formula of Signal-to-Noise-Ratio (SNR) - Part 1

Question 1: Consider an Autoregressive model : \begin{align} y[n] &= y[n-1] + x[n]\\ z[n] &= y[n] + w[n] \end{align} where $y$ is the output observation, $x$ is a random input and $w$ is ...
7
votes
1answer
22k views

How to determine if the system is invertible

Is there any systematic way to determine if the system is invertible? My general approach is first trying to find the inverse system by using mathematical method; that is, solving for the output in ...
4
votes
3answers
2k views

Impulse Response to LTI

I am new to DSP, and I am self-studying using mostly Proakis. I have a question. There are some examples in the text where you will be given the impulse response of an LTI system, and then asked to ...
2
votes
1answer
685 views

In the context of transfer functions, what is the relationship between the terms “proper”, “causal”, and “realizable”?

I am thinking about these terms in the context of linear control. A transfer function is proper if the degree of the numerator is not greater than the degree of the denominator. I've read often that ...
0
votes
1answer
15k views

System Properties; Linear, Causal, Time-Invariant, Stable?

I know the answers of the below question but I dont know why, except linear, could you explain the rest? Small tick: correct, small tick with dash incorrect answer.
6
votes
1answer
3k views

Relationship between Discrete Deconvolution and Toeplitz Matrices

I have 2 vectors, $a$ & $c$, both of length M. I know they are related by $a*b=c$. My goal is to recover $b$. Obviously $b=$deconv$(c,a)$. I am only interested in the first M elements of the ...
6
votes
2answers
19k views

How to obtain impulse response from the differential equation of a system?

This year I'm having trouble with my Signals and Systems class. My major subject is Software Engineering and Electric and Electrical Engineering is my Minor. This question was my previous exam ...
3
votes
3answers
147 views

Determining output of a LTI system

Consider following LTI system $$y[n] - 2y[n-1] = x[n]$$ where $x[n]$ is the input to the system and $y[n]$ is the output. Let $x[n] = \cos[n\frac{\pi}{3}] + 2\cos[n\frac{\pi}{2} + \frac{\pi}{4}]$, ...
3
votes
3answers
163 views

Estimating the Signal by Deconvolution with a Prior on the Filter Coefficients and the Signal Samples

Assume I have signal $y[n]$ which is a result of convolution between channel $h[n]$ and signal $x[n]$. which means: $$y[n] = h[n] \ast x[n]$$ where $\ast$ is the convolution operation The signal $...
2
votes
2answers
664 views

Linear Constant Coefficient Differential Equations: Zero-Input and Zero-State responses

The solution to a linear constant coefficient differential equation of the form $$\sum_{k = 0}^{N} a_k y^{(k)} (t) = \sum_{k = 0}^{M} b_k x^{(k)} (t)$$ can be written as $y(t) = y_{ZI} (t) + y_{ZS} (t)...
5
votes
2answers
209 views

Negative group delay and envelope advance

I am having a doubt reading about delays in signal processing. Let there be an input to a LTI system with frequency response $H(f)$, given signal $x(t) = a(t)\cos(2\pi f_ot)$, where $a(t)$ is a ...
4
votes
2answers
135 views

Running Integral of sine and cosine functions

In typical signal processing course we were taught that the integral of signal $x(t)$ is given by $$y(t) = \int_{-\infty}^{t}x(\tau) d\tau$$ How can we use this definition to evaluate the integrals of ...
4
votes
3answers
1k views

Does scaling property imply superposition?

For a system to be linear,it follow the principles of scaling and superposition.Does scaling imply superposition?If so why are two different conditions given for linearity?If not can u specify an ...
3
votes
3answers
5k views

What is the relationship between poles and system stability?

I see two notions that describe the relationship between poles and system stability. But they are not the same from my understanding The system is BIBO stable if and only if all the poles are in the ...
2
votes
3answers
2k views

Why cosine is not an eigen signal?

According to this website: If the output of a system has the same type as its input signal, then the input signal is referred to as the eigen function of the system. but in this question it is ...
1
vote
2answers
193 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 ...
1
vote
3answers
744 views

Output of discrete-time LTI system guaranteed to be same form as input?

I know that in the continuous-time context, if I supply a complex exponential input to a Linear Time Invariant system, the output will be of the same form as the input - for example, if the input is $...
1
vote
1answer
176 views

Beginner level : Help with terminologies : smooting length, length of channel, channel equalization delay, blind system identification, equalization

Say the channel model is an univariate FIR filter with true coefficients $h=[1,a_1, a_2 ]$. I am learning algorithms for system identification and channel equalization. For this, I am implementing ...
1
vote
2answers
231 views

should this be viewed as an impulse response or step response

I'm trying to teach myself the relation between simple discrete ODE's and the impulse response-step response concept. Getting back to the question: I don't expect anyone to read the whole thing but I'...
1
vote
2answers
201 views

System Identification with a Limited Bandwidth Input Signal and Region of Interest

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 $h$...
-3
votes
1answer
549 views

Ideal BandPass Filter

Let suppose x(t)=$\sum\limits_{k=-∞}^∞ R(t-kT)$ $R(t) = \begin{cases}1 &[0,2T] \\ 0 & \text{otherwise} \end{cases}$ x(t) is the input to an ideal bandpass filter with $\text{BandWidth} = \...
6
votes
1answer
9k views

How to find frequency response, stability, and causality of a linear system?

I have the following transfer function: $$H(s)=\frac{s}{(s+1)(s+2)}$$ How can I find the gain and phase response of the above system? I know the first step has something to do with substituting $s = ...
6
votes
1answer
411 views

Causal system and Physical Systems

According to the Paley-Wiener criterion, a system is causal if satisfies: $$\int\limits_{-\infty }^{+\infty }{\frac{\ln (|H(f)|)}{1+{{f}^{2}}}}df<\infty$$ So I want to know This equation is ...
5
votes
5answers
5k views

What Is the Transfer Function of a Moving Average (FIR Filter)?

To make post-processing easier, I export scope measurements as CSV files, which are then post-processed (mostly in Microsoft Excel, which is not the best tool for the job, but it is all I have at my ...
5
votes
1answer
277 views

Difference between convolving before/after discretizing LTI systems

Suppose I have transfer functions for two continuous causal linear-time invariant (LTI) systems: $F_1(s)$ and $F_2(s)$. Let $D\left\{\cdot\right\}$ denote the function that maps a transfer function ...
3
votes
6answers
419 views

Design a LTI system which returns DC value of the input signal

Suppose that $h(t)$ is the impulse response of LTI system. The input signal $x(t)$ is periodic with period $T$. Determine $h(t)$ so that the output signal $y(t)$ only be the DC component of $x(t)$. Is ...
2
votes
1answer
867 views

Exponential decaying step response in LTI System

I'm attempting to better understand the relationship between step responses, impulse responses, and convolutions. Say that I have a system where if I apply a constant input, my output decays from a ...
2
votes
1answer
576 views

Verifying Linear Time Invariance

I have a system of the form: $$T(x(n))=x(n)+3x(n-2)-5x(n-3)x(2n)$$ I claim that $$T(x(n-k))=x(n-k)+3x(n-k-2)-5x(n-k-3)x(2n-2k),$$ $$y(x(n-k)= x(n-k)+3x(n-k-2)-5x(n-k-3)x(2n-2k)$$ and $$T(...
1
vote
1answer
179 views

$\mathcal Z$-transform ROC

Let's say I have a $\mathcal Z$-transform that represents some transfer function and its has some ROC. My question is how do I know if this system is causal? I know that if the ROC contains the ...
1
vote
1answer
610 views

Hilbert transform properties

Here Its says Hilbert transform is a non-causal,linear,and time-invariant system How can I prove it mathematically? wikipedia says the input output relation like this $$\boxed{y(t)=\frac{1}{\pi}\...
1
vote
2answers
518 views

Alternative to BIBO stability of a system

In DSP textbooks a system is stable in the BIBO (Bounded-Input, Bounded Output) sense if and only if every bounded input sequence produces a bounded output sequence. After stating this definition ...
1
vote
0answers
77 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 ...
1
vote
1answer
642 views

Why is the total signal response response of zero input + zero state, $y(t) = y_0(t) + h(t)\star x(t)$, not an LTI?

For the zero input + zero state response in continuous time linearly time-invariant systems, why is the $y(t)$ equation not "technically" considered an LTI? I read this in a journal and there was no ...