Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

3
votes
1answer
993 views

Determining if the given system is LTI

Problem Given the compound system below, with the input $x(t)=\operatorname{sinc(t)}$, the output of A is $y(t)=\operatorname{sinc(2t)}$ and the output of B is $z(t)=\operatorname{sinc(t)}$, ...
2
votes
2answers
448 views

Is a filter/control transfer function with positive phase “causal”?

In control we often use transfer functions with positive phase, i.e., a "lead filter" has transfer function $$G_c(s) = \frac{\alpha \tau s+1}{\tau s+1}$$ (with $\alpha>1$). Since the zero occurs ...
1
vote
1answer
137 views

Create distortion from basic linear (and non-linear if neccessary) DSP elements

I'm studying mechatronics and I'm intrested in DSP basics. My lecturer said that there are four basic linear DSP elements: Adder (and other mathematical operations) Amplification (shown on diagrams ...
1
vote
1answer
251 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 ...
4
votes
2answers
422 views

Control design: under what conditions can closed-loop poles be placed arbitrarily?

Say we have a single-input linear system $\dot{\mathbf{x}} = A\mathbf{x}+Bu$. With full-state feedback ($u=-G\mathbf{x}$), it is straightforward to arbitrarily place the $n$ closed-loop poles (i.e., ...
0
votes
1answer
99 views

Why is it necessary to average signals with filters?

As someone who is new in working with filters, why do signals need to be averaged? What are the advantages, by performing filters on signals? As I know, there are different types of filters for ...
3
votes
2answers
153 views

Why do we assume zero mean noise in sensor data?

I am reading a paper on measuring respiratory patterns from video data. In defining the model, the authors formulate the problem mathematically as: $x_i(t)=h_i(t) \ast g(t) + n_i(t) $ Where $n_i(t)...
1
vote
0answers
59 views

identifying overshoots in given state space system with step inputs

Suppose I have the following state space system: $$ \dot{x}(t) = Ax(t) + Bu, \quad y(t) = Lx(t), \quad x(0) = x_0 $$ where $A$, $B$ and $L$ are real matrices, $u$ is a constant real vector (so that ...
3
votes
1answer
490 views

Differential equations and LTI systems [duplicate]

I've seen in many textbooks on Signals and Systems that an LTI (Linear Time-Invatiant) system can be described as a constant-coefficient linear differential equation, such as $$\sum_{k=1}^N a_k \frac{...
1
vote
1answer
767 views

Stability of a logarithmic function and system

A system such that $$y[n] = \log_{10}(|x[n]|) $$ was given as a stable one. I am not able to understand it though. When $x[n] = 0$ (which is bounded), the output $y[n]$ tends to get unbounded. So ...
0
votes
1answer
232 views

Linearity and shift-invariance of 2-D system on lattice

I know how to check a 1-D system for these conditions, but am confused about translating this to a 2-D system over a lattice. The system $H$ is define as: $$w[\mathbf x] = H\left\{u[\mathbf x]\right\}...
0
votes
1answer
1k views

Using the linear operator to check for time invariance of a differential equation?

I have a differential equation $$\frac{d^2y(t)}{dt^2}+y(t) = \frac{dx(t)}{dt} + x^2(t)$$ and I need to see if this system with input $x(t)$ and output $y(t)$ is time invariant and linear. I tried to ...
1
vote
3answers
642 views

Deconvolution and Polynomial division

I found this comment inside MATLAB's deconv.m function Deconvolution and polynomial division are the same operations as a digital filter's impulse response $B(z)/A(z)$. What does this statement ...
1
vote
0answers
51 views

Variance and Co-variance of a Linear Forecast

Consider a linear forecasting problem where all shocks $\{\epsilon_i\}_1^n$ are independently distributed with $\epsilon_i\sim N(0,\sigma_i^2)$ for all $i$. Suppose you want to forecast $\theta = \...
0
votes
3answers
340 views

BIBO stable LTI system frequency response for this input signal?

For an LTI system with bounded input and bounded output, I have the input $$x(t) = 5 + \cos(12t+\pi/4)$$ and output $$y(t) = 6\sin(12t)$$ It is said that the magnitude of the frequency response $|H(...
0
votes
0answers
72 views

DSP Homework Help - Proofs and Causality

I'm completing my last semester of university and for my last EE minor course, I had only one option available, an Audio and Speech processing class. Safe to say, it's difficult for me and would like ...
3
votes
2answers
4k views

What is the difference between a lag filter and “PI” control?

A lag filter/compensator has the form $$G_c(s) = \frac{s+z}{s+p}$$ with $-z < -p < 0$. In practice, the effect of lag compensation in feedback control is to increase the DC gain of the open-...
-2
votes
1answer
85 views

Modeling an infinite delay system

If black holes swallow everything and assumes that the black holes do not get out the things from them, then can we consider a black hole as an infinite delay system? It takes the input and it is a ...
0
votes
1answer
311 views

Non invertibility of system $y[n]=x[n]-x[n-1]$ using transform method?

For the system to be invertible, we should have different outputs for different inputs. In terms of constant functions say, $$X_1[n]=3 \quad \forall n \in \mathbb{Z}$$ and $$X_2[n]=4 \quad \forall ...
1
vote
1answer
137 views

Fourier transform relationship

I am having trouble understanding the relationship between a frequency function and it's Inverse Fourier transform. The Frequency function is $$\frac{1+0.8(e^{-j 2\pi f}+e^{j 2\pi f})+0.64}{1+1.4\...
0
votes
0answers
125 views

Fourier transform and duality

Consider continuous even signals $x$ and $h$ with spectra $X$ and $H$. Now consider LTI system $L_1$ characterised by IMPULSE response $h$. When $x$ is the input in $L_1$, the output is $y_1$ with ...
0
votes
1answer
152 views

What is the steady state response of a system of two exponentials to sinusoidal excitation?

What is the two exponential steady state response? Here is an example solution for sinusoidal excitation of a system having a single exponential response to a impulse excitation: Example: ...
0
votes
1answer
370 views

advantage of using impulse response function for LTI systems? [duplicate]

i have a discrete-time LTI system $L$ that takes input signal $x[n]$ and gives the output signal $y[n]$. since $L$ is LTI, $y[n]$ can be derived as a sum of shifted and scaled impulse responses of $L$....
0
votes
1answer
205 views

Impulse response (general form for linear systems as two-variables function $h(t,\tau)$) applied to Time-invariant systems

If a system is linear, then the operator $S$ mapping input signals into output signals - i.e. $y(t)=S\{u(t)\}(t)$ - is the integral of the input weighted by the impulse response: $$ y(t) = \int_{-\...
0
votes
1answer
301 views

Combined impulse responses

I'm looking for a simple way to show that you can form a single impulse response that is the equivalent $M$ other impulse responses, ie: $$ h\left(t\right)=h_0\left(t\right)\star h_1\left(t\right)...\...
1
vote
1answer
69 views

Is blending of colour lights a linear system?

Can we consider recombination process of different light colors a linear system/phenomenon? Like, when you shed two different light sources for example, a blue light and a yellow light on a surface or ...
5
votes
1answer
291 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 ...
0
votes
2answers
92 views

convergence of Fourier transform of $e^{-t}\sin(2\pi ft)u(t)$

As you see Fourier transform function is being divergent for the first statement but it seems to converge. What is my fault? $$ \begin{align} \int\limits_{-\infty }^{+\infty }{{{e}^{-t}}\sin(2\pi ...
0
votes
1answer
318 views

How to shift input for testing shift invariance in a system?

As I know, when we want to test if a system is shift invariant, we define system as $g(x) = H(f(x))$ which $f(x)$ is the input function, $H$ is the system and $g(x)$ is the output function. Then we ...
0
votes
3answers
1k views

How to get the sum of unit steps and unit ramps from a discrete signal?

I have the discrete-time signal depicted below. This signal can be written as follows $$y[n] = \{0, 0, 2, 4, 6, 6 ,6,0,0,\ldots\}\,,$$ or as the sum of shifted impulses as: $$y[n] = 2 \delta [n-2]...
1
vote
3answers
1k views

Signals and systems : why do we study causal signals?

Till now I have read that causal signals are right sided and anti-causal, left sided. Why did we need to classify a signal with respect to its position? What is it's physical interpretation? ...
3
votes
1answer
424 views

Difference equations applied to DSP

I've found almost nothing about difference equations on the internet. Can you please recommend me something like books or pdf online that handle deep this topic? I'm searching also some exercises. ...
0
votes
1answer
127 views

Continuous-time frequency response question

Suppose the CTFT of continuous-time input $x_c(t)$ to an LTI system is $X_c(j\Omega)$ and that of its continuous-time output $y_c(t)$ is $Y_c(j\Omega)$. We have, $$X_c(j\Omega) = 0,\phantom{1}\text{...
0
votes
1answer
2k views

How to find ROC of system when input is two sided and output is one-sided

Given the $\mathcal Z$-transform of input $x[n]$ and output $y[n]$, how can I find the ROC of the system function $H(z) = Y(z)/X(z)$? I have $$X(z) = \frac{2z\left(z-\frac{10}{3}\right)}{\left(z-\frac ...
0
votes
1answer
150 views

Is Z-transform of $\sin(\omega_0n)$ same as that of $\sin(\omega_0n)u[n]$

The Z-transform tables only mention the transform of $\sin(\omega_0n)u[n]$, e.g. #21 at this link: https://en.wikipedia.org/wiki/Z-transform#Table_of_common_Z-transform_pairs But how can I find of Z-...
1
vote
1answer
275 views

Discrete time Signal Transformation

I have a discrete-time system defined by $y[ n]=x[5n+2]$. And an input $x_1[n]=\delta[n-5]$. Is the output $y_1[n]=\delta[25n-5+2]$?
1
vote
3answers
805 views

LTI systems inverse of each other

So my question looks like this: Suppose that we have two LTI systems with impulse responses $$h_1(t) = \frac 12\delta(t-1)\quad\text{and}\quad h_2(t)=2\delta(t+1).$$ Determine whether these ...
1
vote
1answer
154 views

BIBO stability of $1/x(t)$

I realize this is quite basic stuff but I am having trouble with the following question: Determine whether the following system with input $x(t)$ and output $y(t)$ is BIBO stable: $$y(t) = \frac{1}{...
2
votes
1answer
9k views

How to identify causality, stability and ROC from the pole-zero plot?

To preface, this is not a homework related question but purely for self-study purposes. If I am given the following Pole-Zero Plot: (Source: Berkeley Exam1) How would I go about trying to ...
-1
votes
1answer
200 views

Implementing FIR with MATLAB

There are several ways that MATLAB implements FIR. One of them is that convolution function conv directly implements the convolution equation and if the input is ...
2
votes
0answers
142 views

How does the ROC (Region of Convergence) related to a real world application?

In class, we are often given exercises to find the impulse response, output, and Z-transform of a system. In addition, we are often asked to define the Region of Convergence (ROC) depending on where ...
0
votes
1answer
293 views

Equation that describes a gravity drained tank system

I have a system like this: The upper tank has an input ($q_i$) and an output, that is the input for the lower tank ($q_{12}$). The lower tank has an output ($q_o$). Does anyone know how to get the ...
0
votes
1answer
59 views

Beginner LTI system Fourier transform question

This is a question I feel too stupid asking my professor about. I'm having a mental block remembering how this works even though I think I understood it at one point: I know the following properties:...
4
votes
1answer
14k 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 ...
0
votes
2answers
554 views

Textbook question regarding LTI System

I was posed the following homework problem: 2.10 The following input-output pairs have been observed during the operation of a time-invariant system: \begin{align} x_1(n)&=\{\underset{\...
6
votes
1answer
203 views

For a discrete LTI system, does “bounded memory” imply “rational transfer function?”

Every LTI system with a $\mathcal Z$-domain transfer function that is a rational function - aka a quotient of two polynomials in $z$ - can be implemented using a bounded amount of memory. Is the ...
5
votes
1answer
182 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 ...
2
votes
2answers
59 views

Time-shift confusion

Say input-output of a system is defined as: $$ x[n] \longrightarrow x[nM] $$ then what will be the output of $x[n-1]$? will it be: \begin{align} x[n-1] \longrightarrow &x\left[(n-1)M\right] = ...
1
vote
3answers
131 views

Can we see the Fourier transform as a filtering operation? [duplicate]

Is the Fourier Transform (FT) a filtering operation? Is it possible to graphically represent this transform? I know that a signal can be represented in the frequency domain, but I want to know if the ...
0
votes
0answers
65 views

How can I separate three noise sources?

This is the system I am trying to characterize I have three uncorrelated noise sources. I am trying to characterize all three noise sources, and more important find out what the original noise ...