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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
172 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
43 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
48 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 ...
0
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1answer
135 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 ...
4
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3answers
206 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
85 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
35 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
68 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
88 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
576 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 ...
2
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3answers
153 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
71 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
40 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
66 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
98 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
140 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
71 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 ...
1
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1answer
545 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 ...
2
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2answers
121 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 ...
3
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2answers
398 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,...
2
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3answers
1k 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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0answers
24 views

Relationship between signal amplitude and input output correlation

I have a linear system with input $ x(t) $ and output $ y(t)$ given by $$ y(t) = \int_0^\infty K(t')x(t-t')dt', $$ where $K(t)$ is a known kernel, with some parameters. The functional form of $K(t)$ ...
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1answer
49 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
100 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
89 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
419 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
712 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 ...
1
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2answers
96 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 ...
1
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1answer
138 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 ...
0
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1answer
152 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 ...
3
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1answer
252 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
1k 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
75 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
57 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)-\...
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1answer
617 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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0answers
38 views

Recursive systematic convolutional code with TCM

My question is: What are the outputs from the TCM encoder ?
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2answers
1k views

Initial conditions for systems described in state space - LTI or not?

Suppose we have some system given by $$\begin{aligned} \dot{x}(t) &= Ax(t) +Bu(t) \\ y(t) &= Cx(t)+Du(t) \end{aligned}$$ where $x(t)$ are the state variables, $y(t)$ is the output and $u(t)$ ...
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1answer
877 views

Physical Meaning of Negative Group Delay for causal LTI systems

I have implemented in Matlab (with minor variations) the example 5.1.2 "Illustration of Effects of Group Delay and Attenuation" I found in Alan Oppenheim's Discrete-Time Signal Processing 3rd edition. ...
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1answer
129 views

Problem understanding steps towards defining the convolution sum

I have a problem understanding the reasoning behind a step that was taken to characterize a LTI system. So, we were told the following: For each integer $k$, we have the following function: $$ \...
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0answers
69 views

Frequency Response Question on LSI System

Someone please explain me the question its seems very complicated to me. I just want know what the question asking and how to solve it i dont want fully solved solution. Thank you!!
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1answer
141 views

Understanding and Correcting phase shift in General Linear Phase FIR filters

I have been learning about General Linear Phase Filters and the four standard FIR implementations. I was wondering in the case $\beta = \frac{\pi}{2}$ or $\beta = \frac{3\pi}{2}$ if there was a way ...
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3answers
108 views

Causality as applied to capacitors

This question stems from a point of confusion that I still have about the causality, linearity, and time-invariance in LCCDEs. I wanted to use the capacitor as an example. Consider a capacitor with ...
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1answer
564 views

Determining Causality From Discrete Impulse Response

Trying to wrap my mind around the concepts of this one... Consider the following impulse response $h[n]$ for a linear, time-invariant system: $$ h[n] =\left\{\underline{1} , -2, 2, -1\right\} $$ ...
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0answers
27 views

should block diagrams in Direct Form 1/2 contain empty branches to fit structure?

When creating a block diagram in a well-recognized structure (Direct Forms 1&2 in particular), does one remove branches where the coefficient is 0? This picture includes a coefficient of $b_1=0....
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2answers
204 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)...
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1answer
110 views

Time-invariance, causality and stability of $h(t)$ of four given systems

Question: The impulse response functions of four linear systems $S_1,\ S_2,\ S_3,\ S_4$ are given respectively by \begin{align} h_1(t)&=1\\ h_2(t)&=u(t)\\ h_3(t)&=\frac{u(t)}{(t+1)}...
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1answer
978 views

Causality, Linearity, and Time Invariance for Systems Described by Linear Constant Coefficient Differential Equations

I am currently using Signals and Systems by Alan Oppenheim as a reference to learn about LTI systems. Before introducing systems represented by linear constant coefficient differential equations, ...
5
votes
3answers
1k 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 ...
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1answer
593 views

Time shifting and time reversal

For the problem given in the fig. how to solve it? My approach: Let $y(n)$ be the first output (e.g for 1st case output of system $A$ is $y_1(n)$; for 2nd case output of system $B$ is $y_2(n)$, then ...
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3answers
976 views

Solving for impulse response h[n] given input-output pairs

This is a homework problem that I've worked on and I want to confirm some of my reasoning. I'm given two sets of input-output pairs of a particular system, $S$, that we know is linear time-invariant ...