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
34 views

What is the reason behind complex conjugate pairs in Linear Phase FIR filter analysis from the Pole Zero plot

When we say that a Generalized Linear Phase System must satisfy the pole zero plot with the condition that a complex zero not on the unit circle exist's in a pair of 4. Then I understand that I need ...
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0answers
34 views

Augment State Space Model with Derivative of State

Problem How do you augment a state space model with the derivative of a state? I know how to augment a state space model with the integral of a state by doing the following. Given a linear system $$ \...
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1answer
44 views

Determine if system is linear time variant

The system equation is given as: $$y(n)=(n-1)x(n-1)+(n+1)x(n+1)$$ I solved that the system is time variant: \begin{align} y(n-k)&=(n-k-1)x(n-k-1)+(n-k+1)x(n-k+1)\\ H[x(n-k)]&=(n-1)x(n-k-1)+(n+...
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3answers
64 views

How can we show that $|H(e^{j\omega})|=|H(e^{-j\omega})|$?

Let $H(z)$ be the rational system function of an LTI system. How can we show that $|H(e^{j\omega})|=|H(e^{-j\omega})|$?
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30 views

Matched Filter Impulse Response

At the receiver end of a digital communication system, the matched filter is designed with impulse response matched to the effective signal that is given as : $$x_{(z,1)}^{*}(T_p-t)-x_{(z,0)}^{*}(T_p-...
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2answers
73 views

Inverse system of sinc?

I'm doing some self-study for an important exam I'll have in late March and came across the following question: So, using the convolution properties, if I want to find an identity system so that the ...
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0answers
32 views

Calculate system function using pole and zero locations? [closed]

The pole zero of a digital system is given as zeros at $z=+j,z=-j,z=1,z=-1$ and 4 poles at $z=0$. As far my understanding this is a FIR filter. But I am not getting how to solve the system function ...
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1answer
30 views

What is the decay rate (time constant) of two One-Pole LPF's cascaded in series?

If you have a One-Pole LPF, and set the cutoff frequency to $\frac{1}{2ฯ€t}$ where $t$ is the time to $\frac{1}{e}$ amplitude, it will provide an exponential decay in response to a step function (eg. ...
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1answer
23 views

What is the relation between input and output PSDs given system transfer function $H(s)$

If I have the system transfer function $H(s)$ in the complex frequency domain, how would I relate the input/output power spectral densities? I have come across the relation $P_{out}(f) = |H(f)|^2P_{in}...
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1answer
74 views

Why not use the same “standard” exponentials for both continuous and discrete time

In continuous time the standard exponential signal is usually defined as $$ e^{st}, \quad\text{with}\quad s = \sigma+j \omega $$ In discrete time the standard exponential signal is usually defined as ...
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2answers
19 views

Impulse response of forward difference cascaded with one sample delay

Below is the excerpt from Discrete Time Signal Processing by Alan Oppenheim. I don't get how $(\delta[n+1] - \delta[n]) * \delta[n-1])$ becomes $\delta[n] - \delta[n-1]$. The convolution sum operator ...
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1answer
41 views

If the impulse response of a causal LTIC system is $h(t) = \delta(t) + \sin(t)u(t)$, is it marginally stable or unstable (BIBO)?

If you take the $H(s) = \mathcal{L}[h(t)]$ the poles are on the imaginary axis so the system should be marginally stable, but is it?
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1answer
34 views

Linear systems: Square root of input product

Hi guys i'm studying signals and systems, and my professor ask us if this signal is linear or not $$y(t) = \big[x(t โˆ’ 1)x(t + 1)\big]^{\frac 12}$$ the fact that is in the form of $x\cdot x$ told me ...
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2answers
118 views

Determining Causality and Time-Invariance of a system

Consider the following system: $$y(t-1)=\int_{-\infty}^\infty x(๐œ)u(๐œ-t) d๐œ $$ where $u(t)$ is the unit step function, which is zero for $t<0$ and equals $1$ for $t>0$. $(1)$ Is the system ...
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1answer
50 views

Why can you use the one-sided laplace transform to solve differential equation describing a causal LTI-system?

In an example, an equation describing a causal LTI-system is $$ (D^2 + 5D + 6) y(t) = (D+1) x(t) $$ where $y(t) = y_{zs}(t) + y_{zi}(t)$ and the initial conditions are $y(0^-) = 2, \dot{y}(0^-) = 1$. $...
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2answers
34 views

Question about a transient response statement (Oppenheim-Schafer"s discrete-time signal processing book, 3rd ed)

This is the statement from the book: Eq. (129) only ensures that $y_t[n]$ will be bounded for all values of $n$. For example, this may include a case where $y_t[n]$ oscillates and hence never ...
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2answers
34 views

Clarification on eigenfunction property from Oppenheim's Discrete-time signal processing, 3rd ed

Could anybody explain what is meant by the following statement on page 50 of Oppenheim and Schafer's book Discrete-Time Signal Processing (third edition)? The eigenfunction property of complex ...
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2answers
64 views

Laplace Transform: zeros and corresponding impulse response $h(t)$

Poles and the impulse response If our impulse response is in the form : $$h(t) = e^{-\sigma_0 t}\cos(\omega_0 t) \, u(t)$$ (where $u(t)$ is the unit step function) And its Laplace transform is : $$H(s)...
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1answer
45 views

Why can't this system be adequately represented using a z-domain transfer function?

According to this question and answer the following system cannot be adequately captured by a z-transform transfer function. $$y[n] = y[n-1] + F_{\psi}(y[n-1)) + F_{\phi}(x[n-1])$$ where $F_{\alpha}(z)...
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1answer
24 views

RoC and Stability of a Rectangular Signal

If we have a system with an impulse defined as: $$h(t)=u(t)-u(t-2)$$ Then the Laplace Transform of h(t) would be the transfer function: $$H(s)=\frac{1}{s}-\frac{e^{-2s}}{s}, \quad Re(s)>0$$ We also ...
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0answers
29 views

How to find parameters of Kalman filter using matrix information?

I'm trying to understand concepts on Kalman Filters. Consider the overdetermined system $Ax=y$; $$\begin{bmatrix}1 \\ 1 \\ 1 \\1 \end{bmatrix} x = \begin{bmatrix} 3 \\ 5 \\ 4 \\ 8 \end{bmatrix}$$ Let $...
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3answers
147 views

Are all exponential functions eigensignals of LTI systems?

I know that complex exponential functions are eigensignals to LTI systems. Do these include real exponential functions? E.g. $2^t, e^t, ...$ Thanks for the help!
2
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2answers
203 views

How can initial conditions be taken into account to calculate a system's terminal value using the final value theorem or some other technique

I wish to calculate the Final Value of the following system given initial conditions not at rest, and assuming the X will not change from its initial condition value. The transfer function is $$H(z) = ...
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1answer
48 views

Finding impulse response $h(t)$ by impulse matching; problem understanding Example 2.3 in “Linear systems and signals”, Lathi 3rd edition

Before the example it has been stated that in a system described by $$ Q(D)y(t) = P(D)x(t), \quad (1.) \iff \\ (D^N + a_1D^{N-1} + ... + a_{N-1}D + a_N)y(t) = (b_{N-M}D^M + b_{N-M+1}D^{M-1} + ...+b_{N-...
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2answers
74 views

Why this system is linear?

Hi guys i'm studying signals and systems, and my professor told us that $$y(t) = \int\limits_{ t+T }^{t-T/2} {x(a+T/2)}\mathrm{d} a$$ is a linear system. But a primitive of $x$ isn't $ x^2$ ? How it'...
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1answer
114 views

Stability of a system in time-domain

Consider a system for which the input $x(t)$ and output $y(t)$ are related by the differential equation $$\frac{d^2y(t)}{dt^2} + \frac{3}{2}\frac{dy(t)}{dt} - y(t) = x(t) \tag{1}$$Determine $h(t)$ for ...
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1answer
49 views

Discrete time Final Value Theorem applied to feedback system

I wish to calculate the Final Value of systems in which a high pass filter of the output feeds back into the input. A simple example would be: where is a 1st order high pass filter with transfer ...
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3answers
74 views

Linear approximation of a system described by a logarithm - how?

The system described by a logarithm of any base, let's call it $y(t) = \ln(t)$, is non-linear. Is it possible, by any means (transforms, operators etc.), and for small values of $t$ (for example, $t&...
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1answer
33 views

Disjoint ROCs of input and system function

I am trying to understand how the output behaves when the input and the system function do not have a common region of convergence (ROC) for an LSI system. Consider an LSI system with $x[n]$, $h[n]$, $...
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1answer
25 views

Does fading memory mean impulse response with finite support?

Given a continuous-time impulse response $h(t)$, bandlimited to $B$. The discrete-time $h[n]=h(n/(2B))$ is then a unique and perfect representation of $h(t)$ and a discrete-time system $h[n]$ is then ...
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0answers
53 views

Richardson-Lucy iterative deconvolution is producing erroneous deconvoluted images

I am working on producing deconvoluted image from degraded image where the process of degradation is known to be linear and space invariant. I have used Richardson-Lucy iterative deconvolution method ...
4
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2answers
54 views

Moving from deterministic signals to stochastic signals in s-domain (Power Spectral Density)

Assume we have the following system (coming from control systems theory, hence in s-domain) $ Y(s) = H_A (s) \cdot A(s) - H_B (s) \cdot B(s) $ I now wish to consider $a(t)$ and $b(t)$ as white noise ...
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2answers
97 views

Output of a stable LTI system

Let $\mathcal{L}$ be a stable LTI system. Is it true that if input has finite energy then output also has finite energy? I'm not sure about that. We know that $$\int_{-\infty}^{+\infty}|h(t)|dt\lt\...
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1answer
69 views

Find the length of the impulse response for the given output and input

Homework Question: Consider a signal $x[n]=\alpha e^{j \omega_{0} n}+\beta e^{j \omega_{1} n}+\gamma e^{j \omega_{2} n} .$ What is the length of impulse response $h[n]$ of a system (non-trivial) such ...
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0answers
24 views

Discrete Time Systems with cosine()

I am trying to see if y[n] = [cos(ฯ€n)]x[n] is casual, stable, linear and shift-invariant. I came up with the solution that it is not stable since it is not "...
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1answer
27 views

Testing for Linearity and Shift-Invariance

I started school a week and a half ago and we have gotten already a big chunk of material and I wanted to check if what I am currently doing is correct. Here below are the problems to work out that ...
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1answer
60 views

Why isn't option C correct answer for the given question?

W Please answer this question I am getting option c as my answer by putting input frequency of 3 radians in frequency response of system.
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1answer
56 views

Fourier transform of time division

I know that Fourier transform of $t^n f(t)= i^n \frac{d}{d\omega^n} F(\omega)$. But does this work when $n<0$? Is there any direct relation to compute the Fourier transform of $\frac{f(t)}{t}$?
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1answer
74 views

Confusion in initial condition of differential equation using Laplace transform transform

I'm confused in solving linear constant coefficients differential equations (LCCDEs) by Laplace transform if initial conditions are given at time just before $t=0$ just after $t=0$ exactly at $t=0$ ...
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2answers
50 views

Stability of passive LTI system

Is a passive LTI system will always be a stable(bibo and lyapunov) system? Or in other words Passivity of a LTI system implies stability? If above statements are true then there must be some ...
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2answers
153 views

Derivation of the LMMSE (Linear Minimum Mean Squared Error) Estimate and the MMSE Under Gaussian Prior

I am learning estimation theory through Steven M. Kay - Fundamentals of Statistical Signal Processing, Volume 1: Estimation Theory. In the ...
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2answers
79 views

Output of marginally stable systems

In' b.p lathi's book of systems and signal it is written that "marginally stable systems have one important application in the oscillator, which is a system that generates a signal on its own ...
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1answer
67 views

Determining a system's causality using its impulse response

I have the following input-output relation for a system: $$y(t) = Odd Part Of [x(t)]$$ My question is: Is the system causal? What my approach has been: I expressed $y(t)$ alternatively as: $$y(t) = \...
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1answer
39 views

Transients in marginally stable system

I know that if a system is asymptotically stable then after infinite time its output contains only characteristic of input signal and it is also known as steady state and there are no transient ...
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1answer
174 views

Time invariance of a summation sequence

The following sequence is given which is supposed to be time-variant: $$y[n] = \sum_{k=n_0}^n x[k]$$ I'm having difficulties proving the time-variance or finding a counterexample for it being time-...
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2answers
91 views

How to compute transfer function $G(s) = \exp \left( - \sqrt{s} \right)$ in Matlab / Simulink?

How to compute transfer function $$G(s) = \exp \left( - \sqrt{s} \right)$$ in Matlab / Simulink? I am trying to calculate a PID controller for this function. This function describes heat transfer via ...
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0answers
121 views

Time-Invariant (non)autonomous Systems

Regarding potential distinctions between autonomous, non-autonomous, time-invariant and time-varying systems, I have found out opinions supporting that: autonomous systems are time-invariant and non-...
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1answer
36 views

Compute output given input, transfer function and initial conditions

The problem statement is Consider a causal LTI system whose transfer function $H(s)$ is given as $$H(s)=\frac{s+2}{(s+3)(s+4)}$$ Compute the output $y(t)$ for an input $x(t)=e^{-2t}u(t)$ when $y(0)=1$...
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1answer
45 views

Behaviour of integrator at steady state

I wanted to calculate response of integrator of sinusoidal input at steady state via these two methods as mention in image but these two methods give two different answers at steady state, so where ...
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5answers
1k views

Are there any real world applications for complex-valued signals or impulse responses?

I was just curious... $$x[n] {\longrightarrow} \boxed{h[n]} {\longrightarrow} y[n]$$ I've never seen a real world filter where the coefficient of $h[n]$ were complex, or where $x[n]$ was a complex ...

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