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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71 views

Impulse response for an LTI system

I'm new to signal processing and working my way through a textbook. There is an exercise where a causal LTI system is given that responds to a rectangular pulse. I have an exercise where a causal LTI ...
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Under what conditions is the convolution of an input signal with the system's impulse response periodic?

I'm currently solving the following convolution problem from Oppenheimer's book: In the solution, it was stated that "$x(t)$ periodic implies $y(t)$ is periodic" So I wondered if it's ...
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How do i know it is will be periodic

(system is LTI and Casual) 1)If a periodic signal is applied to the input of this system. Does output always have to be periodic ? 2)What conditions are required for this system to be linear? thanks ...
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Output of a stable LTI system (discrete)

Consider an LTI (linear and time invariant) system that is BIBO (bounded input bounded output) stable and is such that x[n] = 0 for all n < 0 (note: this is sometimes referred to as a relaxed ...
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Understanding LTI systems graphically

I'm currently working on this problem from Oppenheimer's book: Given $x_{1}(t)$ and $y_{1}(t)$ I should figure out $y_{2}(t)$ and $y_{3}(t)$ given that the system is LTI. My progress so far: I'm ...
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139 views

How do I check for linearity for the following piecewise-defined system?

The problem at hand: Where I'm currently stuck: I'm not entirely sure about how to move on from this point, I'm trying to find the superposition of the responses of the two individual signals so I ...
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Physically, what does the usage of two variables mean for convolution

My intuition of convolution is that it is just a way to depict multiplication of two signals where each signal is made up of various frequencies and phases. Since it isn't easy to find the value of $\...
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70 views

Scaling the input vs scaling the impulse response for an LTI system

Two different cases: We pass $x(t)$ to an LTI system with impulse response $h(2t)$ and get the output $y(t)$. We pass $x(2t)$ to an LTI system with impulse response $h(t)$ and get the output $z(t)$. ...
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Stable and causal system

How many stable and causal systems with the same magnitude response are there? I know this relates to an all pass system for two rational transfer functions but am not sure about the specifics of this
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Z-transform of an affine function

What is the transfer function of the system described by the following affine input ($x$)-output ($y$) relationship: $$ y[n] = \alpha x[n] + \beta. $$ Using the Z-transform we find: $$ Y[z] = \alpha X[...
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convolution of two exponential signals with imaginary numbers

I can solve problems without imaginary numbers, but when exponential contain imaginary numbers, I can't solve the problem. For example, $x(t)=e^{3jt}+e^{4jt}$, $y(t)=(e^{-3t}-e^{-4t})u(t)$ (where $j$ ...
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Unit impulse response of a cascade interconnection of three discrete-time systems

I am nearly at the end of finishing a problem in my textbook but I couldn't understand something in the answer; I did everything to the point I found the overall response of the system in terms of $...
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Calculate impulse response when output contributes in input

I have an exercise in which I need to find the impulse response for this given system: $$y(n)=\frac{1}{2}y(n−1)+x(n−1)+x(n)$$ As per my knowledge, I need to find the homogenous solution. My homogenous ...
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How do I get a faster system response?

I have this model in simulink (the graph is my output): The step input has amplitude 0.5 m/s, and it steps up after 0.1 seconds. The gain $K_p=5$. The saturation block is to keep the voltage between -...
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Unit impulse response $h[n]$ of a discrete time system with multiple characteristic roots 0

I am trying to obtain the unit impulse response of a system in the form: $$y[n+N]+...+a_{N-1}y[n+1]+a_{N}y[n]=b_0x[n+N]+...+b_{N-1}x[n+1]+b_{N}x[n]$$ $$Q[E]y[n]=P[E]x[n]$$ (where E is the unit advance ...
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Output of a linear time-invariant(LTI) system

I am very confused about one of the questions I received during an exam. How do I solve this question?
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Finding the impulse response given response to another signal

I was trying to solve this question : I respresented $x(t) = u(t+1)-u(t-1)$ writing the convolution as $[u(t+1)-u(t-1)]*h(t) = y(t)$ I then used the property of differentiation to convert from the ...
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49 views

Initial rest condition applied on $x(t)$ vs $h(t)$

Define the LTI system $\mathcal{H} : x\mapsto y$ Define the convolution for continuous-time system : $$ (x*h)(t)=\int_{-\infty}^{\infty}x(\tau)h(t-\tau)\;\text{d}\tau $$ The initial rest condition ...
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Impulse Response of real coefficient, LTI System

I'm trying to obtain the impulse response $h[n]$ of a system whose frequency response is $H(e^{j\omega})=R(\omega)e^{-25j\omega}$. I believed that $h[n]=h[n-25]$, would be the correct answer, however ...
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understanding echo cancellation model

so, here is a very simple model. The CCDE is given, but I was trying to derive it on my own and now I am stuck. first of all, the reverse y[n] when goes through the delayed system, it turns to y[n-M]....
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Applying Superposition Property : $x^{2}[n]$ vs $x[n^{2}]$

If we consider the mapping $\mathcal{H} : x[n]\mapsto y[n]$ and define the following output signal $y_{1}[n]:=\mathcal{H}\{x[n]\}:=x^{2}[n]$, then one can easily verify that such system is non-linear ...
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Kalman filter with multiple sensors

I have been reading more about EKF and I am a bit confused on how you handle predictions with multiple sensors. Ex, I have IMU, GPS, Odom and Stereo Camera. Each can be used to predict location, how ...
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From transfer function to differential equation

I have the below detailed solution (boxed in blue) that I don't understand completely: I can reconstitute the differential equation from: $$ (1+Ts) X(s) = K_v U(s) $$ $$ x(t) + T\dot x(t) = K_v u(t) $...
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How could I approach determining if this 2D system represented as a 2D summation formula is linear?

I have a given 2D system: $$y(m,n) = \sum_{k_1=-\infty}^{m} \sum_{k_2=-\infty}^{n} x(k_1,k_2)$$ My usual approach to determining if a system is linear is to test if it is homogeneous and additive. ...
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471 views

How can convolution be a linear and invariant operation?

I'm having a slight breakdown right now with a seemingly simple question. Say I have a system that convolves an input function with itself to produce an output function: $g(x) = f(x) ∗ f(x)$ I've ...
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33 views

Analyze the transient response of a system

Suppose I have a sytem with the following transfer function : $$H(s) = \frac{N_H(s)}{D_H(s)}$$ I would like a general method which is not dependant on the order of the system to analyze what would be ...
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52 views

time linearity of LTI system

There is this following question. Consider the transformation H {x}[n] = n x [n]. Does H define an LTI system? What I understood from the question is: x[n] -> H -> y[n] or according to the ...
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76 views

Particular Solution to Difference Equation

I have a system given by $$y[n] - \frac{1}{4} y[n-1] - \frac{1}{8} y[n-2] =3x[n] $$ I want to solve for $y[n]$ for $x[n]=(\frac{1}{2})^nu[n]$. The complementary solution evaluates to $[k_1(\frac{1}{2})...
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28 views

Impulse Response for an Anti Causal Linear Shift Invariant System

I have a system given by $$y[n] - \frac{1}{4} y[n-1] - \frac{1}{8} y[n-2] =3x[n] $$ I'm asked to determine the impulse response of both an anti-causal and a causal Linear Shift Invariant System ...
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59 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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36 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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57 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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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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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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100 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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37 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 $\frac{1}{2πt}$ 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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29 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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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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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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52 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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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 $$y(t) = \big[x(t − 1)x(t + 1)\big]^{\frac 12}$$ the fact that is in the form of $x\cdot x$ told me ...
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Determining Causality and Time-Invariance of a system

Consider the following system: $$y(t-1)=\int_{-\infty}^\infty x(𝜏)u(𝜏-t) d𝜏 $$ 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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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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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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36 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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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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49 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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31 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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31 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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165 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!

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