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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Effect of BIBO-Instability on the frequency response of a ideal LPF
I recently came across this post stating that the continuous ideal LPF is BIBO-unstable since the impulse response is not absolutely integrable, and this post stating some examples.
I have been trying ...
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LTI system tradeoff between gain, bandwidth, and delay
For first-order LTI systems, the gain-bandwidth constant is often discussed. I've seen the claim that in general, gain and bandwidth don't directly trade off with each other as much as delay. For an ...
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Routh's stability condition
Assume we have a LTI system which has poles in the half left plane of the s domain.
Before I learnt Routh's stability condition I had imagined that this was enough to decide whether a LTI system was ...
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Verifying Linearity and Shift Invariance Under Summation
I am having some trouble working through verifying linearity and shift invariance when the transformation is under a summation.
The given transformation is as follows:
$$
y(m,n)=\sum_{i=-1}^{i=1}\sum_{...
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linear time-invariant system output question?
Subtract the output of a linear time-invariant system whose shock response is $h \left[ n \right] = \left( \frac{1}{4} \right)^{n} \left( u \left[ n \right] - u \left[ n - 2 \right] \right)$, for ...
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Calculating transfer function of a linear time varying system?
If we excite a LTI system with the Dirac delta $\delta(t)$, the system outputs the impulse response $h(t)$. For a LTI system, it doesn't matter when we excite the system with the Dirac delta, we will ...
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Max input of a system given it's transfer function and an assumed step change (beginner)
I have an exercise that gives me the following transfer function
$$
\frac{0.5}{s+0.5}
$$
and an assumed step change in the target of 20
I am asked to calculate the maximum input for the assumed step ...
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Non-causality of fractional delays
Given a physical system (e.g., loudspeaker and microphone) with DA and AD converters. Playing a single pulse from the loudspeaker, I will most likely receive the pulse at the microphone with a ...
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Proof for LTI Systems [duplicate]
Suppose we have a system $S$.If I know the impulse response $h(t)$ I can predict the output for any input $x(t)$ if a system is linear and time invariant.However how do we prove that statement?
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Impulse response of linear time-varying system
I am confused about linear time-varying system. For a time varying system, the output is given by
\begin{align}
y(t)=\int x(\tau) h_{\tau}(t) d\tau,
\end{align}
where $ h_{\tau}(t)$ is the output of ...
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Power of filtered Bernoulli process
I have some doubt about this exercise.
The Bernoulli random process $X(n)$ with means $p=0.5$ is sent in input to a LTI system with impulse response $h(n)= \cos(\frac{\pi n}{3}) R_3(n+1)$ , where $$...
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LLTV Systems breakdown(2)
In this question I proved that for a linear linearly time varying system $S$ such that if $t_{2} = t_{1}+t_{0}$ then $h(t,t_{2}) = h(t,t_{1})+h(t,t_{0})\rightarrow$ $h(t,t_{0}) = g(t_{0})h(t) $ where $...
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LLTV Systems linearity breakdown
Suppose we have a linear linearly time varying system $S$ such that the output depends on the input + the time at which the system becomes excited so $S(t,t_{0}) \rightarrow g(t_{0})S(t)$ and it is ...
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Linear linearly time varying systems Laplace transform
Suppose that for a system $S$ if we have $t_{2} = t_{1}+t_{0}\rightarrow h(t,t_{2}) =h(t,t_{1})+h(t,t_{0}) $ .Then if we take the double Laplace transform to$t,t_{2}$ we will get:
$$L_{t_{2}}(L_{t}(h(...
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Frequency response of an LTI system described by the diagram below
I am trying to solve the below problem:
To begin with the frequency response of the ideal low pass filter with $\omega_c= \pi/4$ is given by $$
H(\omega) =
\begin{cases}
1 & \text{if -$\pi/4$ $...
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is $y[n] = y[n - 4] + x[n - 4]$ time variant or invariant?
I am confused about a solution because there is feedback.
Let's introduce a delay parameter $k$ and rewrite the system equation as:
$$ y \left[ n \right] = y \left[ n - 4 - k \right] + x \left[ n - 4 -...
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Real & Imaginary part of the frequency response of LTI system
I am looking at the H(f) which is the frequency response of an LTI system. What kind of relationship should I expect between the real and imaginary part of this frequency response? Here is an example ...
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Feedback stabilizes unstable systems?
Can feedback stabilize a unstable system?I guess so since we may get rid of any poles in the positive complex plane by feedback but I am not sure.
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How can an impulse generate an output in the past time frame?
I am studying signal processing and currently doing signals & systems. While going through convolution and especially the impulse response , there are problems where LTI systems wherein the input ...
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Linear Linearly Time Varying Systems properties?
Yesterday I asked a question about LTV Systems and how the impulse response h(t,t0) is found.
I was thinking about it and realized there is a subsclass of LTV systems which I call Linear Linearly Time ...
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Impulse response of an LTI system
I want to find the impulse response of the linear time invariant system given by
$y''(t)+4y'(t)+3y(t)=x'(t)+2x(t)$
using time domain analysis.
As solved in the textbook "Continuous and Discrete ...
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Non-linear external effect in Kalman filter
Let's say I have a Kalman filter with this simple state model:
$$\begin{pmatrix} x^0_{k+1}\\ x^1_{k+1}\\ \end{pmatrix} = \begin{pmatrix} 1 & \Delta t\\ 0 & 1\\ \end{pmatrix} \begin{pmatrix} x^...
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How are LTI systems related to Toeplitz matrices?
I am having trouble understanding why the system matrix of an LTI system is Toeplitz. I am following an Edx online course by Professor Richard Baraniuk of Rice University, named discrete-time signals ...
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A response of an unstable system
I'm dealing with the following question :
The system realizes an accumulator, therefore its impulse response (h[n]) is just the unit step function multiplied by some factor . In the next subsections ...
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Initial conditions of state-space realization
In a discrete-time difference equation (ARX model), the output $y[k]$ is dependent on its past values and inputs, expressed as:
$y[k] = -a_1 y[k-1] - ... - a_N y[k-N] + b_1 u[k-1] + ... + b_M u[k-M]$, ...
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System Characterization: multiply imaginary component by scalar
I tried solving a test question that frankly stumped me. If you could explain to me the solution I’d be really grateful.
Given $a, b \in \mathcal{R}$ the system $R_v$ takes the complex input $a + bj$ ...
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Matrix form of Overlap-add
We know overlap-add of a en-framed signal can be done easily by following code
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Real coefficient linear phase FIR filter design based on least square
I want to build up linear phase filter by least square approach.
I have formulated a least square problem similar to Matt's blog. To approximate the desired complex response $F \times 1 $ vector $d$ ...
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Show that decomposition does not hold for non-linear system
The solution to an inhomogeneous differential equation can be split up into homogeneous solution and a particular solution (forced response).
Another way to split up the solution to an inhomogeneous ...
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Invertible system for the eigenfunction $x[n]=e^{j\omega n}$
I was doing some calculations in my LTI systems course and I stumbled in an interesting question I wasn't really sure how to answer so I'd appreciate any direction or solution you can give me:
I'm ...
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Finding transfer functions from a system of multiple inputs
If I have a system: $$sX(s) = AX(s)+BY(s)+CZ(s)$$ How would I find the transfer functions $\frac{X(s)}{Y(s)}$ and $\frac{X(s)}{Z(s)}$?
Do I simply disregard one of the inputs? I am quite confused and ...
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Classification of a system
I am preparing for an examination and have a study guide that I feel has a couple of errors. The questions concern the classification of discrete time dynamical system. Here are the problems that I am ...
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is this signal is perodic?
What is the time period of
$$ x(t) = 7 e^{\jmath(5t + \pi/2)} + 10^{\jmath(7t + \pi/5)}$$ ?
X(t) is the combination of two functions. one is the natural logarithm base, $e$, and the other is 10.
if ...
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What is the definition of an incrementally linear system?
Even though the concept of an incrementally linear system is mentioned in several textbooks, I haven't found an explicit definition. What exactly do we mean by "incrementally linear"?
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How to perform an operation on a signal which depends on 2 variables?
How do we perform DFT when the input signal depends on 2 variables (like in a image the color of the pixel depends on both $x$ and $y$)
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Expressing the frequency response in a more 'compact' form
I've just been working through the questions in Discrete-Time Signal Processing (Oppenheim and Schafer) and I came across this (Q33):
Consider an LTI system defined by the difference equation $$y[n] =...
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Find $E[Z^2(t)]$ when $Z(t) = X(t) - Y(t)$ where $Y$ is the output of a LTI system with WSS process $X$ as its input
I received this as a practice problem (part b only).
I was able to figure out that $E[Z^2(t)]$ = $R_X(0)+R_Y(0)-2R_\text{XY}(0)$ but did not see how to continue.
Checking the answers, I saw this line ...
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First Order State Space Question
I am trying to understand the state space equation for a simple first-order LTI system. Suppose I have a system with impulse response
$$h(t) = \frac{1}{\tau}e^{\frac{-t}{\tau}} \ \theta(t)$$
In this ...
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Find out if the system is linear and time-invariant
Consider a discrete-time system which in response to input sequence x[n] (n integer)
outputs the sequence y[n] such that
$y[n] = 0, n = −1, −2, −3,...$
and $y[n]=2 (y[n − 1] + nx[n]), n = 0, 1, 2,...$....
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Solving for the Kernel of a system using impulse balancing with diracs delta's derivative
I wanted to practice Impulse balancing and tried using it to solve this IAR system in one go as opposed to using its LTI attributes. The exercise wants me to find the impulse response of the system:
$$...
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Shift invariance of system example
I'm preparing for upcoming exam.
I'm having difficulties with deciding if system is shift invariant.
System is described as
$$y[n] = \sum_{k=-n+1}^\infty x[1 - k]$$
I'm not sure how would output for ...
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LTI-properties of a system summing infinite number of input values
A pop-quiz featured a system analysis of the following system:
$y[n]=\sum_{n=0} ^{-\infty} x[n]$
The professor argued that this system exhibits the following properties:
Linearity
Causality
...
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Are marginally stable LTI systems also BIBO stable?
In the realm of LTI systems (perhaps even in general systems, I am not sure) it is so that if a system is asymptotically stable then it is also BIBO stable. Is that the case when a system is ...
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Does the definition of stable system contradict itself?
A system is said to be stable when any of its poles are <0.
However I don't get why that is the case. Negative poles mean negative angular frequency, and negative angular frequency is equal to ...
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"Instantaneous impulse response" in a linear time-varying system
I have a LTV (linear and time-varying) system. So, $h(\tau, t)$ is the "instantaneous impulse response" at time $t$ such that if the input signal is $x(t) = \delta(t - t_0)$ (an impulse at ...
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When $x(t)$ and the output $y(t)$ are related by linear differential equations, why is the system unstable when $M>N$?
This is two linear differential systems equating each other.
\begin{equation}
\frac{d^Ny(t)}{dt^N}+a_1\frac{d^{N-1}y(t)}{dt^{N-1}}+\cdots+a_{N-1}\frac{dy(t)}{dt}+a_Ny(t)=\\
b_{N-M}\frac{d^Mx(t)}{dt^M}+...
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Converting a SISO LTI System to a MIMO LTI System
I have a SIMO IIR filter represented as a state space system. I'm wondering if there is a canonical way of converting a IIR filter into a MIMO system, so I can calculate $N$ terms as input at once and ...
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How is this system nonlinear
Suppose we have a system S and the relationship between $x(t)$ and $y(t)$ is $y(t) = x(t)+1$
However despite the fact the system is made of linear operators the system is not linear.How is this ...
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How to show that this system is time invariant and linear? [closed]
I have a system described by the following difference equation:
$$y(n)−y(n+1) = 12x(n+2)$$
where $x(n)$ is the input, and $y(n)$ is the output.
How do I show that this system is time invariant and ...
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What is causal inverse of a system?
Let's say that I have a system $H(z)$. What is causal inverse and how do I compute the causal inverse of $H(z)$?
