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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What is meant by a system's "impulse response" and "frequency response?"

Can anyone state the difference between frequency response and impulse response in simple English?
Ant's's user avatar
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Fitting new images from a SVD/PCA calculation

I'm trying to replicate the ideas from the Eigenface page on wikipedia. From a hundred sample images represented by a data matrix $\bf X$ (where each image flattened to a vector of length $n$, thus $\...
Hooked's user avatar
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16 votes
1 answer
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How do I find a system's impulse response from its state-space repersentation using the state transition matrix?

Suppose we have a linear represented in the standard state space notation: $$ \dot{x}(t)=Ax(t)+Bu(t)$$ $$y(t) = Cx(t) + Du(t)$$ In order to get its impulse response, it is possible to take its ...
Phonon's user avatar
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15 votes
6 answers
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What Are Linear and Circular Convolution?

I have some basic understanding of signals and convolution. As far as I know it shows the similarities of two signals. Could I get some explanation in plain English of: what are the linear and ...
Sturm's user avatar
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14 votes
5 answers
7k views

Why do linear systems show sinusoidal fidelity?

I am looking for a proof for sinusoidal fidelity. In DSP we study a lot about linear systems. Linear systems are homogenous and additive. One more condition it satisifies is that if a signal is a sine ...
Hobyist's user avatar
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2 answers
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How to Deduce a Linear System's Impulse Response from a Set of Input and Output Signals?

I want to know how to solve those types of problems.. is it by inspection ? Consider the linear system below. When the inputs to the system $x_1[n]$, $x_2[n]$ and $x_3[n]$, the responses of the ...
Belbesy's user avatar
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13 votes
4 answers
4k views

Are complex exponentials the only eigenfunctions of LTI systems?

Is there an example of an eigenfunction of a linear time invariant (LTI) system that is not a complex exponential? Justin Romberg's Eigenfunctions of LTI Systems says such eigenfuctions do exist, but ...
Vinod's user avatar
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12 votes
5 answers
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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 ...
pipen's user avatar
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10 votes
4 answers
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Why LTI system cannot generate new frequencies?

Why $Y (\omega) = X(\omega)H(\omega)$ implies that an LTI system cannot generate any new frequencies? Why if a system generates new frequencies, then it is not LTI?
USER's user avatar
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2 answers
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What is the difference between Lumped and Distributed systems?

What are the salient differences between Lumped and Distributed systems? In what contexts are distributed systems the appropriate model and in what are lumped systems the appropriate model? Also, ...
Gowtham's user avatar
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10 votes
4 answers
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What is the difference between natural response and zero input response?

I am new to DSP and was going through different responses of a system subjected to an input. My understanding of zero input response is: it is the response/output of the system when the input signal ...
rotating_image's user avatar
10 votes
3 answers
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Is there a way to obtain the impulse response of a discrete system by just knowing it's response to the discrete unit step function?

In continuous time it was possible; $$ u(t){\longrightarrow} \boxed{\quad\textrm{system}\quad} {\longrightarrow} y(t)\implies \delta(t)=\frac{du(t)}{dt}{\longrightarrow}\boxed{\quad\textrm{system}\...
pyler's user avatar
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9 votes
1 answer
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Theory behind Linear Predictive Coding (LPC)

What is the theory behind LPC? Why are(were) certain implementations of LPC said to be more tolerant of transmission or encoding errors quantization than other compressed voice encoding schemes? ...
hotpaw2's user avatar
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9 votes
3 answers
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Is Ideal LPF BIBO unstable?

In one of other discussions : How to find frequency response, stability, and causality of a linear system? I found a comment which was quite strong and definitely caught my attention. An ideal ...
Dipan Mehta's user avatar
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9 votes
7 answers
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Intuition behind commutativity of convolution in LTI systems

Why is convolution commutative, as it seems to treat two signals in a different way in an LTI system? If you imagine $y[n] = x[n] \star h[n]$ with $x[n]$ being an input signal and $h[n]$ being the ...
NightRain23's user avatar
9 votes
2 answers
5k 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 ...
VMMF's user avatar
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9 votes
1 answer
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Ideas on Matrix Factorization / Transformations for $ {L}_{1} $ Minimization

I am starting with a typical $\ell_1$ basis pursuit problem: $$ \min_{\mathbf{x}} \Vert \mathbf{x} \Vert_1 \quad \mathrm{s.t.} \quad \Vert \mathbf{ERx} - \mathbf{y} \Vert_2 \leq \epsilon, $$ where $\...
AnonSubmitter85's user avatar
8 votes
4 answers
8k views

Is the first derivative operation on a signal a causal system?

Please help me sort this issue out. Consider a system whose output $y(t)$ is the first derivative of the input signal $x(t)$. We can write the first derivative of an input signal as follows: $$y(t)...
Nishanth Rao's user avatar
8 votes
1 answer
493 views

What is the adjoint of a linear operator and why is it useful?

The concept of linear operators and their adjoints arises frequently in some corners of signal processing, but is not particularly well documented, at least from a signal processing perspective (you ...
Gillespie's user avatar
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8 votes
2 answers
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Negative group delay and envelope advance

I am having a doubt reading about delays in signal processing. Let there be an input to a LTI system with frequency response $H(f)$, given signal $x(t) = a(t)\cos(2\pi f_ot)$, where $a(t)$ is a ...
Dsp guy sam's user avatar
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8 votes
1 answer
3k views

Problem with the definition of Linearity

From the high school maths we know that y=mx+c is a linear equation. However, in DSP the linear system must satisfy Additivity properties which y=mx+c does not hold because of +c. So, is the ...
Zahid Hasan's user avatar
8 votes
6 answers
920 views

The Least Norm Solution of Under Determined Linear System

Suppose I have a matrix $$A= \begin{pmatrix} 1 & 0 & 1 & 0\\ 0 & 1 & 1& 0\\ \end{pmatrix} $$ where the variables are channel information like assume $X_1$, $X_2$, $X_3$ ...
user59419's user avatar
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8 votes
2 answers
6k 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 $u(t)$ ...
Tendero's user avatar
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8 votes
1 answer
904 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 ...
Ehsan Zakeri's user avatar
7 votes
4 answers
4k views

Are all LTI systems invertible? If not, what is a good counterexample?

I have been trying to figure this out for a while now. Everywhere I have looked I could easily find examples of invertible LTI systems, but I could not find any counterexamples. Can anybody shed some ...
Andrew's user avatar
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7 votes
2 answers
1k views

If the convolution of two signals is a unit impulse, what does this tell us?

I have two discrete-time LTI systems whose transfer functions satisfy $h_1[n] * h_2[n]= \delta[n]$. We also know that system 1 is causal and stable. Does $h_1[n] * h_2[n]= \delta[n]$ tell us anything ...
tonythestark's user avatar
7 votes
5 answers
5k views

Why Does the Kalman Filter Remove Only Gaussian Noise?

What and where in the derivation of the Kalman filter is the assumption of Gaussian noise? Why and how does this assumption help?
IG123's user avatar
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7 votes
3 answers
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What is the relationship between poles and system stability?

I see two notions that describe the relationship between poles and system stability. But they are not the same from my understanding The system is BIBO stable if and only if all the poles are in the ...
Joe's user avatar
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7 votes
5 answers
11k views

What Is the Transfer Function of a Moving Average (FIR Filter)?

To make post-processing easier, I export scope measurements as CSV files, which are then post-processed (mostly in Microsoft Excel, which is not the best tool for the job, but it is all I have at my ...
Sclrx's user avatar
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7 votes
2 answers
36k views

How to obtain impulse response from the differential equation of a system?

This year I'm having trouble with my Signals and Systems class. My major subject is Software Engineering and Electric and Electrical Engineering is my Minor. This question was my previous exam ...
M. Jay's user avatar
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7 votes
2 answers
580 views

Why can adaptive IIR filters result in unstable solutions?

For adaptive filtering, both finite and infinite impulse response (FIR/IIR) filters can be utilized. As an advantage of FIR filters in this context, guaranteed stability is often mentioned, while IIR ...
Jonas Schwarz's user avatar
7 votes
1 answer
4k views

Relationship between discrete deconvolution and Toeplitz matrices

I have 2 vectors, $a$ & $c$, both of length $M$. I know they are related by $a*b=c$. My goal is to recover $b$. Obviously $b = \mbox{deconv} (c,a)$. I am only interested in the first $M$ elements ...
DankMasterDan's user avatar
7 votes
1 answer
30k 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 ...
IgNite's user avatar
  • 175
7 votes
1 answer
6k views

Do Causal Discrete-time systems have proper transfer functions?

In the case of continuous-time systems, if the system is causal, its Laplace transfer function is strictly proper (the degree of the numerator is less than the degree of the denominator). Is this ...
Simone Giorgi's user avatar
7 votes
2 answers
734 views

Eigen Function of Linear Time Invariant (LTI) System

Whenever I try to find the derivation of eigen signal of a LTI system, they always start with assumption that the input signal is complex exponential. Is there any other way to prove that this is the ...
Interception's user avatar
7 votes
4 answers
553 views

Does instability make an otherwise LTI system nonlinear (or time-variant)?

I am spinning this question off from the question from johnny. Matt L. and I have had directly opposite conclusions to johnny's question. I want to decouple the question from issues of causality and ...
robert bristow-johnson's user avatar
7 votes
1 answer
343 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 ...
Mike Battaglia's user avatar
6 votes
3 answers
282 views

Estimating the Signal by Deconvolution with a Prior on the Filter Coefficients and the Signal Samples

Assume I have signal $y[n]$ which is a result of convolution between channel $h[n]$ and signal $x[n]$. which means: $$y[n] = h[n] \ast x[n]$$ where $\ast$ is the convolution operation The signal $...
Gze's user avatar
  • 640
6 votes
3 answers
3k views

How to Prove a System Is Invertible?

what i know is that for a system to be invertibel it should be one-one , but I am confused that if i am given a transfer function of a LTI system how can I prove or verify if it is invertible. ...
Buzz bee's user avatar
  • 175
6 votes
3 answers
1k views

Oscillations in a first-order discrete-time linear systems

According to my understanding, continuous first order systems do not exhibit oscillatory behavior because a first order system has only one energy storage element and thus oscillation is not possible. ...
HaneenSu's user avatar
6 votes
3 answers
3k 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 ...
DarkLink's user avatar
  • 307
6 votes
3 answers
611 views

Deconvolution (Linear Convolution) with an Under Determined System of Equations?

If I have a measured signal $\mathbf{y}$ which is the result of the true signal $\mathbf{x}$ convolved with another function (linear and not circular convolution), I always seem to get an ...
hatmatrix's user avatar
  • 239
6 votes
1 answer
812 views

How to combine a rotation matrix and a stretch matrix into a single matrix for easy Fourier Transform

For full disclosure, this is related to homework. I have to find the Fourier Transform of a function that I've boiled down to the following. I have a function $f(x,y)$ that I can think of as another ...
Mark's user avatar
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6 votes
2 answers
1k 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 ...
McZhang's user avatar
  • 73
6 votes
3 answers
895 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 ...
Curiosity's user avatar
  • 357
6 votes
1 answer
4k 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, ...
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6 votes
1 answer
11k views

How to find frequency response, stability, and causality of a linear system?

I have the following transfer function: $$H(s)=\frac{s}{(s+1)(s+2)}$$ How can I find the gain and phase response of the above system? I know the first step has something to do with substituting $s = ...
Cell-o's user avatar
  • 163
5 votes
4 answers
1k views

Can every type of linear filter be modelled by a convolution?

I have an input time series going through a filter that creates another time series as output. If I assume in first approximation that my filter is linear, does it necessarily mean that I can model ...
Poseidon's user avatar
5 votes
2 answers
2k views

Why does reversing the order of these two transfer functions give me different outputs?

Consider these two systems: \begin{align} &u\ {\longrightarrow}\boxed{s}{\longrightarrow}{\boxed{\frac 1s}}{\longrightarrow}\ y\\ &u\ {\longrightarrow}\boxed{\frac 1s}{\longrightarrow}{\boxed{...
Max's user avatar
  • 352
5 votes
3 answers
775 views

Discrete State Space Model - Why Are We Calculating $ x \left[ k + 1 \right] $ Instead of $ \dot{\boldsymbol{x}} \left( t \right) $?

A continuous state space model is defined as follows. $$ \dot{\boldsymbol{x}}(t) = A \boldsymbol{x}(t)+ B \boldsymbol{u}(t) \\ \boldsymbol{y}(t)= C \boldsymbol{x}(t)+ D \boldsymbol{u}(t) $$ If we ...
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