# 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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### Linear system stability criteria

Suppose we have a closed-loop system $H(s)=\frac{A(s)}{1+A(s)f(s)}=\frac{A(s)}{1+T(s)}$. I've seen the stability of the system stated a couple ways: If $H(s)$ has any poles in the RHP, then it is ...
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### When is a LTI IIR $H(z)$ system minimum phase

EDIT: I failed to mention that the system's inverse also needs to be causal and stable. I cannot wrap my mind around on how when a system and its inverse are both causal and stable and LTI IIR it is ...
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### Please explain the following Linear Prediction estimate graph

I am reading this doc https://librosa.org/doc/latest/generated/librosa.lpc.html#librosa.lpc? I can't understand the graph for linear prediction coefficients. Please help explain it as I a new to this ...
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### Signal separation

This is probably a blind signal separation problem of sorts, but it seems like it should be easier than I am finding it. Let’s say I have N time series, each of length [M x 1] that are a superposition ...
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### Why is particular solution zero for an impulse excitation signal?

We were being taught the impulse response for a series RC Circuit- consisting of simply one resistance, one capacitor, and an impulse excitation all in series. I get that the homogeneous part of the ...
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I have a DSP exam coming up this summer and I got an abysmal mark on an assignment earlier this year. Obviously, the lecture slides are not enough for me to understand DSP, so I am wondering where can ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 ...