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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 ...

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21 views

Expressing a square as product of rect functions

A square is centered at origin and length of one of its side is 'a' units. In terms of rect functions, such a square can be expressed as product of two rect functions, i.e., $f_a(x,y) = A [rect(mx)] ...
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1answer
32 views

Signals and Systems - LTI - Transforms - Impulse Response

I have $x(t)-> LTI -> y(t)$ where $LTI=h(t),H(jw)$. and $H(jw)=ab/((a+jw)(b+jw))$ where a and B are real numbers. I am wanting to find the impulse response $h(t)$ as well as the input/...
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0answers
14 views

Fitting filter coefficients within a narrow band of frequencies, and using them as multi-rate

Intro: I have a vibrating system that I am trying to model. I've input a sine sweep up to 1200Hz and measured the device's input and output into MATLAB. I am interested in finding the inverse filter ...
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1answer
29 views

3-tap FIR filter: simple expression for $H(e^{j\omega})$ using trigonometric identities

We have a linear time-invariant system described by the input-output relation $$y[n] = x[n] + 2x[n - 1] + x[n - 2]$$ Below is my approach to analyze this system. The impulse response of this system ...
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1answer
59 views

How to find the impulse response of this system with complex roots?

I've been working trying to analyze a causal stable system. Hopefully a numeric example such as the one I am working on, and the problems I face during it could be useful to others. I have this ...
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3answers
56 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 ...
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1answer
61 views

Linearity of the given system [closed]

I am given the following system and I am checking the additive property: $$y(t)=x(e^t)$$ where $y(t)$ is the output and $x(t)$ is the input given to the system. Now this is what I did so far: \...
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0answers
42 views

Depending on how a system function is written, will it always have the same zeros and poles?

There are many ways to write a system function. In terms of direct forms, cascade, parallel, transposed. Also the system function depends on if you write the poles and zeros in terms of $1+az^{-1}$ or ...
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1answer
26 views

Why is the impulse response function of this system 0?

Suppose I have an system $ y(t) = t^{2}x(t)$. The impulse response of this system would be: $h(t) = t^{2} \delta(t)$. Since $\delta(t) = 0$ for $t \neq 0 , h(t) = 0$ for $t \neq 0$. And at $t=0, h(...
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0answers
17 views

Correlation between elements of a video

Should there be an expected correlation between the audio signal and the frame pixel data, considering the frames of a video? If so, what is the reason for such correlation? Pondering over the topic ...
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1answer
35 views

classify the system if it's linear , non-linear , time variants or invariants [closed]

classify the system if it's linear , non-linear , time variants or invariants
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2answers
60 views

Conversion from stationarity to non-stationarity

Is there any way to convert a non-stationary signal to a stationary one, perform operations on it meant for a stationary signal and then convert it back to the non-stationary one?
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0answers
16 views

Wiener Filter: spacing of the data

I read in an article that for the discrete version of Wiener filter as proposed by Levinson, the data can be arbitrarily spaced. What is implied by this? I believe that the spacing does matter to ...
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2answers
569 views

What does it mean for a function to have frequencies?

In a lecture, my professor mentioned that $\cos$ has two frequencies. I see that using the inverse Euler's formula we can express $\cos$ as a some linear combination of complex exponentials, each with ...
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3answers
69 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 ...
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1answer
28 views

Linear combination of DT unit impulse

Came across this example in class but I'm not sure how the expression $x(n)$ was derived.
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1answer
34 views

Transfer function of open loop system

Suppose i have the system equation $Y(s) = G(s)X(s)+ 3T(s)$ Then what is the transfer function of the system? I know that the transfer function is $Y(s)/X(s)$, but i can't get that expression.
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13 views

Are eigenvalues of A poles of H(z)? [duplicate]

So I got this discrete LTI system of which the transfer function is defined by H(z). Given the state space model: x[k+1] = A x[k] + B u[k] and y[k = C x[k] + D u[k] the Z-transform can be written ...
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0answers
47 views

Trying to prove BIBO stability implies the boundedness of the impulse response of an LTI system without a counterexample

I have tried to prove that BIBO stability implies that the impulse response of the LTI system, can someone please check if the proof is correct?
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1answer
47 views

Damped Harmonic Oscillation as an LTI

The goal is to create an LTI filter which is exactly, or approximates, damping of harmonic modes. The equation of course is: $$\frac{d^2 x}{dt^2} + 2 \xi \omega \frac{dx}{dt}+\omega^2x=0$$ This can ...
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3answers
89 views

Discrete state space model - Why are we calculating $x[k+1]$ instead of $\dot{\textbf{x}}(t)$?

A continuous state space model is defined as follows. $$ \dot{\textbf{x}}(t)=\textbf{A}\textbf{x}(t)+\textbf{B}\textbf{u}(t) \\ \textbf{y}(t)=\textbf{C}\textbf{x}(t)+\textbf{D}\textbf{u}(t) $$ If we ...
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1answer
50 views

Calculate the Output of Linear Time Invariant System Given it Impulse Response [closed]

A filter is defined as $ h \left[ n \right] = \delta \left[ n \right] - \delta \left[ n - 1 \right] $. Given a signal $ h \left[ n \right] $ defined as: $$ x \left [ n \right ] = \begin{cases} 1 &...
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2answers
57 views

Discrete filter $y[n] = \frac{1}{3} x[n] + \frac{1}{3} x[n-1] + \frac{1}{3} x[n-2]$

Consider the filter which equation can be represented by $y[n] = \frac{1}{3}x[n] + \frac{1}{3}x[n-1] + \frac{1}{3}x[n-2]$, in $x[n]$ and $y[n]$ are sequence of input and output of the system ...
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1answer
79 views

Determine if $ y[n] = ny[n-1] + x[n]$ is linear time invariant and BIBO stable

Check if the following system is linear time invariant and BIBO stable.. $$ y[n] = ny[n-1] + x[n] $$ for $n\ge 0$. We are also given that the system is at rest (i.e. $y[−1] = 0$). I know that to ...
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2answers
75 views

Frequency response of marginally stable LTI systems

The frequency response of a system is defined as: $$\int_0^\infty{h(t)e^{-j\omega t}dt}$$ where $h(t)$ is the impulse response. But in marginally stable systems, $h(t)$ does not decay so the integral ...
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2answers
178 views

Are discrete systems defined by LCCDE always LTI?

Suppose a discrete-time system is defined by linear constant-coefficient difference equation $$\sum_{k=0}^{N} a_k y[n-k] = \sum_{k=0}^{M} b_k x[n-k]$$ where at least two different coefficients $a_i,...
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3answers
156 views

Proof of linearity

I have this system: $$y[n] − 4y[n − 1] + 4y[n − 2] = 20x[n] + 10x[n − 1]$$ I have no idea how to prove if the system is linear because it depends on future outputs.
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0answers
17 views

Relationship between signal amplitude and input output correlation

I have a linear system with input $ x(t) $ and output $ y(t)$ given by $$ y(t) = \int_0^\infty K(t')x(t-t')dt', $$ where $K(t)$ is a known kernel, with some parameters. The functional form of $K(t)$ ...
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1answer
44 views

Explicit a succession using z inverse transform

Is it possible to explicit $y(n)$ of this mathematical succession in recursive form using z inverse transform?: $ y(0) = 1 \\ y(n+1) = 2y(n) + 3 $ I can't write ...
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3answers
94 views

Least Squares with blocks/updates

I have a continuous-time system that I want to fit via least squares. I just send $N$ digital samples $x[n]$ through the system and receive (via analog signal chain, ADC etc) $N$ digital samples $y[n]$...
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1answer
85 views

Is this two input discrete system linear? [closed]

Given two inputs $\: x_1[n]\: x_2[n]\:$Is the system $\:y[n]=x_1[n]\times x_2[n]\:$ linear ? My Approach: $(x_1\times x_2)[n]=S_1[n]\rightarrow Y_1[n]$ $(x_3\times x_4)[n]=S_2[n]\rightarrow Y_2[n]$ $...
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3answers
110 views

Distinguishing FIR and IIR from difference equation

Find the transfer function of the difference equation $$y_n = x_n + 1.2y_{n-1}$$ I fail to understand how one can distinguish between an FIR filter and an IIR filter by looking at the equation given ...
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34 views

Relationship of the cross correlation and the amplitude of the signal

I would like to characterise the maximum amplitude $|x(t)|$ (or similar quantity $|x(t)|$ or $|x(t)|^2$) of the following linear system $$ x(t) = \int_0^\infty K(t')S(t-t')dt',$$ where $ S(t) $ is ...
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1answer
227 views

How to conclude LTI, causality and BIBO stability of a system represented by a differential equation?

I have started to learn about systems represented by differential equations in Oppenheim's Signals & Systems, and I got really confused about it. I am trying to understand how I can show that a ...
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25 views

Complete Response Linear System

I couldn't find out the answer for item d.
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1answer
64 views

Eigenfunction property for LTI sinusoidal and the sinusoidal steady-state response

All LTI systems possess the eigenfunction property for complex exponential inputs. That is (restricting our attention to periodic complex exponentials), if $e^{j\omega_k t}$ is an input to the LTI ...
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1answer
41 views

Time-invariant and Time-varying Systems

Determine whether the following system is time-invariant or not: $y(t)=x(t)\sin 10\pi t$ Solution: Given: $y(t)=x(t)\sin 10\pi t$ $y(t)=T[x(t)]=x(t)\sin 10\pi t$ The output due to input delayed ...
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1answer
199 views

Signals and systems book with linear algebra approach

I am currently taking a course on discrete-time signals and systems. I am using B. P. Lathi's Signal Processing and Linear Systems, which I don't like at all, as it doesn't draw any paralells to ...
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2answers
249 views

Why $y[n] = x[-n]$ is not time-invariant?

I followed these steps, but the answer still says that this system is time-invariant let: $x_2[n] = x[n-k]$ $$\begin{align} y_2[n] &= x_2[-n] \\ &= x[-(n-k)] \\ &= x[k-n] \\ \end{align}...
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1answer
72 views

Is it possible to replace an integrator system with an equivalent differentiator?

I have a system whose input-output relation is as follows $$y(t)=x(t)+\int_{-\infty }^{t} x(\tau) \,\mathrm d \tau$$ Can I create an equivalent system by using differentiators rather than ...
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1answer
51 views

Finding the output of a system where the input is a sum of complex exponentials

So, I have to find $H\{ x(t)\})$ (which is an LTI system), where $$x(t) = \sum_{k=0}^{\infty} a_ke^{ \ jw_kt}$$ and where the impulse response of the system is given by: $$h(t) = \frac{\delta(t+\tau)-\...
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1answer
187 views

Does “improper” imply that a system cannot be stable and causal?

This answer and the comments in it made me wonder whether the following statement is true: If a transfer function is improper, then that system cannot be causal and stable at the same time. I had ...
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31 views

Recursive systematic convolutional code with TCM

My question is: What are the outputs from the TCM encoder ?
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2answers
588 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 $u(t)$ ...
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1answer
393 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 3rd edition. ...
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1answer
52 views

Problem understanding steps towards defining the convolution sum

I have a problem understanding the reasoning behind a step that was taken to characterize a LTI system. So, we were told the following: For each integer $k$, we have the following function: $$ \...
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46 views

Frequency Response Question on LSI System

Someone please explain me the question its seems very complicated to me. I just want know what the question asking and how to solve it i dont want fully solved solution. Thank you!!
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1answer
78 views

Understanding and Correcting phase shift in General Linear Phase FIR filters

I have been learning about General Linear Phase Filters and the four standard FIR implementations. I was wondering in the case $\beta = \frac{\pi}{2}$ or $\beta = \frac{3\pi}{2}$ if there was a way ...
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3answers
91 views

Causality as applied to capacitors

This question stems from a point of confusion that I still have about the causality, linearity, and time-invariance in LCCDEs. I wanted to use the capacitor as an example. Consider a capacitor with ...
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1answer
237 views

Determining Causality From Discrete Impulse Response

Trying to wrap my mind around the concepts of this one... Consider the following impulse response $h[n]$ for a linear, time-invariant system: $$ h[n] =\left\{\underline{1} , -2, 2, -1\right\} $$ ...