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 ...
0
votes
0answers
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)] ...
0
votes
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/...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
4
votes
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 ...
-1
votes
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:
\...
0
votes
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 ...
0
votes
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(...
0
votes
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 ...
-2
votes
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
0
votes
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?
0
votes
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 ...
0
votes
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 ...
2
votes
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 ...
1
vote
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.
0
votes
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.
0
votes
0answers
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 ...
0
votes
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?
0
votes
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 ...
0
votes
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 ...
-1
votes
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:
$$ x \left [ n \right ] = \begin{cases}
1 &...
0
votes
2answers
57 views
Discrete filter $y[n] = \frac{1}{3} x[n] + \frac{1}{3} x[n-1] + \frac{1}{3} x[n-2]$
Consider the filter which equation can be represented by $y[n] = \frac{1}{3}x[n] + \frac{1}{3}x[n-1] + \frac{1}{3}x[n-2]$, in $x[n]$ and $y[n]$ are sequence of input and output of the system ...
1
vote
1answer
79 views
Determine if $ y[n] = ny[n-1] + x[n]$ is linear time invariant and BIBO stable
Check if the following system is linear time invariant and BIBO stable..
$$
y[n] = ny[n-1] + x[n]
$$
for $n\ge 0$. We are also given that the system is at rest (i.e. $y[−1] = 0$).
I know that to ...
2
votes
2answers
75 views
Frequency response of marginally stable LTI systems
The frequency response of a system is defined as:
$$\int_0^\infty{h(t)e^{-j\omega t}dt}$$
where $h(t)$ is the impulse response. But in marginally stable systems, $h(t)$ does not decay so the integral ...
3
votes
2answers
178 views
Are discrete systems defined by LCCDE always LTI?
Suppose a discrete-time system is defined by linear constant-coefficient difference equation
$$\sum_{k=0}^{N} a_k y[n-k] = \sum_{k=0}^{M} b_k x[n-k]$$
where at least two different coefficients $a_i,...
2
votes
3answers
156 views
Proof of linearity
I have this system:
$$y[n] − 4y[n − 1] + 4y[n − 2] = 20x[n] + 10x[n − 1]$$
I have no idea how to prove if the system is linear because it depends on future outputs.
0
votes
0answers
17 views
Relationship between signal amplitude and input output correlation
I have a linear system with input $ x(t) $ and output $ y(t)$ given by $$ y(t) = \int_0^\infty K(t')x(t-t')dt', $$ where $K(t)$ is a known kernel, with some parameters.
The functional form of $K(t)$ ...
0
votes
1answer
44 views
Explicit a succession using z inverse transform
Is it possible to explicit $y(n)$ of this mathematical succession in recursive form using z inverse transform?:
$ y(0) = 1 \\ y(n+1) = 2y(n) + 3 $
I can't write ...
0
votes
3answers
94 views
Least Squares with blocks/updates
I have a continuous-time system that I want to fit via least squares. I just send $N$ digital samples $x[n]$ through the system and receive (via analog signal chain, ADC etc) $N$ digital samples $y[n]$...
1
vote
1answer
85 views
Is this two input discrete system linear? [closed]
Given two inputs $\: x_1[n]\: x_2[n]\:$Is the system $\:y[n]=x_1[n]\times x_2[n]\:$ linear ?
My Approach:
$(x_1\times x_2)[n]=S_1[n]\rightarrow Y_1[n]$
$(x_3\times x_4)[n]=S_2[n]\rightarrow Y_2[n]$
$...
0
votes
3answers
110 views
Distinguishing FIR and IIR from difference equation
Find the transfer function of the difference equation $$y_n = x_n + 1.2y_{n-1}$$
I fail to understand how one can distinguish between an FIR filter and an IIR filter by looking at the equation given ...
0
votes
0answers
34 views
Relationship of the cross correlation and the amplitude of the signal
I would like to characterise the maximum amplitude $|x(t)|$ (or similar quantity $|x(t)|$ or $|x(t)|^2$) of the following linear system
$$ x(t) = \int_0^\infty K(t')S(t-t')dt',$$
where $ S(t) $ is ...
-1
votes
1answer
227 views
How to conclude LTI, causality and BIBO stability of a system represented by a differential equation?
I have started to learn about systems represented by differential equations in Oppenheim's Signals & Systems, and I got really confused about it. I am trying to understand how I can show that a ...
0
votes
0answers
25 views
1
vote
1answer
64 views
Eigenfunction property for LTI sinusoidal and the sinusoidal steady-state response
All LTI systems possess the eigenfunction property for complex exponential inputs. That is (restricting our attention to periodic complex exponentials), if $e^{j\omega_k t}$ is an input to the LTI ...
0
votes
1answer
41 views
Time-invariant and Time-varying Systems
Determine whether the following system is time-invariant or not:
$y(t)=x(t)\sin 10\pi t$
Solution:
Given: $y(t)=x(t)\sin 10\pi t$
$y(t)=T[x(t)]=x(t)\sin 10\pi t$
The output due to input delayed ...
4
votes
1answer
199 views
Signals and systems book with linear algebra approach
I am currently taking a course on discrete-time signals and systems. I am using B. P. Lathi's Signal Processing and Linear Systems, which I don't like at all, as it doesn't draw any paralells to ...
0
votes
2answers
249 views
Why $y[n] = x[-n]$ is not time-invariant?
I followed these steps, but the answer still says that this system is time-invariant
let: $x_2[n] = x[n-k]$
$$\begin{align}
y_2[n] &= x_2[-n] \\
&= x[-(n-k)] \\
&= x[k-n] \\
\end{align}...
1
vote
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 ...
0
votes
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: $$h(t) = \frac{\delta(t+\tau)-\...
5
votes
1answer
187 views
Does “improper” imply that a system cannot be stable and causal?
This answer and the comments in it made me wonder whether the following statement is true:
If a transfer function is improper, then that system cannot be causal and stable at the same time.
I had ...
0
votes
0answers
31 views
Recursive systematic convolutional code with TCM
My question is:
What are the outputs from the TCM encoder ?
6
votes
2answers
588 views
Initial conditions for systems described in state space - LTI or not?
Suppose we have some system given by
$$\begin{aligned}
\dot{x}(t) &= Ax(t) +Bu(t) \\
y(t) &= Cx(t)+Du(t)
\end{aligned}$$
where $x(t)$ are the state variables, $y(t)$ is the output and $u(t)$ ...
7
votes
1answer
393 views
Physical Meaning of Negative Group Delay for causal LTI systems
I have implemented in Matlab (with minor variations) the example 5.1.2 "Illustration of Effects of Group Delay and Attenuation" I found in Alan Oppenheim's Discrete-Time Signal Processing 3rd edition. ...
0
votes
1answer
52 views
Problem understanding steps towards defining the convolution sum
I have a problem understanding the reasoning behind a step that was taken to characterize a LTI system.
So, we were told the following:
For each integer $k$, we have the following function: $$ \...
0
votes
0answers
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!!
0
votes
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 ...
0
votes
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 ...
0
votes
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:
$$
h[n] =\left\{\underline{1} , -2, 2, -1\right\}
$$
...
