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 ...
4
votes
0answers
50 views
Practical impulse response estimation using NumPy
I would like to estimate the impulse response of a simple environment (room, two speakers, one microphone). I am aware of the issues/limitations of working with "near" LTI systems. That said, I have a ...
3
votes
1answer
77 views
Impulse Response Question
For the differential equation
$$\frac{d^2y(t)}{dt^2}+\frac{3dy(t)}{dt} + 2y = x(t),$$
I was able to find the frequency response as $$H(j\omega)= \frac{1}{-\omega^2+3j\omega+2}.$$ However, I am not ...
1
vote
1answer
53 views
How can frequency dispersion in a vibrating string be modelled using an LTI filter?
According to JOS on stiff-string synthesis, stiff strings (like on a piano) introduce inharmonicity (i.e. the harmonics of the tone are not all in tune) due to different frequency components of the ...
3
votes
1answer
58 views
How to perform model fitting for system identification
I am having a really hard time in understanding how to formulate a model say linear AR model to represent a communication channel or maybe any motion. I have the experimental data representing the ...
4
votes
1answer
106 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 ...
1
vote
1answer
62 views
Calculating a delayed response of a system
Suppose I have a system:
$$y[n] = x[n] + x[n-1] + x[0]$$
To see if this system is time-variant, we can compare $H\{x[n-1]\}$ to $y[n-1]$ and check that they are different.
I think the delayed (by ...
3
votes
1answer
109 views
Group Delay of a system
Say I have an LTI system whose impulse response h[n] is:
$$
h[n] = \delta[n] + \delta[n-4]
$$
I want to find the group delay of this system. I know the group delay ($grd$) is defined as:
$$
...
2
votes
1answer
77 views
Why is the state space representation, $dx/dt = A x(t) + B u(t)$ linear?
One thing puzzles me about the state space representation and that is why is it claimed that such systems are linear and time-invariant but particularly why are they claimed to be linear? The ...
7
votes
0answers
197 views
How to deduce a linear system's impulse response from a set of input/output signals?
I want to know how to solve those types of problems.. is it by inspection ?
Consider the linear system below. When the inputs to the system $x_1[n]$, $x_2[n]$ and $x_3[n]$, the responses of the ...
4
votes
0answers
61 views
Concept of finding forced response from difference equation?
I know how the process works, first from difference equation make a homogeneous equation and then find natural response , then find a particular response (in the form of input) substitute it back in ...
3
votes
1answer
52 views
ARMA Process Representation as AR
Let's say we have an ARMA (Auto-regressive Moving Average Model) process where the transfer function is a minimum phase system (Namely Invertible).
By Wold's Decomposition it is guaranteed to have MA ...
0
votes
1answer
105 views
Inverse Discrete Time Fourier Transform of a given Function using the definition
I would like to compute the Inverse Fourier Transform of a given function using the DTFT definition
$$H(e^{jω})=\frac{1}{(2−e^{jω})(2−e^{−jω})}$$
But I have to use the definition of the DTFT
...
3
votes
1answer
147 views
How do I compute the impulse response of a robot?
I am trying to model a robot (my question in robotics.SE), so that I may train it using some reinforcement learning techniques. I was wondering if I can compute an impulse response of a four wheel ...
0
votes
1answer
135 views
Calculating the system output using frequency response
Given an input signal $$x(n)=\cos(6\pi n +\frac{\pi}{6})$$ and system $$y(n)=0.5x(n)-0.1x(n-1)$$. In this case, the coefficients of the difference equation are $a_0=1$, $b_0=0.5$, and $b_1=$. The ...
1
vote
1answer
82 views
Verifying Linear Time Invariance
I have a system of the form:
$$T(x(n))=x(n)+3x(n-2)-5x(n-3)x(2n)$$
I claim that
$$T(x(n-k))=x(n-k)+3x(n-k-2)-5x(n-k-3)x(2n-2k),$$
$$y(x(n-k)= x(n-k)+3x(n-k-2)-5x(n-k-3)x(2n-2k)$$
and
...
7
votes
5answers
642 views
Why do linear systems show sinusoidal fidelity?
I am looking for a proof for sinusoidal fidelity. In DSP we study a lot about linear systems. Linear systems are homogenous and additive. One more condition it satisifies is that if a signal is a sine ...
7
votes
1answer
272 views
Theory behind Linear Predictive Coding (LPC)
What is the theory behind LPC? Why are(were) certain implementations of LPC said to be more tolerant of transmission or encoding errors quantization than other compressed voice encoding schemes?
...
12
votes
1answer
136 views
Fitting new images from a SVD/PCA calculation
I'm trying to replicate the ideas from the Eigenface page on wikipedia. From a hundred sample images represented by a data matrix $\bf X$ (where each image flattened to a vector of length $n$, thus ...
4
votes
1answer
333 views
Finding the Wiener filter transfer function
I am trying to understand the mechanism of finding the transfer function for a Wiener function.
...
6
votes
1answer
200 views
How to combine a rotation matrix and a stretch matrix into a single matrix for easy Fourier Transform
For full disclosure, this is related to homework. I have to find the Fourier Transform of a function that I've boiled down to the following.
I have a function $f(x,y)$ that I can think of as another ...
5
votes
2answers
339 views
Is Ideal LPF BIBO unstable?
In one of other discussions : How to find frequency response, stability, and causality of a linear system?
I found a comment which was quite strong and definitely caught my attention.
An ideal ...
6
votes
1answer
2k views
How to find frequency response, stability, and causality of a linear system?
I have the following transfer function:
$$H(s)=\frac{s}{(s+1)(s+2)}$$
How can I find the gain and phase response of the above system? I know the first step has something to do with substituting $s = ...
16
votes
4answers
7k views
What is meant by a system's “impulse response” and “frequency response?”
Can anyone state the difference between frequency response and impulse response in simple English?
7
votes
1answer
645 views
How do I find a system's impulse response from its state-space repersentation using the state transition matrix?
Suppose we have a linear represented in the standard state space notation:
$$ \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 ...
