Questions tagged [dirac-delta-impulse]
The Dirac delta impulse is a mathematical idealization which is used in the theory of DSP, e.g., for the description and analysis of sampling, impulse responses, Fourier transform of pure tones, etc.
46
questions
0
votes
4
answers
86
views
Why use white noise over an impulse signal (or vice versa) to understand how a system would behave?
An impulse signal and white noise both have a flat spectrum. When they are passed into a system, the response will indicate how each frequency component will be modified by the system. Is there any ...
- 55
0
votes
3
answers
144
views
Rigorous derivation of autocorrelation of white noise
It is said that the autocorrelation of white noise is the dirac delta function $\delta(\tau)$, but I don't know how to derive that...
Since white noise is a function with constant power spectral ...
1
vote
2
answers
122
views
Why is white noise uncorrelated for any two different samples?
Consider white noise with spectrum density $N_0/2$, it is known that the autocorrelation is given by:
$$R(\tau) = \frac{N_0}{2}\delta(\tau)$$
Meanwhile the definition of delta function in the ...
- 123
2
votes
1
answer
109
views
Convolution with Kronecker delta
I know that convolution with delta shifts a signal. As for example, $x \!\left[ n \right] * \delta \!\left[ n - 2 \right] = x \!\left[ n - 2 \right]$. How to do convolution with $x \!\left[ -n \right]$...
- 21
1
vote
1
answer
109
views
Impulse response of a causal LTI system without using Laplace transform
I have this differential equation that models a causal LTI system:
$$
\ddot{v}(t) - \dot{v}(t) - 2v(t) = \ddot{u}(t) + 2\dot{u}(t) + u(t)
$$
I was asked to find the impulse response both by using ...
- 13
1
vote
1
answer
158
views
How to find impulse response for the given system?
How can I find the impulse response for the following system in time domain? I actually would like to find my mistake in my attempt. Below is what I have tried according to the answer given for this ...
- 113
3
votes
2
answers
376
views
Why is sampling a signal equivalent with multiplying with a Dirac comb?
Given a continuous time signal $f(t)$, we can sample it signal by multiplying with a Dirac comb (impulse train)
$$\bar{f}(t) = \sum_{n=-\infty}^{\infty} f(nT) \delta(t-nT) \tag{1}$$
where each impulse ...
- 480
0
votes
1
answer
88
views
Multiplication of function with Dirac impulses: sketch signal
I should sketch the signal below:
I think I should use delta dirac function sampling theorem but I don't know how in this special case
I know that $x(t)δ(t-t_{0})=x(t_{0})δ(t-t_{0})$ but I don't have ...
0
votes
0
answers
54
views
What is the expected amplitude of the flat frequency response of a Kronecker delta function? Does sample rate affect it?
I was instructed here:
In the sampled-time domain a Kronecker delta, $\delta(n)$, has a perfectly flat spectrum. If you look at a table of discrete-time Fourier transform pairs, this is at the top ...
- 533
0
votes
1
answer
68
views
What happens to the frequency response as a single sample Kronecker delta impulse widens into a square pulse, and then into two distinct steps?
1) Kronecker
I was instructed here that a single sample Kronecker delta unit impulse function (goes from 0 to $A$ to 0 again in single sample) has a white noise type frequency response - all ...
- 533
2
votes
1
answer
121
views
Fourier Transform of $|t|$
I was going through Papoulis' book (The Fourier Integral and its Applications) when I came across the Fourier Transform for $|t|$. To find it he writes $|t|$ as (I am not sure how):
$$|t| = -\frac{1}{\...
- 1,593
1
vote
3
answers
223
views
Phase of a Complex Exponential
It is known that the discrete-time Fourier transform (DTFT) of a complex exponential
$$
x[n] = e^{j\omega_0 n}
$$
is
$$
X(e^{j\omega}) = 2\pi \sum_{k = -\infty}^{\infty} \delta(\omega - \omega_0 + 2\...
2
votes
1
answer
166
views
Fourier transform of $|x_\mathrm{a}(t)|^2$
Let $x_\mathrm{a}(t)$ be the analytic signal for real signal $x(t)$. I want to find an expression for $\mathscr{F}\{|x_\mathrm{a}(t)|^2\}(f)$ in terms of $x(t)$. The analytic signal can be written as $...
- 726
1
vote
1
answer
96
views
Why are convolutions between those functions equivalent (signal processing for theoretical neuroscience)?
I'm reading a book on theoretical neuroscience [1], in which the following definitions are given:
$\rho(t)=\sum_{i=1}^n \delta(t-t_i)$
where $\delta$ is Dirac's delta and the $t_i$ are timestamps at ...
- 111
0
votes
1
answer
142
views
Sampling a pulse train with a controllable square wave
I have an issue regarding a sampling process of a pulse train in an image sensor based on events. Basically, these are a family of image sensors in which each pixel outputs a train of pulses, and the ...
- 1
2
votes
3
answers
756
views
How can I find the following Fourier Transform without directly using FT pairs?
The question is to find the CTFT of $$x(t) = e^{-t}u(t)\cdot \sum_{n=-\infty}^{\infty} \delta\left(t-\frac{n}{2}\right)$$
Now I know that multiplication in time means convolution in the Fourier Domain ...
user67157
4
votes
2
answers
474
views
Fourier transform of a propagating Dirac delta
I have a spatio-temporal signal $f(x, t)$ that propagates at a constant velocity $v$. To represent that propagation I'm reading in multiple papers that we can represent it in the Fourier domain with ...
- 141
2
votes
3
answers
80
views
Unit impulse added to non-zero interval of a function - graphical representation?
Followup to this question, let the ramp be $r_0(t)$. We seek to plot $x(t) = r_0(t) + \delta(t + 1)$. Should it be as in left, right, or neither? Is there a convention?
Note, in either case we could ...
- 9,054
1
vote
1
answer
111
views
Intuitive or physical explanation of DTFT$\{1\}=2\pi\delta(\omega)$
I am trying to understand the fact that
"The DTFT of 1 (an infinite discrete sequence of unit impulses from from $-\infty$ to $+\infty$) is $2\pi\delta(\omega)$"
in an intuitive or physical ...
3
votes
1
answer
268
views
What is the reason the Kronecker delta function is noted this way in DSP?
I'm trying to make sense of different notations used in DSP. This is sometimes quite hard since notation brackets like in x[n] are used both to denote a single value or a sequence.
In mathematics, the ...
- 473
3
votes
1
answer
464
views
Integral of product of two shifted Dirac impulses
In this question the multiplication of two impulse functions is discussed
However, what if I have two shifted impulses say:
$z(t) = \delta(t-\tau)\cdot \delta(t-\beta)$
Now if I want to integrate $z(t)...
- 1,593
4
votes
1
answer
445
views
Fourier Transform of impulse train proof in Julius O Smith's textbook
$\DeclareMathOperator{\sinc}{sinc}\DeclareMathOperator{\asinc}{asinc}$I'm trying to follow the proof of the Fourier Transform for an Impulse train given in Julius Smith's textbook. I come across the ...
- 225
4
votes
2
answers
981
views
Product of Doublet and Arbitrary Function
We know that the product of the delta function and another function samples the latter function. That is,
$$
\delta(t-\tau)f(t)=\delta(t-\tau)f(\tau)
$$
Does the doublet function retain this same ...
- 143
0
votes
1
answer
61
views
How can I get rid of this unexpected minus sign on my inverse Fourier transform of two impulse functions?
I'm trying to find the inverse Fourier transform of two impulse functions, which correspond to the Fourier transform of the function $h(t)=A\sin(2πf_0t)$.
The Fourier transform of the above sine ...
16
votes
6
answers
77k
views
What is the first derivative of Dirac delta function?
Could you please help me in a simple way, what is the first derivative of a Dirac delta function?
I found this answer:
The informal answer is a positive Delta function immediately followed by a ...
- 413
0
votes
3
answers
2k
views
What is the Effect of Multiplying a Function by the Unit Impulse Function in the Frequency Domain?
I know about the the shifting property of the impulse function in the time domain as can be seen in equation $(1)$.
$$
\int_{-\infty}^{\infty} f(x)\delta(x - a)dx = f(a)\tag{1}
$$
But what is the ...
- 1
2
votes
3
answers
8k
views
Time scaling and shifting of delta function
Learning signals and systems.
Solving time scaling and shifting problems.
For the question
$$x(t) = u(2t - 1)$$
First we shift by 1 to the right side and then we do time scaling , i.e divide by 2 ...
- 25
3
votes
3
answers
447
views
How is $\delta(at+b)=\frac{1}{|a|}\delta(t+b/a)$?
This result has been used in the second last line of the pic. I don't know why it's true. Both functions are zero for $t$ not equal to $-b/a$. But at $t=-b/a$, a scaling factor $1/|a|$ has been ...
user45317
2
votes
2
answers
1k
views
Confusion in deriving formula for fourier tansform of impulse train
I was trying to derive fourier transform for impulse train :
I know how to solve for this using using properties of fourier transform. But now I wanted to use a brute force approach to it so I did ...
- 55
0
votes
1
answer
248
views
Inverse Fourier Transform Dirac impulse with scaled argument
Currently, I am dealing with the sampling problems and I don't understand how to calculate inverse Fourier transform of a scaling impulse function
$\textrm{IFT}\{\delta(\Omega T)\} = ?$, $T$ is ...
- 3
1
vote
2
answers
577
views
Help regarding property of unit impulse function
We know that using properties of unit impulse function it can be shown that
$$\int_{t_1}^{t_2} x(t) \delta^{(n)}(t-t_0) dt=(-1)^nx^{(n)}(t_0),\quad t_1<t_0<t_2$$ (source: Continuous and ...
- 75
1
vote
1
answer
3k
views
Unit Impulse : Bounded or Unbounded?
As the title suggests, I have a confusion here. In a Systems text I am going through, it mentions of the Unit Impulse as an unbounded signal. Yes, the unit impulse "height" is unbounded, but the "...
- 47
0
votes
1
answer
520
views
Fourier transform of dirac comb with function: The scaling factor
Multiplication in the time domain corresponds to convolution in the frequency domain:
$$
f(t) \cdot x(t) \iff F(j \omega) * X( j \omega) \tag*{No scaling factor}
$$
I know the fourier transform of ...
- 53
0
votes
3
answers
582
views
Definition of sampling using delta or indicator function?
I just came from a class where the professor showed a slide with the definition of sampling:
But I do not understand how we can multiply a signal $x(t)$ with the delta function $\delta(t)$, as the $\...
- 103
1
vote
2
answers
1k
views
Fourier transform and impulse function $\delta(\omega)$
Why does impulse function $\delta(\omega)$ keep occurring in the Fourier transform expression of standard functions like $\sin(t)$, $\cos(t)$, constant function, unit step $u(t)$ etc? (can someone ...
- 29
-1
votes
1
answer
5k
views
What is the product of Dirac Delta function and $t$? [closed]
$t\cdot \delta (t)$
Does it equal to zero? If so, how can we prove it?
- 133
2
votes
3
answers
20k
views
Multiplication of two impulse function $\delta(t)\cdot \delta(t)=?$
If a impulse function is mulplied with a function $f(x)$ then the formula will be apply
$$f(x)\delta(x-a)=f(a)\delta(x-a)$$ so from this formula mulplication of two impulse function will be
$$\...
- 578
9
votes
4
answers
11k
views
Is the Dirac delta (impulse) signal a power signal or an energy signal?
I'm a beginner so sorry if this question is very fundamental. Dirac impulse has finite area i.e = 1. But I've heard that $|\delta(t)|^2$ is undefined. So area under $|\delta(t)|^2$ is also undefined ...
1
vote
2
answers
4k
views
Laplace transform of product of signal and impulse train
I'm reading 'Discrete Time Control Systems' book by Ogata and came across a few statements (specifically, (3-1) and (3-2)) which I have not been able to understand.
It is said that any continuous ...
- 133
5
votes
5
answers
14k
views
Where does the delta function come from if we derive autocorrelation directly?
I am reading a book "Creating Noise" written by Hollos & Hollos and have a question about the autocorrelation function of the Gaussain white noise when reading the following passage:
From the ...
- 189
0
votes
2
answers
917
views
Zero-order interpolation problem
Let $x_c(t)=\cos(\omega_0t)$. This signal is sampled with $\omega_s$, which is greater than the Nyquist rate. It is then interpolated with a zero-order interpolator. The signal obtained is $y_c(t)$. ...
- 5,040
1
vote
2
answers
342
views
Can an Delta Dirac function (not response) ever be sampled?
This is purely theoretical question but can an actual delta-dirac function ever be sampled? To make it clear again, I'm not talking about an impulse response from the impulse function. Lets say we ...
- 13
0
votes
1
answer
115
views
Periodic Signal Fourier Transform
If I have a signal x(t) as follow
x(t)=$\sum\limits_{k=-∞}^∞ R_{T}(t-2kT)$
$R_{T}(t) = \begin{cases}1 ;&[\dfrac{-T}{2},\dfrac{T}{2}] \\ 0; & \text{otherwise} \end{cases}$
$\mathfrak{F}\{R_{...
- 13
17
votes
4
answers
15k
views
Why is the Fourier transform of a Dirac comb a Dirac comb?
This doesn't make sense to me, because the Heisenberg inequality states that $\Delta t\Delta \omega$ ~ 1.
Therefore when you have something perfectly localized in time, you get something completely ...
- 631
0
votes
1
answer
672
views
Adding Dirac Delta with Unit Step
What does the graph of the function $x(t)=-\delta(t)+u(t)$ look like?
$$\delta(t)\ldots\text{ Dirac delta impulse}\\
u(t)\ldots\text{unit step function}$$
Will the impulse at the origin start ...
- 265
9
votes
3
answers
2k
views
Sampling the Dirac function
I would like to ask a theoretical question concerning the Dirac function. The Fourier Transform of the Dirac function is the value 1 (DC) for every frequency. If we consider the Sampling Theorem, we ...
- 193