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.

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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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}{\...
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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\...
Sami Al-Dalahmah's user avatar
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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 $...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
Madavan Viswanathan's user avatar
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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 ...
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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)...
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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 ...
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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 ...
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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 ...
Daniel Rodriguez's user avatar
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 "...
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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 ...
Nash Brewer's user avatar
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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 $\...
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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 ...
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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?
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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 $$\...
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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 ...
Hendry Newman's user avatar
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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 ...
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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 ...
user5280911's user avatar
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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 $y_c(t)$. ...
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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 ...
user26569's user avatar
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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_{...
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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 ...
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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 ...
Undertherainbow's user avatar
9 votes
3 answers
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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 ...
George Tseres's user avatar