# 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
Filter by
Sorted by
Tagged with
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
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
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
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
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
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
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
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 ...
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
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
121 views

166 views

• 1,593
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
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
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 ...
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
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
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
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 ...
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
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 ...
1 vote
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
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
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 ...