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35 votes
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What are advantages of having higher sampling rate of a signal?

Sampling at a higher frequency will give you more effective number of bits (ENOB), up to the limits of the spurious free dynamic range of the Analog to Digital Converter (ADC) you are using (as well ...
Dan Boschen's user avatar
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22 votes
Accepted

Why doesn't sampling a periodic continuous-time signal yield a periodic discrete-time signal?

If the ratio between your sampling frequency and the frequency of your signal is irrational, you will not have a periodic discrete signal. Assuming you have a 1-kHz sine wave and you sample at 3000*...
Ben's user avatar
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22 votes
Accepted

What is the first derivative of Dirac delta function?

If you imagine a Dirac delta impulse as the limit of a very narrow very high rectangular impulse with unit area centered at $t=0$, then it's clear that its derivative must be a positive impulse at $0^-...
Matt L.'s user avatar
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15 votes

What is the first derivative of Dirac delta function?

First of all the dirac delta is NOT a function, it's a distribution. See for example http://web.mit.edu/8.323/spring08/notes/ft1ln04-08-2up.pdf Treating it as a conventional function can lead to ...
Hilmar's user avatar
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13 votes

Fourier Transform Identities

You are correct. Your last equation is simply a fancy way of saying that $X(f)$ is real valued. In general: if it's real in one domain, it's conjugate symmetric in the other.
Hilmar's user avatar
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12 votes

What is the first derivative of Dirac delta function?

Maybe a picture is worth a thousand words? Here's how a Gaussian pulse of variable width and its derivatives look like: As others have said, Dirac is a distribution, hence the Gaussian pulse, and its ...
a concerned citizen's user avatar
12 votes

What information does the Hilbert transform give?

If the OP is interested in what is the practical purpose of the Hilbert Transform, the rest of this post applies. Similar to the Fourier, Laplace, and Z transforms, the Hilbert Transform can be used ...
Dan Boschen's user avatar
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10 votes

Is Fourier series a sampled version of Fourier transform?

There are 4 versions of Fourier transforms that are all close cousins. It's all due to the basic property that "sampling in one domain corresponds to periodicity in the other domain". If a ...
Hilmar's user avatar
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10 votes

How can I find the following Fourier Transform without directly using FT pairs?

The results shown in the two answers provided by Ahsan Yousaf don't agree. This answer is about explaining why the two solutions are different, and how to arrive at the correct solution. Note that ...
Matt L.'s user avatar
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9 votes

About Fourier transform of periodic signal

We can figure out what's going on if we first understand a simple identity and then just compute the Fourier transform of the periodic function. A useful identity First let's prove that $$D(\omega - ...
DanielSank's user avatar
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9 votes
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What is the difference between continuous, discrete, analog and digital signal?

A signal is indeed a function. Given a signal $f(x)$, according to whether continuous or discrete for both the variable $x$ and the function $f(x)$, there are four types of combinations: (1) $\mathbf{...
lxg's user avatar
  • 465
9 votes

Understanding the definition of mean/autocorrelation

The definition of the autocorrelation function $R_x(\tau)$ depends on the nature of your $x$. If $x$ is a deterministic signal with finite energy then: $$R_x(\tau)=\int_{-\infty}^{+\infty}x(t)x^*(t-\...
Learn_and_Share's user avatar
8 votes

How to do continuous signal processing (i.e without windowing)?

Yes, it's possible to analyse sound the way ears do. For example, you could compute the DFT of a signal continuously using several Goertzel filters. $$ y_k[n] = e^{j2\pi k/N} y_k[n-1] + x[n] $$ ...
Peter K.'s user avatar
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8 votes

How does shift and scaling inside of a function affect its Fourier Transform?

More generally, we have: $$y(t) = x(at + b)$$ You may want to combine both properties listed by Dilip Sarwate into one equation. Using the definition of the Fourier transform, we can insert our input ...
Mook's user avatar
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8 votes

Fourier Transform Identities

Yes, if eqs. (2) and (3) hold for any "type of signal" (which they do), then (5) must hold. Inserting (4) into (2) we get $$ \mathscr{F}\big\{x^*(t)\big\} = X(-f) $$ and using (3) $$ X(-f) = X^*(-f) ...
Deve's user avatar
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8 votes
Accepted

why total energy of a finite duration continuous signal becomes infinite after sampling

If you multiply a continuous-time finite energy signal $f(t)$ with an impulse train you get $$\tilde{f}(t)=\sum_{n=-\infty}^{\infty}f(nT)\delta(t-nT)\tag{1}$$ where $T$ is the sampling interval and $\...
Matt L.'s user avatar
  • 90.6k
8 votes
Accepted

DC value of $x(t)=\frac{1}{t}$

You need to be careful with the definition of the DC value of a signal. The actual time average, which is often called DC value is given by $$\overline{s(t)}=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{...
Matt L.'s user avatar
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8 votes
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the $L^2$-norm of a signal is also applied as its energy!

Yes, the square of the $L_2$ norm of a signal is also by definition its energy $\mathcal{E}_x$. The concept of signal energy : $$ \mathcal{E}_x = \int_{-\infty}^{ \infty } x(t)^2 dt\tag{1} $$ is ...
Fat32's user avatar
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8 votes
Accepted

Does $\cos(bt)\cdot u(t)$ have a Fourier Transform?

The integral doesn't converge in the conventional sense, so you can't solve it with standard methods. Assuming that you know (or can look up) the Fourier transform of the unit step function $u(t)$, it ...
Matt L.'s user avatar
  • 90.6k
7 votes

When does the convolution of $2$ signals equal zero?

Time-domain convolution is frequency-domain multiplication. If at all frequencies at least one of the signals is zero-valued in frequency domain, then the convolution of the two signals will be zero-...
Olli Niemitalo's user avatar
7 votes

Convolution of two exponential signals

Just in case if you look for a complete solution based on integration: $$y(t)=\int_{-\infty}^{+\infty}e^{-4\tau}u(\tau)\times 6e^{-(t-\tau)}u(t-\tau)d\tau$$ Consider these facts: Both step functions ...
msm's user avatar
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7 votes
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Is the system represented by the equation $y(t) = x(2t)$ time invariant?

From your solution: I followed the following algorithm: $$ y(t) =x(2t) $$ $$ y_1(t) = x_1(2t)$$ Let $$x_2(t) = x_1(t-t_0) ~~~\text{and}~~~ y_2(t) = x_2(2t) $$ On this following step (time ...
Fat32's user avatar
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7 votes

Fourier Transform Identities

The answers by @Deve and @Hilmar are technically perfect. I would like to provide some additional insights, with a few questions. First, do you know of a signal satisfying this reversed-time/...
Laurent Duval's user avatar
7 votes

Effects of linear interpolation of a time series on its frequency spectrum

Duane Wise and i wrote a paper back in the 90s that we presented to an AES convention that spelled out how to model time-domain polynomial interpolation (of which linear interpolation is an example) ...
robert bristow-johnson's user avatar
7 votes

the $L^2$-norm of a signal is also applied as its energy!

From physics, energy is a term often used as a quantitative property. In other words, energy is a quantity that is preserved under some actions, transformations, etc. In signal processing (where ...
Laurent Duval's user avatar
7 votes

Are the output functions of a continuous-time LTI system necessarily continuous (in the calculus sense) for any given input functions?

Consider the identity system $y(t) = x(t)$. This system is LTI. If the input $x(t)$ is discontinuous, then the output $y(t)$ will be discontinuous too.
MBaz's user avatar
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7 votes

Are the output functions of a continuous-time LTI system necessarily continuous (in the calculus sense) for any given input functions?

To add an even worse example to MBaz (best possible) counterexample: The derivative $\frac{\mathrm d}{\mathrm dt}$ is an LTI system. $f(t)=|t|$ is a continuous function. $\frac{\mathrm d}{\mathrm dt}(|...
Marcus Müller's user avatar
7 votes

Trying to understand how to get this basic Fourier Series

This is indeed a gnarly integral to do by hand, so it may be easier to leverage some properties of the Fourier Transform applied to simpler signals. Your signal is a trapezoid and it can be created ...
Hilmar's user avatar
  • 46.2k
6 votes

For an LTI system, why does the Fourier transform of the impulse response give the frequency response?

The intuitive answer is that an impulse in time at t=0 contains all frequencies of equal magnitude, so applying an impulse to an LTI system is the same as applying all frequencies at once, thus the ...
Pat Eblen's user avatar
6 votes

The difference between convolution and cross-correlation from a signal-analysis point of view

There is a lot of subtlety between the meanings of convolution and correlation. Both belong to the broader idea of inner products and projections in linear algebra, i.e. projecting one vector onto ...
orodbhen's user avatar
  • 511

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