40
votes
Accepted
The difference between convolution and cross-correlation from a signal-analysis point of view
In signal processing, two problems are common:
What is the output of this filter when its input is $x(t)$? The answer is given by $x(t)\ast h(t)$, where $h(t)$ is a signal called the "impulse ...
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31
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 ...
- 36.1k
21
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*...
- 3,544
19
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^-...
- 79.2k
18
votes
The difference between convolution and cross-correlation from a signal-analysis point of view
The two terms convolution and cross-correlation are implemented in a very similar way in DSP.
Which one you use depends on the application.
If you are performing a linear, time-invariant filtering ...
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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 ...
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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.
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10
votes
If low frequency travels longer distance, why our speech is not travelling longer distance?
Well, first of all the Sound Level Pressure decreases by $6 \; \mathtt{dB}$ when doubling the distance - this plays a big role. We do also have sound attenuation coming from our medium - air. Let's ...
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10
votes
Accepted
Proof of time-invariance of continuous-time system
Let $y_1(t)$ be the response to the signal $x_1(t)$:
$$y_1(t)=x_1(-t)\tag{1}$$
Now let $x_2(t)$ be a shifted version of $x_1(t)$:
$$x_2(t)=x_1(t-T)\tag{2}$$
The response to $x_2(t)$ is
$$y_2(t)=...
- 79.2k
9
votes
Is Fourier series a sampled version of Fourier transform?
There are 4 versions of Fourier transforms that are all close cousins.
It's all due to the basic property that "sampling in one domain corresponds to periodicity in the other domain". If a time ...
- 30.7k
9
votes
What is the first derivative of Dirac delta function?
Maybe a picture is worth a thousand words? Here's how a Gaussian pulse of variable width and its derivatives look like:
As others have said, Dirac is a distribution, hence the Gaussian pulse, and its ...
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8
votes
About Fourier transform of periodic signal
We can figure out what's going on if we first understand a simple identity and then just compute the Fourier transform of the periodic function.
A useful identity
First let's prove that
$$D(\omega - ...
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8
votes
The difference between convolution and cross-correlation from a signal-analysis point of view
@MathBgu I have read all above given answers, all are very informative one thing I want to add for your better understanding, by considering the formula of convolution as follows
$$f(x)*g(x)=\int\...
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8
votes
Accepted
What is the difference between continuous, discrete, analog and digital signal?
A signal is indeed a function. Given a signal $f(x)$, according to whether continuous or discrete for both the variable $x$ and the function $f(x)$, there are four types of combinations:
(1) $\mathbf{...
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8
votes
Fourier Transform Identities
Yes, if eqs. (2) and (3) hold for any "type of signal" (which they do), then (5) must hold.
Inserting (4) into (2) we get
$$
\mathscr{F}\big\{x^*(t)\big\} = X(-f)
$$
and using (3)
$$
X(-f) = X^*(-f)
...
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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 $\...
- 79.2k
8
votes
Accepted
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 ...
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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 ...
- 79.2k
7
votes
Analog signal can be discrete time?
This sounds like a confusion in terminology. See http://en.wikipedia.org/wiki/Digitizing
Digitization basically involves two steps:
Discretization: Sampling the signal at discrete times
...
- 30.7k
7
votes
$2\pi$ periodicity of discrete-time Fourier transform
The argument does not work in continuous time. In discrete time the argument is that
$$e^{j\omega n}=e^{j(\omega+2\pi)n},\qquad n\in\mathbb{Z}\tag{1}$$
This is true because by definition $n$ is an ...
- 79.2k
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-...
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7
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.
$$
y_k[n] = e^{j2\pi k/N} y_k[n-1] + x[n]
$$
...
- 21.7k
7
votes
Accepted
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 ...
- 26.6k
7
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: $$R_x(\tau)=\int_{-\infty}^{+\infty}x(t)x^*(t-\...
- 408
7
votes
Fourier Transform Identities
The answers by @Deve and @Hilmar are technically perfect. I would like to provide some additional insights, with a few questions.
First, do you know of a signal satisfying this reversed-time/...
- 29.7k
7
votes
Effects of linear interpolation of a time series on its frequency spectrum
Duane Wise and i wrote a paper back in the 90s that we presented to an AES convention that spelled out how to model time-domain polynomial interpolation (of which linear interpolation is an example) ...
- 16.8k
7
votes
How to Generate White Gaussian Noise with Known PSD in MATLAB
The specific PSD is basically white noise which was filtered with a filter which its Magnitude is the same as the PSD.
This is a result of the Wiener Khinchin Theorem.
So if you have the shape of ...
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7
votes
Are the output functions of a continuous-time LTI system necessarily continuous (in the calculus sense) for any given input functions?
Consider the identity system $y(t) = x(t)$. This system is LTI. If the input $x(t)$ is discontinuous, then the output $y(t)$ will be discontinuous too.
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7
votes
Are the output functions of a continuous-time LTI system necessarily continuous (in the calculus sense) for any given input functions?
To add an even worse example to MBaz (best possible) counterexample:
The derivative $\frac{\mathrm d}{\mathrm dt}$ is an LTI system. $f(t)=|t|$ is a continuous function.
$\frac{\mathrm d}{\mathrm dt}(|...
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6
votes
How does shift and scaling inside of a function affect its Fourier Transform?
More generally, we have:
$$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 ...
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