New answers tagged fourier-transform
2
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Why is the CT inverse Fourier Transform also an integral?
Integration and differentiation are inverse operations only in the case of either indefinite integrals or integrals with the independent variable as upper bound:
$$G(x)=\int_a^xg(t)dt\quad\...
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1
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Implementation of dispersion compensation of lamb waves
In order to help you I need you to expand the functions:
get_velocity_at_freq(),
wp.theoretical_group_phase_vel()
that are not ...
4
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Why is the time domain low-pass filter the "sinc" shape?
One way to think about it is the requirement of what a filter does, and what is the relation between the time domain and frequency domain plots of the signal or the filter.
This also requires to know ...
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20
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Why is the time domain low-pass filter the "sinc" shape?
It is a good way to understand the lowpass behavior of sinc function (as well as the convolution) through visualization. I've made some modification on this animated convolution project and here are ...
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5
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Why is the time domain low-pass filter the "sinc" shape?
Perhaps one way to see the sinc is as a special moving average filter. As you noted, the lower the cutoff frequency (filtering out higher frequencies), the wider the sinc mainlobe. This corresponds to ...
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Fourier transform magnitude of the sum of two signals
In general we have
$$\big|X_1(f)+X_2(f)\big|\neq \big|X_1(f)\big|+\big|X_2(f)\big|\tag{1}$$
However, the condition $X_1(f)X_2(f)=0$ $\forall f$ implies that for any $f$, either $X_1(f)$ or $X_2(f)$ or ...
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0
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Frequency content of a noisy signal
Couple of mistakes!
#1 Take the fourier transform of its power signal (square the noisy signal)
That's for anything but a constant zero signal not the same as the PSD estimate you get from your #2.
...
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Why does multiplying a real signal by a random complex phase term result in "spreading" in the Fourier domain?
I'm not sure I correctly understand what you mean by DC term, but will assume you mean fundamental frequency $f_0$ of some periodic signal, i.e. an unmodulated carrier.
So your real-valued signal $f(x)...
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2
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Accepted
Relationship between fourier transform and fourier series
Transform looks right, but the logic afterwards needs some correction.
$$
X(\omega) =
-i \pi A \left(e^{i \alpha } \delta \left(\omega -2 \pi f_0\right)-e^{-i \alpha } \delta \left(2 \pi f_0+\...
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0
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Finding Discrete Fourier Transform (DFT) for different DFT size
This exercise is aimed at showing that zero-padding in the time domain interpolates the frequency domain. Since this is homework, I'll give you the beginning of the solution.
Just like you did, start ...
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Accepted
Is it useful to think of a Fourier Transform as writing out a signal in terms of a basis?
Question 1: Yes. In fact, the Fourier series becomes countable, although infinite, under restriction, say $[0,2\pi]$, with some exceptions. See this: Schauder basis: Relation to fourier series.
...
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Accepted
Alternative way to find fourier transform
There are two main strategies to simplify the calculation of the Fourier Transform.
Use Fourier Transform properties
"Deconstruct" the time domain functions into other functions with easier ...
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Alternative way to find fourier transform
With such a simple function it is probably easiest to directly solve the Fourier integral. It's also wise to commit such simple Fourier identities to memory.
Another relatively simple method I can ...
- 88.8k
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What is the reason of existence of Fourier transform? (Why we use Fourier transform?)
The Taylor series attempts to model a limited piece of a function or a data series using a weighted sum of polynomials. This only works for relatively small values of x or t.
The Fourier ...
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“The Fourier transform cannot measure two phases at the same frequency.” Why not?
If you change the phase at a constant frequency, FFT calculates a different result. Unfortunately, it is common practice to erase the phase by forming and displaying the magnitude (see line 7 in the ...
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Why is the DC component of discrete fourier transform not the same as the signal's arithmetic mean?
The question you cite proofs this for Continuous Fourier Transform, your code however uses the Discrete Fourier Transform which has significantly different properties.
The DFT is typically defined as
$...
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Difference between DC component and zero frequency component of signal
A brief answer.
Rect does NOT contain any DC component = No component at frequency = 0.
But the fourier transform does tell there is a DC component at frequency = 0, and the rect function associated ...
2
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Accepted
Fourier transform of $|x_\mathrm{a}(t)|^2$
Your result is correct, but it can be simplified by noticing that the two integrals in your solution are actually identical, which for $f>0$ leads to
$$\mathscr{F}\left\{|x_\mathrm{a}(t)|^2\right\}=...
- 88.8k
0
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Accepted
On FFT, interpolating signal vs extending signal in time
Interpolating in time causes Frequncy domain to scale completly, also the boundaries of frequency domain changes accordingly, since nyquist frequiency will be changed.
On time extentioning, Nyquist ...
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Accepted
Why does applying Fourier Transform on point Spread Function yield h(t) which is complex-valued
The Fourier transform doesn't move only from time to frequency (or space etc.), it moves between these domains. The only difference between the Fourier transform and its inverse is a sign in the ...
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