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23 votes

Is there any practical application for performing a double Fourier transform? ...or an inverse Fourier transform on a time-domain input?

No, taking the Fourier transform twice is equivalent to time inversion (or inversion of whatever dimension you're in). You just get $x(-t)$ times a constant which depends on the type of scaling you ...
Matt L.'s user avatar
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17 votes

Is there any practical application for performing a double Fourier transform? ...or an inverse Fourier transform on a time-domain input?

Whilst taking the Fourier transform directly twice in a row just gives you a trivial time-inversion that would be much cheaper to implement without FT, there is useful stuff that can be done by taking ...
leftaroundabout's user avatar
16 votes
Accepted

What is the Fourier Transform of a constant signal?

I'll complete a bit the answer given in a comment above. Intuitively first, to which frequency corresponds a signal constant in time, for exemple $x(t) = 1$ $\forall t$ ? Such a signal shows no ...
anpar's user avatar
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16 votes
Accepted

Is there any practical application for performing a double Fourier transform? ...or an inverse Fourier transform on a time-domain input?

"Is there any practical application?" Definitely yes, at least to check code, and bound errors. Especially for huge data or a large number of iterations "In theory, theory and practice ...
Laurent Duval's user avatar
13 votes

Qualitative Explanation of Fourier Transform

Bottom Line of the Qualitative Explanation: The Fourier Transform is a correlation of our arbitrary signal $x(t)$ with all frequencies, with each frequency given as a complex exponential: $e^{j\omega ...
Dan Boschen's user avatar
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12 votes

Is there any practical application for performing a double Fourier transform? ...or an inverse Fourier transform on a time-domain input?

2D Fourier transform (2D DFT) is used in image processing since an image can be seen as a 2D signal. E.g. for a grayscale image $I$, $I(x,y)=z$, that means that at the coordinates $x$ and $y$ the ...
SheppLogan's user avatar
6 votes

Is there any practical application for performing a double Fourier transform? ...or an inverse Fourier transform on a time-domain input?

To answer the second question, in digital communications there is a technique in use in cellphones right now that makes good use of applying the IFFT to a time-domain signal. OFDM applies an IFFT to ...
myeslf's user avatar
  • 61
6 votes

Qualitative Explanation of Fourier Transform

Because the best $\phi$ from your (1) can be found by solving $$\begin{aligned} f_i(\omega) = \int_t f(t) \cos(2\pi(\omega t))dt \\ f_q(\omega) = \int_t f(t) \sin(2\pi(\omega t))dt \end{aligned} \tag ...
TimWescott's user avatar
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5 votes
Accepted

Is sampling at double the desired reproduction frequency accurate?

There is not only one sampling rate that theoretically allows exact reproduction. The sampling theorem states that it is sufficient for perfect reconstruction if the sampling frequency $f_s$ is ...
Matt L.'s user avatar
  • 90.5k
5 votes

Why is the reference pressure for dB SPL 20uPa?

It's complicated. Let's take it step by step: Sound pressure is a physical quantity that's related to the intensity of the sound field where you measure the pressure. It's measured in Pascal $Pa = \...
Hilmar's user avatar
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4 votes

Is it theoretically possible to perfectly quantize a continuous signal?

I'd like to point out Heisenberg Uncertainty principle, based on which theoretical achievable precision is limited. It states that one can not measure two complementary qualities (e.t. here time and ...
MimSaad's user avatar
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4 votes
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explanation of Hybrid systems?

If you're not interested in specific practical aspects of A/D conversion, but if you want to learn basic theory concerning sampling and digital (discrete-time) processing of analog (continuous-time) ...
Matt L.'s user avatar
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3 votes

Derivation of Nyquist Frequency and Sampling Theorem

It really boils down to aliasing. In continuous-time, if you have any two signals $x_1(t) = \sin(2 \pi F_1 t)$ and $x_2(t) = \sin(2 \pi F_2 t)$, then as long as $F_1$ and $F_2$ are distinct, the ...
TimWescott's user avatar
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3 votes

Relationship between the IDFT of a sampled DTFT and its discrete-time domain signal

For simplicity, 1D notation can be used without losing generality. IDFT associated with uniform (frequency) samples of a (valid) DTFT $X(e^{j \omega})$ of $x[n]$: Case-1: finite length $x[n]$ of ...
Fat32's user avatar
  • 28.3k
3 votes

Is it theoretically possible to perfectly quantize a continuous signal?

No, and the reason is not so much a question of how fast one can sample a continuous-time signal (as the accepted answer and another one says) but rather the impossibility of representing a real ...
Dilip Sarwate's user avatar
3 votes

Is it theoretically possible to perfectly quantize a continuous signal?

Is it theoretically possible to perfectly quantize a continuous signal? No. A quantization has an information content obviously countable as bits. Now, if you have a continuously distributed 1D ...
Marcus Müller's user avatar
3 votes
Accepted

What really means stochastic in field of signal processing

Well, getting a bit linguistic, according to the Oxford dictionary: stochastic (adj.): Having a random probability distribution or pattern that may be analysed statistically but may not be ...
Tendero's user avatar
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3 votes

Is there any new scientific capability to build non-causal filters in real world?

is there any new scientific theory which could yield to making none causal FIR Filters in the real world? No. If anything, scientific theories seem to be challenging determinism itself, which is what ...
A_A's user avatar
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3 votes

Is there any new scientific capability to build non-causal filters in real world?

In discrete-time systems, causality is a requirement only when processing (filtering) signals in real time; Too long for a comment: In practice this is overstating the importance of "causality". ...
Hilmar's user avatar
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3 votes
Accepted

How is time used in this autocorrelation expression?

The autocorrelation function of a random process $x(t)$ is defined as $$R(t,\tau) = {\mathbb E}\{x(t) x(t+\tau)\}.$$ For a stationary random process, this function does not depend on $t$, i.e., we ...
Florian's user avatar
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3 votes

Are all exponential functions eigensignals of LTI systems?

Complex exponentials are eigenfunctions of LTI systems because they are eigenfunctions of the convolution operator: $$\begin{align}e^{j\omega_0t}\star h(t)&=\int_{-\infty}^{\infty}h(\tau)e^{j\...
Matt L.'s user avatar
  • 90.5k
3 votes

Qualitative Explanation of Fourier Transform

If you want to learn more, you must start with Fourier series. They are an essential prerequisite, and mathematically and conceptually much simpler than Fourier transforms. At least you can learn the ...
user59855's user avatar
3 votes

Why is sampling not “idempotent”?

Sampling” actually means multiplying by a Dirac comb No. Sampling means turning a continuous function into a discrete set of numbers, so the process is simply $$x[n] = x(nT_\mathrm{s})$$ where $T_\...
Hilmar's user avatar
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3 votes

Why is sampling not “idempotent”?

Here is what I think is a nice pictoral way to think through this question. Sadly there is signal processing literature out there that does equate multiplication by a Dirac comb with sampling. But ...
Georg Essl's user avatar
2 votes

What really means stochastic in field of signal processing

I never saw the second definition, but the example of SGD actually fits the first definition. SGD works with a random estimator of the gradient, and just GD is using the real gradient (calculated). To ...
Cherny's user avatar
  • 471
2 votes

Is sampling at double the desired reproduction frequency accurate?

This is related to 'Three points determine a circle'. (i.e. The sine wave is a circle - i.e. draw a circle in (r,theta) and transform that to x-y with the x axis being the distance travelled along the ...
giwyni's user avatar
  • 121
2 votes

Is sampling at double the desired reproduction frequency accurate?

No, it is not accurate in the finite real world. Note that any finite length signal has infinite support in the frequency domain, thus is not strictly bandlimited. This is one reason why sampling ...
hotpaw2's user avatar
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2 votes

What really means stochastic in field of signal processing

I wanted to chime in a little only because I think I see why someone would have given you the obviously flawed definition of sample-by-sample. Adaptive algorithms like the LMS filter are based on a ...
hops's user avatar
  • 1,422
2 votes
Accepted

LTI system and initial conditions

The system is described by a linear difference equation with constant coefficients and as such, it is described in the same way as a linear time-invariant system. It is just the non-zero initial ...
Matt L.'s user avatar
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