Questions tagged [fourier-transform]

The Fourier transform is a mathematical operation that decomposes a function into its constituent frequencies, known as a frequency spectrum.

Filter by
Sorted by
Tagged with
143 votes
8 answers
176k views

Why is the Fourier transform so important?

Everyone discusses the Fourier transform when discussing signal processing. Why is it so important to signal processing and what does it tell us about the signal? Does it only apply to digital signal ...
jcolebrand's user avatar
  • 1,549
116 votes
12 answers
99k views

What is the physical significance of negative frequencies?

This has been one of the holes in my cheddar cheese block of understanding DSP, so what is the physical interpretation of having a negative frequency? If you have a physical tone at some frequency ...
Spacey's user avatar
  • 9,667
55 votes
4 answers
22k views

What is the sparse Fourier transform?

MIT has been making a bit of noise lately about a new algorithm that is touted as a faster Fourier transform that works on particular kinds of signals, for instance: "Faster Fourier transform ...
nibot's user avatar
  • 3,773
43 votes
6 answers
163k views

Difference between Discrete Time Fourier Transform and Discrete Fourier Transform

I have read many articles about DTFT and DFT but am not able to discern the difference between the two except for a few visible things like DTFT goes till infinity while DFT is only till $N-1$. Can ...
BaluRaman's user avatar
  • 679
38 votes
7 answers
7k views

What is the most lucid, intuitive explanation for the various FTs - CFT, DFT, DTFT and the Fourier Series?

Even after having studied these for quite sometime, I tend to forget (if I'm out of touch for a while) how they are related to each other and what each stands for (since they have such similar ...
Vighnesh's user avatar
  • 489
26 votes
4 answers
62k views

What effect does a delay in the time domain have in the frequency domain?

If I have a signal that is time limited, say a sinusoid that only lasts for $T$ seconds, and I take the FFT of that signal, I see the frequency response. In the example this would be a spike at the ...
gallamine's user avatar
  • 693
26 votes
3 answers
46k views

Intuitive explanation of cross-correlation in frequency domain

According to the cross-correlation theorem : the cross-correlation between two signals is equal to the product of fourier transform of one signal multiplied by complex conjugate of fourier transform ...
silver surfer's user avatar
26 votes
1 answer
9k views

How to circularly shift a signal by a fraction of a sample?

The shift theorem says: Multiplying $x_n$ by a linear phase $e^{\frac{2\pi i}{N}n m}$ for some integer m corresponds to a circular shift of the output $X_k$: $X_k$ is replaced by $X_{k-m}$, where ...
endolith's user avatar
  • 15.7k
25 votes
1 answer
3k views

Is the Laplace transform redundant?

The Laplace transform is a generalization of the Fourier transform since the Fourier transform is the Laplace transform for $s = j\omega$ (i.e. $s$ is a pure imaginary number = zero real part of $s$). ...
Vinz's user avatar
  • 361
24 votes
3 answers
12k views

Tips for improving pitch detection

I'm working on a simple web app that allows the user to tune his/her guitar. I'm a real beginner in signal processing, so please don't judge me too harshly if my question is inappropriate. So, I ...
Rad'Val's user avatar
  • 443
22 votes
2 answers
11k views

Inverse Short Time Fourier Transform algorithm described in words

I'm trying to conceptually understand what is happening when the forward and inverse Short Time Fourier Transforms (STFT) are applied to a discrete time-domain signal. I've found the classic paper by ...
Nicholas Kinar's user avatar
21 votes
3 answers
1k views

How were windows originally conceived?

I am aware of the common types of windows, (Hamming, Hanning, Kaiser, Tukey, etc etc). However while many books describe them - almost none tell me just how exactly they were derived. What is so ...
TheGrapeBeyond's user avatar
20 votes
3 answers
12k views

What's wrong with this code for tomographic reconstruction by the Fourier method?

I've been playing around with tomographic reconstruction algorithms recently. I already have nice working implementations of FBP, ART, a SIRT/SART-like iterative scheme and even using straight linear ...
timday's user avatar
  • 538
20 votes
3 answers
50k views

Meaning of Real and Imaginary part of Fourier Transform of a signal

Say $f$ is a signal of time $t$, $F$ its Fourier transform of the variable $v$. It is known that in polar coordinate, $|F(v)|$ tells us how much the frequency $v$ is present over the signal, and $Arg(...
user2682877's user avatar
19 votes
3 answers
7k views

FFT with asymmetric windowing?

Common non-rectangular window functions all seem to be symmetric. Is there ever a case when one would want to use a non-symmetric window function before an FFT? (Say if the data on one side of the ...
hotpaw2's user avatar
  • 35.2k
18 votes
2 answers
18k views

Why do we say that "zero-padding doesn't really increase frequency resolution"

Here is a sinusoid of frequency f = 236.4 Hz (it is 10 milliseconds long; it has N=441 points at sampling rate ...
Basj's user avatar
  • 1,267
18 votes
3 answers
19k views

When to use the DTFT vs the DFT (and their inverses) in analysis?

In many of my readings, whenever some author speaks about working in the frequency (transform) domain (of a digital signal), they often times take the DFT, or the DTFT, (and of course their ...
TheGrapeBeyond's user avatar
18 votes
5 answers
20k views

How do I optimize the window lengths in STFT?

I have many EEG signals and I want to analyze them using linear methods such as STFT (Short Time Fourier Transform). In STFT , How can I optimize the analysis window length, to reflect the frequency ...
Maen's user avatar
  • 181
17 votes
6 answers
13k views

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

In mathematics you can take the double derivative, or double integral of a function. There are many cases where performing a double derivative models a practical real-world situation, like finding the ...
tjwrona1992's user avatar
17 votes
8 answers
21k views

Why Does the DFT Assume the Transformed Signal Is Periodic?

In many signal processing books, it is claimed that the DFT assumes the transformed signal to be periodic (and that this is the reason why spectral leakage for example may occur). Now, if you look at ...
user10839's user avatar
  • 171
17 votes
4 answers
15k views

Why is the Fourier transform of a Dirac comb a Dirac comb?

This doesn't make sense to me, because the Heisenberg inequality states that $\Delta t\Delta \omega$ ~ 1. Therefore when you have something perfectly localized in time, you get something completely ...
Fraïssé's user avatar
  • 601
17 votes
2 answers
2k views

Why real part of FFT converts image into rotation + original?

I have read this image: taken its FFT (2D) and then Inverse FFT to get exactly the image back. Code is provided for reference: ...
Failed Scientist's user avatar
17 votes
2 answers
6k views

Fourier transform 4 times = original function (from Bracewell book)

I was glancing through "The Fourier Transform & Its Applications" by Ronald N. Bracewell, which is a good intro book on Fourier Transforms. In it, he says that if you take the Fourier ...
sambajetson's user avatar
16 votes
4 answers
27k views

Extracting frequencies from FFT

I performed 512 point FFT on a signal. I got another set of 512 Numbers. I understand that those numbers represent amplitude of the various sine and cosine waves having different frequencies. If my ...
gpuguy's user avatar
  • 1,370
16 votes
2 answers
1k views

How do you measure "detail" of a signal?

I have an image and I would like to measure the amount of detail in it. Another way to look at it is to measure how blurry an image is. One way is to analyse the high frequency components in the ...
Patrik's user avatar
  • 263
16 votes
2 answers
34k views

Image Reconstruction:Phase vs. Magnitude

Figure 1.(c) shows the Test image reconstructed from MAGNITUDE spectrum only. We can say that the intensity values of LOW frequency pixels are comparatively more than HIGH frequency pixels. Figure 1.(...
sagar's user avatar
  • 549
15 votes
5 answers
7k views

“The Fourier transform cannot measure two phases at the same frequency.” Why not?

I have read that the Fourier transform cannot distinguish components with the same frequency but different phase. For example, in Mathoverflow, or xrayphysics, where I got the title of my question ...
Antoni Parellada's user avatar
15 votes
1 answer
27k views

What are the units of my data after an FFT?

Magnetometer measures the derivative of the magnetic field, or dB/dt, with an output in microvolts (mV). The Sampling rate is 128 Hz, so if we collect data for 2 minutes, $2 \times 60 \times 128=...
MikiBelavista's user avatar
15 votes
3 answers
6k views

When can we write Heisenberg uncertainty Principle as a equality?

We know that Heisenberg uncertainty Principle states that $$\Delta f \Delta t \geq \frac{1}{4 \pi}.$$ But (in many case for Morlet wavelet) I have seen that they changed the inequality to an equality....
SAH's user avatar
  • 1,307
14 votes
7 answers
4k views

Fourier transform is an isomorphism...but we don’t get when each frequency appears?

Statistician here who wants to get some DSP knowledge for time series analysis. I’ve known for years that if we hit a function with a Fourier transform, we have an inverse Fourier transform that will ...
Dave's user avatar
  • 251
14 votes
5 answers
33k views

Fastest implementation of fft in C++?

I have a MATLAB program that uses fft and ifft a lot. Now I want to translate it to C++ for production. I used OpenCV but I ...
Saeid's user avatar
  • 255
14 votes
7 answers
3k views

Why does a longer observation time improve DFT resolution, but repeating a signal does not?

As was proven here: https://math.stackexchange.com/questions/228614/why-doesnt-repeating-a-signal-give-rise-to-a-finer-resolution-of-dft-fft repeating a certain sequence does not improve DFT frequency ...
Ariane's user avatar
  • 141
14 votes
4 answers
2k views

Is the discrete Gaussian kernel an eigenfunction of the DFT?

So the Gaussian function is an eigenfunction of the Fourier transform because it transforms into itself, right? But this isn't true for the sampled Gaussian in the DFT because the tails of the ...
endolith's user avatar
  • 15.7k
14 votes
1 answer
16k views

Comparison between Fourier transform, short-time Fourier transform and wavelets

What is the difference between the Fourier transform, short-time Fourier transform and wavelets?
student's user avatar
  • 141
14 votes
4 answers
9k views

Why is a negative exponent present in Fourier and Laplace transform?

could anyone explain why there is a need of negative exponent in fourier and laplace transform.I looked through the web but I couldn't get anything.Does anything happen if a positive exponent is ...
justin's user avatar
  • 259
14 votes
7 answers
11k views

The difference between DFT and DFS

In the literature, I've found that DFS and DFT are one and the same. If they are one and the same why to use two different names for them? If there is really a difference what is it and what is the ...
phanitej's user avatar
  • 450
13 votes
4 answers
2k views

Why is the time domain low-pass filter the "sinc" shape?

Consider: I'm looking at low-pass filters, and I see that the time domain representation of an "ideal" filter resembles the shape above whereas the frequency domain is a box. I also get the ...
thepman's user avatar
  • 133
13 votes
2 answers
16k views

What are the statistics of the discrete Fourier transform of white Gaussian noise?

Consider a white Gaussian noise signal $ x \left( t \right) $. If we sample this signal and compute the discrete Fourier transform, what are the statistics of the resulting Fourier amplitudes?
DanielSank's user avatar
  • 1,026
13 votes
5 answers
7k views

Discrete-time Fourier transform

I am a junior high school student who has a general fascination for electronics, programming, and the like. Recently, I have been learning about signal processing. Unfortunately, I haven't done much ...
ElectroNerd's user avatar
13 votes
1 answer
2k views

Which transform most closely mimics the human auditory system?

The Fourier transform is commonly used for frequency analysis of sounds. However, it has some disadvantages when it comes to analyzing the human perception of sound. For example, its frequency bins ...
user76284's user avatar
  • 233
13 votes
2 answers
13k views

Discrete-time Fourier Transform of the unit step sequence $u[n]$

From text books we know that the DTFT of $u[n]$ is given by $$U(\omega)=\pi\delta(\omega)+\frac{1}{1-e^{-j\omega}},\qquad -\pi\le\omega <\pi\tag{1}$$ However, I haven't seen a DSP textbook that ...
Matt L.'s user avatar
  • 88.9k
13 votes
2 answers
5k views

Choices of convention and notation for the Fourier transform?

The definitions of the Fourier transform and inverse Fourier transform I learned in college were $$ F(j\omega) = \int_{-\infty}^{\infty} f(t) e^{-j\omega t}\ dt $$ $$ f(t)=\frac{1}{2\pi}\int_{-\...
rtollert's user avatar
  • 470
12 votes
2 answers
10k views

Centering zero frequency for Discrete Fourier Transform

I am working on a image processing application which uses a discrete fourier transform to implement blurring/sharpening. The application is more or less working, but something about the mechanics is ...
dizzy's user avatar
  • 243
12 votes
9 answers
1k views

Where is the flaw in this derivation of the DTFT of the unit step sequence $u[n]$?

This question is related to this other question of mine where I ask for derivations of the discrete-time Fourier transform (DTFT) of the unit step sequence $u[n]$. During my search for derivations I ...
Matt L.'s user avatar
  • 88.9k
12 votes
3 answers
914 views

Other end of Nyquist limit

Say I perform FFT on some data. If the underlying (measurement) sampling rate is not twice the highest frequency, I will almost assuredly get aliasing. This limit on sampling we call the Nyquist limit....
Davey's user avatar
  • 272
12 votes
5 answers
1k views

What is the frequency representation of nonuniform sampling?

Uniform sampling can be thought of as multiplication of a function $x(t)$ with a Dirac comb function: $$\text{III}_T(t) = \sum_{k=-\infty}^{\infty}\delta(t-kT)$$ Multiplication of $x(t)$ with $\text{...
Gillespie's user avatar
  • 1,662
12 votes
3 answers
4k views

Slow Down Music Playing While Maintaining Frequency

Playing a piece of music audio at a slower speed would lower its pitch (frequency). Is there a tool and theory to slow down the song playing while keep the frequency the same? I suppose one can do ...
Hans's user avatar
  • 223
12 votes
3 answers
10k views

Can edge detection be done in the frequency domain?

Can we take advantage of the fact that high frequency components in the FFT of an image generally correspond to edges, to implement an edge detection algorithm in the fourier domain? I did try ...
rounak's user avatar
  • 263
12 votes
5 answers
2k views

Inconsistency between the units of power spectral density and the definition that people often give

Perhaps someone can help me resolve something - this is my understanding: In deterministic signal analysis, for a continuous signal $x(t)$ the signal energy is defined by $$E_{\textrm{s}} = \int^{+\...
teeeeee's user avatar
  • 681
12 votes
1 answer
578 views

Recognizing math functions within songs

I'm new to DSP, and just discovered this StackExchange, so apologies if this isn't the right place to post this question. Is there a resource that describes genres in a more mathematical terms? For ...
XSL's user avatar
  • 671

1
2 3 4 5
44