# Questions tagged [fourier-transform]

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

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### 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 ...
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 ...
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 ...
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 ...
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### 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 ...
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 ...
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 ...
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 ...
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### 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 $s$). ...
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### 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 ...
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 ...
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### 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 ...
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 ...
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### 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....
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### 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 ...
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 ...
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### 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 ...
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### 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 ...
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?
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 ...
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### 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 ...
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 ...
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?
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 ...
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### 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 ...
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 ...