# 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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### What is difference between terms $X(j \omega) ,X(\ e^{j \omega })$ and $X(\omega)$?

While studying frequency transforms ,I get confused with the terms like $X(j \omega) ,X(\ e^{j \omega })$ and $X(\omega)$ ,where $\omega = 2 \pi f$. So what is the difference between them ?
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### Implementing Convolution in Frequency Domain?

Suppose, we have a bitmap image represented as a 2D integer array, int [,] image2D; whose FFT is Complex[,] fftImage2D; ...
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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 ...
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### Variance of periodogram estimate of the power spectrum

I have been reading chapter 13.4. ("Power Spectrum Estimation Using the FFT") of the Numerical Recipies Book. Some things related to the expectation value of the "periodogram estimate of the power ...
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### How to combine a rotation matrix and a stretch matrix into a single matrix for easy Fourier Transform

For full disclosure, this is related to homework. I have to find the Fourier Transform of a function that I've boiled down to the following. I have a function $f(x,y)$ that I can think of as another ...
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### RMS calculation in frequency domain after windowing

I can calculate RMS in frequency domain as derived from Parseval's Theorem. But what if I have applied a windowing function before doing the FFT (in my case a Hann window)?. Now the RMS values are ...
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### How do I construct input to neural network from audio signals?

Input: Microphone recordings of digits from 0 to 9 from different speakers. Output: The digit from 0 to 9. I am doing this for fun. So first I will train my neural network using some samples and ...
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### How to compute the main lobe width of generalized Hamming windows (i.e. Hann, Hamming, etc.)?

Is there a way to compute the main lobe width of windows of the generalized Hamming window family (i.e. Hann, Hamming, etc.)? By main lobe I mean the first zeros left and right of the center of the ...
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### Removing values from FFT result same as filtering?

I don't quite understand why the textbooks say it is impossible to implement an ideal low pass filter. If I was to take the FFT of a discrete signal x[n], with Matlab's fft function I'd be returned ...
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### Why idft(dft(a) * dft(b)) not equal to convolve(a, b)?

I'm a little confused... I always thought the DFT of a convolution was equal to a product of DFTs, but when I tried this in Python: ...
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### Effects of linear interpolation of a time series on its frequency spectrum

Situation In order to synchonisize different time series i have to apply linear interpolation on them. After the interpolation and synchronization the signal is transferred into its frequency domain ...
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### Why do we get different imaginary parts of a zero centered Gaussian for the the same number of data points N?

Suppose we have a total number N= 2048 points in a data and we wish to have zero centered Gaussian. There are two possibilities that we use the x-axis as ...
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### Amplitude after Fourier transform

How to obtain the correct amplitude after the numerical Fourier transform of a signal? Example: consider an exponential decaying wave $y(x)=e^{-x}\sin(100\pi x)$ with Fourier transform $y_f(x_f)$ ...
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### How do I convert a real baseband signal to a complex baseband signal?

I have radio telescope observations that have resulted in two real-valued signals (corresponding to the right- and left-handed circular polarizations). The signals are sampled at rate $2B$, and ...
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### Image Processing and applicability of 2D Fourier Transform

As a newbie in the world of signal processing, I am having a hard time in appreciating image 2-D fourier transforms. I am fully able to appreciate the concept of 1-D Fourier transform. Essentially, ...
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### 2-d circularly symmetric low-pass filter

For a square pixel grid, the ideal 2-d low-pass filter with a horizontal and a vertical cut-off angular frequency $\omega_c$ in radians has an impulse response (kernel) $h_{\small\square}(x, y)$ that ...
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### About Discrete Fourier Transform vs. Discrete Fourier Series

I am new to the field of signal processing. I am wondering what is the difference between DFS(Fourier Series) vs. DFT(Fourier Transform). For common applications, usually we get a segment(length <...
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### Is Fourier series a sampled version of Fourier transform?

I recently learned about dtft and how dft/dfs is the sampled version of dtft. I was wondering if Fourier series is also obtainable by sampling Fourier transform? I am a noob in the subject so sorry if ...