116 lines
5.8 KiB
Markdown
116 lines
5.8 KiB
Markdown
# [FFT](https://numpy.org/doc/stable/reference/routines.fft.html)
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{:.no_toc}
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<nav markdown="1" class="toc-class">
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* TOC
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{:toc}
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</nav>
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## The goal
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Fouriert transformations are an important part of data analysis.
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Questions to [David Rotermund](mailto:davrot@uni-bremen.de)
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## [Numpy](https://numpy.org/doc/stable/reference/routines.fft.html) vs [scipy](https://docs.scipy.org/doc/scipy/tutorial/fft.html#fourier-transforms-scipy-fft)
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```shell
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pip install scipy
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```
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Numpy says [itself](https://numpy.org/doc/stable/reference/routines.fft.html#discrete-fourier-transform-numpy-fft):
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> The SciPy module scipy.fft is a more comprehensive superset of numpy.fft, which includes only a basic set of routines.
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## fft vs rfft
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### [numpy.fft.fft](https://numpy.org/doc/stable/reference/generated/numpy.fft.fft.html)
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```python
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fft.fft(a, n=None, axis=-1, norm=None)[source]
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```
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> Compute the one-dimensional discrete Fourier Transform.
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>
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> This function computes the one-dimensional n-point discrete Fourier Transform (DFT) with the efficient Fast Fourier Transform (FFT) algorithm [CT].
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### [numpy.fft.rfft](https://numpy.org/doc/stable/reference/generated/numpy.fft.rfft.html)
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```python
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fft.rfft(a, n=None, axis=-1, norm=None)[source]
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```
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> Compute the one-dimensional discrete Fourier Transform for real input.
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>
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> This function computes the one-dimensional n-point discrete Fourier Transform (DFT) of a real-valued array by means of an efficient algorithm called the Fast Fourier Transform (FFT).
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### Comparison
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If the input array is **real-valued** (i.e. no complex numbers) then use **rfft**. Otherwise use **fft**. However, you can always use **fft** if you want but you might need to add extra steps to remove the complex noise from the results. E.g. if x is real-valued ifft(fft(x)) can be complex, due to numerical noise.
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The test signal:
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```python
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import numpy as np
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import matplotlib.pyplot as plt
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# Test signal
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f: float = 10.0
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t = np.linspace(0, 10, 10000)
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x = np.sin(t * f * 2 * np.pi)
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plt.plot(t, x)
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plt.ylabel("sin(x)")
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plt.xlabel("sin(x)")
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plt.show()
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```
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```python
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fft_result = np.fft.fft(x)
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print(fft_result.shape) # -> (10000,)
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rfft_result = np.fft.rfft(x)
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print(rfft_result.shape) # -> (5001,)
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```
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## [Discrete Fourier Transform (numpy.fft)](https://numpy.org/doc/stable/reference/routines.fft.html#discrete-fourier-transform-numpy-fft)
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## [Standard FFTs](https://numpy.org/doc/stable/reference/routines.fft.html#standard-ffts)
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|[fft(a[, n, axis, norm])](https://numpy.org/doc/stable/reference/generated/numpy.fft.fft.html#numpy.fft.fft)|Compute the one-dimensional discrete Fourier Transform.|
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|[ifft(a[, n, axis, norm])](https://numpy.org/doc/stable/reference/generated/numpy.fft.ifft.html#numpy.fft.ifft)|Compute the one-dimensional inverse discrete Fourier Transform.|
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|[fft2(a[, s, axes, norm])](https://numpy.org/doc/stable/reference/generated/numpy.fft.fft2.html#numpy.fft.fft2)|Compute the 2-dimensional discrete Fourier Transform.|
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|[ifft2(a[, s, axes, norm])](https://numpy.org/doc/stable/reference/generated/numpy.fft.ifft2.html#numpy.fft.ifft2)|Compute the 2-dimensional inverse discrete Fourier Transform.|
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|[fftn(a[, s, axes, norm])](https://numpy.org/doc/stable/reference/generated/numpy.fft.fftn.html#numpy.fft.fftn)|Compute the N-dimensional discrete Fourier Transform.|
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|[ifftn(a[, s, axes, norm])](https://numpy.org/doc/stable/reference/generated/numpy.fft.ifftn.html#numpy.fft.ifftn)|Compute the N-dimensional inverse discrete Fourier Transform.|
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## [Real FFTs](https://numpy.org/doc/stable/reference/routines.fft.html#real-ffts)
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|[rfft(a[, n, axis, norm])](https://numpy.org/doc/stable/reference/generated/numpy.fft.rfft.html#numpy.fft.rfft)|Compute the one-dimensional discrete Fourier Transform for real input.|
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|[irfft(a[, n, axis, norm])](https://numpy.org/doc/stable/reference/generated/numpy.fft.irfft.html#numpy.fft.irfft)|Computes the inverse of rfft.|
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|[rfft2(a[, s, axes, norm])](https://numpy.org/doc/stable/reference/generated/numpy.fft.rfft2.html#numpy.fft.rfft2)|Compute the 2-dimensional FFT of a real array.|
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|[irfft2(a[, s, axes, norm])](https://numpy.org/doc/stable/reference/generated/numpy.fft.irfft2.html#numpy.fft.irfft2)|Computes the inverse of rfft2.|
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|[rfftn(a[, s, axes, norm])](https://numpy.org/doc/stable/reference/generated/numpy.fft.rfftn.html#numpy.fft.rfftn)|Compute the N-dimensional discrete Fourier Transform for real input.|
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|[irfftn(a[, s, axes, norm])](https://numpy.org/doc/stable/reference/generated/numpy.fft.irfftn.html#numpy.fft.irfftn)|Computes the inverse of rfftn. |
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## [Hermitian FFTs](https://numpy.org/doc/stable/reference/routines.fft.html#hermitian-ffts)
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|[hfft(a[, n, axis, norm])](https://numpy.org/doc/stable/reference/generated/numpy.fft.hfft.html#numpy.fft.hfft)|Compute the FFT of a signal that has Hermitian symmetry, i.e., a real spectrum.|
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|[ihfft(a[, n, axis, norm])](https://numpy.org/doc/stable/reference/generated/numpy.fft.ihfft.html#numpy.fft.ihfft)|Compute the inverse FFT of a signal that has Hermitian symmetry.|
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## [Helper routines](https://numpy.org/doc/stable/reference/routines.fft.html#helper-routines)
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|[fftfreq(n[, d])](https://numpy.org/doc/stable/reference/generated/numpy.fft.fftfreq.html#numpy.fft.fftfreq)|Return the Discrete Fourier Transform sample frequencies.|
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|[rfftfreq(n[, d])](https://numpy.org/doc/stable/reference/generated/numpy.fft.rfftfreq.html#numpy.fft.rfftfreq)|Return the Discrete Fourier Transform sample frequencies (for usage with rfft, irfft).|
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|[fftshift(x[, axes])](https://numpy.org/doc/stable/reference/generated/numpy.fft.fftshift.html#numpy.fft.fftshift)|Shift the zero-frequency component to the center of the spectrum.|
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|[ifftshift(x[, axes])](https://numpy.org/doc/stable/reference/generated/numpy.fft.ifftshift.html#numpy.fft.ifftshift)|The inverse of fftshift.|
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