numpy.fft ========= Functions related to Fourier transforms can be called by prepending them with ``numpy.fft.``. The module defines the following two functions: 1. `numpy.fft.fft <#fft>`__ 2. `numpy.fft.ifft <#ifft>`__ ``numpy``: https://docs.scipy.org/doc/numpy/reference/generated/numpy.fft.ifft.html fft --- Since ``ulab``\ ’s ``ndarray`` does not support complex numbers, the invocation of the Fourier transform differs from that in ``numpy``. In ``numpy``, you can simply pass an array or iterable to the function, and it will be treated as a complex array: .. code:: # code to be run in CPython fft.fft([1, 2, 3, 4, 1, 2, 3, 4]) .. parsed-literal:: array([20.+0.j, 0.+0.j, -4.+4.j, 0.+0.j, -4.+0.j, 0.+0.j, -4.-4.j, 0.+0.j]) **WARNING:** The array returned is also complex, i.e., the real and imaginary components are cast together. In ``ulab``, the real and imaginary parts are treated separately: you have to pass two ``ndarray``\ s to the function, although, the second argument is optional, in which case the imaginary part is assumed to be zero. **WARNING:** The function, as opposed to ``numpy``, returns a 2-tuple, whose elements are two ``ndarray``\ s, holding the real and imaginary parts of the transform separately. .. code:: # code to be run in micropython from ulab import numpy as np x = np.linspace(0, 10, num=1024) y = np.sin(x) z = np.zeros(len(x)) a, b = np.fft.fft(x) print('real part:\t', a) print('\nimaginary part:\t', b) c, d = np.fft.fft(x, z) print('\nreal part:\t', c) print('\nimaginary part:\t', d) .. parsed-literal:: real part: array([5119.996, -5.004663, -5.004798, ..., -5.005482, -5.005643, -5.006577], dtype=float) imaginary part: array([0.0, 1631.333, 815.659, ..., -543.764, -815.6588, -1631.333], dtype=float) real part: array([5119.996, -5.004663, -5.004798, ..., -5.005482, -5.005643, -5.006577], dtype=float) imaginary part: array([0.0, 1631.333, 815.659, ..., -543.764, -815.6588, -1631.333], dtype=float) ulab with complex support ~~~~~~~~~~~~~~~~~~~~~~~~~ If the ``ULAB_SUPPORTS_COMPLEX``, and ``ULAB_FFT_IS_NUMPY_COMPATIBLE`` pre-processor constants are set to 1 in `ulab.h `__ as .. code:: c // Adds support for complex ndarrays #ifndef ULAB_SUPPORTS_COMPLEX #define ULAB_SUPPORTS_COMPLEX (1) #endif .. code:: c #ifndef ULAB_FFT_IS_NUMPY_COMPATIBLE #define ULAB_FFT_IS_NUMPY_COMPATIBLE (1) #endif then the FFT routine will behave in a ``numpy``-compatible way: the single input array can either be real, in which case the imaginary part is assumed to be zero, or complex. The output is also complex. While ``numpy``-compatibility might be a desired feature, it has one side effect, namely, the FFT routine consumes approx. 50% more RAM. The reason for this lies in the implementation details. ifft ---- The above-mentioned rules apply to the inverse Fourier transform. The inverse is also normalised by ``N``, the number of elements, as is customary in ``numpy``. With the normalisation, we can ascertain that the inverse of the transform is equal to the original array. .. code:: # code to be run in micropython from ulab import numpy as np x = np.linspace(0, 10, num=1024) y = np.sin(x) a, b = np.fft.fft(y) print('original vector:\t', y) y, z = np.fft.ifft(a, b) # the real part should be equal to y print('\nreal part of inverse:\t', y) # the imaginary part should be equal to zero print('\nimaginary part of inverse:\t', z) .. parsed-literal:: original vector: array([0.0, 0.009775016, 0.0195491, ..., -0.5275068, -0.5357859, -0.5440139], dtype=float) real part of inverse: array([-2.980232e-08, 0.0097754, 0.0195494, ..., -0.5275064, -0.5357857, -0.5440133], dtype=float) imaginary part of inverse: array([-2.980232e-08, -1.451171e-07, 3.693752e-08, ..., 6.44871e-08, 9.34986e-08, 2.18336e-07], dtype=float) Note that unlike in ``numpy``, the length of the array on which the Fourier transform is carried out must be a power of 2. If this is not the case, the function raises a ``ValueError`` exception. ulab with complex support ~~~~~~~~~~~~~~~~~~~~~~~~~ The ``fft.ifft`` function can also be made ``numpy``-compatible by setting the ``ULAB_SUPPORTS_COMPLEX``, and ``ULAB_FFT_IS_NUMPY_COMPATIBLE`` pre-processor constants to 1. Computation and storage costs ----------------------------- RAM ~~~ The FFT routine of ``ulab`` calculates the transform in place. This means that beyond reserving space for the two ``ndarray``\ s that will be returned (the computation uses these two as intermediate storage space), only a handful of temporary variables, all floats or 32-bit integers, are required. Speed of FFTs ~~~~~~~~~~~~~ A comment on the speed: a 1024-point transform implemented in python would cost around 90 ms, and 13 ms in assembly, if the code runs on the pyboard, v.1.1. You can gain a factor of four by moving to the D series https://github.com/peterhinch/micropython-fourier/blob/master/README.md#8-performance. .. code:: # code to be run in micropython from ulab import numpy as np x = np.linspace(0, 10, num=1024) y = np.sin(x) @timeit def np_fft(y): return np.fft.fft(y) a, b = np_fft(y) .. parsed-literal:: execution time: 1985 us The C implementation runs in less than 2 ms on the pyboard (we have just measured that), and has been reported to run in under 0.8 ms on the D series board. That is an improvement of at least a factor of four.