163 lines
4.6 KiB
ReStructuredText
163 lines
4.6 KiB
ReStructuredText
None
|
||
Fourier transforms
|
||
==================
|
||
|
||
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)
|
||
|
||
|
||
|
||
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.
|
||
|
||
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
|
||
|
||
import ulab as np
|
||
from ulab import vector
|
||
from ulab import fft
|
||
|
||
x = np.linspace(0, 10, num=1024)
|
||
y = vector.sin(x)
|
||
|
||
@timeit
|
||
def np_fft(y):
|
||
return 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.
|