Libraries like NumPy
NumPy Fourier Transforms
These notes cover the use of NumPy’s functions for performing Fourier Transforms. NumPy provides efficient implementations for these crucial signal processing operations.
Key functions:
numpy.fft.fft(): Computes the one-dimensional discrete Fourier Transform (DFT).
import numpy as np
x = np.array(1.0, 2.0, 1.0, -1.0]])
transformed_x = np.fft.fft(x)
print(transformed_x)
numpy.fft.ifft(): Computes the inverse DFT, reconstructing the original signal from its Fourier transform.
original_x = np.fft.ifft(transformed_x)
print(original_x)
numpy.fft.fft2(),numpy.fft.ifft2(): Two-dimensional DFT and inverse DFT, useful for image processing.
image = np.random.rand(64, 64)
transformed_image = np.fft.fft2(image)
numpy.fft.fftn(),numpy.fft.ifftn(): N-dimensional DFT and inverse DFT, generalizing to higher dimensions.
Important Considerations:
-
Normalization: The scaling of the output of
fft()andifft()might need adjustment depending on the application. Refer to the NumPy documentation for details on normalization factors. -
Frequency Interpretation: The output of
fft()represents the frequency components of the input signal. Understanding how the frequency axis is mapped is crucial for interpretation. Frequency Axis Interpretation -
Real and Imaginary Parts: The output of
fft()is generally complex, with real and imaginary parts. These represent the magnitude and phase of each frequency component respectively. Complex Numbers in FFT -
Efficiency: NumPy’s FFT implementation uses highly optimized algorithms (typically FFTW) which are significantly faster than naive implementations.
Related Notes:
Example: Simple signal analysis
import matplotlib.pyplot as plt
# Generate a simple sine wave
t = np.linspace(0, 1, 128, endpoint=False)
sig = np.sin(2*np.pi*10*t)
# Add some noise
sig += np.random.randn(128)*0.1
# Compute and plot the FFT
sig_fft = np.fft.fft(sig)
plt.plot(np.abs(sig_fft))
plt.show()