Signal processing and optimization using SciPy (Part 1)

Introduction

Signal processing is the study of modifying, controlling, and extracting information from data signals embedded in noise. SciPy is a Python library that contains tools to do scientific computation, including routines for signal processing and optimization.

This blog post will briefly discuss what signal processing is and how it relates to optimization, followed by three examples of using SciPy to solve different optimization problems. Finally, this post will conclude with a discussion on the development workflow for using SciPy with large scale problems. If you want to know more about Data Science and methodologies you can check this link Data Science courses in Kochi.

SciPy

SciPy is a Python package that provides tools for scientific computing, including data analysis and optimization. We will use SciPy in our example below.

Optimization

Optimization is a difficult task and it is a difficult problem to solve. There are many ways to apply optimization problems, but the most common approach has been the use of linear algebra. This can be an expensive method of optimization if the number of iterations is large and the model highly structured.

A more efficient approach would be to convert these linear algebra problems into a convex objective function so that they can be efficiently solved using convex optimization techniques. A typical approach for converting these functions is via Taylor expansions or using Lagrange multipliers as both allow for efficient summation over an entire spectrum of variables.

However, these approaches only work for the case where the model is dense in the variables such as linear models. In general, when an optimization problem has a number of non-linear terms, these methods are not able to be applied due to lack of convergence.

Using SciPy for optimization

In this blog post, we will demonstrate how to use scipy to optimize a signal using the fourier transform method. We will use a simple sinusoidal signal with precise frequency of 1Hz as an example.

The analysis is mostly divided into two portions: continuous-time and discrete-time. In continuous-time, we’ll be looking at the Fourier Transform and the Laplace Transform for basic spectral information. Using these transforms, we’ll find where in the frequency domain our minimum value will occur in terms of power density. Discrete-time will be using the FFT to get the spectrum at a specific instant in time. We’ll use Laplace Transform to compute the impulse responses of various algorithms and compare them to find which one is better in terms of minimal power density.

For this post, we won’t be using any of the signal processing libraries other than scipy . This is because scipy has a robust set of functions that cover all the needs for signal processing, thus we believe it is easier to understand at first glance and less likely to require extended reading on similar topics.

Creating a Signal

Sine waves are sometimes called pure tones because they represent a single frequency. You use sine waves to create audio because they create different peaks in the resulting frequency spectrum.

Another great thing about sine waves is that they are generated directly with NumPy. Here is some code that generates a sine wave:

import numpy as np
from matplotlib import pyplot as plt
SAMPLE_RATE = 44100  # Hertz
DURATION = 5  # Seconds
def generate_sine_wave(freq, sample_rate, duration):
    x = np.linspace(0, duration, sample_rate * duration, endpoint=False)
    frequencies = x * freq
    # 2pi because np.sin takes radians
    y = np.sin((2 * np.pi) * frequencies)
    return x, y
# Generate a 2 hertz sine wave that lasts for 5 seconds
x, y = generate_sine_wave(2, SAMPLE_RATE, DURATION)
plt.plot(x, y)
plt.show()

(to be continued…)

Conclusion

We have discussed some aspects about signal processing and optimization using scipy. In the upcoming post we will continue the discussion on signal processing and optimization. If you want to know more about data science then you can check Data Science training institute in Kerala.