Spectral leakage concept in Signal Processing - Mini Project: Build & Apply

Understanding Spectral Leakage in Signal Processing

📖 Scenario: You are analyzing a simple sound wave signal using Python. You want to understand how spectral leakage happens when you perform a Fourier transform on a signal that does not fit perfectly into the sample window.This is important because spectral leakage can blur the frequency information, making it harder to identify the true frequencies in a signal.

🎯 Goal: You will create a simple sine wave signal, set a sample window length, compute the Fourier transform, and observe spectral leakage by printing the frequency spectrum.

📋 What You'll Learn

Create a sine wave signal with a frequency that does not fit exactly into the sample window

Set the sample window length

Compute the Fourier transform using numpy

Print the magnitude of the frequency spectrum to observe spectral leakage

💡 Why This Matters

🌍 Real World

Spectral leakage affects audio processing, communications, and any field where frequency analysis of signals is important.

💼 Career

Understanding spectral leakage helps data scientists and engineers improve signal analysis accuracy in real-world applications like speech recognition and radar.

1

Create a sine wave signal

Create a numpy array called t representing time points from 0 to 1 second with 500 samples. Then create a sine wave signal called signal with frequency 5.5 Hz using np.sin(2 * np.pi * 5.5 * t).

Signal Processing

import numpy as np
# Create time array t with 500 points from 0 to 1 second
# Create sine wave signal with frequency 5.5 Hz
# Your code here

Hint

Use np.linspace to create the time array and np.sin for the sine wave.

2

Set the sample window length

Create a variable called window_length and set it to 500, which is the number of samples in the signal.

Signal Processing

import numpy as np

t = np.linspace(0, 1, 500, endpoint=False)
signal = np.sin(2 * np.pi * 5.5 * t)
# Set window_length to 500
# Your code here

Hint

The window length is the total number of samples in the signal.

3

Compute the Fourier transform

Use np.fft.fft on the signal to compute its Fourier transform and store it in a variable called fft_result. Then compute the frequency bins using np.fft.fftfreq with window_length and store it in freqs.

Signal Processing

import numpy as np

t = np.linspace(0, 1, 500, endpoint=False)
signal = np.sin(2 * np.pi * 5.5 * t)

window_length = 500
# Compute fft_result using np.fft.fft on signal
# Compute freqs using np.fft.fftfreq with window_length
# Your code here

Hint

Use np.fft.fft to get the frequency components and np.fft.fftfreq to get the corresponding frequencies.

4

Print the magnitude of the frequency spectrum

Print the magnitude of the first 10 values of fft_result using np.abs(fft_result[:10]) to observe spectral leakage.

Signal Processing

import numpy as np

t = np.linspace(0, 1, 500, endpoint=False)
signal = np.sin(2 * np.pi * 5.5 * t)

window_length = 500

fft_result = np.fft.fft(signal)
freqs = np.fft.fftfreq(window_length, d=1/window_length)
# Print the magnitude of the first 10 fft_result values
# Your code here

Hint

Use np.abs() to get the magnitude of the complex FFT results.