Rectangular window limitations in Signal Processing - Mini Project: Build & Apply

Rectangular Window Limitations in Signal Processing

📖 Scenario: Imagine you are analyzing a short sound clip to find its main frequencies. You use a rectangular window to cut the clip into a segment before analyzing it. This project will help you see how the rectangular window affects the frequency analysis.

🎯 Goal: You will create a simple signal, apply a rectangular window, compute its frequency spectrum using the Fast Fourier Transform (FFT), and observe the limitations of the rectangular window by plotting the results.

📋 What You'll Learn

Create a simple sine wave signal with a known frequency

Create a rectangular window of the same length as the signal

Apply the rectangular window to the signal

Compute the FFT of the windowed signal

Plot the magnitude spectrum to observe the rectangular window effects

💡 Why This Matters

🌍 Real World

Windowing is used in audio processing, communications, and radar to analyze signals in small segments.

💼 Career

Understanding window effects helps data scientists and engineers improve signal analysis accuracy in real-world applications.

1

Create a sine wave signal

Create a numpy array called t with 500 points linearly spaced between 0 and 1 second. Then create a sine wave signal called signal with frequency 5 Hz using t.

Signal Processing

import numpy as np
# Create time array t with 500 points from 0 to 1
# Create sine wave signal with frequency 5 Hz
# Your code here

Hint

Use np.linspace(0, 1, 500) to create t. Use np.sin(2 * np.pi * 5 * t) for the sine wave.

2

Create a rectangular window

Create a numpy array called rect_window of length 500 filled with ones to represent the rectangular window.

Signal Processing

import numpy as np

t = np.linspace(0, 1, 500)
signal = np.sin(2 * np.pi * 5 * t)
# Create rectangular window of length 500
# Your code here

Hint

Use np.ones(500) to create the rectangular window.

3

Apply the rectangular window and compute FFT

Multiply signal by rect_window and store in windowed_signal. Then compute the FFT of windowed_signal using np.fft.fft and store it in fft_result.

Signal Processing

import numpy as np

t = np.linspace(0, 1, 500)
signal = np.sin(2 * np.pi * 5 * t)

rect_window = np.ones(500)
# Multiply signal by rectangular window
# Compute FFT of windowed signal
# Your code here

Hint

Multiply arrays element-wise with *. Use np.fft.fft() to get the frequency spectrum.

4

Plot the magnitude spectrum

Import matplotlib.pyplot as plt. Compute the magnitude spectrum by taking the absolute value of fft_result and store it in magnitude. Plot the first 50 points of magnitude using plt.plot(). Label the x-axis as 'Frequency bins' and y-axis as 'Magnitude'. Finally, call plt.show() to display the plot.

Signal Processing

import numpy as np
import matplotlib.pyplot as plt

t = np.linspace(0, 1, 500)
signal = np.sin(2 * np.pi * 5 * t)

rect_window = np.ones(500)

windowed_signal = signal * rect_window
fft_result = np.fft.fft(windowed_signal)
# Compute magnitude spectrum
# Plot first 50 points
# Label axes
# Show plot
# Your code here

Hint

Use np.abs() for magnitude. Use plt.plot(), plt.xlabel(), plt.ylabel(), and plt.show().