Rectangular window limitations in Signal Processing - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to create a rectangular window of length 50 using NumPy.

Signal Processing

import numpy as np
window = np.[1](50)

Drag options to blanks, or click blank then click option'

Ablackman

Bhanning

Cones

Dhamming

Attempts:

3 left

💡 Hint

Common Mistakes

Using other window functions like hamming or hanning instead of ones.

Confusing the window length with the window type.

The rectangular window is created by an array of ones, so np.ones is used.

2fill in blank

medium

Complete the code to compute the Fourier Transform of the rectangular window using NumPy's FFT.

Signal Processing

import numpy as np
window = np.ones(50)
fft_window = np.fft.[1](window)

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Afft

Bfftshift

Crfft

Difft

Attempts:

3 left

💡 Hint

Common Mistakes

Using inverse FFT instead of FFT.

Using FFT shift which only rearranges the FFT output.

The Fourier Transform is computed using np.fft.fft.

3fill in blank

hard

Fix the error in the code to plot the magnitude of the FFT of the rectangular window using matplotlib.

Signal Processing

import numpy as np
import matplotlib.pyplot as plt
window = np.ones(50)
fft_window = np.fft.fft(window)
plt.plot(np.abs(fft_window[1]))
plt.title('Magnitude of FFT of Rectangular Window')
plt.show()

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A[:25]

B[0:25]

C[0:50]

D[:50]

Attempts:

3 left

💡 Hint

Common Mistakes

Using incorrect slice notation like [0:25] which is valid but less common.

Trying to slice with [0:50] which is the full length and unnecessary.

To plot the first half of the FFT magnitude, slice with [:25].

4fill in blank

hard

Fill in the blank to create a dictionary of window lengths and their main lobe widths, assuming main lobe width is 2 divided by window length.

Signal Processing

window_lengths = [20, 40, 80]
main_lobe_widths = {length: 2 [1] length for length in window_lengths}

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A+

B/

C*

D-

Attempts:

3 left

💡 Hint

Common Mistakes

Using multiplication instead of division for main lobe width.

Not iterating over the list correctly.

The main lobe width is calculated as 2 divided by each window length, so the expression is 2 / length. The dictionary comprehension iterates over window_lengths.

5fill in blank

hard

Fill all three blanks to create a plot comparing the rectangular window and its FFT magnitude normalized by the maximum value.

Signal Processing

import numpy as np
import matplotlib.pyplot as plt
N = 50
window = np.ones(N)
fft_window = np.fft.fft(window)
fft_magnitude = np.abs(fft_window) / np.[1](np.abs(fft_window))
plt.plot(window, label='Rectangular Window')
plt.plot(fft_magnitude[:N//2], label='Normalized FFT Magnitude')
plt.[2]('Window Samples')
plt.[3]('Amplitude')
plt.legend()
plt.show()

Drag options to blanks, or click blank then click option'

Amin

Bxlabel

Cylabel

Dmax

Attempts:

3 left

💡 Hint

Common Mistakes

Using min instead of max for normalization.

Mixing up xlabel and ylabel functions.

The FFT magnitude is normalized by its maximum value using np.max. The x-axis label is set with plt.xlabel and the y-axis label with plt.ylabel.

Practice

1. What is a main limitation of using a rectangular window in signal processing?

easy

A. It causes frequency spreading and high side lobes

B. It completely removes noise from the signal

C. It increases the signal amplitude

D. It shortens the signal duration

Solution

1. Step 1: Understand the rectangular window effect

   The rectangular window is a simple window that cuts the signal abruptly, causing spectral leakage.

2. Step 2: Identify the impact on frequency analysis

   This abrupt cut causes frequency spreading and high side lobes, making it hard to distinguish close frequencies.

3. Final Answer:

   It causes frequency spreading and high side lobes -> Option A

4. Quick Check:

   Rectangular window = frequency spreading and side lobes [OK]

Hint: Rectangular window causes side lobes and spreading [OK]

Common Mistakes:

• Thinking it removes noise completely
• Believing it increases signal amplitude
• Assuming it shortens signal duration

2. Which of the following is the correct way to apply a rectangular window to a signal array sig in Python using NumPy?

easy

A. windowed = sig + np.zeros(len(sig))

B. windowed = sig * np.ones(len(sig))

C. windowed = sig / np.ones(len(sig))

D. windowed = sig - np.ones(len(sig))

Solution

1. Step 1: Recall rectangular window definition

   A rectangular window is an array of ones with the same length as the signal.

2. Step 2: Apply window by element-wise multiplication

   Multiplying the signal by an array of ones keeps the signal unchanged but applies the window.

3. Final Answer:

   windowed = sig * np.ones(len(sig)) -> Option B

4. Quick Check:

   Rectangular window = multiply by ones array [OK]

Hint: Multiply signal by ones array for rectangular window [OK]

Common Mistakes:

• Adding or subtracting ones instead of multiplying
• Dividing by ones which is unnecessary
• Using zeros which nullifies the signal

3. Given the following Python code, what will be the approximate side lobe level in dB when using a rectangular window on a signal?

import numpy as np
N = 64
rect_win = np.ones(N)
# Side lobe level for rectangular window is about -13 dB

medium

A. -3 dB

B. -60 dB

C. -40 dB

D. -13 dB

Solution

1. Step 1: Recall typical side lobe level for rectangular window

   The rectangular window has a main lobe and side lobes around -13 dB relative to the main lobe.

2. Step 2: Match the known value to options

   Among the options, -13 dB matches the known side lobe level for rectangular windows.

3. Final Answer:

   -13 dB -> Option D

4. Quick Check:

   Rectangular window side lobes ≈ -13 dB [OK]

Hint: Rectangular window side lobes near -13 dB [OK]

Common Mistakes:

• Confusing side lobe level with main lobe width
• Choosing much lower side lobe levels like -40 dB
• Assuming side lobes are negligible

4. You have this code snippet to apply a rectangular window but it causes unexpected frequency leakage:

import numpy as np
signal = np.sin(2 * np.pi * 5 * np.arange(100) / 100)
window = np.ones(50)
windowed_signal = signal * window

What is the main error?

medium

A. Window length does not match signal length

B. Window contains zeros instead of ones

C. Signal frequency is too high for window

D. Multiplication should be addition

Solution

1. Step 1: Check window and signal length

   The signal length is 100, but the window length is only 50, causing a length mismatch.

2. Step 2: Understand effect of length mismatch

   Multiplying arrays of different lengths causes broadcasting or unintended results, leading to frequency leakage.

3. Final Answer:

   Window length does not match signal length -> Option A

4. Quick Check:

   Window length must equal signal length [OK]

Hint: Window length must match signal length [OK]

Common Mistakes:

• Ignoring length mismatch
• Thinking zeros in window cause leakage
• Confusing multiplication with addition

5. You want to analyze two close frequencies in a signal but the rectangular window causes them to blur together. Which approach best reduces this limitation?

hard

A. Apply rectangular window twice in a row

B. Increase the signal amplitude before applying rectangular window

C. Use a window with lower side lobes like Hamming or Hann

D. Shorten the signal length to reduce frequency resolution

Solution

1. Step 1: Identify the problem with rectangular window

   Rectangular window has high side lobes causing frequency spreading, blurring close frequencies.

2. Step 2: Choose a better window to reduce side lobes

   Windows like Hamming or Hann have lower side lobes, improving frequency resolution and reducing blur.

3. Final Answer:

   Use a window with lower side lobes like Hamming or Hann -> Option C

4. Quick Check:

   Lower side lobes = clearer frequency separation [OK]

Hint: Choose low side lobe windows to separate close frequencies [OK]

Common Mistakes:

• Increasing amplitude does not fix side lobes
• Shortening signal reduces resolution, worsening problem
• Applying rectangular window twice has no benefit