Spectral leakage concept in Signal Processing - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to create a simple sine wave signal.

Signal Processing

import numpy as np
fs = 1000  # Sampling frequency
f = 5     # Frequency of the sine wave
t = np.arange(0, 1, 1/fs)
signal = np.sin(2 * np.pi * [1] * t)

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

Afs

Bf

Ct

Dnp.pi

Attempts:

3 left

💡 Hint

Common Mistakes

Using sampling frequency fs instead of signal frequency f.

The sine wave frequency is given by f, so it should multiply 2 * np.pi * f * t.

2fill in blank

medium

Complete the code to compute the FFT of the signal.

Signal Processing

from numpy.fft import fft
N = len(signal)
fft_values = fft([1])

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

At

BN

Csignal

Dfs

Attempts:

3 left

💡 Hint

Common Mistakes

Passing the time vector t instead of the signal.

The FFT should be computed on the signal data, which is stored in signal.

3fill in blank

hard

Fix the error in the code to correctly compute the frequency bins for the FFT.

Signal Processing

freqs = np.fft.fftfreq([1], d=1/fs)

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

AN

Bfs

Csignal

Dt

Attempts:

3 left

💡 Hint

Common Mistakes

Passing sampling frequency fs instead of number of samples N.

The fftfreq function needs the number of samples N and the sample spacing d=1/fs.

4fill in blank

hard

Fill both blanks to create a windowed signal and compute its FFT.

Signal Processing

window = np.[1](N)
windowed_signal = signal * window
fft_windowed = np.fft.[2](windowed_signal)

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

Ahanning

Bhamming

Cfft

Difft

Attempts:

3 left

💡 Hint

Common Mistakes

Using ifft instead of fft.

Using hanning when the question expects hamming.

Applying a Hamming window reduces spectral leakage. Then compute FFT with fft.

5fill in blank

hard

Fill both blanks to create a dictionary of frequency magnitudes above a threshold.

Signal Processing

threshold = 10
freq_magnitudes = {freq: abs(fft_windowed[i]) [1] threshold for i, freq in enumerate(freqs) if freq >= 0 and abs(fft_windowed[i]) [2] threshold}

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

A:

B>

C>=

D,

Attempts:

3 left

💡 Hint

Common Mistakes

Using comma , instead of colon : in dictionary comprehension.

Using >= instead of > inconsistently.

The dictionary comprehension syntax requires a colon : between key and value. We filter frequencies and magnitudes greater than the threshold using >.

Practice

1. What is spectral leakage in signal processing?

easy

A. A type of noise added to the signal during recording

B. Energy spreading to nearby frequencies when signal frequency doesn't match FFT bins

C. Loss of signal amplitude due to filtering

D. A method to increase signal frequency resolution

Solution

1. Step 1: Understand FFT bin matching

   Spectral leakage occurs when the signal frequency does not align exactly with FFT frequency bins.

2. Step 2: Identify effect on energy distribution

   This mismatch causes the signal's energy to spread into nearby frequency bins, not just one bin.

3. Final Answer:

   Energy spreading to nearby frequencies when signal frequency doesn't match FFT bins -> Option B

4. Quick Check:

   Spectral leakage = Energy spreading [OK]

Hint: Mismatch in frequency and bins causes energy spread [OK]

Common Mistakes:

• Confusing spectral leakage with noise
• Thinking spectral leakage increases amplitude
• Assuming spectral leakage is a filtering method

2. Which of the following is the correct way to reduce spectral leakage in FFT analysis?

easy

A. Use a shorter signal segment

B. Decrease the sampling rate

C. Use a window function before FFT

D. Ignore frequencies outside the main peak

Solution

1. Step 1: Recall methods to reduce spectral leakage

   Applying a window function smooths the signal edges, reducing leakage effects.

2. Step 2: Evaluate other options

   Decreasing sampling rate or shortening signal length worsens frequency resolution and leakage; ignoring frequencies doesn't reduce leakage.

3. Final Answer:

   Use a window function before FFT -> Option C

4. Quick Check:

   Windowing reduces leakage [OK]

Hint: Windowing smooths edges to reduce leakage [OK]

Common Mistakes:

• Thinking shorter signals reduce leakage
• Believing lower sampling rate helps leakage
• Ignoring leakage by discarding frequencies

3. Given this Python code snippet using NumPy FFT, what will the plot show regarding spectral leakage?

import numpy as np
import matplotlib.pyplot as plt
fs = 1000
T = 1.0
N = 1000
x = np.linspace(0.0, T, N, endpoint=False)
freq = 50.5
signal = np.sin(2*np.pi*freq*x)
fft_vals = np.fft.fft(signal)
fft_freq = np.fft.fftfreq(N, 1/fs)
plt.plot(fft_freq[:N//2], np.abs(fft_vals)[:N//2])
plt.show()

medium

A. A sharp peak at 50.5 Hz with no spreading

B. No visible peaks due to noise

C. Multiple peaks at integer frequencies only

D. A peak near 50.5 Hz with energy spread to nearby frequencies

Solution

1. Step 1: Analyze signal frequency vs FFT bins

   The signal frequency 50.5 Hz is not an integer multiple of the FFT bin frequency (fs/N = 1 Hz), causing mismatch.

2. Step 2: Understand effect on FFT magnitude plot

   This mismatch causes spectral leakage, spreading energy around 50.5 Hz instead of a sharp peak.

3. Final Answer:

   A peak near 50.5 Hz with energy spread to nearby frequencies -> Option D

4. Quick Check:

   Non-integer frequency causes leakage spread [OK]

Hint: Non-integer frequency causes spread, not sharp peak [OK]

Common Mistakes:

• Expecting a perfect sharp peak at 50.5 Hz
• Thinking peaks appear only at integer frequencies
• Assuming noise hides the peak

4. Identify the error in this code that causes unexpected spectral leakage behavior:

import numpy as np
fs = 500
T = 1
N = fs * T
x = np.linspace(0, T, N)
signal = np.sin(2 * np.pi * 60 * x)
fft_vals = np.fft.fft(signal)
fft_freq = np.fft.fftfreq(N, 1/fs)

medium

A. Using np.linspace with endpoint=True causes off-by-one sample

B. Frequency 60 Hz is too high for sampling rate 500 Hz

C. FFT length N is not an integer

D. Signal frequency must be less than 50 Hz due to Nyquist

Solution

1. Step 1: Check np.linspace usage

   By default, np.linspace includes the endpoint, so x has N samples from 0 to T inclusive, causing one extra sample period.

2. Step 2: Understand effect on FFT

   This causes the signal to be non-periodic in the sampled window, increasing spectral leakage.

3. Final Answer:

   Using np.linspace with endpoint=True causes off-by-one sample -> Option A

4. Quick Check:

   Endpoint inclusion causes leakage [OK]

Hint: Use endpoint=False in linspace for periodic signals [OK]

Common Mistakes:

• Confusing Nyquist limit with leakage cause
• Ignoring endpoint parameter in linspace
• Thinking frequency 60 Hz is invalid for 500 Hz sampling

5. You have a signal with frequency 123.4 Hz sampled at 2000 Hz. To minimize spectral leakage in FFT analysis, which approach is best?

hard

A. Use a longer signal duration and apply a window function

B. Use a shorter signal duration without windowing

C. Increase sampling rate to 5000 Hz without windowing

D. Round frequency to nearest integer before FFT

Solution

1. Step 1: Understand spectral leakage causes

   Frequency 123.4 Hz is not an integer multiple of FFT bins, causing leakage.

2. Step 2: Apply best practices to reduce leakage

   Using a longer signal increases frequency resolution; applying a window reduces leakage by smoothing edges.

3. Step 3: Evaluate other options

   Shorter signals worsen resolution; increasing sampling rate alone doesn't fix leakage; rounding frequency distorts signal.

4. Final Answer:

   Use a longer signal duration and apply a window function -> Option A

5. Quick Check:

   Longer + window reduces leakage [OK]

Hint: Longer signal + windowing reduces leakage best [OK]

Common Mistakes:

• Thinking higher sampling rate alone fixes leakage
• Using shorter signals to reduce leakage
• Rounding frequency instead of handling leakage