Frequency spectrum interpretation in Signal Processing - Mini Project: Build & Apply

Frequency Spectrum Interpretation

📖 Scenario: You have recorded a simple sound signal and want to understand its frequency components. This is like listening to a song and figuring out which musical notes are playing.

🎯 Goal: Build a small program that creates a signal, sets a frequency threshold, finds frequencies above that threshold, and prints them out.

📋 What You'll Learn

Create a dictionary with frequency values and their amplitudes

Set a threshold amplitude value

Use a dictionary comprehension to select frequencies with amplitude above the threshold

Print the filtered frequencies and their amplitudes

💡 Why This Matters

🌍 Real World

Frequency spectrum analysis helps in music, engineering, and science to understand signals and sounds.

💼 Career

Data scientists and engineers use frequency analysis to process audio, vibrations, and other signals in many industries.

1

Create the frequency spectrum dictionary

Create a dictionary called frequency_spectrum with these exact entries: 100: 0.5, 200: 1.2, 300: 0.9, 400: 0.3, 500: 1.5

Signal Processing

# Create the frequency_spectrum dictionary with given frequencies and amplitudes
# Your code here

Hint

Use curly braces {} to create a dictionary with frequency keys and amplitude values.

2

Set the amplitude threshold

Create a variable called threshold and set it to 1.0

Signal Processing

frequency_spectrum = {100: 0.5, 200: 1.2, 300: 0.9, 400: 0.3, 500: 1.5}
# Set the threshold variable to 1.0
# Your code here

Hint

Just assign the number 1.0 to the variable threshold.

3

Filter frequencies above the threshold

Use a dictionary comprehension to create a new dictionary called filtered_frequencies that contains only the frequencies and amplitudes from frequency_spectrum where the amplitude is greater than threshold

Signal Processing

frequency_spectrum = {100: 0.5, 200: 1.2, 300: 0.9, 400: 0.3, 500: 1.5}
threshold = 1.0
# Create filtered_frequencies using dictionary comprehension
# Your code here

Hint

Use {freq: amp for freq, amp in frequency_spectrum.items() if amp > threshold} to filter the dictionary.

4

Print the filtered frequencies

Write a print statement to display the filtered_frequencies dictionary

Signal Processing

frequency_spectrum = {100: 0.5, 200: 1.2, 300: 0.9, 400: 0.3, 500: 1.5}
threshold = 1.0
filtered_frequencies = {freq: amp for freq, amp in frequency_spectrum.items() if amp > threshold}
# Print the filtered_frequencies dictionary
# Your code here

Hint

Use print(filtered_frequencies) to show the result.

Practice

1. What does a peak in a frequency spectrum represent in signal processing?

easy

A. A strong presence of a specific frequency in the signal

B. The time duration of the signal

C. The amplitude of the original time signal

D. The noise level in the signal

Solution

1. Step 1: Understand frequency spectrum meaning

   The frequency spectrum shows how much of each frequency is present in the signal.

2. Step 2: Interpret peaks in the spectrum

   A peak means that frequency has a strong presence or high amplitude in the signal.

3. Final Answer:

   A strong presence of a specific frequency in the signal -> Option A

4. Quick Check:

   Peak = strong frequency presence [OK]

Hint: Peaks show strong frequencies in the signal [OK]

Common Mistakes:

• Confusing time duration with frequency
• Thinking amplitude in time domain equals peak in spectrum
• Assuming noise causes peaks

2. Which of the following is the correct way to compute the frequency spectrum of a signal in Python using numpy?

easy

A. np.fft.freq(signal)

B. np.fft.ifft(signal)

C. np.fft.fft(signal)

D. np.fft.spectrum(signal)

Solution

1. Step 1: Recall numpy FFT functions

   np.fft.fft() computes the Fourier Transform to get frequency spectrum.

2. Step 2: Check other options

   np.fft.ifft() computes inverse FFT, others do not exist.

3. Final Answer:

   np.fft.fft(signal) -> Option C

4. Quick Check:

   FFT = np.fft.fft() [OK]

Hint: Use np.fft.fft() for frequency spectrum [OK]

Common Mistakes:

• Using inverse FFT instead of FFT
• Using non-existent numpy functions
• Confusing frequency and time domain functions

3. Given the Python code:

import numpy as np
signal = np.sin(2 * np.pi * 5 * np.linspace(0, 1, 100))
spectrum = np.fft.fft(signal)
peak_freq_index = np.argmax(np.abs(spectrum))
print(peak_freq_index)

What is the printed output?

medium

A. 50

B. 95

C. 0

D. 5

Solution

1. Step 1: Understand the signal frequency

   The signal is a sine wave at 5 Hz sampled 100 times over 1 second.

2. Step 2: Find peak frequency index in FFT

   FFT peak index corresponds to frequency 5, so index 5 is max magnitude.

3. Final Answer:

   5 -> Option D

4. Quick Check:

   Peak index = frequency = 5 [OK]

Hint: Peak index matches sine wave frequency [OK]

Common Mistakes:

• Confusing index with frequency value
• Ignoring sampling rate and length
• Mistaking negative frequency indices

4. The following code attempts to find the main frequency in a signal but gives wrong results:

import numpy as np
signal = np.sin(2 * np.pi * 10 * np.linspace(0, 1, 100))
spectrum = np.fft.fft(signal)
peak = np.argmax(spectrum)
print(peak)

What is the error?

medium

A. Using np.fft.fft instead of np.fft.ifft

B. Using np.argmax on complex spectrum instead of magnitude

C. Signal frequency is too high for sampling rate

D. Not normalizing the spectrum before argmax

Solution

1. Step 1: Check how peak is found

   np.argmax(spectrum) finds max of complex numbers, which is incorrect.

2. Step 2: Correct method to find peak

   Use np.argmax(np.abs(spectrum)) to find peak magnitude index.

3. Final Answer:

   Using np.argmax on complex spectrum instead of magnitude -> Option B

4. Quick Check:

   Peak must be on magnitude, not complex values [OK]

Hint: Use magnitude (abs) before argmax [OK]

Common Mistakes:

• Argmax on complex array directly
• Confusing FFT and IFFT
• Ignoring magnitude for peak detection

5. You have a signal composed of two sine waves at 3 Hz and 7 Hz sampled at 50 Hz for 2 seconds. Which code snippet correctly identifies both frequencies from the frequency spectrum?

hard

A. import numpy as np
t = np.linspace(0, 2, 100)
signal = np.sin(2*np.pi*3*t) + np.sin(2*np.pi*7*t)
spectrum = np.fft.fft(signal)
freqs = np.fft.fftfreq(len(t), d=1/50)
peaks = freqs[np.argsort(np.abs(spectrum))[-2:]]
print(np.sort(np.abs(peaks)))

B. import numpy as np
t = np.linspace(0, 2, 100)
signal = np.sin(2*np.pi*3*t) + np.sin(2*np.pi*7*t)
spectrum = np.fft.ifft(signal)
freqs = np.fft.fftfreq(len(t), d=1/50)
peaks = freqs[np.argmax(np.abs(spectrum))]
print(peaks)

C. import numpy as np
t = np.linspace(0, 2, 50)
signal = np.sin(2*np.pi*3*t) + np.sin(2*np.pi*7*t)
spectrum = np.fft.fft(signal)
freqs = np.fft.fftfreq(len(t), d=1/50)
peaks = freqs[np.argsort(np.abs(spectrum))[-2:]]
print(np.sort(np.abs(peaks)))

D. import numpy as np
t = np.linspace(0, 2, 100)
signal = np.sin(2*np.pi*3*t) + np.sin(2*np.pi*7*t)
spectrum = np.fft.fft(signal)
freqs = np.fft.fftfreq(len(t), d=1/100)
peaks = freqs[np.argsort(np.abs(spectrum))[-2:]]
print(np.sort(np.abs(peaks)))

Solution

1. Step 1: Check sampling and signal length

   Sampling at 50 Hz for 2 seconds means 100 samples, so t has 100 points.

2. Step 2: Verify FFT and frequency calculation

   Use np.fft.fft(signal) and np.fft.fftfreq with d=1/50 for correct frequency bins.

3. Step 3: Identify two largest peaks

   Sort magnitudes and pick last two indices for peaks, then print sorted absolute frequencies.

4. Final Answer:

   Code snippet D correctly identifies both frequencies -> Option A

5. Quick Check:

   Correct sampling, FFT, freq bins, and peak selection [OK]

Hint: Match sample count and rate, use fftfreq with correct d [OK]

Common Mistakes:

• Using inverse FFT instead of FFT
• Wrong sample count or sampling interval
• Incorrect frequency bin calculation