Frequency spectrum interpretation in Signal Processing - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to compute the frequency spectrum of a signal using FFT.

Signal Processing

spectrum = np.fft.[1](signal)

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

Aifft

Bfft

Crfft

Dfftshift

Attempts:

3 left

💡 Hint

Common Mistakes

Using inverse FFT (ifft) instead of FFT.

Using fftshift which only shifts zero frequency component.

The fft function computes the frequency spectrum of the signal.

2fill in blank

medium

Complete the code to calculate the magnitude spectrum from the FFT result.

Signal Processing

magnitude = np.abs([1])

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

Aspectrum

Bsignal

Cfreqs

Dphase

Attempts:

3 left

💡 Hint

Common Mistakes

Taking absolute value of the original signal instead of FFT result.

Using frequency array instead of spectrum.

The magnitude spectrum is the absolute value of the FFT result stored in spectrum.

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(len(signal), d=[1])

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

A1/fs

Bfs

Clen(signal)

D1

Attempts:

3 left

💡 Hint

Common Mistakes

Using sampling frequency fs directly instead of its inverse.

Using fixed value 1 which assumes 1 second sample spacing.

The d parameter is the sample spacing, which is the inverse of sampling frequency fs.

4fill in blank

hard

Fill both blanks to create a dictionary of frequencies and their magnitudes for frequencies above zero.

Signal Processing

freq_magnitude = {f: m for f, m in zip(freqs, magnitude) if f [1] 0 and m [2] 0}

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

A>

B<

C!=

D==

Attempts:

3 left

💡 Hint

Common Mistakes

Using less than zero for frequency which excludes positive frequencies.

Using equality instead of inequality for magnitude.

We want frequencies greater than zero and magnitudes not equal to zero.

5fill in blank

hard

Fill all three blanks to create a filtered dictionary of frequencies and magnitudes where magnitude is above threshold and frequency is positive.

Signal Processing

filtered = { [1]: [2] for [3] in zip(freqs, magnitude) if [1] > 0 and [2] > threshold }

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

Af

Bm

Cf, m

Dfreq, mag

Attempts:

3 left

💡 Hint

Common Mistakes

Using variable names not unpacked from zip.

Mixing keys and values in the dictionary comprehension.

Use f and m as keys and values, and unpack f, m from the zipped frequencies and magnitudes.

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