Fourier Transform intuition in Signal Processing - Practice Problems & Coding Challenges

Challenge - 5 Problems

🧠 Conceptual

intermediate

Understanding the frequency components of a signal

Imagine you have a sound wave that is a mix of two pure tones: one at 5 Hz and another at 20 Hz. What does the Fourier Transform of this signal show?

A It shows a continuous spread of frequencies from 5 Hz to 20 Hz.

B It shows a single spike at 25 Hz, the sum of the two frequencies.

C It shows two spikes at 5 Hz and 20 Hz representing the two tones.

D It shows no spikes because the signal is a mix and not pure.

💡 Hint

Think about how Fourier Transform breaks down signals into their basic frequency parts.

The Fourier Transform decomposes a signal into its frequency components. Since the signal is a mix of two pure tones, the transform shows spikes at those exact frequencies.

❓ Predict Output

intermediate

Output of Fourier Transform on a simple sine wave

What is the output magnitude array of the Fourier Transform for a pure sine wave of frequency 1 Hz sampled at 10 Hz for 10 samples?

Signal Processing

import numpy as np

fs = 10  # sampling frequency
f = 1    # sine wave frequency
N = 10   # number of samples
t = np.arange(N) / fs
signal = np.sin(2 * np.pi * f * t)
ft = np.fft.fft(signal)
magnitude = np.abs(ft)
print(magnitude)

A [0.0, 5.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 5.0]

B [0.0, 0.0, 5.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 5.0]

C [0.0, 0.0, 0.0, 5.0, 0.0, 0.0, 0.0, 0.0, 0.0, 5.0]

D [5.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]

💡 Hint

Recall that the FFT output is symmetric and the magnitude peaks at the frequency bins corresponding to the sine wave frequency.

The FFT magnitude shows peaks at indices corresponding to the sine wave frequency and its symmetric counterpart. For 1 Hz with 10 samples at 10 Hz sampling, peaks appear at indices 1 and 9 (which is 10-1).

❓ data_output

advanced

Fourier Transform of a noisy signal

You have a signal composed of a 10 Hz sine wave plus random noise. After applying the Fourier Transform, what would you expect the frequency spectrum to look like?

A A clear spike at 10 Hz with a noisy background spread across other frequencies.

B A single spike at 0 Hz representing the average value.

C A flat line with no spikes because noise cancels the sine wave.

D Multiple clear spikes at random frequencies with no dominant peak.

💡 Hint

Think about how noise affects the frequency spectrum and how a sine wave appears in it.

The sine wave produces a clear spike at its frequency (10 Hz). Noise adds random components across frequencies, creating a noisy background but not hiding the main spike.

❓ visualization

advanced

Interpreting the Fourier Transform plot

Given a plot of a Fourier Transform magnitude showing two peaks at 4 Hz and 15 Hz, what can you infer about the original time signal?

A The original signal is random noise with no clear frequencies.

B The original signal is a single frequency at 19 Hz, the sum of the peaks.

C The original signal is a constant value with no variation.

D The original signal contains two main frequency components at 4 Hz and 15 Hz.

💡 Hint

Remember that peaks in the Fourier Transform correspond to frequencies present in the signal.

Peaks in the Fourier Transform magnitude indicate the presence of those frequencies in the original signal. Two peaks mean two main frequency components.

🔧 Debug

expert

Identifying the error in Fourier Transform frequency indexing

Consider this code snippet that tries to find the frequency corresponding to the maximum magnitude in an FFT result. What is the error?

import numpy as np
fs = 100
N = 1000
t = np.arange(N) / fs
signal = np.sin(2 * np.pi * 5 * t)
ft = np.fft.fft(signal)
magnitude = np.abs(ft)
max_index = np.argmax(magnitude)
freq = max_index * fs / N
print(freq)

A The sampling frequency fs is used incorrectly; it should be divided by N squared.

B The code does not handle negative frequencies; max_index may correspond to a mirrored frequency.

C The np.argmax function returns the minimum value index, causing wrong frequency calculation.

D The FFT magnitude should be squared before finding the max index.

💡 Hint

Think about how FFT output is symmetric and how frequency bins correspond to positive and negative frequencies.

The FFT output is symmetric. The maximum magnitude might appear at a mirrored index representing a negative frequency. Without shifting or limiting to positive frequencies, the frequency calculation can be incorrect.

Practice

1. What does the Fourier Transform do to a signal?

easy

A. Breaks it into simple waves of different frequencies

B. Changes the signal into a random noise

C. Removes all frequencies except the highest

D. Converts the signal into a single number

Solution

1. Step 1: Understand the purpose of Fourier Transform

   The Fourier Transform breaks a complex signal into simpler parts called sine and cosine waves.

2. Step 2: Identify what the transform reveals

   It shows the different frequencies that make up the original signal.

3. Final Answer:

   Breaks it into simple waves of different frequencies -> Option A

4. Quick Check:

   Fourier Transform = signal broken into frequencies [OK]

Hint: Fourier Transform splits signals into frequency waves [OK]

Common Mistakes:

• Thinking it removes frequencies instead of showing them
• Believing it converts signal to a single number
• Confusing it with noise generation

2. Which of the following is the correct syntax to compute the Fourier Transform of a signal array signal using NumPy in Python?

easy

A. np.fft.fft(signal)

B. np.fft.transform(signal)

C. np.fourier(signal)

D. np.fft.fourier_transform(signal)

Solution

1. Step 1: Recall NumPy FFT function name

   NumPy uses np.fft.fft() to compute the Fourier Transform of a signal.

2. Step 2: Check other options for correctness

   Other options are not valid NumPy functions for Fourier Transform.

3. Final Answer:

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

4. Quick Check:

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

Hint: Use np.fft.fft() for Fourier Transform in NumPy [OK]

Common Mistakes:

• Using incorrect function names like np.fft.transform
• Confusing Fourier Transform with other NumPy functions
• Misspelling 'fft'

3. Given the Python code:

import numpy as np
signal = np.array([1, 0, -1, 0])
ft = np.fft.fft(signal)
print(np.round(ft, 2))

What is the output?

medium

A. [0.+0.j 0.+0.j 0.+0.j 0.+0.j]

B. [0.+0.j 0.+2.j 0.+0.j 0.-2.j]

C. [0.+0.j 2.+0.j 0.+0.j 2.+0.j]

D. [0.+0.j 1.-1.j 0.+0.j 1.+1.j]

Solution

1. Step 1: Compute FFT of the signal [1, 0, -1, 0]

   The FFT of this 4-point signal results in complex values representing frequency components.

2. Step 2: Calculate or recall FFT output

   The FFT output is [0.+0.j, 0.+2.j, 0.+0.j, 0.-2.j] after rounding to 2 decimals.

3. Final Answer:

   [0.+0.j 0.+2.j 0.+0.j 0.-2.j] -> Option B

4. Quick Check:

   FFT of [1,0,-1,0] = [0, 2j, 0, -2j] [OK]

Hint: FFT output shows complex frequency components [OK]

Common Mistakes:

• Ignoring imaginary parts in FFT output
• Mixing up real and imaginary values
• Assuming FFT output is all real numbers

4. The following code attempts to compute the Fourier Transform but gives an error:

import numpy as np
signal = [1, 2, 3, 4]
ft = np.fft.fft(signal, n='4')
print(ft)

What is the error and how to fix it?

medium

A. Error: print statement syntax wrong, fix by adding parentheses

B. Error: signal must be a NumPy array, fix by converting signal to np.array

C. Error: np.fft.fft does not accept 'n' parameter, remove it

D. Error: 'n' must be an integer, fix by removing quotes: n=4

Solution

1. Step 1: Identify the error cause

   The parameter n expects an integer, but a string '4' is given causing a type error.

2. Step 2: Fix the error

   Remove quotes around 4 to pass an integer: n=4.

3. Final Answer:

   Error: 'n' must be an integer, fix by removing quotes: n=4 -> Option D

4. Quick Check:

   Parameter types must match function signature [OK]

Hint: Pass integer, not string, for numeric parameters [OK]

Common Mistakes:

• Passing string instead of integer for 'n'
• Not converting list to NumPy array (not error here)
• Misreading error as print syntax issue

5. You have a noisy audio signal sampled at 1000 Hz. You want to remove frequencies above 300 Hz using Fourier Transform. Which steps correctly describe this process?

hard

A. Remove frequencies above 300 Hz directly from time signal, then compute FFT

B. Apply inverse FFT first, then remove frequencies above 300 Hz

C. Compute FFT, zero out frequencies above 300 Hz, then apply inverse FFT

D. Compute FFT, keep only frequencies above 300 Hz, then apply inverse FFT

Solution

1. Step 1: Use FFT to convert time signal to frequency domain

   Compute the Fourier Transform to see all frequency components.

2. Step 2: Filter frequencies above 300 Hz

   Set frequency components above 300 Hz to zero to remove noise.

3. Step 3: Convert back to time domain

   Apply inverse FFT to get the cleaned time signal without high frequencies.

4. Final Answer:

   Compute FFT, zero out frequencies above 300 Hz, then apply inverse FFT -> Option C

5. Quick Check:

   Filter in frequency domain, then inverse FFT [OK]

Hint: Filter frequencies in FFT, then inverse transform [OK]

Common Mistakes:

• Filtering in time domain instead of frequency domain
• Applying inverse FFT before filtering
• Keeping unwanted frequencies instead of zeroing them