2D FFT (fft2) in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of 2D FFT on a simple matrix

What is the output of the following code using scipy.fft.fft2 on a 2x2 matrix?

SciPy

import numpy as np
from scipy.fft import fft2

matrix = np.array([[1, 2], [3, 4]])
result = fft2(matrix)
print(np.round(result, 2))

A

[[10.+0.j  2.+0.j]
 [ 4.+0.j  0.+0.j]]

B

[[10.+0.j  2.+0.j]
 [-4.+0.j  0.+0.j]]

C

[[10.+0.j -2.+0.j]
 [ 4.+0.j  0.+0.j]]

D

[[10.+0.j -2.+0.j]
 [-4.+0.j  0.+0.j]]

💡 Hint

Recall that 2D FFT sums over rows and columns with complex exponentials.

The 2D FFT of the matrix [[1,2],[3,4]] results in [[10, -2], [-4, 0]] with zero imaginary parts. This matches option D.

❓ data_output

intermediate

Shape of 2D FFT result

If you apply scipy.fft.fft2 to a 5x3 matrix, what will be the shape of the output?

SciPy

import numpy as np
from scipy.fft import fft2

matrix = np.random.rand(5,3)
result = fft2(matrix)
print(result.shape)

A (8, 8)

B (3, 5)

C (5, 3)

D (1, 1)

💡 Hint

The FFT output shape matches the input shape unless specified otherwise.

The 2D FFT output shape is the same as the input matrix shape by default.

❓ visualization

advanced

Visualizing magnitude spectrum of 2D FFT

Which option correctly plots the magnitude spectrum of the 2D FFT of a 256x256 image stored in image?

SciPy

import numpy as np
import matplotlib.pyplot as plt
from scipy.fft import fft2, fftshift

# image is a 256x256 numpy array
f_transform = fft2(image)
f_shifted = fftshift(f_transform)
magnitude_spectrum = np.abs(f_shifted)
plt.imshow(np.log1p(magnitude_spectrum), cmap='gray')
plt.title('Magnitude Spectrum')
plt.show()

A Use fft2, plot magnitude spectrum directly without fftshift or log scale.

B Use fft2, then fftshift, then plot log of magnitude spectrum with gray colormap.

C Use fftshift first, then fft2, then plot magnitude spectrum with jet colormap.

D Plot the real part of fft2 output without any transformation or scaling.

💡 Hint

Centering the zero frequency with fftshift and using log scale helps visualize details.

Option B correctly applies fft2, shifts zero frequency to center, takes magnitude, applies log scale, and plots in grayscale for clear visualization.

🧠 Conceptual

advanced

Effect of zero-padding on 2D FFT resolution

What is the main effect of zero-padding a 2D signal before applying fft2?

A It increases frequency resolution by interpolating the FFT output.

B It changes the original frequency content of the signal.

C It removes noise from the signal before FFT.

D It reduces the computation time of the FFT.

💡 Hint

Think about how zero-padding affects the FFT output size and frequency bins.

Zero-padding adds more points, increasing the number of frequency bins, which interpolates the FFT and improves frequency resolution without changing original frequencies.

🔧 Debug

expert

Identify the error in 2D FFT magnitude calculation

What error will this code raise when calculating the magnitude of a 2D FFT result?

SciPy

import numpy as np
from scipy.fft import fft2

matrix = np.array([[1, 2], [3, 4]])
result = fft2(matrix)
magnitude = np.abs(result.real)
print(magnitude)

A No error, but magnitude is incorrect because it uses only real part.

B TypeError because np.abs cannot be applied to real numbers.

C ValueError due to shape mismatch in np.abs call.

D SyntaxError due to missing parentheses.

💡 Hint

Check how magnitude should be calculated from complex numbers.

The code runs without error but magnitude is wrong because it takes absolute value of only the real part, ignoring imaginary parts.

Practice

1. What does the scipy.fft.fft2 function do with a 2D array?

easy

A. It calculates the sum of all elements in the 2D array.

B. It converts the 2D data from the spatial domain to the frequency domain.

C. It sorts the 2D array in ascending order.

D. It normalizes the 2D array values between 0 and 1.

Solution

1. Step 1: Understand the purpose of fft2

   The fft2 function computes the 2-dimensional discrete Fourier Transform, which changes data from spatial to frequency domain.

2. Step 2: Compare options with function behavior

   Only It converts the 2D data from the spatial domain to the frequency domain. correctly describes this transformation. Other options describe unrelated operations.

3. Final Answer:

   It converts the 2D data from the spatial domain to the frequency domain. -> Option B

4. Quick Check:

   2D FFT = spatial to frequency transform [OK]

Hint: Remember fft2 changes space data into frequency patterns [OK]

Common Mistakes:

• Confusing fft2 with sorting or summing functions
• Thinking fft2 normalizes data
• Assuming fft2 works only on 1D data

2. Which of the following is the correct way to import and use fft2 from scipy?

easy

A. import fft2 from scipy
result = fft2(data)

B. import scipy.fft2
result = scipy.fft2(data)

C. from scipy import fft2
result = fft2(data)

D. from scipy.fft import fft2
result = fft2(data)

Solution

1. Step 1: Recall correct import syntax for fft2

   The fft2 function is in the scipy.fft module, so the correct import is from scipy.fft import fft2.

2. Step 2: Check usage correctness

   Calling fft2(data) after import is correct. Other options have wrong import syntax or module names.

3. Final Answer:

   from scipy.fft import fft2 result = fft2(data) -> Option D

4. Quick Check:

   Use 'from scipy.fft import fft2' [OK]

Hint: Import fft2 from scipy.fft module directly [OK]

Common Mistakes:

• Trying to import fft2 directly from scipy
• Using incorrect import syntax like 'import fft2 from scipy'
• Assuming fft2 is a standalone package

3. What is the output of this code snippet?

import numpy as np
from scipy.fft import fft2

arr = np.array([[1, 2], [3, 4]])
result = fft2(arr)
print(np.round(result, 2))

medium

A. [[1.+0.j 2.+0.j]
[3.+0.j 4.+0.j]]

B. [[10.+0.j 2.+0.j]
[ 4.+0.j 0.+0.j]]

C. [[10.+0.j -2.+0.j]
[-4.+0.j 0.+0.j]]

D. [[0.+0.j 0.+0.j]
[0.+0.j 0.+0.j]]

Solution

1. Step 1: Calculate 2D FFT of the array

   The 2D FFT of [[1,2],[3,4]] results in [[10, -2], [-4, 0]] with complex zeros in imaginary parts.

2. Step 2: Round and check output

   Rounding to 2 decimals keeps values same. [[10.+0.j -2.+0.j] [-4.+0.j 0.+0.j]] matches the expected complex array output.

3. Final Answer:

   [[10.+0.j -2.+0.j] [-4.+0.j 0.+0.j]] -> Option C

4. Quick Check:

   fft2([[1,2],[3,4]]) = [[10,-2],[-4,0]] [OK]

Hint: Remember fft2 sums all elements for [0,0] frequency [OK]

Common Mistakes:

• Confusing fft2 output with original array
• Ignoring complex number format
• Mixing up signs in output

4. Identify the error in this code that tries to compute the 2D FFT:

import numpy as np
from scipy.fft import fft2

arr = [[1, 2], [3]]
result = fft2(arr)
print(result)

medium

A. The array shape is invalid for fft2.

B. The input array must be a NumPy array, not a list of lists.

C. The print statement syntax is incorrect.

D. fft2 function is not imported correctly.

Solution

1. Step 1: Check the input array construction

   The arr = [[1, 2], [3]] creates a ragged array (rows have different lengths: 2 and 1), which cannot be converted to a proper NumPy 2D array.

2. Step 2: Identify the error cause

   fft2(arr) fails because np.asarray internally raises ValueError due to invalid shape.

3. Final Answer:

   The array shape is invalid for fft2. -> Option A

4. Quick Check:

   ragged lists cause shape error in fft2 [OK]

Hint: Use consistent lengths in 2D lists before fft2 [OK]

Common Mistakes:

• Creating lists with uneven row lengths
• Not validating array shape before FFT
• Assuming fft2 accepts any list structure

5. You have a 2D image array and want to filter out high-frequency noise using fft2. Which sequence of steps correctly applies this?

hard

A. Apply fft2 to image, zero out high-frequency components, then apply ifft2 to get filtered image.

B. Apply ifft2 to image, zero out low-frequency components, then apply fft2 to get filtered image.

C. Apply fft2 to image, multiply all frequencies by zero, then apply ifft2 to get filtered image.

D. Apply fft2 to image, add random noise to frequency components, then apply ifft2.

Solution

1. Step 1: Understand filtering in frequency domain

   To remove high-frequency noise, transform image with fft2, then zero out high-frequency parts (usually edges of frequency matrix).

2. Step 2: Reconstruct filtered image

   After modifying frequency data, apply inverse fft2 (ifft2) to get back filtered spatial image.

3. Final Answer:

   Apply fft2 to image, zero out high-frequency components, then apply ifft2 to get filtered image. -> Option A

4. Quick Check:

   Filter noise by zeroing high frequencies in fft2 domain [OK]

Hint: Filter noise by zeroing high frequencies, then inverse fft2 [OK]

Common Mistakes:

• Applying ifft2 before fft2
• Zeroing all frequencies removes entire image
• Adding noise instead of removing it