FFT with np.fft module in NumPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of FFT on a simple signal

What is the output of the following code that computes the FFT of a simple signal?

NumPy

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

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.+0.j 0.+0.j 1.+0.j]

💡 Hint

Recall that FFT output is complex and symmetric for real inputs.

The FFT of the signal [1, 0, -1, 0] results in complex values with imaginary parts at certain frequencies. The output shows non-zero imaginary parts at indices 1 and 3.

❓ data_output

intermediate

Length of FFT output array

Given a real input array of length 8, what is the length of the output array from np.fft.fft?

NumPy

import numpy as np
x = np.arange(8)
fft_x = np.fft.fft(x)
print(len(fft_x))

A 4

B 16

C 7

D 8

💡 Hint

FFT output length equals input length for np.fft.fft.

The FFT output array length is the same as the input array length when using np.fft.fft.

🔧 Debug

advanced

Identify the error in FFT frequency calculation

What error does the following code raise when trying to compute FFT frequencies for a signal of length 5? import numpy as np freqs = np.fft.fftfreq(5, d=0) print(freqs)

NumPy

import numpy as np
freqs = np.fft.fftfreq(5, d=0)
print(freqs)

A ZeroDivisionError

B TypeError

C ValueError

D No error, prints frequencies

💡 Hint

Check the meaning of the parameter d in fftfreq.

The parameter d is the sample spacing. Setting d=0 causes division by zero inside fftfreq, raising ZeroDivisionError.

🚀 Application

advanced

Using FFT to find dominant frequency

Given a noisy signal composed of two sine waves at 5 Hz and 20 Hz sampled at 100 Hz, which code snippet correctly identifies the dominant frequency using FFT?

NumPy

import numpy as np
fs = 100
T = 1/fs
t = np.linspace(0, 1, fs, endpoint=False)
signal = np.sin(2*np.pi*5*t) + 0.5*np.sin(2*np.pi*20*t) + 0.1*np.random.randn(fs)

A

fft_vals = np.fft.fft(signal)
freqs = np.fft.fftfreq(len(signal), T)
dominant_freq = freqs[np.argmax(np.abs(fft_vals))]
print(round(dominant_freq, 1))

B

fft_vals = np.fft.fft(signal)
freqs = np.fft.fftfreq(len(signal), 1)
dominant_freq = freqs[np.argmax(np.abs(fft_vals))]
print(round(dominant_freq, 1))

C

fft_vals = np.fft.fft(signal)
freqs = np.fft.fftfreq(len(signal))
dominant_freq = freqs[np.argmax(np.abs(fft_vals))]
print(round(dominant_freq, 1))

D

fft_vals = np.fft.fft(signal)
freqs = np.fft.fftfreq(len(signal), 0)
dominant_freq = freqs[np.argmax(np.abs(fft_vals))]
print(round(dominant_freq, 1))

💡 Hint

Sample spacing d should be the inverse of sampling frequency fs.

The sample spacing d must be 1/fs to get correct frequency values. Option A uses d=T=1/fs correctly.

🧠 Conceptual

expert

Interpretation of FFT output symmetry

For a real-valued input signal of length N, which statement about the FFT output from np.fft.fft is true?

A The FFT output is always purely real.

B The FFT output contains only positive frequencies.

C The FFT output is symmetric: the second half is the complex conjugate of the first half reversed.

D The FFT output length is half the input length.

💡 Hint

Think about properties of FFT for real inputs.

For real inputs, FFT output is symmetric: the second half is the complex conjugate of the first half reversed. This is called Hermitian symmetry.