FFT computation (fft) in SciPy - 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?

SciPy

import numpy as np
from scipy.fft import fft

signal = np.array([1, 0, 1, 0])
result = fft(signal)
print(result)

A [1.+0.j 0.+0.j 1.+0.j 0.+0.j]

B [2.+0.j 1.+0.j 0.+0.j 1.+0.j]

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

D [4.+0.j 0.+0.j 0.+0.j 0.+0.j]

💡 Hint

Recall that FFT output length matches input and sums frequency components.

The FFT of [1,0,1,0] results in [2,0,2,0] in complex form, representing frequency components.

❓ 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 scipy.fft.fft?

SciPy

import numpy as np
from scipy.fft import fft

x = np.arange(8)
y = fft(x)
print(len(y))

A 4

B 8

C 16

D 7

💡 Hint

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

The FFT output array length is the same as the input array length for scipy.fft.fft.

🔧 Debug

advanced

Identify the error in FFT usage

What error does this code raise when trying to compute FFT on a list of strings?

SciPy

from scipy.fft import fft
signal = ['1', '0', '1', '0']
result = fft(signal)

A TypeError: ufunc 'add' did not contain a loop with signature matching types

B ValueError: Input contains NaN

C SyntaxError: invalid syntax

D IndexError: list index out of range

💡 Hint

FFT expects numeric input, not strings.

Passing a list of strings to fft causes a TypeError because FFT operations require numeric types.

🚀 Application

advanced

Using FFT to find dominant frequency

Given a sampled signal with 1000 points and sampling rate 1000 Hz, which code snippet correctly finds the frequency with the highest amplitude?

SciPy

import numpy as np
from scipy.fft import fft, fftfreq

sampling_rate = 1000
N = 1000
x = np.linspace(0, 1, N, endpoint=False)
signal = np.sin(2*np.pi*50*x) + 0.5*np.sin(2*np.pi*120*x)

# Find dominant frequency code here

A

yf = fft(signal)
xf = fftfreq(N, 1/sampling_rate)
idx = np.argmax(np.abs(yf))
print(xf[idx])

B

yf = fft(signal)
xf = fftfreq(N, sampling_rate)
idx = np.argmin(np.abs(yf))
print(xf[idx])

C

yf = fft(signal)
xf = fftfreq(N, 1/sampling_rate)
idx = np.argmin(np.abs(yf))
print(xf[idx])

D

yf = fft(signal)
xf = fftfreq(N, 1/sampling_rate)
idx = np.argmax(yf)
print(xf[idx])

💡 Hint

Use fftfreq with correct sample spacing and find index of max amplitude.

The dominant frequency corresponds to the index of the maximum absolute FFT value. fftfreq needs sample spacing 1/sampling_rate.

🧠 Conceptual

expert

Understanding FFT symmetry for real inputs

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

A The FFT output length is half the input length for real inputs.

B The FFT output is always purely real for real inputs.

C The FFT output contains only positive frequencies for real inputs.

D The FFT output is symmetric, so the second half is the complex conjugate of the first half reversed.

💡 Hint

Think about conjugate symmetry property of FFT for real signals.

For real inputs, FFT output has Hermitian symmetry: second half is complex conjugate of first half reversed.