Real FFT (rfft) in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of rfft on a simple signal

What is the output of the following code that uses scipy.fft.rfft on a simple signal array?

SciPy

import numpy as np
from scipy.fft import rfft
signal = np.array([1, 2, 3, 4])
result = rfft(signal)
print(result)

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

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

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

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

💡 Hint

Remember that rfft returns the positive frequency terms including zero frequency.

The rfft of [1,2,3,4] sums to 10 at zero frequency. The other terms are computed as complex numbers representing frequency components. The correct output matches option A.

❓ data_output

intermediate

Length of rfft output array

Given a real input array of length 10, what is the length of the output array from scipy.fft.rfft?

SciPy

import numpy as np
from scipy.fft import rfft
signal = np.arange(10)
result = rfft(signal)
print(len(result))

A 6

B 9

C 5

D 10

💡 Hint

The length of rfft output is N//2 + 1 for input length N.

For input length 10, rfft returns 10//2 + 1 = 6 complex numbers representing frequencies from 0 to Nyquist.

❓ visualization

advanced

Visualizing magnitude spectrum from rfft

Which code snippet correctly plots the magnitude spectrum of a real signal using scipy.fft.rfft and matplotlib?

A

import numpy as np
import matplotlib.pyplot as plt
from scipy.fft import rfft
signal = np.sin(2*np.pi*5*np.linspace(0,1,100))
freqs = np.fft.fftfreq(len(signal))
spectrum = np.abs(rfft(signal))
plt.plot(freqs, spectrum)
plt.xlabel('Frequency (Hz)')
plt.ylabel('Magnitude')
plt.show()

B

import numpy as np
import matplotlib.pyplot as plt
from scipy.fft import rfft
signal = np.sin(2*np.pi*5*np.linspace(0,1,100))
freqs = np.fft.fftfreq(len(signal), 1/100)
spectrum = np.abs(rfft(signal))
plt.plot(freqs, spectrum)
plt.xlabel('Frequency (Hz)')
plt.ylabel('Magnitude')
plt.show()

C

import numpy as np
import matplotlib.pyplot as plt
from scipy.fft import rfft
signal = np.sin(2*np.pi*5*np.linspace(0,1,100))
freqs = np.fft.rfftfreq(len(signal))
spectrum = rfft(signal)
plt.plot(freqs, spectrum)
plt.xlabel('Frequency (Hz)')
plt.ylabel('Magnitude')
plt.show()

D

import numpy as np
import matplotlib.pyplot as plt
from scipy.fft import rfft
signal = np.sin(2*np.pi*5*np.linspace(0,1,100))
freqs = np.fft.rfftfreq(len(signal), 1/100)
spectrum = np.abs(rfft(signal))
plt.plot(freqs, spectrum)
plt.xlabel('Frequency (Hz)')
plt.ylabel('Magnitude')
plt.show()

💡 Hint

Use rfftfreq with the correct sample spacing and plot the magnitude (absolute value) of rfft output.

Option D correctly uses np.fft.rfftfreq with sample spacing 1/100, computes magnitude with np.abs, and plots frequency vs magnitude. Other options misuse frequency calculation or plot complex values directly.

🔧 Debug

advanced

Error raised by incorrect rfft usage

What error will this code raise when run?

SciPy

from scipy.fft import rfft
signal = 'not an array'
result = rfft(signal)

A ValueError: object too deep for desired array

B TypeError: unsupported operand type(s) for *: 'str' and 'int'

C TypeError: loop of ufunc does not support argument 0 of type str which has no callable rfft method

D TypeError: float() argument must be a string or a number, not 'str'

💡 Hint

rfft expects a numeric array, not a string.

Passing a string to rfft causes a TypeError when numpy tries to multiply string by int internally during FFT computation.

🧠 Conceptual

expert

Understanding output symmetry of rfft

Which statement correctly describes the output of scipy.fft.rfft compared to the full FFT of a real-valued input?

A rfft returns only the negative frequency terms excluding zero frequency, halving the output size.

B rfft returns the full complex FFT output including negative frequencies, doubling the output size.

C rfft returns only the positive frequency terms including zero and Nyquist frequency, exploiting symmetry to reduce output size.

D rfft returns the magnitude spectrum only, discarding phase information.

💡 Hint

Think about how real input signals produce symmetric FFT outputs.

rfft returns N//2+1 complex numbers representing frequencies from zero to Nyquist frequency, using symmetry of real inputs to reduce output size.