Real FFT (rfft) in SciPy - Mini Project: Build & Apply

Real FFT (rfft) Analysis of a Signal

📖 Scenario: You have recorded a simple signal that changes over time. You want to find out which frequencies are present in this signal. This is useful in many areas like music, engineering, and science.

🎯 Goal: You will create a list of signal values, set up the sampling rate, use the real FFT function to find the frequencies, and then print the frequency values.

📋 What You'll Learn

Create a list called signal with exact values: 0, 1, 0, -1, 0, 1, 0, -1

Create a variable called sampling_rate and set it to 8

Use scipy.fft.rfft on signal to get the frequency components

Print the frequencies using scipy.fft.rfftfreq with len(signal) and 1/sampling_rate

💡 Why This Matters

🌍 Real World

Real FFT is used in audio processing, vibration analysis, and any field where understanding the frequency content of real signals is important.

💼 Career

Data scientists and engineers use FFT to analyze signals, detect patterns, and extract features for machine learning and diagnostics.

1

Create the signal data

Create a list called signal with these exact values: 0, 1, 0, -1, 0, 1, 0, -1

SciPy

# Create the signal list with the exact values
# Your code here

Hint

Use square brackets to create a list and separate values with commas.

2

Set the sampling rate

Create a variable called sampling_rate and set it to 8

SciPy

signal = [0, 1, 0, -1, 0, 1, 0, -1]
# Set the sampling_rate variable to 8
# Your code here

Hint

Sampling rate is how many samples per second the signal has.

3

Calculate the real FFT of the signal

Import scipy.fft and use scipy.fft.rfft on signal to get the frequency components. Store the result in a variable called freq_components

SciPy

signal = [0, 1, 0, -1, 0, 1, 0, -1]
sampling_rate = 8
# Import scipy.fft and calculate freq_components using rfft on signal
# Your code here

Hint

Use import scipy.fft and then call scipy.fft.rfft(signal).

4

Print the frequency values

Use scipy.fft.rfftfreq with len(signal) and 1/sampling_rate to get the frequencies. Print the frequencies.

SciPy

import scipy.fft

signal = [0, 1, 0, -1, 0, 1, 0, -1]
sampling_rate = 8

freq_components = scipy.fft.rfft(signal)
# Calculate frequencies using rfftfreq and print them
# Your code here

Hint

Use scipy.fft.rfftfreq(len(signal), 1/sampling_rate) to get the frequency values.

Practice

1. What does the scipy.fft.rfft function compute when applied to a real-valued signal?

easy

A. The full complex Fourier transform including negative frequencies

B. The positive frequency components of the signal's Fourier transform

C. The inverse Fourier transform of the signal

D. The time-domain signal filtered by a low-pass filter

Solution

1. Step 1: Understand the input type

   rfft is designed for real-valued input signals.

2. Step 2: Analyze the output of rfft

   It returns only the positive frequency components because the negative frequencies are mirror images for real signals.

3. Final Answer:

   The positive frequency components of the signal's Fourier transform -> Option B

4. Quick Check:

   Real input -> half spectrum output [OK]

Hint: Real signals -> use rfft for positive frequencies only [OK]

Common Mistakes:

• Confusing rfft with full fft output
• Thinking rfft returns time-domain data
• Assuming rfft returns negative frequencies

2. Which of the following is the correct way to import and use rfft from SciPy's FFT module?

easy

A. import scipy.fft as rfft
result = rfft.rfft(signal)

B. import scipy.rfft
result = scipy.rfft(signal)

C. from scipy.fft import rfft
result = rfft(signal)

D. from scipy import rfft
result = rfft.fft(signal)

Solution

1. Step 1: Recall correct import syntax

   The rfft function is in scipy.fft, so import it directly from there.

2. Step 2: Check usage

   Calling rfft(signal) after import is the correct usage.

3. Final Answer:

   from scipy.fft import rfft\nresult = rfft(signal) -> Option C

4. Quick Check:

   Correct import and call syntax [OK]

Hint: Import rfft directly from scipy.fft for simple use [OK]

Common Mistakes:

• Trying to import rfft from scipy directly
• Using wrong module names
• Calling rfft as a method of scipy.fft without import

3. What is the output of the following code?

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

medium

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

B. [0.+0.j 0.+0.j 0.+0.j]

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

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

Solution

1. Step 1: Understand the input signal

   The signal is [1, 0, -1, 0], a simple 4-point real array.

2. Step 2: Compute rfft manually or recall pattern

   rfft returns N/2+1 = 3 complex values for length 4 input. The values correspond to DC, positive frequency, and Nyquist frequency components. The result is approximately [0.+0.j, 2.+0.j, 0.+0.j].

3. Final Answer:

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

4. Quick Check:

   rfft length 4 -> 3 outputs, middle value 2 [OK]

Hint: rfft output length = input//2 + 1 [OK]

Common Mistakes:

• Expecting output length same as input
• Ignoring complex number format
• Misreading the signal values

4. Identify the error in this code snippet using rfft and fix it:

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

medium

A. The parameter n is smaller than signal length; use n=4 instead

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

C. rfft does not accept the parameter n

D. The import statement is incorrect; use import scipy.fftpack

Solution

1. Step 1: Check the n parameter usage

   The n parameter sets the length of the FFT. Here, n=2 is less than the signal length 4, so the signal is truncated.

2. Step 2: Fix the parameter to match or exceed signal length

   Using n=4 ensures the full signal is transformed without truncation.

3. Final Answer:

   The parameter n is smaller than signal length; use n=4 instead -> Option A

4. Quick Check:

   FFT length must cover signal length [OK]

Hint: Set n ≥ signal length to avoid truncation [OK]

Common Mistakes:

• Passing list instead of array (allowed but less efficient)
• Ignoring effect of n parameter truncation
• Wrong import module for rfft

5. You have a real sensor signal sampled at 100 Hz with 1000 points. You want to find the frequency with the highest amplitude using rfft. Which code snippet correctly computes this frequency?

hard

A. import numpy as np
from scipy.fft import rfft, rfftfreq
signal = np.random.rand(1000)
fft_vals = rfft(signal)
freqs = rfftfreq(len(signal), 1/100)
peak_freq = freqs[np.argmax(np.abs(fft_vals))]
print(peak_freq)

B. import numpy as np
from scipy.fft import fft
signal = np.random.rand(1000)
fft_vals = fft(signal)
freqs = np.fft.fftfreq(len(signal), 1/100)
peak_freq = freqs[np.argmax(np.abs(fft_vals))]
print(peak_freq)

C. import numpy as np
from scipy.fft import rfft
signal = np.random.rand(1000)
fft_vals = rfft(signal)
peak_freq = np.argmax(np.abs(fft_vals))
print(peak_freq)

D. import numpy as np
from scipy.fft import rfft, rfftfreq
signal = np.random.rand(1000)
fft_vals = rfft(signal)
freqs = rfftfreq(len(signal), 100)
peak_freq = freqs[np.argmax(np.abs(fft_vals))]
print(peak_freq)

Solution

1. Step 1: Use rfft and rfftfreq correctly

   For real signals, rfft computes the positive frequencies, and rfftfreq gives the corresponding frequency bins. The sampling interval is 1/100 seconds.

2. Step 2: Find the peak frequency

   Calculate the absolute values of FFT coefficients, find the index of the max, then map to frequency using freqs.

3. Final Answer:

   Code snippet A correctly computes the peak frequency -> Option A

4. Quick Check:

   Use rfft + rfftfreq with correct sampling interval [OK]

Hint: Use rfftfreq with 1/sampling_rate for correct frequencies [OK]

Common Mistakes:

• Using fft and fftfreq for real signals (less efficient)
• Wrong sampling interval in rfftfreq
• Not mapping peak index to frequency