Fast Fourier Transform (FFT) algorithm in Signal Processing

Introduction

The Fast Fourier Transform (FFT) helps us quickly find the different frequencies inside a signal. It changes a signal from time view to frequency view so we can understand its parts better.

When you want to find the main tones in a music recording.

When checking the heartbeat signal to find any unusual patterns.

When analyzing vibrations in a machine to detect problems early.

When compressing images or sounds by focusing on important frequencies.

When filtering noise out of a recorded sound or signal.

Syntax

Signal Processing

FFT_result = FFT(signal)

# signal: list or array of numbers representing the signal
# FFT_result: complex numbers showing frequency components

The input signal length is often a power of two for fastest results.

The output shows both strength and phase of each frequency.

Examples

This example finds the frequency parts of a simple wave signal.

Signal Processing

import numpy as np
signal = [1, 0, -1, 0]
FFT_result = np.fft.fft(signal)
print(FFT_result)

This example finds the strength of frequencies in a 5 Hz sine wave.

Signal Processing

import numpy as np
signal = np.sin(2 * np.pi * 5 * np.linspace(0, 1, 100, endpoint=False))
FFT_result = np.fft.fft(signal)
print(abs(FFT_result))

Sample Program

This program creates a signal with two known frequencies, finds its frequency parts using FFT, prints the main frequencies detected, and shows a graph of frequency strengths.

Signal Processing

import numpy as np
import matplotlib.pyplot as plt

# Create a signal with two frequencies: 5 Hz and 20 Hz
sampling_rate = 100  # samples per second
T = 1.0 / sampling_rate
x = np.linspace(0.0, 1.0, sampling_rate, endpoint=False)
signal = 0.7 * np.sin(2.0 * np.pi * 5.0 * x) + 0.3 * np.sin(2.0 * np.pi * 20.0 * x)

# Compute FFT
FFT_result = np.fft.fft(signal)

# Compute frequencies for x-axis
freqs = np.fft.fftfreq(len(signal), T)

# Take only positive frequencies and their magnitudes
positive_freqs = freqs[:len(freqs)//2]
magnitudes = np.abs(FFT_result)[:len(freqs)//2]

# Print main frequencies found
main_freqs = positive_freqs[magnitudes > 10]
print('Main frequencies detected (Hz):', main_freqs)

# Plot the magnitude spectrum
plt.plot(positive_freqs, magnitudes)
plt.title('Frequency components of the signal')
plt.xlabel('Frequency (Hz)')
plt.ylabel('Magnitude')
plt.grid(True)
plt.show()

Important Notes

FFT output contains complex numbers; use abs() to get strength.

FFT works best when signal length is a power of two, but libraries handle other lengths too.

Frequency bins correspond to specific frequencies based on sampling rate and signal length.

Summary

FFT quickly finds frequency parts inside a signal.

It helps analyze sounds, vibrations, and other signals in frequency view.

Use libraries like NumPy to easily apply FFT and understand signal frequencies.

Practice

1. What is the main purpose of the Fast Fourier Transform (FFT) algorithm in signal processing?

easy

A. To compress the signal data for storage

B. To convert a signal from time domain to frequency domain

C. To filter out noise from the signal

D. To increase the amplitude of the signal

Solution

1. Step 1: Understand FFT role and compare options

   FFT transforms time-domain signal to frequency domain. A matches exactly; B is compression, C filtering, D amplification.

2. Final Answer:

   To convert a signal from time domain to frequency domain -> Option B

3. Quick Check:

   FFT = Time to Frequency conversion [OK]

Hint: FFT changes signal view from time to frequency fast [OK]

Common Mistakes:

• Confusing FFT with noise filtering
• Thinking FFT compresses data
• Assuming FFT amplifies signals

2. Which of the following is the correct way to import the FFT function from NumPy in Python?

easy

A. from numpy import fft

B. import numpy.fft as fft

C. import fft from numpy

D. from numpy.fft import fft

Solution

1. Step 1: Recall and match Python import syntax

   Correct for specific function: 'from numpy.fft import fft' (C). A imports submodule, B aliases submodule (requires fft.fft()), D invalid.

2. Final Answer:

   from numpy.fft import fft -> Option D

3. Quick Check:

   Correct import syntax = from numpy.fft import fft [OK]

Hint: Use 'from module import function' to import FFT [OK]

Common Mistakes:

• Using 'import fft from numpy' which is invalid syntax
• Importing whole numpy.fft as fft causing confusion
• Trying 'from numpy import fft' which imports the module, not function

3. Given the Python code:

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

What is the output?

medium

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

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

Solution

1. Step 1: Compute and verify FFT output

   np.fft.fft([1,0,-1,0]) corresponds to a sine wave component, resulting in imaginary parts at k=1 and k=3. The output is [0.+0.j 0.+2.j 0.+0.j 0.-2.j], matching [0.+0.j 0.+2.j 0.+0.j 0.-2.j].

2. Final Answer:

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

3. Quick Check:

   FFT sine wave = imaginary components at k=1,3 [OK]

Hint: Sine-like signals yield imaginary FFT components [OK]

Common Mistakes:

• Confusing cosine signal with sine (imaginary output)
• Confusing zeros with non-zero frequency components
• Rounding incorrectly or not at all

4. Consider this Python code snippet:

import numpy as np
signal = [1, 2, 3, 4]
fft_result = fft(signal)
print(fft_result.real)

It raises an error. What is the most likely cause?

medium

A. signal should be a NumPy array, not a list

B. np.fft.fft does not have a real attribute

C. Missing import statement for fft

D. print statement syntax is incorrect

Solution

1. Step 1: Trace code execution for error

   fft(signal) raises NameError: 'fft' undefined. Import is numpy as np (use np.fft.fft); no direct fft import (e.g., from numpy.fft import fft). A, B, D unrelated.

2. Final Answer:

   Missing import statement for fft -> Option C

3. Quick Check:

   Undefined fft = missing import [OK]

Hint: Import fft properly: from numpy.fft import fft [OK]

Common Mistakes:

• Not importing the fft function when using fft()
• Thinking .real is invalid on FFT output
• Confusing print syntax errors

5. You have a noisy signal sampled at 1000 Hz stored in a NumPy array data. You want to remove frequencies above 100 Hz using FFT. Which approach correctly achieves this?

hard

A. Compute FFT, zero out components with frequency > 100 Hz, then inverse FFT

B. Compute FFT, keep only first 100 components, then inverse FFT

C. Apply FFT and multiply all values by 0.1 to reduce noise

D. Use FFT to find max frequency and subtract it from signal

Solution

1. Step 1: Outline correct FFT low-pass filtering

   FFT to freq domain (use fftfreq for bins), zero coeffs where |freq| > 100 (fs=1000, cutoff index ~100/500=0.2*N), ifft back. Matches B precisely; A ignores freq mapping, C/D ineffective.

2. Final Answer:

   Compute FFT, zero out components with frequency > 100 Hz, then inverse FFT -> Option A

3. Quick Check:

   Zero high freq + ifft = low-pass filter [OK]

Hint: Zero unwanted FFT frequencies, then inverse transform [OK]

Common Mistakes:

• Confusing component count with frequency
• Scaling FFT values instead of zeroing
• Misusing FFT to subtract frequencies directly