Why Fourier transforms reveal frequencies in SciPy - See It in Action

Why Fourier transforms reveal frequencies

📖 Scenario: Imagine you recorded a sound that is a mix of two musical notes played together. You want to find out which notes (frequencies) are in the sound.

🎯 Goal: You will create a simple signal with two frequencies, use Fourier transform to find these frequencies, and then display the result.

📋 What You'll Learn

Create a time array for the signal

Create a signal with two sine waves of different frequencies

Use Fourier transform to find frequency components

Print the frequencies found in the signal

💡 Why This Matters

🌍 Real World

Fourier transforms help us find hidden frequencies in signals like music, sound, or sensor data.

💼 Career

Data scientists use Fourier analysis in fields like audio processing, communications, and vibration analysis.

1

Create a time array for the signal

Create a variable called t that is a numpy array of 500 points evenly spaced from 0 to 1 second.

SciPy

import numpy as np
# Create the time array t with 500 points from 0 to 1
# Your code here

Hint

Use np.linspace(0, 1, 500) to create the time array.

2

Create a signal with two sine waves

Create a variable called signal that is the sum of two sine waves: one with frequency 5 Hz and one with frequency 20 Hz, both using the time array t. Use np.sin() and multiply the frequency by 2 * np.pi * t.

SciPy

import numpy as np

t = np.linspace(0, 1, 500)
# Create signal as sum of 5 Hz and 20 Hz sine waves
# Your code here

Hint

Use np.sin(2 * np.pi * frequency * t) for each sine wave and add them.

3

Apply Fourier transform to find frequencies

Import fft and fftfreq from scipy.fft. Create a variable called fft_values by applying fft to signal. Create a variable called frequencies by applying fftfreq to the length of signal and the sample spacing 1/500.

SciPy

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

t = np.linspace(0, 1, 500)
signal = np.sin(2 * np.pi * 5 * t) + np.sin(2 * np.pi * 20 * t)
# Compute fft_values and frequencies
# Your code here

Hint

Use fft(signal) and fftfreq(len(signal), 1/500) to get frequency components.

4

Print the main frequencies found

Print the frequencies where the absolute value of fft_values is greater than 100. Use a for loop with variables freq and value iterating over zip(frequencies, fft_values). Print freq rounded to 1 decimal place.

SciPy

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

t = np.linspace(0, 1, 500)
signal = np.sin(2 * np.pi * 5 * t) + np.sin(2 * np.pi * 20 * t)

fft_values = fft(signal)
frequencies = fftfreq(len(signal), 1/500)
# Print frequencies with large fft_values
# Your code here

Hint

Check absolute fft values and print frequencies where value is large.

Practice

1. Why does a Fourier transform help us understand a signal better?

easy

A. Because it averages the signal values over time

B. Because it removes noise from the signal

C. Because it changes the signal into a longer time series

D. Because it shows the different frequencies inside the signal

Solution

1. Step 1: Understand what a Fourier transform does

   A Fourier transform breaks a signal into its frequency parts, showing which frequencies are present.

2. Step 2: Compare options to this understanding

   Only Because it shows the different frequencies inside the signal correctly states that it reveals the different frequencies inside the signal.

3. Final Answer:

   Because it shows the different frequencies inside the signal -> Option D

4. Quick Check:

   Fourier transform = frequency breakdown [OK]

Hint: Fourier transform = signal to frequency parts [OK]

Common Mistakes:

• Thinking Fourier transform removes noise
• Confusing time and frequency domains
• Believing it averages signal values

2. Which of the following is the correct way to import the Fourier transform function from scipy?

easy

A. import scipy.fftpack.fft

B. from scipy.fft import fft

C. from scipy import fourier_transform

D. import fft from scipy

Solution

1. Step 1: Recall the correct scipy module for Fourier transforms

   The modern scipy module for Fourier transforms is scipy.fft, and the function is fft.

2. Step 2: Check each import statement

   Only 'from scipy.fft import fft' is valid Python syntax and correct module usage.

3. Final Answer:

   from scipy.fft import fft -> Option B

4. Quick Check:

   Correct import = from scipy.fft import fft [OK]

Hint: Use scipy.fft for Fourier transforms in recent versions [OK]

Common Mistakes:

• Using scipy.fftpack which is older
• Wrong import syntax causing errors
• Assuming function name is fourier_transform

3. What is the output of this code snippet?

import numpy as np
from scipy.fft import fft

signal = np.array([1, 0, -1, 0])
result = fft(signal)
print(np.round(result, 2))

medium

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

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

C. [0.+0.j 1.+0.j 0.+0.j 1.+0.j]

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

Solution

1. Step 1: Understand the input signal and FFT output

   The signal is [1, 0, -1, 0], a simple wave. FFT returns complex numbers representing frequency components.

2. Step 2: Calculate FFT and round results

   FFT of this signal is [0.+0.j 2.+0.j 0.+0.j 2.+0.j]. Rounding to 2 decimals keeps these values.

3. Final Answer:

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

4. Quick Check:

   FFT output matches [0.+0.j 2.+0.j 0.+0.j 2.+0.j] [OK]

Hint: FFT output shows complex frequency components [OK]

Common Mistakes:

• Ignoring complex part signs
• Confusing zeros with small imaginary parts
• Misreading array values

4. Identify the error in this code that tries to compute the Fourier transform:

import numpy as np
from scipy.fft import fft

signal = [1, 2, 3, 4]
result = fft(signal, n=2)
print(result)

medium

A. fft does not accept the parameter n

B. The signal must be a numpy array, not a list

C. The parameter n=2 truncates the signal incorrectly

D. The import statement is incorrect

Solution

1. Step 1: Check fft function parameters

   The fft function accepts n, which sets the length of the transform. Setting n=2 truncates or pads the signal to length 2.

2. Step 2: Understand effect of n=2 on signal length

   The original signal length is 4, but n=2 means fft uses only first 2 elements, effectively truncating the signal.

3. Final Answer:

   The parameter n=2 truncates the signal incorrectly -> Option C

4. Quick Check:

   n parameter changes transform length [OK]

Hint: n parameter sets FFT length, can truncate signal [OK]

Common Mistakes:

• Thinking fft requires numpy array input
• Believing fft does not accept n parameter
• Assuming import is wrong

5. You have a noisy signal composed of two sine waves at 5 Hz and 20 Hz. How can Fourier transform help you separate these frequencies?

hard

A. By converting the signal to frequency domain, peaks at 5 Hz and 20 Hz appear clearly

B. By filtering the signal in time domain before transform

C. By averaging the signal values over time to remove noise

D. By applying inverse Fourier transform directly

Solution

1. Step 1: Understand the purpose of Fourier transform on noisy signals

   Fourier transform converts time signals into frequency domain, showing frequency peaks clearly.

2. Step 2: Identify how frequency peaks help separate components

   In frequency domain, the 5 Hz and 20 Hz sine waves appear as distinct peaks, allowing separation despite noise.

3. Final Answer:

   By converting the signal to frequency domain, peaks at 5 Hz and 20 Hz appear clearly -> Option A

4. Quick Check:

   Frequency peaks reveal signal components [OK]

Hint: Fourier transform reveals frequency peaks in noisy signals [OK]

Common Mistakes:

• Thinking noise is removed before transform
• Confusing time and frequency domain filtering
• Using inverse transform without frequency analysis