FFT-based filtering in SciPy

Introduction

FFT-based filtering helps us remove unwanted parts of a signal or data by changing it into frequencies, then filtering, and changing it back.

You want to remove noise from a sound recording.

You need to smooth a temperature sensor's data to see trends.

You want to highlight certain patterns in an image by filtering frequencies.

You want to separate slow and fast changes in a signal.

Syntax

SciPy

from scipy.fft import fft, ifft

# fft_data = fft(original_data)
# filtered_fft = fft_data * filter_mask
# filtered_data = ifft(filtered_fft)

Use fft to convert data to frequency space.

Multiply by a filter mask to keep or remove frequencies.

Examples

This example shows a basic FFT and inverse FFT without changing frequencies.

SciPy

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

# Create a simple signal
signal = np.array([1, 2, 3, 4])

# FFT
freq_data = fft(signal)

# Simple filter: keep all frequencies
filtered_freq = freq_data * np.array([1, 1, 1, 1])

# Inverse FFT
filtered_signal = ifft(filtered_freq)

This example removes high frequencies by zeroing them out in the frequency domain.

SciPy

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

signal = np.array([1, 2, 3, 4])
freq_data = fft(signal)

# Filter: zero out high frequencies
filter_mask = np.array([1, 1, 0, 0])
filtered_freq = freq_data * filter_mask
filtered_signal = ifft(filtered_freq)

Sample Program

This program creates a signal with slow and fast parts, removes fast noise using FFT filtering, and shows both signals on a plot.

SciPy

from scipy.fft import fft, ifft
import numpy as np
import matplotlib.pyplot as plt

# Create a noisy signal: slow wave + fast noise
np.random.seed(0)
time = np.linspace(0, 1, 500)
slow_wave = np.sin(2 * np.pi * 5 * time)  # 5 Hz
noise = 0.5 * np.sin(2 * np.pi * 50 * time)  # 50 Hz noise
signal = slow_wave + noise

# FFT
freq_data = fft(signal)

# Frequency array
freq = np.fft.fftfreq(len(signal), d=time[1] - time[0])

# Create filter mask: keep frequencies below 10 Hz
filter_mask = np.abs(freq) < 10

# Apply filter
filtered_freq = freq_data * filter_mask

# Inverse FFT to get filtered signal
filtered_signal = ifft(filtered_freq)

# Plot original and filtered signals
plt.figure(figsize=(10, 6))
plt.plot(time, signal, label='Original Signal')
plt.plot(time, filtered_signal.real, label='Filtered Signal', linewidth=2)
plt.xlabel('Time (seconds)')
plt.ylabel('Amplitude')
plt.title('FFT-based Filtering Example')
plt.legend()
plt.show()

Important Notes

FFT output is complex numbers; use .real to get real part after inverse FFT.

Frequency filtering works by zeroing or reducing unwanted frequency components.

Make sure your filter mask matches the FFT frequency array size.

Summary

FFT-based filtering changes data to frequency space to remove unwanted parts.

Use fft and ifft from scipy.fft for this process.

Apply a mask to keep or remove frequencies, then convert back to original space.

Practice

1. What is the main purpose of using FFT-based filtering in data analysis?

easy

A. To transform data into frequency domain for easier noise removal

B. To sort data values in ascending order

C. To calculate the average of data points

D. To convert data into text format

Solution

1. Step 1: Understand FFT transformation

   FFT converts data from time domain to frequency domain, revealing frequency components.

2. Step 2: Purpose of filtering in frequency domain

   Filtering in frequency domain allows easy removal of unwanted noise or frequencies.

3. Final Answer:

   To transform data into frequency domain for easier noise removal -> Option A

4. Quick Check:

   FFT filtering = frequency domain noise removal [OK]

Hint: FFT helps remove noise by working in frequency domain [OK]

Common Mistakes:

• Confusing FFT with sorting or averaging
• Thinking FFT changes data to text
• Assuming FFT directly cleans data without filtering

2. Which of the following is the correct way to compute the FFT of a 1D numpy array data using scipy?

easy

A. fft_data = scipy.fft_transform(data)

B. fft_data = scipy.fft(data)

C. fft_data = scipy.fft.fft(data)

D. fft_data = scipy.fft_fft(data)

Solution

1. Step 1: Identify correct scipy FFT function

   The scipy FFT module uses fft() to compute the Fast Fourier Transform.

2. Step 2: Check syntax correctness

   The correct syntax is scipy.fft.fft(data), other options are invalid function names.

3. Final Answer:

   fft_data = scipy.fft.fft(data) -> Option C

4. Quick Check:

   Correct FFT call = scipy.fft.fft(data) [OK]

Hint: Use scipy.fft.fft() to compute FFT of array [OK]

Common Mistakes:

• Using incorrect function names like fft_transform
• Omitting the .fft part after scipy.fft
• Confusing with numpy.fft functions

3. Given the code below, what will be the output of filtered_signal?

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

data = np.array([1, 2, 3, 4])
fft_data = fft(data)
mask = np.array([1, 0, 0, 1])
filtered_fft = fft_data * mask
filtered_signal = ifft(filtered_fft).real
print(np.round(filtered_signal, 2))

medium

A. [1. 0. 0. 4.]

B. [2. 2. 3. 3.]

C. [1.5 0. 0. 3.5]

D. [2. 0. 0. 2.]

Solution

1. Step 1: Compute FFT and apply mask

   FFT of data is computed, then multiplied by mask [1,0,0,1], zeroing middle frequencies.

2. Step 2: Compute inverse FFT and round

   Inverse FFT of masked data gives filtered signal; rounding to 2 decimals yields [2. 0. 0. 2.].

3. Final Answer:

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

4. Quick Check:

   Mask zeros middle frequencies, inverse FFT smooths signal [OK]

Hint: Mask zeros frequencies, inverse FFT returns filtered signal [OK]

Common Mistakes:

• Forgetting to multiply FFT by mask
• Not taking real part after inverse FFT
• Rounding incorrectly or not at all

4. Identify the error in the following FFT-based filtering code snippet:

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

data = np.array([1, 2, 3, 4])
fft_data = fft(data)
mask = np.array([1, 0, 0])
filtered_fft = fft_data * mask
filtered_signal = ifft(filtered_fft).real
print(filtered_signal)

medium

A. Data array should be complex type

B. Mask length does not match FFT data length

C. Using ifft instead of fft

D. Missing import for numpy

Solution

1. Step 1: Check mask and FFT data lengths

   FFT of data has length 4, but mask length is 3, causing shape mismatch error.

2. Step 2: Identify error cause

   Multiplying arrays of different lengths raises an error; mask must match FFT length.

3. Final Answer:

   Mask length does not match FFT data length -> Option B

4. Quick Check:

   Mask length must equal FFT length [OK]

Hint: Mask length must match FFT output length [OK]

Common Mistakes:

• Using mask shorter than FFT result
• Confusing ifft with fft usage
• Assuming data must be complex initially

5. You have a noisy signal sampled at 1000 Hz stored in signal. You want to remove frequencies above 100 Hz using FFT-based filtering. Which code snippet correctly implements this filter?

hard

A. import numpy as np
from scipy.fft import fft, ifft
N = len(signal)
freqs = np.fft.fftfreq(N, 1/1000)
fft_signal = fft(signal)
mask = np.abs(freqs) <= 100
filtered_fft = fft_signal * mask
filtered_signal = ifft(filtered_fft).real

B. import numpy as np
from scipy.fft import fft, ifft
N = len(signal)
freqs = np.fft.fftfreq(N)
fft_signal = fft(signal)
mask = freqs < 100
filtered_fft = fft_signal * mask
filtered_signal = ifft(filtered_fft).real

C. import numpy as np
from scipy.fft import fft, ifft
N = len(signal)
freqs = np.fft.fftfreq(N, 1/1000)
fft_signal = fft(signal)
mask = freqs > 100
filtered_fft = fft_signal * mask
filtered_signal = ifft(filtered_fft).real

D. import numpy as np
from scipy.fft import fft, ifft
N = len(signal)
freqs = np.fft.fftfreq(N, 1/1000)
fft_signal = fft(signal)
mask = np.abs(freqs) >= 100
filtered_fft = fft_signal * mask
filtered_signal = ifft(filtered_fft).real

Solution

1. Step 1: Calculate frequency bins correctly

   Use np.fft.fftfreq(N, 1/1000) to get frequencies in Hz for sampling rate 1000 Hz.

2. Step 2: Create mask to keep frequencies ≤ 100 Hz

   Mask should be np.abs(freqs) <= 100 to keep low frequencies and remove higher ones.

3. Step 3: Apply mask and inverse FFT

   Multiply FFT by mask, then apply inverse FFT and take real part for filtered signal.

4. Final Answer:

   mask = np.abs(freqs) <= 100 with freqs = np.fft.fftfreq(N, 1/1000) -> Option A

5. Quick Check:

   Mask keeps frequencies ≤ 100 Hz [OK]

Hint: Use np.abs(freqs) ≤ cutoff for low-pass mask [OK]

Common Mistakes:

• Ignoring sampling rate in frequency calculation
• Using wrong comparison operators in mask
• Not taking absolute value of frequencies