FFT-based filtering in SciPy - Step-by-Step Execution

Concept Flow - FFT-based filtering

Input signal

↓

Apply FFT -> frequency domain

↓

Create filter mask

↓

Multiply FFT by filter mask

↓

Apply inverse FFT -> filtered signal

↓

Output filtered signal

The signal is transformed to frequency domain using FFT, filtered by multiplying with a mask, then transformed back to time domain.

Execution Sample

SciPy

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

x = np.array([1, 2, 3, 4, 5, 6, 7, 8])
X = fft(x)
mask = np.array([1, 1, 0, 0, 0, 0, 0, 0])
Y = X * mask
y = ifft(Y)

This code applies a simple low-pass filter by zeroing out high frequency components using FFT and inverse FFT.

Execution Table

Step	Action	Variable	Value/Result
1	Input signal defined	x	[1 2 3 4 5 6 7 8]
2	Apply FFT to x	X	[36.+0.j -4.+9.65685425j -4.+4.j -4.+1.65685425j -4.+0.j -4.-1.65685425j -4.-4.j -4.-9.65685425j]
3	Create filter mask	mask	[1 1 0 0 0 0 0 0]
4	Multiply FFT by mask	Y	[36.+0.j -4.+9.65685425j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j]
5	Apply inverse FFT to Y	y	[4. +0.j 0.5+1.20710678j 0. +0.j 0.5-1.20710678j 0. +0.j 0. +0.j 0. +0.j 0. +0.j]
6	Output filtered signal	real part of y	[4. 0.5 0. 0.5 0. 0. 0. 0.]

💡 Filtering complete after inverse FFT; high frequencies zeroed out by mask.

Variable Tracker

Variable	Start	After Step 2	After Step 3	After Step 4	After Step 5	Final
x	[1 2 3 4 5 6 7 8]	[1 2 3 4 5 6 7 8]	[1 2 3 4 5 6 7 8]	[1 2 3 4 5 6 7 8]	[1 2 3 4 5 6 7 8]	[1 2 3 4 5 6 7 8]
X	N/A	[36.+0.j -4.+9.65685425j -4.+4.j -4.+1.65685425j -4.+0.j -4.-1.65685425j -4.-4.j -4.-9.65685425j]	[36.+0.j -4.+9.65685425j -4.+4.j -4.+1.65685425j -4.+0.j -4.-1.65685425j -4.-4.j -4.-9.65685425j]	[36.+0.j -4.+9.65685425j -4.+4.j -4.+1.65685425j -4.+0.j -4.-1.65685425j -4.-4.j -4.-9.65685425j]	[36.+0.j -4.+9.65685425j -4.+4.j -4.+1.65685425j -4.+0.j -4.-1.65685425j -4.-4.j -4.-9.65685425j]	[36.+0.j -4.+9.65685425j -4.+4.j -4.+1.65685425j -4.+0.j -4.-1.65685425j -4.-4.j -4.-9.65685425j]
mask	N/A	N/A	[1 1 0 0 0 0 0 0]	[1 1 0 0 0 0 0 0]	[1 1 0 0 0 0 0 0]	[1 1 0 0 0 0 0 0]
Y	N/A	N/A	N/A	[36.+0.j -4.+9.65685425j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j]	[36.+0.j -4.+9.65685425j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j]	[36.+0.j -4.+9.65685425j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j]
y	N/A	N/A	N/A	N/A	[4. +0.j 0.5+1.20710678j 0. +0.j 0.5-1.20710678j 0. +0.j 0. +0.j 0. +0.j 0. +0.j]	[4. +0.j 0.5+1.20710678j 0. +0.j 0.5-1.20710678j 0. +0.j 0. +0.j 0. +0.j 0. +0.j]

Key Moments - 3 Insights

Why do some values in the filtered signal have imaginary parts after inverse FFT?

The inverse FFT can produce small imaginary parts due to numerical errors, but the real part is the meaningful filtered signal (see step 5 in execution_table).

Why do we multiply the FFT by a mask instead of filtering the original signal directly?

Multiplying in frequency domain corresponds to filtering specific frequencies, which is more efficient and precise than time-domain filtering (see step 4 in execution_table).

Why are some frequency components set to zero in the mask?

Zeroing frequency components removes those frequencies from the signal, acting as a filter to keep only desired frequencies (see step 3 and 4 in execution_table).

Visual Quiz - 3 Questions

Look at the execution_table at step 2. What is the FFT value of the first frequency component?

A -4 + 9.65j

B 36 + 0j

C 0 + 0j

D 4 + 0j

💡 Hint

Check the 'Value/Result' column for variable X at step 2.

At which step does the frequency mask get applied to the FFT result?

A Step 4

B Step 2

C Step 3

D Step 5

💡 Hint

Look for the step where variable Y is defined as X multiplied by mask.

If the mask was all ones, what would be the output signal after inverse FFT?

A Only high frequencies remain

B A zero signal

C The original input signal

D Only low frequencies remain

💡 Hint

Refer to variable_tracker for mask and its effect on Y and y.

Concept Snapshot

FFT-based filtering steps:
1. Transform signal to frequency domain with FFT.
2. Create a frequency mask to keep or remove frequencies.
3. Multiply FFT by mask to filter.
4. Use inverse FFT to get filtered signal back.
Key: Multiplying in frequency domain filters signal components.

Full Transcript

FFT-based filtering transforms a signal from time domain to frequency domain using FFT. Then, a filter mask is created to select which frequencies to keep or remove. The FFT result is multiplied by this mask, zeroing out unwanted frequencies. Finally, inverse FFT converts the filtered frequency data back to time domain, producing the filtered signal. Imaginary parts after inverse FFT are small numerical errors and the real part is the filtered output. This method efficiently filters signals by frequency components.