FFT-based filtering in SciPy - Mini Project: Build & Apply

FFT-based filtering

📖 Scenario: Imagine you have a noisy signal from a sensor measuring temperature over time. You want to remove the high-frequency noise to see the smooth trend clearly.

🎯 Goal: You will create a simple signal, set a frequency threshold, apply FFT-based filtering to remove noise, and then display the filtered signal.

📋 What You'll Learn

Create a time series signal with noise using numpy arrays

Set a frequency threshold to separate noise from the main signal

Use FFT to transform the signal to frequency domain

Filter out frequencies above the threshold

Use inverse FFT to get the filtered signal back

Print the filtered signal array

💡 Why This Matters

🌍 Real World

FFT-based filtering is used in many fields like audio processing, sensor data cleaning, and image processing to remove unwanted noise and improve signal quality.

💼 Career

Understanding FFT and filtering is important for data scientists and engineers working with time series data, signal processing, and any application where noise reduction is needed.

1

Create a noisy signal

Create a numpy array called time with 100 points from 0 to 1 using np.linspace(0, 1, 100). Then create a numpy array called signal that is sin(2 * pi * 5 * time) plus random noise using np.random.normal(0, 0.5, 100).

SciPy

import numpy as np
# Create time array and noisy signal
# Your code here

Hint

Use np.linspace to create time. Use np.sin and np.random.normal to create signal.

2

Set frequency threshold

Create a variable called freq_threshold and set it to 10. This will be the cutoff frequency to keep in the signal.

SciPy

import numpy as np

time = np.linspace(0, 1, 100)
signal = np.sin(2 * np.pi * 5 * time) + np.random.normal(0, 0.5, 100)
# Set frequency threshold
# Your code here

Hint

Just create a variable freq_threshold and assign it the value 10.

3

Apply FFT and filter frequencies

Use np.fft.fft(signal) to get the FFT of signal and store it in fft_signal. Use np.fft.fftfreq(len(signal), d=1/100) to get the frequency bins and store in freqs. Then set all values in fft_signal to zero where the absolute value of freqs is greater than freq_threshold.

SciPy

import numpy as np

time = np.linspace(0, 1, 100)
signal = np.sin(2 * np.pi * 5 * time) + np.random.normal(0, 0.5, 100)
freq_threshold = 10
# Compute FFT and filter frequencies
# Your code here

Hint

Use np.fft.fft and np.fft.fftfreq. Then zero out frequencies above the threshold.

4

Get filtered signal and print

Use np.fft.ifft(fft_signal) to get the filtered signal and store it in filtered_signal. Then print filtered_signal.real to show the real part of the filtered signal.

SciPy

import numpy as np

time = np.linspace(0, 1, 100)
signal = np.sin(2 * np.pi * 5 * time) + np.random.normal(0, 0.5, 100)
freq_threshold = 10

fft_signal = np.fft.fft(signal)
freqs = np.fft.fftfreq(len(signal), d=1/100)

fft_signal[np.abs(freqs) > freq_threshold] = 0
# Compute inverse FFT and print filtered signal
# Your code here

Hint

Use np.fft.ifft and print the real part of the result.

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