How to Design FIR Filter Using Hamming Window - Simple Guide

To design a FIR filter using the Hamming window, first define the ideal filter's impulse response, then multiply it by the Hamming window to reduce ripples. This method smooths the filter coefficients and controls side lobes for better frequency response.

Syntax

The basic steps to design a FIR filter using the Hamming window are:

• Define the filter order N and cutoff frequency fc.
• Calculate the ideal impulse response h_ideal[n] for a low-pass filter.
• Create the Hamming window w[n] of length N+1.
• Multiply the ideal response by the window: h[n] = h_ideal[n] * w[n].

python

N = 50  # Filter order
fc = 0.3  # Normalized cutoff frequency (0 to 0.5)
n = np.arange(N+1)
# Ideal impulse response for low-pass filter
h_ideal = 2 * fc * np.sinc(2 * fc * (n - N/2))
# Hamming window
w = 0.54 - 0.46 * np.cos(2 * np.pi * n / N)
# Windowed filter coefficients
h = h_ideal * w

Example

This example shows how to design a low-pass FIR filter with cutoff frequency 0.3 (normalized) and order 50 using the Hamming window. It plots the filter coefficients and frequency response.

python

import numpy as np
import matplotlib.pyplot as plt
from scipy.signal import freqz

N = 50  # Filter order
fc = 0.3  # Normalized cutoff frequency (0 to 0.5)
n = np.arange(N+1)

# Ideal impulse response of low-pass filter
h_ideal = 2 * fc * np.sinc(2 * fc * (n - N/2))

# Hamming window
w = 0.54 - 0.46 * np.cos(2 * np.pi * n / N)

# Windowed filter coefficients
h = h_ideal * w

# Frequency response
w_freq, h_freq = freqz(h, worN=8000)

# Plot filter coefficients
plt.figure(figsize=(12,5))
plt.subplot(1,2,1)
plt.stem(n, h, use_line_collection=True)
plt.title('FIR Filter Coefficients (Hamming Window)')
plt.xlabel('Sample')
plt.ylabel('Amplitude')

# Plot frequency response
plt.subplot(1,2,2)
plt.plot(w_freq/np.pi, 20 * np.log10(np.abs(h_freq)))
plt.title('Frequency Response')
plt.xlabel('Normalized Frequency (×π rad/sample)')
plt.ylabel('Magnitude (dB)')
plt.grid(True)
plt.tight_layout()
plt.show()

Output

A plot showing the FIR filter coefficients as stem plot and the frequency response in dB with a clear low-pass shape and smooth roll-off.

Common Pitfalls

Common mistakes when designing FIR filters with the Hamming window include:

• Using an incorrect cutoff frequency scale (should be normalized between 0 and 0.5).
• Not centering the ideal impulse response around N/2, which causes phase distortion.
• Forgetting to multiply by the window, which leads to high ripple in the frequency response.
• Choosing too low filter order, resulting in poor frequency selectivity.

python

import numpy as np

# Wrong: cutoff frequency not normalized
fc_wrong = 3000  # Should be normalized

# Wrong: impulse response not centered
N = 50
n = np.arange(N+1)
h_ideal_wrong = 2 * 0.3 * np.sinc(2 * 0.3 * n)  # Missing centering

# Correct approach
fc = 0.3
h_ideal_correct = 2 * fc * np.sinc(2 * fc * (n - N/2))

Quick Reference

Steps to design FIR filter using Hamming window:

• Choose filter order N (higher means sharper cutoff).
• Set normalized cutoff frequency fc (0 to 0.5).
• Calculate ideal impulse response: h_ideal[n] = 2 * fc * sinc(2 * fc * (n - N/2)).
• Generate Hamming window: w[n] = 0.54 - 0.46 * cos(2πn/N).
• Multiply: h[n] = h_ideal[n] * w[n].
• Use h[n] as FIR filter coefficients.

Key Takeaways

Design FIR filters by multiplying the ideal impulse response with the Hamming window to reduce ripples.

Always center the ideal impulse response around half the filter order to maintain linear phase.

Normalize cutoff frequency between 0 and 0.5 before designing the filter.

Higher filter order improves frequency selectivity but increases computation.

The Hamming window smooths coefficients and reduces side lobes in frequency response.