Frequency Sampling Method for FIR: Explanation and Example

The frequency sampling method for FIR filters designs the filter by specifying desired frequency responses at selected points and then uses inverse discrete Fourier transform to find filter coefficients. It creates a filter that matches these frequency samples exactly, making it easy to control the filter's behavior at specific frequencies.

How It Works

The frequency sampling method designs an FIR filter by first choosing specific frequencies where you want to control the filter's response. Imagine you want to build a filter that behaves a certain way at some key frequencies, like allowing some frequencies to pass and blocking others.

At these chosen frequencies, you specify the desired output values (like 1 for pass, 0 for stop). Then, the method uses a mathematical tool called the inverse discrete Fourier transform (IDFT) to convert these frequency values back into time-domain filter coefficients. These coefficients define the filter's behavior for all frequencies.

This approach is like setting checkpoints on a map and then drawing the path that connects them smoothly. It is simple and intuitive, especially when you want exact control at certain frequencies.

Example

This example shows how to design a simple low-pass FIR filter using the frequency sampling method in Python with NumPy.

python

import numpy as np
import matplotlib.pyplot as plt

# Number of filter taps (length)
N = 21

# Define frequency samples (normalized from 0 to 1, where 1 is Nyquist frequency)
freq_samples = np.linspace(0, 1, N, endpoint=False)

# Desired frequency response: pass low frequencies (0 to 0.3), stop others
H = np.array([1 if f <= 0.3 else 0 for f in freq_samples])

# Compute filter coefficients using inverse DFT
h = np.fft.ifft(H).real

# Shift coefficients to center the filter (make it symmetric)
h = np.roll(h, N//2)

# Plot filter coefficients
plt.stem(h, use_line_collection=True)
plt.title('FIR Filter Coefficients (Frequency Sampling Method)')
plt.xlabel('Coefficient Index')
plt.ylabel('Amplitude')
plt.show()

# Plot frequency response
w = np.linspace(0, np.pi, 512)
H_freq = np.abs(np.fft.fft(h, 512))
plt.plot(w/np.pi, H_freq)
plt.title('Frequency Response of Designed FIR Filter')
plt.xlabel('Normalized Frequency (×π rad/sample)')
plt.ylabel('Magnitude')
plt.grid(True)
plt.show()

Output

Two plots: 1) Stem plot showing filter coefficients centered around the middle tap. 2) Line plot showing frequency response with magnitude close to 1 for frequencies up to 0.3 and near 0 after.

When to Use

The frequency sampling method is useful when you want to design FIR filters with specific frequency responses at certain points, especially if you want exact control at those frequencies. It is simple and fast for filters with moderate length.

Real-world uses include audio equalizers where you want to boost or cut specific frequency bands, or communication systems where filters must meet precise frequency specifications. However, it may produce ripples or less smooth responses between sampled frequencies, so it is best when exact frequency points matter more than smoothness.

Key Points

• The method designs FIR filters by specifying desired frequency values at selected points.
• It uses inverse discrete Fourier transform to find filter coefficients.
• Gives exact frequency response at sampled points but may have ripples elsewhere.
• Good for filters needing precise control at specific frequencies.
• Simple to implement and understand.

Key Takeaways

Frequency sampling method designs FIR filters by specifying desired frequency responses at chosen points.

It uses inverse discrete Fourier transform to convert frequency samples into filter coefficients.

This method provides exact control at sampled frequencies but may cause ripples between them.

Ideal for applications needing precise frequency control like audio equalizers and communication filters.

It is simple and fast for moderate-length FIR filter design.