HIGH PASS FILTER USING HAMMING WINDOW FUNCTION

HIGH PASS FILTER USING HAMMING WINDOW FUNCTION

Summary:
A high-pass filter (HPF) allows signals with frequencies higher than a certain cutoff frequency to pass through while attenuating lower frequencies. In digital signal processing, such a filter can be designed using the window method, where an ideal filter’s impulse response is truncated and shaped by a window function. The Hamming window is commonly used for this purpose due to its good frequency response—offering a balance between main-lobe width and side-lobe attenuation. This process involves designing an ideal HPF in the time domain, multiplying it by the Hamming window to control leakage, and then implementing it digitally. A step-by-step design example is provided at the end.

---

What Is a High-Pass Filter?

A high-pass filter passes high-frequency components of a signal and suppresses frequencies lower than a specified cutoff $f_c$. It is used in applications like noise removal, speech processing, and edge detection in images.

---

Ideal vs. Real Filters

• Ideal HPF has a sharp transition and infinite impulse response—non-causal and impractical.

• Practical HPF has a finite impulse response (FIR), achieved by truncating the ideal response and applying a window function like Hamming.

---

Why Use the Hamming Window?

Truncating an ideal filter leads to Gibbs phenomenon—ripples in the frequency response. A Hamming window smooths the edges of the truncated filter to reduce these ripples.

The Hamming window is defined as:

$$w[n] = 0.54 - 0.46\cos\left(\frac{2\pi n}{N-1}\right), \quad 0 \leq n \leq N-1$$

Here, $N$ is the length of the filter.

---

The Filtering Process

1. Design ideal HPF in time-domain:

   $$h_d[n] = \delta[n] - \frac{\sin(2\pi f_c (n - \frac{N}{2}))}{\pi(n - \frac{N}{2})}$$

   where $f_c$ is the normalized cutoff frequency (0 < $f_c$ < 0.5).

2. Apply Hamming window:

   $$h[n] = h_d[n] \cdot w[n]$$

3. Result: FIR high-pass filter with controlled sidelobes and smoother frequency transition.

---

Key Parameters

• Cutoff Frequency $f_c$: Normalized to sampling rate (e.g., 0.3 means 0.3 * fs / 2).

• Filter Length $N$: Determines sharpness of the cutoff and computational cost.

• Window Type: Hamming is used for its smooth spectral behavior.

Comparison of Windowed and Ideal Filters

Feature	Ideal HPF	Hamming-Windowed HPF
Impulse Response	Infinite	Finite
Frequency Transition	Sharp (ideal)	Smooth (realistic)
Sidelobes	High (ripples)	Lowered (less leakage)

Illustrative Example: Design a High-Pass FIR Filter

Design goal: Block frequencies below 500 Hz and allow above.

Given:

• Sampling rate $f_s = 2000$ Hz

• Cutoff $f_c = 500$ Hz → normalized $f_c = 500 / 2000 = 0.25$

• Filter length $N = 21$

Steps:

1. Compute the ideal high-pass filter $h_d[n]$ centered around $n = 10$ (since N is odd).

2. Generate the Hamming window $w[n]$ of length 21.

3. Multiply $h_d[n] \cdot w[n]$ element-wise to get the final FIR coefficients.

First few coefficients:

h[0] ≈ -0.013,  h[1] ≈ -0.017,  h[2] ≈ -0.022, ...
h[10] ≈ 0.894 (center),  h[11] ≈ -0.022, ...

This filter now attenuates signals below 500 Hz and passes those above, with minimal distortion due to the Hamming window.

Here's the frequency response of the high-pass FIR filter using a Hamming window. You can clearly see that frequencies below 500 Hz are attenuated, while higher frequencies pass through, with a smooth transition around the cutoff due to the windowing effect.