Multi-rate Signal Processing - Half-band Filter

Jul 14, 2025#数字通信2

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The half-band filter is a type of FIR filter characterized by having nearly half of its coefficients equal to zero, which reduces the computational load to half that of a standard FIR filter. Key features include: - The passband width $ w_p $ (cutoff frequency) is equal to the stopband width $ \pi - w_s $ (where $ w_s $ is the start frequency of the stopband), and both the passband and stopband ripples are equal. - The frequency response satisfies the condition: $$ H(\mathrm{e}^{\mathrm{j}w}) + H(\mathrm{e}^{\mathrm{j}(\pi-w)}) = 1 $$ - The unit impulse response is defined as: $$ h(n) = \begin{cases} 0 & \text{if } n - \frac{N-1}{2} \text{ is even} \\ \frac{1}{2} & \text{if } n = \frac{N-1}{2} \end{cases} $$ where $ N $ is the filter length and must be odd.

A halfband filter is essentially an FIR filter, but with nearly half of its coefficients being zero, thus reducing the computational load to half that of a regular FIR filter.

The halfband filter has the following characteristics:

The passband width $w_p$ (passband cutoff frequency) of the halfband filter is equal to the stopband width $\pi - w_s$ (where $w_s$ is the stopband starting frequency), and the passband ripple is equal to the stopband ripple.
The frequency response of the halfband filter satisfies

$H(\mathrm{e}^{\mathrm{j}w}) + H(\mathrm{e}^{\mathrm{j}(\pi-w)}) = 1$
```
 The unit impulse response satisfies
```

h(n) = \begin{cases}0 & \text{when } n - \frac{N-1}{2} \text{ is even} \\ \frac{1}{2} & \text{when } n = \frac{N-1}{2} \text{ is true}\end{cases}

N is the length of the filter and must be odd.