The Cascade Integrator Comb (CIC) filter is an efficient digital filter used in multi-rate signal processing, characterized by its low-pass filtering properties and several advantages, including the use of all coefficients equal to 1, which eliminates the need for storage and multipliers during filtering.
### Integrator
The integrator can be represented in the time domain as:
$$
y_1(n) = x(n) + y_1(n-1)
$$
In the frequency domain, its transfer function is:
$$
H_1(e^{jw}) = \frac{1}{1 - e^{-jw}}
$$
The integrator has poles at integer multiples of \(2\pi\) and infinite gain for DC signals.
### Comb Filter
The comb filter is defined in the time domain as:
$$
y_C(n) = x(n) - x(n - RM)
$$
Its frequency domain representation is:
$$
H_C(z) = 1 - z^{-RM}
$$
This filter has only zeros and no poles.
### CIC Filter Characteristics
For a single-stage CIC filter, the amplitude spectrum is given by:
$$
|H_{CIC}(e^{jw})| = \left|\frac{\sin\left(\frac{RM}{2}w\right)}{\sin\left(\frac{w}{2}\right)}\right|
$$
The filter achieves zero-pole cancellation at specific frequencies, with a maximum gain of \(RM\) at DC. The first sidelobe level can be calculated, and as \(R\) approaches infinity, sidelobe suppression reaches approximately 13.46 dB.
### Performance Metrics
For a multi-stage CIC filter, sidelobe suppression, stopband attenuation, and passband ripple can be expressed as:
$$
\begin{cases}
A_N = 13.46N \, \text{dB} \\
\alpha_N = -20N \log b \\
