Cascaded integrator–comb filter

At the high sampling rate of ${\displaystyle f_{s}}$, a CIC's transfer function in the z-domain is:

${\displaystyle {\begin{aligned}H(z)&=\left[\sum _{k=0}^{RM-1}z^{-k}\right]^{N}\\&=\left({\frac {1-z^{-RM}}{1-z^{-1}}}\right)^{N}\end{aligned}}}$

where:

${\displaystyle R}$ is the decimation or interpolation ratio,

${\displaystyle M}$ is the number of samples per stage (usually 1 but sometimes 2), and

${\displaystyle N}$ is the order: the number of comb-integrator pairs.

• The numerator comes from multiplying ${\displaystyle N}$ negative feedforward comb stages (each is simply multiplication by ${\displaystyle 1-z^{-RM}}$ in the z-domain).
• The denominator comes from multiplying ${\displaystyle N}$ integrator stages (each is simply multiplication by ${\displaystyle {\tfrac {1}{1-z^{\text{-1}}}}}$ in the z-domain).

An integrator–comb filter is an efficient implementation of a simple 1^st-order moving-average FIR filter, with division by ${\displaystyle RM}$ omitted.[note 1] To see this, consider how a simple moving average filter can be implemented recursively by adding the newest sample ${\displaystyle x[n]}$ to the previous result ${\displaystyle y[n-1]}$ and subtracting the oldest sample ${\displaystyle x[n-RM]}$:

${\displaystyle {\begin{aligned}y[n]&=\sum _{k=0}^{RM-1}x[n-k]\\y[n]&=y[n-1]+\underbrace {x[n]-x[n-RM]} _{{\text{comb filter }}c[n]}.\end{aligned}}}$

The second equality corresponds to a comb filter (${\displaystyle c[n]=x[n]-x[n-RM]}$) that gets integrated (${\displaystyle y[n]=y[n-1]+c[n]}$).

Higher-order CIC structures are obtained by cascading ${\displaystyle N}$ identical simple moving average filters, then rearranging the sections to place all integrators first (decimator) or combs first (interpolator). Such rearrangement is possible because both the combs, the integrators, and the entire structure are linear time-invariant (LTI) systems.

In the interpolating CIC, its upsampler (which normally precedes an interpolation filter) is passed through the comb sections using a Noble identity, reducing the number of delay elements needed by a factor of ${\displaystyle R}$. Similarly, in the decimating CIC, its downsampler (which normally follows a decimation filter) is moved before the comb sections.

1st, 2nd, 3rd-order CIC filters (RM=8) normalized frequency response. Top plot in linear gain, bottom plot in dB.

In the z-domain, each integrator contributes one pole at DC (${\displaystyle z{=}1}$) and one zero at the origin (${\displaystyle z{=}0}$). Each comb contributes ${\displaystyle RM}$ poles at the origin and ${\displaystyle RM}$ zeroes that are equally-spaced around the z-domain's unit circle, but its first zero at DC cancels out with each integrator's pole. N^th-order CIC filters have N times as many poles and zeros in the same locations as the 1^st-order.

Thus, the 1^st-order CIC's frequency response is a crude low-pass filter. Typically the gain is normalized by diving by ${\displaystyle (RM)^{N}}$ so DC has the peak of unity gain. The main lobes drop off as it reaches the next zero, and is followed by a series of successive lobes that have smaller and smaller peaks, separated by the subsequent zeros. This approximates at large ${\displaystyle R}$ a sinc-in-frequency.

An N^th-order CIC's shape corresponds to multiplying that sinc shape on itself N times, resulting in successively greater attenuation. Thus, N^th-order CIC filters are called sinc^N filters. The first sidelobe is attenuated ~13N dB.

The CIC filter's possible range of responses is limited by this shape. Larger amounts of stopband rejection can be achieved by increasing the order, but that increases attenuation in the passband and requires increased bit width for the integrator and comb sections. For this reason, many real-world filtering requirements cannot be met by a CIC filter alone.

A short to moderate length FIR or infinite impulse response (IIR) filter can compensate for the falling slope of a CIC filter's shape.[5] Multiple interpolation and decimation rates can reuse the same set of compensation FIR coefficients, since the shape of the CIC's main lobe changes very little when the decimation ratio is changed.[5]^{Figure 11(b)}