Cascaded integrator–comb filter

In digital signal processing, a cascaded integrator–comb (CIC) is an optimized class of finite impulse response (FIR) filter combined with an interpolator or decimator. A CIC filter consists of one or more integrator and comb filter pairs. In the case of a decimating CIC, the input signal is fed through one or more cascaded integrators, then a down-sampler, followed by one or more comb sections (equal in number to the number of integrators). An interpolating CIC is simply the reverse of this architecture, with the down-sampler replaced with a zero-stuffer (up-sampler). CIC filters were invented by Eugene B. Hogenauer, and are a class of FIR filters used in multi-rate digital signal processing. The CIC filter finds applications in interpolation and decimation. Unlike most FIR filters, it has a decimator or interpolator built into the architecture. The figure at the right shows the Hogenauer architecture for a CIC interpolator. The system function for the composite CIC filter referenced to the high sampling rate, fs is: Where: R = decimation or interpolation ratio M = number of samples per stage (usually 1 but sometimes 2) N = number of stages in filter Characteristics of CIC Filters Linear phase response; Utilize only delay and addition and subtraction; that is, it requires no multiplication operations; A CIC filter is an efficient implementation of a moving-average filter. To see this, consider how a moving average filter can be implemented recursively by adding the newest sample to the previous result and subtracting the oldest sample. Omitting the division by , we have: The second equality corresponds to a comb () followed by an integrator (). The conventional CIC structure is obtained by cascading identical moving average filters, then rearranging the sections to place all integrators first (decimator) or combs first (interpolator). Such rearrangement is possible because both combs and integrators are LTI. For an interpolator, the upsampler which normally precedes the interpolation filter can be passed through the comb sections using a Noble identity, reducing the number of delay elements needed by a factor of .