The frequency response of a CIC filter is obtained by evaluating the equation in General Design Guidelines at:
where f is the discrete-time frequency, normalized to the higher frequency in a rate-changing filter — input sampling frequency in a CIC decimation, or output sampling frequency in a CIC interpolator. Evaluating the previous equation in the z-plane at those sample points defined by the preceding equation, gives a magnitude frequency response as shown in the following equation.
This magnitude response is low-pass. In the design process of a CIC filter implementation, the parameters R, M, and N are selected to provide adequate passband characteristics over the frequency range from zero to a predetermined cutoff frequency fc. The following figure shows the frequency response of a 3-stage (N = 3) CIC filter with unity differential delay (M = 1) and a sample rate change R = 7.
According to the preceding equation and as seen in the following figure, there are nulls in the magnitude response (transfer function zeros) at integer multiples of f = 1/(RM). Thus, the differential delay parameter M can be used as a design parameter to control the placement of the nulls.
The following figure shows the effect of the differential delay M on the magnitude response of a filter with three stages (N = 3) and a sample rate change R = 7. Besides the effect on the placement of the response nulls, increasing M also increases the amount of attenuation in the side lobes of the magnitude response.
The rate change parameter R can also be used to control the frequency response of the CIC filter. The effect of R on the magnitude response can be seen in the following figure. In essence, increasing the rate change increases the length of the cascaded unit-amplitude, rectangular window of length R*M. This results in an increase in attenuation and a decrease in the width of the response side lobes.
The number of stages parameter N can also be used to affect the CIC filter magnitude response. This effect can be understood from the fundamental concept of a cascade of N filtering stages, each with an impulse response of a unit-amplitude, rectangular window. The larger the number of cascaded stages, the more attenuated the magnitude response side lobes become. This can be seen in the following figure.
Increasing N has the effect of increasing the order of the zeros in the frequency response, which in turn increases the attenuation at frequencies in the locality of the zero. This effect is clearly shown in the preceding figure where there is increasing attenuation of the filter side lobes as N is increased.
As the order of the zeros increases, the passband droop also increases, thus narrowing the filter bandwidth. The droop is frequently corrected using an additional (non-CIC-based) stage of filtering after the CIC decimator. In the case of a CIC interpolator, the signal can be precompensated using a compensation filter (not part of the CIC Compiler) to flatten the passband frequency response. For a CIC decimator, the compensation filter operates at the decimated sample rate and provides (x/sin(x))^N shaping. An example of a third-order (N = 3) R = 64 compensated CIC system is shown in the following figure. The plot shows the uncompensated CIC frequency response, the compensation filter frequency response, and the compensated CIC. In this case, because the number of CIC stages is three, the compensation filter has a cubic response of the form (x/sin(x))3 .
The compensation filter coefficients employed were [1, 4, 16, 32, 64, 136, 352, 1312, 352, 136, 64, 32, 16, 4, 1]. The following figure provides an exploded view of the compensated filter passband.