When the output of the filter given by the equation in General Design Guidelines is decimated (down-sampled) by a factor R, the response of the filter referenced to the lower, down-sampled output rate is expressed in the following equation as:
This response can be viewed as a cascade of N integrators and N comb filters.
The following figure shows a block diagram of a realization of this response. There are two sections to the CIC decimator filter: an integrator section with N integrator stages that processes input data samples at a sampling rate fs, and a comb section that operates at the lower sampling rate fs/R. This comb section consists of N comb stages with a differential delay of M samples per stage. The down-sampling operation decimates the output of the integrator section by passing only every Rth sample to the comb section of the filter.
Referring back to Figure 1, when the CIC filter is employed as a decimator, certain frequency bands alias back into the filter passband. Care must be taken to ensure that the integrated side lobe levels do not impact the intended application. The following figure shows an example of a CIC decimator response before down-sampling to show the effect of aliasing. In the following figure, the ideal response of a decimator with sampling rate change of R = 8, number of stages N = 3, and differential delay M = 1 is shown. The spectrum of the decimator input is also shown containing energy in the intended passband (low frequencies up to a cutoff frequency fc = 1/32 cycles/sample) and in the stopband (around 1/4 cycles/sample). The output of the decimator (without down-sampling) is shown to demonstrate the attenuation produced by this CIC filter. The dashed vertical lines in the following figure indicate the frequency ranges that alias to the passband when down-sampling. In this figure, the frequency axis is normalized to the (higher) sampling frequency before down-sampling.
The following figure shows the output spectrum of the CIC decimator example. In the following figure, the frequency axis is normalized relative to the lower sampling rate obtained after down-sampling. Because of this re-normalization of frequencies, the plots in the following figure can be conceptualized as a zoomed view of the frequency range from 0 to 1/(2*R) = 1/16 cycles/sample of the preceding figure.
- The solid red plot shows the CIC output spectrum if no aliasing occurred.
- The dashed red plot shows the stopband output spectrum when aliased due to down-sampling. This aliased spectrum affects the final output of the CIC decimator by contributing additively to the output spectrum.
- The solid blue plot is the actual output of the CIC decimator, which clearly shows the contribution of the aliased spectrum from down-sampling.
Again, care must be taken to ensure that the CIC decimator parameters are properly chosen to avoid detrimental effects from aliasing.