11.5 CIC Interpolator Example
We now examine an example of the use of the Hogenauer (resampled CIC) filter. The example is part of a modulator that shapes and up samples in a polyphase filter and then interpolates to a higher output sample rate with a CIC filter. Figure 11.27 presents an example in which the shaping is performed by a 1-to-5 polyphase filter and the additional interpolation is performed in a 3-stage 1-to-16 CIC filter. The CIC is implemented as a Hogenauer filter with three resampled comb filters at the input, a 1-to-16 resampler, and three integrators at the output.
Figure 11.27 Cascade of 1-to-5 Shaping Filter and 1-to-16 CIC Interpolating Filter
Figure 11.28 presents the time and frequency response of the output series obtained by applying an impulse to the input of the shaping filter. The time axis is normalized to input symbol rate of unity, and the frequency axis is normalized to unit 3-dB bandwidth or unity symbol bandwidth.
Figure 11.28 Time and Frequency Response of Time Series from 1-to-5 Shaping Filter
We gain insight into the operation of the CIC filter by tracking the impulse response of the shaping filter through the CIC filter. By this process, we obtain the impulse response of the composite shaping and interpolating filter. Figure 11.29 follows the response from the shaping filter through the three input comb filters at the input rate and then follows the 1-to-16 up-sampled sequence through the three output integrators. We can clearly see the equivalent zero order hold (ZOH) effect of the 1-to-16 up-sampling operation at the output of the first integrator.
Figure 11.29 Shaping Pulse Response of Three-stage 1-to-16 Hogenauer Filter
Figure 11.30 presents the frequency response obtained from the output of the composite polyphase shaping filter and 3-stage CIC interpolating filter. We can clearly see the effect of the CIC spectral zeros on the up-sampled input time series. The maximum residual spectral level brackets the first spectral zeros of the CIC filter. This level is seen to be -60-dB, which is the target level of 60-dB attenuation. These spectral levels fall below the noise floor of a 10-bit digital to analog converter (DAC) and will be concealed by the system noise. An interesting note is that if the shaping filter were to operate as a 1-to-4 up-sampling filter, the 3-stage CIC would not obtain the desired 60-dB attenuation. Thus the choice is to replace the 3-stage CIC with a 4-stage CIC or to raise the output rate of the shaping filter from 1-to-4 to 1-to-5. This second option is more cost effective since the addition of a fourth CIC stage would not only add another integrator but would also increase the bi- field width of the other three integrators.
Figure 11.30 Spectrum of Composite Impulse Response with CIC Spectral Overlay