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11.6 Coherent and Incoherent Gain in CIC Integrators

Figures 11.31a and b present curves showing the maximum coherent gain between input and integrators of up-sampling CIC filters of order 2 through 5 for up-sampling rates from 2 through 100. These curves were generated in a manner similar to the process described in the up sampling CIC example that examined the specific case of up sampling by 20 with a 4th order CIC. The input sequences that probed the CIC are of the form shown in Figure 11.18. These sequences maximize the peak amplitude at each integrator. The curves contain the information required to determine the required bit width for each integrator in an up-sampling CIC operating at a specific resampling rate.

11fig32.gifFigure 11.31a Coherent Bit Growth of Integrators of 2nd and 3rd Order CIC Filters as Function of Rate Change M

11fig33.gifFigure 11.31b Coherent Bit Growth of Integrators of 4th and 5th Order CIC Filters as Function of Rate Change M

Figures 11.32a and b present curves showing the maximum incoherent gain between integrators and output of down-sampling CIC filters of order 2 through 5 for down-sampling rates from 2 through 100. These curves were generated in a manner similar to the process described in the down sampling CIC example that examined the specific case of down sampling by 20 with a 4th order CIC. A single impulse at each integrator probed the CIC to form the impulse response from which the sum of squares was computed to determine the incoherent gain. The curves contain the information required to determine the position in a bit field that can be pruned to reduce the bit width for successive integrators in a down sampling CIC operating at a specific resampling rate. Figures 11.33a and b present the same curves shown in Figures 11.28 and 11.29 for a wider range of up- and down-sampling rates from 2 to 1000.

11fig34.gifFigure 11.32a Incoherent Bit Growth of Integrators of 2nd and 3rd Order CIC Filters as Function of Rate Change M

11fig35.gifFigure 11.32b Incoherent Bit Growth of Integrators of 4th and 5th Order CIC Filters as Function of Rate Change M

11fig36.gifFigure 11.33a Coherent Bit Growth of Integrators of 2nd, 3rd, 4th, and 5th Order CIC Filters as Function of Rate Change M

11fig37.gifFigure 11.33b Incoherent Bit Growth of Integrators of 2nd, 3rd, 4th, and 5th Order CIC Filters as Function of Rate Change M

References

Chu, S. and Burrus, C. S.,

"Multirate Filter Design Using Comb Filters,"
IEEE Trans. on Circuits and Systems, Vol. 31, Nov. 1984, pp. 913 - 924.

Crochiere, Ronald and Lawrence Rabiner,

"Multirate Signal Processing,"
Englewood Cliffs, NJ, Prentice-Hall, Inc., 1983.

Fliege, Norbert, Multirate Digital Signal Processing: Multirate Systems, Filter Banks, Wavelets, West Sussex, John Wiley & Sons, Ltd., 1994.

Hentschel, Tim, Sample Rate Conversion in Software Configurable Radios, Norwood, MA, Artech House, Inc., 2002

Hogenauer, E. B.,

"An Economical Class of Digital Filters for Decimation and Interpolation,"
IEEE Trans. Acoustics. Speech Signal Proc., Vol. ASSP-29, April 1981, pp. 155 - 162

Jovanovic-Dolecek, Gordana, Multirate Systems: Design and Applications, London, Idea Group, 2002.

Mitra, Sanjit, Digital Signal Processing: A Computer-Based Approach, 2nd ed., New York, McGraw-Hill, 2001.

Mitra Sanjit and James Kaiser, Handbook for Digital Signal Processing, New York, John Wiley & Sons, 1993.

Vaidyanathan, P. P.,

"Multirate Systems and Filter Banks,"
Englewood Cliffs, NJ, Prentice-Hall, Inc., 1993.

Problems

11.1

Program a 20-tap version of the three forms of the boxcar integrator shown in Figure 11.5 and determine the impulse response of all three versions. Note the state of the integrator in the two forms of the CIC.

11.2

Program a 20-tap version of the three forms of the boxcar integrator shown in Figure 11.5 and determine the step response of all three versions. Note the state of the integrator in the two forms of the CIC.

11.3

Program a 20-tap version of the three forms of the boxcar integrator shown in Figure 11.5 using integer arithmetic with a 5-bit 2's-compliment accumulator and with a 4-bit 2's-compliment accumulator, and then determine the step response of all three versions. Note the state of the integrator in the two forms of the CIC for the two different width accumulators.

11.4

The spectrum of a P-stage CIC filter exhibits multiple zeros at multiples of 1/Mth of the sample rate. Determine the attenuation available at an offset of 1/(4M) from the zero at 1/M; i.e., at (3/4M). This is the frequency that aliases into the bandwidth of 1/(4M), the bandwidth of the final cascade of the CIC, and the 4-to-1 FIR filter following the CIC filter. In particular, how many stages of CIC are required to obtain 60-dB, 80-dB, 100-dB, and 120-dB attenuation at the edge of the first Nyquist zone to alias to baseband?

11.5

A P-stage CIC filter of length M has a steady state DC gain of MP. For an M = 100 and P = 5, determine the width required of the accumulators when the input data is 16 bits. Repeat for M = 1,000.

11.6

For the block diagram in Figure 11.17, imagine an input signal consisting of samples of a unity amplitude sine wave of normalized frequency 0.1. Determine the amplitude of the sinusoid observed at the 4 integrator output ports as a function of M. In particular, what is the set of amplitudes for M = 100? Repeat for normalized frequency of 0.01.

11.7

For the block diagram in Figure 11.24, imagine injecting an input signal consisting of samples of a unity amplitude sine wave of normalized frequency 0.1 in any of the 4 input ports of the integrator train. Determine the amplitude of the sinusoid observed at the output port as a function of M. In particular, what is the set of amplitudes for M = 100? Repeat for normalized frequency of 0.01.

11.8

Program the filter chain consisting of a 1-to-4 polyphase Nyquist filter followed by a 4-stage CIC filter that will up sample the filter output by a factor of 20. The Nyquist filter has a 50% excess bandwidth and a side-lobe level 60-dB below pass band response. Apply an impulse to the input and record and plot the time response of the filter chain at each stage as well as the frequency response at the final output. Repeat this programming task but replace the Nyquist filter with a 1-to-5 polyphase filter and a 4-stage CIC filter that will up sample by a factor of 16. What is the peak output amplitude of the final output for the two filter implementations? Can either implementation be realized with a 3-stage CIC? If so, which one, and what is the peak output amplitude for that case?

11.9

A CIC filter of length M does not have to be down sampled by M-to-1. An interesting option is to resample by M/2. When we operate the filter in this manner, the folding frequency for the filter output is fs/M rather than fs/(2M). Program a 3rd-order CIC filter of length 2M operating at an M-to-1 down sample rate. In this test form a 1-to-4 polyphase Nyquist filter with 50% excess bandwidth and 60-dB side-lobes. Zero pack this signal 1-to-20 and process it with a 3rd-order CIC filter of length 40. Examine the frequency response of the filter. Compare the response to a 1-to-20 zero-packed signal processed with a 3rd-order CIC of length 20. Also examine the response of 1-to-20 zero-packed signal processed with a 2nd-order CIC filter of length 40. Comment on the performance differences and the workload difference of the three options.

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