- A Multiply-free Filter
- Binary Integers and Overflow
- Multistage CIC
- Hogenauer Filter
- CIC Interpolator Example
- Coherent and Incoherent Gain in CIC Integrators

## 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.

**Figure 11.31a Coherent Bit Growth of Integrators of 2nd and 3rd Order CIC Filters as Function of Rate Change M**

**Figure 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.

**Figure 11.32a Incoherent Bit Growth of Integrators of 2nd and 3rd Order CIC Filters as Function of Rate Change M**

**Figure 11.32b Incoherent Bit Growth of Integrators of 4th and 5th Order CIC Filters as Function of Rate Change M**

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

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

#### References

##### "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

##### "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.

##### "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 |

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 |

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 |

11.4 |
The spectrum of a P-stage |

11.5 |
A P-stage |

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 |

11.9 |
A |