Cascaded Integrator Comb Filter

National Instruments has partnered with Prentice Hall to bring you large portions of in-depth technical topics from several PTR RF and Communications books, including Digital Communications: Fundamentals and Applications, 2nd Edition. This series of content is designed for a broad range of audiences, from experts who want to review a specific topic to students who need easy-to-understand documentation for their projects.

For the complete list of RF topics, please visit the RF and Communications Resource Main Page.

3.2.6 Cascaded Integrator Comb Filter

In software radio systems, sample rate changes can be very large, with changes from many

tens of MHz to around 100 kHz being common. Of course, such a requirement leads to

large order and high-rate digital filters, which can easily become a bottleneck in the overall

system design. A cascaded integrator comb (CIC) filter⁴ can be used to reduce the computational

demands. A CIC filter is what the name indicates: a cascade of simple integrators

(accumulators) and a cascade of comb filters (delay and subtract from current sample).

These basic building blocks are shown in Figure 3.33. The CIC filter can implement an

interpolation or decimation filter (see Figure 3.34) that uses only delay and add operations

and thus is well-suited for FPGA and ASIC implementation. Furthermore, the same basic

filter structure can be used to handle variable sample rate conversion.

                  Figure 3.33: Building Blocks of a CIC Filter.

 

                      

Basic Structures

CIC DecimationF ilter

The CIC implementation of a decimation filter is the cascade of an integrator stage, a decimation

procedure, and a comb stage as shown in Figure 3.35. To analyze the CIC filter’s

response, combining the integrator and comb stages into a single transfer function are important

to reduce the complexity of the analysis. However, to reduce the computational

expense of the operation, the implementation of the filter is performed in two separate

sections, before and after the decimation.

The integrator section is the cascade ofN ideal digital integrator stages operating at the

high sampling rate of F_s, where the transfer function is defined as

The comb sections consist of N comb stages operating at the low sampling rate of ,where R is the

integer-rate change factor. The comb sections have a differential delay of M samples per stage, where

the system function is defined as

and M is typically chosen as 1 or 2.

The cascade of the integrator and comb sections of the CIC filter requires a decimation

process to be implemented between these two filters. It is possible to apply the Noble

Identities to switch the positions of the comb stages and the decimation procedure (see

Section 3.2.3) to simplify the analysis. Figure 3.36 shows a block diagram of this change.

The order of the comb filter and the decimation procedure can be switched without causing

any change in the end results of the filtering operation.

The integrator and comb stages of the filter can be combined into a single transfer

function, thus simplifying analysis. The transfer function of the cascaded integrator comb

filter before decimation is then

and the frequency response (with respect to the higher input sample rate) is

This filter is a cascade of N copies of an RMth-length FIR filter whose coefficients

specify a rectangular time-domain window, so it is indeed an LPF. Furthermore, because of

the symmetry, this is a linear phase filter. Therefore, the CIC decimation filter is actually

an alternative implementation of a general decimation filter. The key is that, in this CIC

structure, the comb stage operates at the low sampling rate, resulting in a more efficient

implementation of the decimation filters.

Note that the frequency response H(ω) is with respect to the original sample rate. By

lettingω´ = ωR, the frequency response with respect to the decimated sample rate is found.

Note that a null in the frequency response occurs at 1/M for both the decimated and nondecimated

filters. Furthermore, if R is large, then H(ω) can be approximated as

The gain of the filter is (RM)^N, which can become very large, necessitating the need for

large registers.

As an example, the frequency response of Equation 3.48 is plotted in Figure 3.37 for

N = 4,M = 1, R = 7 for a cutoff frequency f_c = 1/8. The input sample rate is 7 and

 

Figure 3.37: Frequency Response of the CIC Filter.

SOURCE: E. B. Hogenauer, “An Economical Class of Digital Filters for Decimation

and Interpolation,” [38]. © IEEE, 1981. Used by Permission.

 

the output sample rate is 1. Note the bands at multiples of 1 Hz will be aliased into the

passband region about 0 Hz. At multiples of 1 Hz, there are nulls in the frequency response

and thus aliasing will be minimal in the passband. The filter is designed to attentuate the

signal so that there are acceptable levels of aliasing at the passband edge, f_c. There is, of

course, a droop in the passband, but this droop can be compensated for by a short FIR filter

at the output of the CIC filter operating at the reduced sample rate or using the sharpening

technique proposed by Kwentus [40].

CIC Interpolation Filter

The structure of the interpolationCIC filter, as seen Figure 3.38, uses a comb stage followed

by an upsampler and an integrator stage. Using the Noble Identities, the CIC interpolator

can be analyzed using the equivalent structure shown in Figure 3.39.

Obviously Figure 3.38 has a more attractive implementation since comb filter stages

operate at a lower rate. The analysis of the interpolator CIC filter follows that of the decimator

CIC filter. One difference here is that the purpose of the CIC filter for the interpolator

is to suppress replicated images that occur due to zero-insertion, while the CIC filter for the

decimator is used to suppress aliasing.

Economics of CIC Filters

The CIC filter presents several advantages over more basic implementations of decimation

and interpolation filters:

• no multipliers are required,

• no storage is required for filter coefficients,

• the structure of CIC filters is very regular, consisting of two simple building blocks,

• very little external or complicated control is needed.

Two primary problems are encountered in the implementation of CIC filters. First, the

register widths can become large, especially for large values of R. Secondly, the frequency

response is fully determined by only three integer parameters, M, N, and R, resulting in a

limited range of filter configurations [38–40]. While a detailed description of this register

growth problem is beyond the scope of this discussion, Hogenauer [38] provides details

that show the number of output bits B_out is given by

where B_in is the number of input bits. Strategies exist for truncating and rounding the

number of bits throughout the filtering process to reduce the number of bits [38].

Given the advantages and disadvantages of using CIC filters, they are ideal for applications

in which high sampling rates make the use of multipliers a computationally expensive

option. This technique is especially useful for FPGA design where multipliers are avoided

because of the large silicon area required. The use of CIC filters is also important in applications

in which large rate change factors require large amounts of coefficient storage

or fast impulse response generation and the memory is either unavailable or too slow to

perform the desired application.

 

                               

                                        

Relevant NI products

Customers interested in this topic were also interested in the following NI products:

• RF and Communication Hardware and Software
  
• Other Modular Instruments (digital multimeters, digitizers, switching, etc...)
• LabVIEW Graphical Programming Environment

For the complete list of tutorials, return to the NI RF and Communications Fundamentals Main page.

Buy the Book

Purchase Software-defined Radio from Prentice Hall Professional through this link and receive the following:

• Between 15% and 30% Off
• Free Shipping and Handling

 

Reader Comments | Submit a comment »