Comb filters are developed from the structures based on the moving average (boxcar) filter. The combbased filter has unity-valued coefficients and, therefore, can be implemented without multipliers. This filter class can operate at high frequencies and is suitable for a single-chip VLSI implementation. The main applications are in communication systems such as software radio and satellite communications. In this chapter, we introduce first the concept of the basic comb filter and discuss its properties. Then, we present the structures of the comb-based decimators and interpolators, discuss the corresponding frequency responses, and demonstrate the overall two-stage decimator constructed as the cascade of a comb decimator and an FIR decimator. In the next section, we expose the application of the polyphase implementation structure, which is aimed to reduce the power dissipation. We consider techniques for sharpening the original comb filter magnitude response and emphasize an approach that modifies the filter transfer function in a manner to provide a sharpened filter operating at the lowest possible sampling rate. Finally, we give a brief presentation of the modified comb filter based on the zero-rotation approach. Chapter concludes with several MATLAB Exercises for the individual study. The reference list at the end of the chapter includes the topics of interest for further research.
TopComb-Based Filter Sections
The simplest lowpass FIR filter is a boxcar or a moving-average (MA) filter whose impulse response is of a rectangle shape,
, (1) where N is an integer. The z-domain representation results in the following transfer function GC(z),
.
(2)Equation (11.2) is recognized as the first N terms of the geometric series, whose closed form expression,
(3) is fond suitable for an efficient implementation. According to equation (3), the filter
G(
z) can be implemented by cascading the
comb section (1−
z−N) and the
integrator section 1/(1−
z−1) thus leading to an extremely efficient device which performs the filtering task employing only two additions regardless of the filter length
N. The term
CIC filter (cascade-integrator-comb) is frequently used for this filter class. As will be shown later in this chapter, the various realization structures have been developed for the implementation of the transfer function (3). In this chapter, we generally use terms comb filter or comb-based filter for the class of digital filters based on the transfer function (3). The term CIC filter we use only for the cascade-integrator-comb implementation scheme.
The frequency response GC(ejω) following from equation (3) is given by,
.
(4)Hence, the comb filter section is a linear-phase lowpass filter, which exhibits sin(Nx)/sin(x) amplitude characteristic. Due to its particular amplitude response, this filter is also called the sinc filter.
Figure 1 shows the gain response of the comb-based filter for N = 10. As Figure 1 illustrates, the GC(ejω) exhibits the comb-like magnitude response. It has N/2 natural nulls distributed along the normalized frequency axis at integer multiples of (2/N). Notice that for N being odd, the number of natural nulls is (N–1)/2. The filter has a very wide transition band and the stop band attenuation at the first side-lobe amounts only to 13 dB.