Comb Filters Characteristics and Applications

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Introduction

A comb filter is a Linear Time-Invariant (LTI) digital filter, where linear means that its output to a scaled sum of input digital signals is equal to the scaled sum of the outputs to every one of these input signals (i.e., the filter satisfies the superposition principle) and time-invariant means that, for any input signal that has a given delay, the output undergoes the same delay as the input (Antoniou, 2006). The name comb is derived by the fact that its magnitude response resembles the teeth of a comb. Since there are several filters having magnitude responses with such characteristic, the term comb filter is rather general. The duration of the impulse response of comb filters can be either finite or infinite, i.e., there are Finite Impulse Response (FIR) comb filters and Infinite Impulse Response (IIR) comb filters (Zölzer, 2008).

The simplest FIR comb filter has the following transfer function,

H_a(z) = 1 + z^–M. (1)

This filter adds to a signal a version of that signal delayed by M sample periods, and it is the basic building block to introduce echo effects in audio signals (Zölzer, 2008). Moreover, if the addition in (1) is replaced by a subtraction, the resulting comb filter is a useful building block to remove DC and harmonics (Diniz, Da Silva & Neto, 2010). The unintentional delay of an audio signal due to the environment is also modeled as a comb filter (Toole, Shaw, Daigle & Stinson, 2001), and this effect may be undesirable in many cases. Similarly, a simple IIR comb filter has the following transfer function,H_b(z) = 1/(1 – az^–M), (2) with a < 1. This filter is a basic building block to model and create reverberation effects or, in general, to artificially reproduce the acoustics of a room (Zölzer, 2008).

One of the most important comb filters for several Digital Signal Processing (DSP) applications is the one based in the FIR filter where all the samples of its impulse response have values equal to one. Unlike the aforementioned comb filters described by H_a(z) in (1) and H_b(z) in (2), this comb filter has a low-pass characteristic, which makes it useful to pass a baseband signal and remove unwanted high-frequency spectra (Milic, 2009). The rest of this article is dedicated to this particular filter, which will be referred as the comb filter hereafter. The main characteristics of the comb filter, as well as its advantages and disadvantages will be highlighted. Moreover, we will present the selected methods commonly used to decrease the disadvantages of the comb filters with minimum affectation of its advantages.

In the efficient implementation of the comb filter, a comb filter with transfer function based in (1) (just with the addition replaced by a subtraction) is employed. In order to avoid confusion, that filter will be referred hereafter as comb differentiator, since it is based on a simple first-order differentiator (Regalia, 1993).

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Background

Consider a simple FIR filter that has the following transfer function (Milic, 2009),

, (3) where M is the filter order. The coefficients that multiply the variable z are all equal to 1. Thus, the non-recursive implementation of this filter does not require multipliers for its coefficient’s values. The impulse response of this filter is h(k) = 1 for 0 <k<M and 0 for other values of k. The scaling factor (1/M) is included to provide a normalized gain of 0 dB at frequency equal to zero.

The transfer function of the comb filter arises from expressing the transfer function given in (3) in recursive form as follows (Lyons, 2004),

. (4)

Key Terms in this Chapter

Computational Complexity: The number of arithmetic operations that a filter performs to obtain an output sample. The computational complexity increases, the power consumption of the filter and decreases its speed of operation. Thus, this complexity must be diminished in practical applications.

Complex Number: A number composed by the sum of a real number a and an imaginary number i × b , where b is a real number and i is defined as the square root of –1.

Digital Signal: A sequence of numbers, so-called samples, where every sample lasts T s seconds. The time T s is called sampling period and the number produced in the n -th sampling period (i.e., after n times T s seconds) is denoted as x ( nT s ). The values x ( nT s ) are constrained to belong to a finite set of possibilities (for example, in a 1-bit signal, these values are just 0 or 1). Additionally, it is usual to consider that x ( nT s ) = 0 for negative values of n . For analysis purposes, it is usual to consider T s = 1.

Impulse Response: The response of a Linear Time-Invariant (LTI) digital filter to an input signal x ( nT s ) that has the value 1 for n = 0 and 0 for other values of n . The samples of the impulse response are usually denoted as h ( nT s ).

Transfer Function: A mathematical representation of the relation of a Linear Time-Invariant (LTI) digital filter between its input and its output. The transfer function is expressed in terms of the variable z , which can take complex values. If the values for z are constrained to be complex numbers with unitary magnitude, the transfer function becomes the frequency response. The transfer function is usually denoted as H ( z ).

Complex Function: Function whose values are complex numbers.

Scaling: The change of the range of values that a signal can take. For example, if a signal can take values from –1 to 1 and that signal is scaled by 5, the scaled signal can take values from –5 to 5.

Complex Exponential Signal: Signal whose samples are complex numbers, where the real and imaginary parts of the samples form, respectively, a cosine wave and a sine wave, both with the same frequency.

Digital Filter: A system that receives a digital signal and returns another digital signal with modified characteristics.

Multiplier: A component of a digital filter that takes the samples at its inputs and produces the multiplication of these values at its output. A multiplier is an expensive an power-consuming element, and therefore the efficient design of digital filters consists on reduce the number of required multipliers or even avoid them.

Magnitude Response: A function of the angular frequency ? where every value is obtained as the magnitude of the complex value of the frequency response in that frequency ? . If the value of the frequency response in ? is a complex number of the form a ( ? ) + i × b ( ? ), the magnitude of that number is given by {[ a ( ? )] 2 + [ b ( ? )] 2 } 1/2 . When the samples of the impulse are real numbers, the magnitude response is symmetric and hence it is described just in the interval of ? from 0 to p .

Adder: A component of a digital filter that produces at its output the sum of the values of the samples present at its inputs.

Phase Response: A function of the angular frequency ? where every value is obtained as the phase of the complex value of the frequency response in that frequency ? . If the value of the frequency response in ? is a complex number of the form a ( ? ) + i × b ( ? ), the phase of that number is given by arctan { b ( ? )/ a ( ? )}. When the samples of the impulse are real numbers, the phase response is anti-symmetric and hence it is described just in the interval of ? from 0 to p .

Digital Frequency: A value that expresses how much of a cycle of a sinusoidal wave is represented by a sample. This value is denoted by f . For example, f = 1/2 means that 2 samples of a digital signal correspond to a sinusoidal cycle. If, instead of cycles we use radians, we obtain the angular digital frequency denoted as ? , which is equal to 2 pf .

Delay: The retention of a sample of a digital signal during a sampling period T s and the subsequent release of that sample at the next sampling period. This is done with a one-input one-output device called delay. In other words, for a signal x ( nT s ), its delayed version is y ( nT s ) = x ( nT s – T s ). In a block diagram, this element is represented as z –1 . A block of M delays is represented as z – M .