DIGITAL INFINITE-IMPULSE RESPONSE (IIR) FILTERS-I

EDITOR: B. SOMANATHAN NAIR

1. INTRODUCTION

In the previous few blogs, we have been discussing the design procedure of analog filters. In the next few blogs, we shall be discussing the design of digital filters.

Digital filters are mainly classified as infinite-impulse response (IIR) and finite-impulse response (FIR) filters. The major difference between the two types is that there is feedback connection in IIR filters, whereas there is no feedback connection in FIR filters. IIR filters have analog counterparts, whereas there are no equivalent counterparts for FIR filters. IIR filters can be of the Butterworth or  Chebyshev type, but in the FIR category, we have no Butterworth or  Chebyshev filters. In this blog, we shall discuss the design of Butterworth IIR filters.   

Digital IIR filters follow the same steps of analog filter design upto obtaining the transfer function H(s) of the desired filter. That is, to design a desired digital filter, we first develop the expression for H(s) based on the procedures in the analog filter design, and then modify it in an appropriate fashion to suit the digital domain. As an example, consider the expression for H(s) of the second-order analog filter

                                    H(s) = ω_c²/(s²+2δω_c+ ω_c²)  (1)

where ω_c = cutoff frequency, and δ = damping factor. To get the transfer function of the desired digital filter, we first take the inverse Laplace transform of (1) to convert the frequency-domain expression of H(s) into its time-domain equivalent h(t). After h(t) is obtained, we use z transform techniques to convert it into H(z), the desired digital filter-transfer function  in the z domain. Then using H(z), we construct the required digital filter. This type of design is known as the impulse-invariant design.

            We may also convert H(s) directly from s-domain into H(z) in z domain by means of what is known as the bilinear-transformation technique. Once H(z) is obtained, we proceed in the same way as in the case of the impulse-invariant-design method to construct the desired digital filter. There are several other methods for the conversion of s-domain expressions into z domain. In this chapter, we discuss the infinite impulse-response (IIR) filters.

2. INFINITE IMPULSE-RESPONSE (IIR) FILTERS

Consider the digital-filter-transfer function H(z), given by the expression

                                                                                                 

                                    H(z) = Y(z)/X(z) = (1+bz^‒1)/ (1+az^‒1+ z^‒2)  (2)

Equation (2) may be written as:

 (1+az^‒1+ z^‒2)Y(z)= (1+bz^‒1)X(z)   (3)

Equation (10.3) may be expanded further as

Y(z) + az^‒1Y(z) + z^‒2Y(z) = X(z)+bz^‒1X(z) (4)  

Taking the inverse z-transform of (4), we obtain

                                                        

                                    y(n) + ay(n ‒ 1)+ y(n ‒ 2) = x(n )+ bx(n ‒ 1)   (5)

From (5) we get the desired output y(n) as

                                   

                                    y(n) = ‒ay(n ‒ 1) ‒y(n ‒ 2) + x(n )+ bx(n ‒ 1)   (6)

             

Equation (6) can be implemented by using the required number of delay lines, summers, and multipliers.

            This type of filter, which makes use of feedback connections to get the desired filter implementation, is known as recursive (feedback) filters. In the case of recursive filters, we find that the impulse response h(n) is not limited by any constraint of time for its existence. In other words, we say that it exists for an infinite duration. Hence, such filters are also called infinite-duration impulse-response, or simply, infinite impulse-response (IIR) filters.

            We have a second class of digital filters for which we do not employ any kind of feedback connections. Such filters are known as non-recursive filters. The impulse response of these non-recursive filters is to be  limited for their  implementation; hence they are also known as finite-duration impulse-response, or simply, finite impulse-response (FIR) filters.