Introduction to the IIR Difference Equation and System Function

IIR Filter Structures

IIR Filter Design with Bilinear Transformation

Typical Analog Filters

Procedure

I. Introduction to the IIR Difference Equation and System Function

The most general form of the relationship between the input and output of physically realizable discrete-time LTI systems can be specified by using the difference equation:

(1)

where {a_l}, called feedback coefficients, and {b_k}, called feedforward coefficients, are the filter coefficients that specify the system and are independent of x(n) and y(n). Taking z-transform on both sides of Eq. (1) gives the corresponding system function for the above difference equation in the form of

(2)

If the {a_l} are non-zero in Eq. (2), then   H(z) has one or more poles so that h[n] contains exponential terms which are of infinite duration. Systems of this form are therefore called infinite impulse response (IIR) systems.

II. IIR Filter Structures

1. Direct form I structure:

This is a common and simple structure. From the following figure, we see that the block-diagram dirctly comes from the system function H(z)=(1/A(z))*B(z), so the system implemented in this way is called the Direct Form I implementation.



            Feed-forward Part                    Feedback Part 
                 B(z)                                1/A(z)

             (First-order IIR filter direct form I sructure)

2. Direct form II structure:

For an LTI cascade system, we can change the order of the system without changing the overall system response. Of couse, we can write the first-order IIR filter system function as H(z)=B(z)*(1/A(z)), from which we get a new block-diagram dircectly (see the following figure). Notice that the direct form II requires half as many delay elements as the direct form I.



              Feedback Part            Feed-forward Part                    
                  1/A(z)                     B(z)   
                  
             (First-order IIR filter direct form II sructure)

3. Direct form II transposed structure:

The direct form II transposed structure is the modified version of the direct form II and requires the same number of delay elements. The following three steps yield a transposed structure from dircet form II:

1. Reverse all the arrows with multipliers being unchanged in value and location;
2. Reverse the roles of the input and output (input <-> output);
3. Redraw the structure and the branching point and the summing point are interchanged.



             
             (First-order IIR filter direct form II transposed sructure)

4. Cascade structure:

An overall transfer function can be represented with cascade transfer functions H(z)=H₁(z)*H₂(z)*H₃(z)*...*H_r(z). So this implies that two or more systems can be cascaded in either order to obtain the same overall system response.



             
             (IIR filter cascade form sructure)

III. IIR Filter Design with Bilinear Transformation

The most commonly used techniques for the design of IIR filters are based on tranformations of continous-time IIR systems into the discrete-time IIR systems. Since analog filter design techniques have been in use for many years, transformations were developed that map the s-plane poles and zeros into the z-plane to achieve the desired digital filter characteristics. One of the most effective ways of converting an analog filter into a digital equivalent is by means of a Bilinear Transformation.

The Bilinear transformation is a one-to-one mapping from the analog s-plane to the digital z-plane and it allows the following:

1. The left region in the s-plane maps to the inside the unit circle in the z-plane;
2. The right region in the s-plane maps to the outside the unit circle in the z-plane;
3. The imaginary jw axis in the s-plane maps to the unit circle in the z-plane.

The transformation maps each of the analog poles and zeros in the s-plane into unique poles and zeros in the z-plane based on a numerical integration technique used to simulate analog filters. The transformation between the analog domain and the discrete time domain is nonlinear since maps onto therefore restricting this technique to cases where the corresponding frequency warping is acceptable. The bilinear transformation corresponds to replacing s by

where T_d=1/f_s is the sampling period. With H_a(s) denoting the analog system function and H(z) the discrete-time system function we get that

(5)

A useful property of the bilinear transformation is that it maps the left half s-plane onto the inside of the unit circle in the z-plane. Since all analog filter design methods give rise to stable and causal transfer functions H_a(s), this property guarantees that the digital filter obtained will be stable and causal. By replacing z=e^jwand we can derive a relationship between the analog frequency and the discrete time frequency

Other versions of the bilinear transformation which are appropriate for designing highpass, bandpass, or bandstop digital filters by starting with an equivalent lowpass filter are given in Table 1, where c is derived based on frequency response specifications of the bandpass and bandstop filters.

Transformation Type	s
Lowpass:
Highpass:
Bandpass:
Bandstop:

IV. Typical Analog Filters

The approximation problem in the design of IIR digital filters is usually solved by using Butterworth, Chebyshev, or Elliptic analog filter approximations.

• Analog Lowpass Butterworth Filters

Butterworth filters offer one way of approximating an ideal rectangular’response characteristic with unity transmission in the passband and zero transmission in the stopband. They have the property that the magnitude response is maximally flat in the passband. The lowpass Butterth filter is characterized by the magnitude squared frequency response



where N is the order of the filter and is the cutoff frequency. The stopband attenuation at the stopband frequency can be used to derive the filter order where



The poles of occur on a circle of radius at equally spaced points in the s-plane. The poles of the lowpass Butterworth filter can be obtained from



• Analog Lowpass Chebyshev Filters

The distinguishing characterstic of Chebyshev filter are that the magnitude of the frequency response is either equipripple in the passband and monotonic in the stopband (type I) or monotonic in the passband and equiripple in the stopband (type II). For a given order, the stopband performance is superior to the Butterworth design with a sharper transition band. It is for this reason that the Chebyshev filter is generally preferred when some passband ripple is acceptable.

A type I Chebyshev filter is characterized by the magnitude squared response



where is called the Chebyshev polynomial of order N and is a passband ripple parameter where the passband ripple . The stopband attenuation is given by at the stopband frequency and the order of the filter N can be expressed as



where . The poles of the Chebyshev lowpass filter lie on an ellipse in the s-plane and have coordinates given by



where ; k = 1, 2, ......, N

V. The Procedure

    a).  Ha(s) = 1/[(s2+s+0.5)*(s+1)]

    b).  Hb(s) = s/(s+2)

    c).  Hc(s) = (s2+0.5271)/(s2+0.096s+0.5271)