CHAPTER 11 Introduction to the Design of Discrete Filters
11.4.1 Transformation Design of IIR Discrete Filters
To take advantage of well-understood analog filter design, a common practice is to design dis- crete filters by means of analog filters and mappings of thes-plane into thez-plane. Two mappings used are:
n The sampling transformationz=esTs.
n The bilinear transformation,
s=K1−z−1 1+z−1
Recall the transformationz=esTswas found when relating the Laplace transform of a sampled signal with its Z-transform. Using this transformation, we convert the analog impulse responseha(t)of an analog filter into the impulse responseh[n] of a discrete filter and obtain the corresponding transfer function. The resulting design procedure is called the impulse-invariant method. Advantages of this method are:
n It preserves the stability of the analog filter.
n Given the linear relation between the analog and the discrete frequencies the specifications for the discrete filter can be easily transformed into the specifications for the analog filter.
Its drawback is possible frequency aliasing. Sampling of the analog impulse response requires that the analog filter be band limited, which might not be possible to satisfy in all cases. Due to this we will concentrate on the approach based on the bilinear transformation.
Bilinear Transformation
The bilinear transformation results from the trapezoidal rule approximation of an integral. Suppose thatx(t)is the input andy(t)is the output of an integrator with transfer function
H(s)= Y(s) X(s)= 1
s (11.16)
Sampling the input and the output of this filter using a sampling periodTs, we have that the integral at timenTsis
y(nTs)=
nTs
Z
(n−1)Ts
x(τ)dτ +y((n−1)Ts) (11.17)
wherey((n−1)Ts)is the integral at time(n−1)Ts. Consider then the approximation of the integral. If Tsis very small, the integral between(n−1)TsandnTscan be approximated by the area of a trapezoid
11.4 IIR Filter Design 655
with basesx((n−1)Ts)andx(nTs)and heightTs(this is called the trapezoidal rule approximation of an integral):
y(nTs)≈[x(nTs)+x((n−1)Ts)]Ts
2 +y((n−1)Ts) (11.18)
with a Z-transform given by
Y(z)=Ts(1+z−1) 2(1−z−1)X(z) The discrete transfer function is thus
H(z)=Y(z) X(z)= Ts
2
1+z−1
1−z−1 (11.19)
which can be obtained directly fromH(s)by letting s= 2
Ts
1−z−1
1+z−1 (11.20)
The resulting transformation is linear in both numerator and denominator, and thus it is called thebilinear transformation. Thinking of the above transformation as a transformation from thezto thesvariable, solving for the variablezin that equation, we obtain a transformation from thesto the zvariable:
z= 1+(Ts/2)s
1−(Ts/2)s (11.21)
The bilinear transformation:
z- to s-plane: s=K1−z−1
1+z−1 K= 2 Ts s- to z-plane: z= 1+s/K
1−s/K (11.22)
maps
n Thejaxis in thes-plane into the unit circle in thez-plane.
n The open left-hands-planeRe[s]<0into the inside of the unit circle in thez-plane, or|z|<1.
n The open right-hands-planeRe[s]>0into the outside of the unit circle in thez-plane, or|z|>1.
Thus, as shown in Figure 11.10, for pointA,s=0 or the origin of thes-plane is mapped intoz=1 on the unit circle; for pointsBandB0,s= ±j∞are mapped intoz= −1 on the unit circle; for pointC, s= −1 is mapped intoz=(1−1/K)/(1+1/K) <1, which is inside the unit circle; and finally for pointD,s=1 is mapped intoz=(1+1/K)/(1−1/K) >1, which is located outside the unit circle.
FIGURE 11.10
Bilinear transformation mapping ofs-plane into z-plane.
A B
B'
σ jΩ
s-plane
C D B A
B'
C z-plane
D
In general, by lettingK = T2s,z=rejωands=σ +jin Equation (11.21), we obtain
r=
s(1+σ/K)2+(/K)2 (1−σ/K)2+(/K)2 ω=tan−1
/K 1+σ/K
+tan−1
/K 1−σ/K
(11.23) From this we have that:
n In thejaxis of thes-plane (i.e., whenσ =0 and−∞< <∞), we obtainr=1 and−π ≤ ω < π, which correspond to the unit circle of thez-plane.
n On the open left-hands-plane, or equivalently whenσ <0 and−∞< <∞, we obtainr<1 and−π≤ω < π, or the inside of the unit circle in thez-plane.
n Finally, on the open right-hand s-plane, or equivalently when σ >0 and −∞< <∞, we obtainr>1 and−π ≤ω < π, or the outside of the unit circle in thez-plane.
The above transformation can be visualized by thinking of a giant who puts a nail in the origin of the s-plane and then grabs the plus and minus infinity extremes of thejaxis and pulls them together to make them agree into one point, getting a magnificent circle, keeping everything in the left plane inside, and keeping out the rest. If our giant lets go, we get back the originals-plane!
Remarks The bilinear transformation maps the whole s-plane into the whole z-plane, differently from the transformation z=esTs that only maps a slab of the s-plane into the z-plane (see Chapter 9 on the Z- transform). Thus, a stable analog filter with poles in the open left-hand s-plane will generate a discrete filter that is also stable as it has poles inside the unit circle.
Frequency Warping
A minor drawback of the bilinear transformation is the nonlinear relation between the analog and the discrete frequencies. Such a relation creates a warping that needs to be taken care of when specifying the analog filter using the discrete filter specifications.
The analog frequencyand the discrete frequencyωaccording to the bilinear transformation are related by
=Ktan(ω/2) (11.24)
11.4 IIR Filter Design 657
FIGURE 11.11
Relation betweenandω forK=1.
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8
−10
−8
−6
−4
−2 0 2 4 6 8 10
Ω(rad/sec)
ω/π
which when plotted displays a linear relation around the low frequencies but it warps as we get into large frequencies (see Figure 11.11).
The relation between the frequencies is obtained by lettingσ =0 in the second equation in Equa- tion (11.23). The linear relationship at low frequencies can be seen using the expansion of the tan(.) function
=K ω
2 +ω3 24+ ã ã ã
≈ ω Ts
for small values ofωorω≈Ts. As frequency increases the effect of the terms beyond the first one makes the relation nonlinear. See Figure 11.11.
To compensate for the nonlinear relation between the frequencies, or the warping effect, the following steps to design a discrete filter are followed:
1. Using the frequency warping relation (Eq. 11.24) the specified discrete frequenciesωpandωstare transformed into specified analog frequenciesp andst. The magnitude specifications remain the same in the different bands—only the frequency is being transformed.
2. Using the specified analog frequencies and the discrete magnitude specifications, an analog filter HN(s)that satisfies these specifications is designed.
3. Applying the bilinear transformation to the designed filterHN(s), the discrete filter HN(z)that satisfies the discrete specifications is obtained.