6.7. IIR Filter Design

• Typically, we convert standard continuous-time (analog) filters to obtain discrete-time IIR filters by applying appropriate transformations.

6.7.1. Prototype Analog Filters

• Let’s start by introducing a few common lowpass analog filter prototypes.

6.7.1.1. Butterworth Filter

• A Butterworth filter of order $N$ is an all-pole filter characterized by the following square-magnitude response:

  (6.16)$$|H(\omega ){|}^2=\frac{1}{1+{\epsilon }^2{\left(\frac{\omega }{{\omega }_p}\right)}^{2N}}=\frac{1}{1+{\left(\frac{\omega }{{\omega }_c}\right)}^{2N}}$$

  where ${\omega }_p$ is the passband edge and ${\omega }_c={\epsilon }^{-\frac{1}{N}}{\omega }_p$ is the $3$dB cutoff frequency.

• The transfer function $H(s)$, i.e. the Laplace transform of the impulse response $h(t)$, of the Butterworth filter satisfies:

  (6.17)$$H(s)H(-s)=\frac{1}{1+{\left(\frac{s}{j{\omega }_c}\right)}^{2N}}.$$

  Tip

  Note that $|H(\omega ){|}^2=H(s)H(-s){|}_{s=j\omega }$.

  The roots of the denominator polynomial on the RHS of (6.17) are ${\tilde{s}}_k={\omega }_c{e}^{j\frac{2k+1+N}{2N}}$ for $k=0,1,\dots ,2N-1$. That is, they are $2N$ evenly spaced points on the circle of radius ${\omega }_c$ centered at the origin on the $s$-plane. Selecting all the roots on the left-half plane, we obtain a stable filter with transfer function:

  $$H(s)=\frac{K}{\prod _{k=0}^{N-1}(s-{\tilde{s}}_k)}$$

  which gives our lowpass Butterworth filter prototype.

• Note that $|H(\omega ){|}^2$ in (6.16) is monotone decreasing in (positive) $\omega$. Thus there are no ripples in either the passband or stopband for the Butterworth filter. We often say that the Butterworth filter has flat passband and stopband.

6.7.1.2. Type-I Chebyshev Filter

• A type-I Chebyshev filter of order $N$ is an all-pole filter characterized by:

  (6.18)$$|H(\omega ){|}^2=\frac{1}{1+{\epsilon }^2{T}_N^2\left(\frac{\omega }{{\omega }_p}\right)}$$

  where

  $$T_N(x)=\{\begin{array}{ll}\cos (N{\cos }^{-1}(x))& \text{if }|x|\le 1\\ \cosh (N{\cosh }^{-1}(x))& \text{if }|x|>1\end{array}$$

  is the Chebyshev polynomial of degree $N$.

  Tip

  One may also get the Chebyshev polynomials using the following recursion:

  $$\begin{array}{rl}{T}_0(x)& =1\\ T_1(x)& =x\\ T_{n+1}(x)& =2x{T}_n(x)-T_{n-1}(x)& \text{for }n\ge 1.\end{array}$$

• The poles of $H(s)H(-s)$ of the type-I Chebyshev filter lie on an ellipse centered at the origin on the $s$-plane with major and minor axis radii ${\displaystyle r_1=\frac{{\omega }_p({\beta }^2+1)}{2\beta }}$ and ${\displaystyle r_2=\frac{{\omega }_p({\beta }^2-1)}{2\beta }}$, where ${\displaystyle \beta ={\left(\frac{\sqrt{1+{\epsilon }^2}+1}{\epsilon }\right)}^{\frac{1}{N}}}$. Choosing all poles on the left half plane gives a stable

  $$H(s)=\frac{K}{\prod _{k=0}^{N-1}(s-s_k)}$$

  where $s_k=r_2\cos {\varphi }_k+j{r}_1\sin {\varphi }_k$ and ${\varphi }_k=\frac{2k+1+N}{2N}$ for $k=0,1,\dots ,N-1$.

• The type-I Chebyshev filter is equiripple in the passband and flat in the stopband.

6.7.1.3. Type-II Chebyshev Filter

• A type-II Chebyshev filter of order $N$ is a filter with $N$ zeros and $N$ poles characterized by:

  (6.19)$$|H(\omega ){|}^2=\frac{1}{1+{\epsilon }^2\frac{T_N^2\left(\frac{{\omega }_s}{{\omega }_p}\right)}{T_N^2\left(\frac{{\omega }_s}{\omega }\right)}}$$

  where ${\omega }_p$ and ${\omega }_s$ are the passband and stopband edge frequency, respectively.

• For a stable filter, the transfer function of the type-II Chebyshev filter is given by

  $$H(s)=\frac{K\prod _{k=0}^{N-1}(s-z_k)}{\prod _{k=0}^{N-1}(s-p_k)}$$

  where $z_k=\frac{j{\omega }_s}{\sin {\varphi }_k}$ and $p_k=\frac{{\omega }_s{s}_k}{\sqrt{r_2^2{\cos }^2{\varphi }_k+r_1^2{\sin }^2{\varphi }_k}}$ for $k=0,1,\dots ,N-1$.

• The type-II Chebyshev filter is equiripple in the stopband and flat in the passband.

6.7.1.4. Elliptic Filter

• An elliptic filter of order $N$ is a filter with $N$ zeros and $N$ poles characterized by:

  (6.20)$$|H(\omega ){|}^2=\frac{1}{1+{\epsilon }^2{U}_N^2\left(\frac{\omega }{{\omega }_p}\right)}$$

  where $U_N(\cdot )$ is the Jacobian elliptic function of order $N$ (see [Orf06] for details).

• The zeros of the transfer function $H(s)$ of the elliptic filter are on the $j\omega$-axis.

• The elliptic filter is equiripple in both the passband and stopband.

• The elliptic filter has a lower order than the Butterworth and Chebyshev filters for the same specification.

6.7.1.5. Analog Filter Design

• To design an analog lowpass filter with specification $({\omega }_p,{\omega }_s,{\delta }_1,{\delta }_2)$:

  1. Pick a filter type from above (Butterworth, type-I Chebyshev, type-II Chebyshev, or elliptic).

  2. Use (6.16), (6.18), (6.19), or (6.20) to determine the value $\epsilon$ from ${\delta }_1$.

  3. Find the minimum filter order $N$ such that the specifications of ${\delta }_1$ and ${\delta }_2$ are satisfied in the passband and stopband, respectively.

• After obtaining the transfer function $H(s)$ of a lowpass prototype design with passband edge frequency ${\omega }_p$, one may use the transformations given in the table below to get the transfer function $H^{\prime }(s)$ of another type of filter:

  New filter type	New band edge(s)	Transformation
  Lowpass	${\omega }_p^{\prime }$	$H^{\prime }(s)=H\left(\frac{{\omega }_p}{{\omega }_p^{\prime }}s\right)$
  Highpass	${\omega }_p^{\prime }$	${\displaystyle H^{\prime }(s)=H\left(\frac{{\omega }_p{\omega }_p^{\prime }}{s}\right)}$
  Bandpass	${\omega }_l,{\omega }_u$	${\displaystyle H^{\prime }(s)=H\left({\omega }_p\frac{s^2+{\omega }_u{\omega }_l}{s({\omega }_u-{\omega }_l)}\right)}$
  Bandstop	${\omega }_l,{\omega }_u$	${\displaystyle H^{\prime }(s)=H\left({\omega }_p\frac{s({\omega }_u-{\omega }_l)}{s^2+{\omega }_u{\omega }_l}\right)}$

6.7.2. Impulse Invariance Method

• Sample the impulse response $h(t)$ of an analog lowpass filter prototype at sampling rate $f_s$ to obtain the impulse response ${\displaystyle h[n]=\frac{1}{f_s}h\left(\frac{n}{f_s}\right)}$ of the target discrete-time lowpass IIR filter.

• If $h(t)$ is bandlimited and we oversample $h(t)$ to get $h[n]$, then from that the Poisson sum formula (3.6), the frequency response of $h[n]$ is

  $$\begin{array}{rlr}H(e^{j\hat{\omega }})& =H(\hat{\omega }{f}_s)& \text{ for }-\pi \le \hat{\omega }<\pi \end{array}$$

  where $H(\omega )$ is the frequency response of $h(t)$. Thus, the design specification $({\hat{\omega }}_p,{\hat{\omega }}_s,{\delta }_1,{\delta }_2)$ of the discrete-time lowpass IIR filter will be met if we set ${\omega }_p={\hat{\omega }}_p{f}_s$, ${\omega }_s={\hat{\omega }}_s{f}_s$, and design the continuous-time lowpass filter prototype to satisfy the specification $({\omega }_p,{\omega }_s,{\delta }_1,{\delta }_2)$.

• Often, we use the Butterworth, Chebyshev, and elliptic filters described above as prototypes for the analog filter design.

  Tip

  Since the magnitude responses of the Butterworth, Chebyshev, and elliptic (see (6.16), (6.18), (6.19), and (6.20)) are functions of the ratio $\frac{\omega }{{\omega }_p}$ or $\frac{{\omega }_s}{\omega }$, we may simply set $f_s=1$ in the design process without any loss of generality.

  Caution

  As none of the Butterworth, Chebyshev, and elliptic filters are strictly bandlimited, the impulse invariance design obtained by sampling the impulse response of the analog filter prototype may suffer from aliasing, typically causing larger stopband ripples. To overcome this problem, we may need to use a tighter values for ${\delta }_1$ and ${\delta }_2$ so that the achieved ripples in the passband and stopband are within the original specification.

• Note that the analog filter prototypes described in Section 6.7.1 above all have single-order poles. Hence their continuous-time transfer functions have the following partial fraction expansion:

  $$H(s)=A_0+\sum _{k=0}^{N-1}\frac{A_k}{s-p_k}$$

  where $p_0,p_1,\dots ,p_{N-1}$ are the poles. Taking inverse Laplace transform to get the causal impulse response

  $$h(t)=A_0\delta (t)+\sum _{k=0}^{N-1}{A}_k{e}^{p_{k}t}u(t).$$

  Sampling $h(t)$ at $f_s=1$ gives

  $$h[n]=A_0\delta [n]+\sum _{k=0}^{N-1}{A}_k{e}^{p_{k}n}u[n].$$

  Taking $z$-transform on $h[n]$, we get

  $$H(z)=A_0+\sum _{k=0}^{N-1}\frac{A_k}{1-e^{p_k}{z}^{-1}}$$

  Since $H(s)$ is stable by construction, $\text{Re}(p_k)<0$ for $k=0,1,\dots ,N-1$. Thus the poles $e^{p_k}$ of the transfer function $H(z)$ of the resulting discrete-time IIR filter are all strictly inside the unit circle, i.e., $H(z)$ is also stable.

• MATLAB Example 7:

  Consider again the same design specification as in Examples 1 and 4 in Section 6.6, except in this example we want to design a lowpass IIR filter with the specification $(0.3\pi ,0.35\pi ,0.01,0.001)$ based on an analog type-I Chebyshev filter prototype. Using the impulse invariance method with $f_s=1$, the required specifications of the analog type-I Chebyshev filter prototype are ${\omega }_p=0.3\pi$, ${\omega }_s=0.35\pi$, ${\delta }_1=0.01$, and ${\delta }_2=0.001$.

  Next, we employ (6.18) to determine the value of $\epsilon$ and the filter order $N$ required to meet the specifications. We can be done using the MATLAB function cheb1ord:

  >> Rp = -20*log10(1-0.01);
  >> Rs = -20*log10(0.001);
  >> [N, wp] = cheb1ord(0.3*pi, 0.35*pi, Rp, Rs, 's')
  
  N =
  
      17
  
  
  wp =
  
      0.9425
  

  to get that the filter required order is $N=17$. Note that the value of $\epsilon$ depends only on ${\delta }_1$ and ${\omega }_p$. Then we can use the MATLAB function cheby1 to obtain the analog type-I Chebyshev filter prototype:

  >> [bc, ac]= cheby1(N, Rp, wp, 's');
  >> freqs(bc, ac, [0:0.0001:pi]);
  

  We can see that the analog filter prototype meets its specifications. Finally, we may use the MATLAB function impinvar to apply the impulse invariance method to obtain the target discrete-time IIR filter:

  >> [b, a] = impinvar(bc, ac);
  >> fvtool(b, a);
  

  We can check that the original specification $(0.3\pi ,0.35\pi ,0.01,0.001)$ is met.

  Tip

  Note that the order of the IIR filter obtained from the impulse invariance method is much smaller than those of the FIR filters obtained in Examples 1 and 4 with the same specification. However, the group delay of the IIR filter is not a constant over the passband.

• MATLAB Example 8:

  Repeat Example 7 using an analog elliptic filter prototype instead:

  >> Rp = -20*log10(1-0.01);
  >> Rs = -20*log10(0.001);
  >> [N, wp] = ellipord(0.3*pi, 0.35*pi, Rp, Rs, 's')
  
  N =
  
      9
  
  
  wp =
  
      0.9425
  
  >> [bc, ac] = ellip(N, Rp, Rs, wp, 's');
  >> freqs(bc, ac, [0:0.0001:pi]);
  

  We can see that this analog filter prototype also meets its specifications. However, applying the impulse invariance method:

  >> [b, a] = impinvar(bc, ac);
  >> fvtool(b, a);
  

  We see that the resulting IIR filter violates its specifications in both the passband and stopband. Redoing the design with more stringent values for ${\delta }_1$ and ${\delta }_2$:

  >> Rp = -20*log10(1-0.01);
  >> Rs = -20*log10(0.001);
  >> [N, wp] = ellipord(0.3*pi, 0.35*pi, Rp*0.75, Rs+15, 's')
  
  N =
  
      10
  
  
  wp =
  
      0.9425
  
  
  >> [bc, ac] = ellip(N, Rp*0.75, Rs+15, wp, 's');
  >> [b, a] = impinvar(bc, ac);
  >> fvtool(b, a);
  

  gives an IIR filter that satisfies the specification of $(0.3\pi ,0.35\pi ,0.01,0.001)$. Note that the order of this IIR filter is $N=10$, smaller than that of the IIR filter obtained from the type-I Chebyshev prototype.

6.7.3. Bilinear Transform Method

• Obtain the transfer function $H(z)$ of the discrete-time IIR filter directly from the transfer function $H(s)$ of an anlog prototype filter by the bilinear transformation:

  (6.21)$$s=2f_s\left(\frac{1-z^{-1}}{1+z^{-1}}\right)$$

  where $f_s$ is the sampling rate.

• Let $s=\sigma +j\omega$. Inverting the bilinear transform gives

  $$\begin{array}{rl}z& =\frac{1+\frac{s}{2f_s}}{1-\frac{s}{2f_s}}\\ & =\frac{1+\frac{\sigma }{2f_s}+j\frac{\omega }{2f_s}}{1-\frac{\sigma }{2f_s}-j\frac{\omega }{2f_s}}.\end{array}$$

  Thus:

  1. If $\sigma <0$, then $|z|<1$. That is, the left half $s$-plane is mapped onto the inside of the unit circle in the $z$-plane by the bilinear transform.

  2. If $\sigma >0$, then $|z|>1$. That is, the right half $s$-plane is mapped onto the outside of the unit circle in the $z$-plane by the bilinear transform.

  3. If $\sigma =0$, then $|z|=1$. That is, the $j\omega$ axis of $s$-plane is mapped onto the unit circle of the $z$-plane by the bilinear transform.

• From observation 1, the bilinear transform (6.21) maps causal stable analog filter to a causal stable discrete-time IIR filter.

• Based on observation 3, substituting $s=j\omega$ and $z=e^{j\hat{\omega }}$ into (6.21) gives

  (6.22)$$\begin{array}{r}\begin{array}{rl}\omega & =\frac{2f_s}{j}\left(\frac{1-e^{-j\hat{\omega }}}{1+e^{-j\hat{\omega }}}\right)\\ & =2f_s\tan \frac{\hat{\omega }}{2}\end{array}\end{array}$$

  for $-\pi \le \hat{\omega }<\pi$, which shows that the mapping between the $j\omega$ axis of $s$-plane and the unit circle of the $z$-plane, i.e., between the angular frequency $\omega$ of the analog filter and the normalized radian frequency $\hat{\omega }$ of the discrete-time filter, induced by the bilinear transform (6.21) is one-to-one. As a result, the bilinear transform (6.21) can be employed to turn lowpass, highpass, bandpass, and bandstop analog filter prototypes to their respective discrete-time counterparts.

• The design steps of the bilinear tranform method are then simply:

  1. Starting from the specification of the discrete-time IIR filter, use (6.22) to convert the specification to one for the analog filter prototype. For example, the specification $({\hat{\omega }}_p,{\hat{\omega }}_s,{\delta }_1,{\delta }_2)$ for the case of a lowpass discrete-time filter is converted to the specification $({\omega }_p,{\omega }_s,{\delta }_1,{\delta }_2)$ of a lowpass analog filter prototype, where ${\omega }_p$ and ${\omega }_s$ are obtained from ${\hat{\omega }}_p$ and ${\hat{\omega }}_s$ using (6.22), respectively.

  2. Design an analog filter prototype that satisfies the specification obtained in step 1.

  3. Use the bilinear transform (6.21) to obtain the transfer function $H(z)$ of the target discrete-time IIR filter from the transfer function $H(s)$ of the analog filter prototype in step 2.

  Tip

  As in the impulse invariance method, since the magnitude responses of the Butterworth, Chebyshev, and elliptic (see (6.16), (6.18), (6.19), and (6.20)) are functions of the ratio $\frac{\omega }{{\omega }_p}$ or $\frac{{\omega }_s}{\omega }$, we may set $f_s=1$ in the design process above without any loss of generality.

• MATLAB Example 9:

  Repeat Example 7 above, except using the bilinear transform method with $f_s=1$ to obtain the discrete-time IIR filter from an analog type-I Chebyshev filter prototype. First, use (6.22) to obtain ${\omega }_p$ and ${\omega }_s$ from the IIR filter’s specification in order to design the analog prototype:

  >> Rp = -20*log10(1-0.01);
  >> Rs = -20*log10(0.001);
  >> wp = 2*tan(0.3*pi/2);
  >> ws = 2*tan(0.35*pi/2);
  >> [N, wp] = cheb1ord(wp, ws, Rp, Rs, 's')
  N =
  
      16
  
  wp =
  
      1.0191
  
  >> [bc, ac] = cheby1(N, Rp, wp, 's');
  >> freqs(bc, ac, [0:0.0001:pi]);
  

  Then, we can use the MATLAB function bilinear to apply the bilinear transformation in (6.21) to get the discrete-time IIR filter:

  >> [b, a] = bilinear(bc, ac, 1);
  >> fvtool(b, a);
  

  Tip

  One may more conveniently use the MATLAB function cheby1 to directly perform the steps above:

  >> [N, wp] = cheb1ord(0.3, 0.35, Rp, Rs)
  
  N =
  
      16
  
  wp =
  
      0.3000
  
  >> [b, a] = cheby1(N, Rp, wp);
  >> fvtool(b, a);
  

• MATLAB Example 10:

  Design a discrete-time highpass IIR filter with passband $[0.7\pi ,\pi ]$ and ripple tolerance ${\delta }_1=0.01$, and stopband $[0,0.65\pi ]$ and ripple tolerance ${\delta }_2=0.001$ as in Example 5 in Section 6.6. Following Example 9, we use the type-I Chebyshev filter as our prototype analog filter to perform the design.

  The following MATLAB commands:

  >> Rp = -20*log10(1-0.01);
  >> Rs = -20*log10(0.001);
  >> [N, wp] = cheb1ord(0.7, 0.65, Rp, Rs)
  
  N =
  
      16
  
  wp =
  
      0.7000
  
  >> [b, a] = cheby1(N, Rp, wp, 'high');
  >> fvtool(b, a);
  

  design an analog lowpass type-I Chebyshev filter with ${\omega }_p=1$ radian per second satisfying the ${\delta }_1$ and ${\delta }_2$ specifications, transform the lowpass prototype to a highpass prototype using the table in Section 6.7.1.5, and finally apply the bilinear transformation to obtain the desired discrete-time highpass IIR filter.

• MATLAB Example 11:

  Design a discrete-time bandpass IIR filter with passband $[0.5\pi ,0.7\pi ]$ and ripple tolerance ${\delta }_1=0.01$, and stopband $[0,0.45\pi ]\cup [0.75\pi ,\pi ]$ and ripple tolerance ${\delta }_2=0.001$. Again, we use the type-I Chebyshev filter as our prototype analog filter to perform the design as in the following MATLAB commands:

  >> Rp = -20*log10(1-0.01);
  >> Rs = -20*log10(0.001);
  >> [N, wp] = cheb1ord([0.5 0.7], [0.45 0.75], Rp, Rs)
  
  N =
  
      10
  
  wp =
  
      0.5000    0.7000
  
  >> [b, a] = cheby1(N, Rp, wp);
  >> fvtool(b, a);
  

  to obtain the desired discrete-time bandpass IIR filter. Note that the order of the filter obtained is $N=20$.

6.7.4. Frequency Transformation of IIR filters

• In Section 6.5.4, we discuss how to transform a lowpass FIR filter to a highpass one by frequency shifting. The same frequency-shifting transformation clearly applies to IIR filters also. As a matter of fact, a more general class of transformations that map the unit circle (on the $z$-plane) onto itself can be applied to convert a lowpass IIR filter to another IIR filter fo other types.

• Write the mapping discussed above as $z^{-1}\mapsto G(z^{-1})$. Since the unit circle is mapped onto itself, $|G(z^{-1})|=1$. That means, we can think of $G(z^{-1})$ as the transfer function of an allpass filter of the general form (6.5) with $b_0=\pm 1$, i.e.,

  $$G(z^{-1})=\pm \prod _{k=1}^N\frac{-a_k^{\ast }+z^{-1}}{1-a_k{z}^{-1}}.$$

  For example, the frequency shifting by $\pi$ considered in Section 6.5.4 corresponds to the mapping $z^{-1}\mapsto G(z^{-1})=-z^{-1}$.

• Other useful mappings of this form are summarized in the table below. The listed mappings can be employed to convert a lowpass IIR filter with passband edge frequency ${\hat{\omega }}_p$ to another lowpass filter, a highpass filter, a bandpass filter, and a bandstop filter, respectively.

  New filter type	New band edge(s)	Transformation	Parameter(s)
  Lowpass	${\hat{\omega }}_p^{\prime }$	${\displaystyle z^{-1}\mapsto \frac{-a+z^{-1}}{1-a{z}^{-1}}}$	${\displaystyle a=\frac{\sin \frac{{\hat{\omega }}_p-{\hat{\omega }}_p^{\prime }}{2}}{\sin \frac{{\hat{\omega }}_p+{\hat{\omega }}_p^{\prime }}{2}}}$
  Highpass	${\hat{\omega }}_p^{\prime }$	${\displaystyle z^{-1}\mapsto -\frac{-a+z^{-1}}{1-a{z}^{-1}}}$	${\displaystyle a=\frac{\cos \frac{{\hat{\omega }}_p+{\hat{\omega }}_p^{\prime }}{2}}{\cos \frac{{\hat{\omega }}_p-{\hat{\omega }}_p^{\prime }}{2}}}$
  Bandpass	${\hat{\omega }}_l,{\hat{\omega }}_u$	${\displaystyle z^{-1}\mapsto -\frac{a_2-a_1{z}^{-1}+z^{-2}}{1-a_1{z}^{-1}+a_2{z}^{-2}}}$	$\begin{array}{rl}{a}_1& =\frac{2\alpha K}{K+1}\\ a_2& =\frac{K-1}{K+1}\\ \alpha & =\frac{\cos \frac{{\hat{\omega }}_u+{\hat{\omega }}_l}{2}}{\cos \frac{{\hat{\omega }}_u-{\hat{\omega }}_l}{2}}\\ K& =\cot \frac{{\hat{\omega }}_u-{\hat{\omega }}_l}{2}\tan \frac{{\hat{\omega }}_p}{2}\end{array}$
  Bandstop	${\hat{\omega }}_l,{\hat{\omega }}_u$	${\displaystyle z^{-1}\mapsto \frac{a_2-a_1{z}^{-1}+z^{-2}}{1-a_1{z}^{-1}+a_2{z}^{-2}}}$	$\begin{array}{rl}{a}_1& =\frac{2\alpha }{K+1}\\ a_2& =\frac{1-K}{1+K}\\ \alpha & =\frac{\cos \frac{{\hat{\omega }}_u+{\hat{\omega }}_l}{2}}{\cos \frac{{\hat{\omega }}_u-{\hat{\omega }}_l}{2}}\\ K& =\tan \frac{{\hat{\omega }}_u-{\hat{\omega }}_l}{2}\tan \frac{{\hat{\omega }}_p}{2}\end{array}$

  The MATLAB functions iirlp2?? implement the frequency transformations in the table above.

• MATLAB Example 12:

  We apply the bandpass transformation to obtain a bandpass IIR filter with passband $[0.5\pi ,0.7\pi ]$ from the lowpass IIR filter in Example 9 above:

  >> Rp = -20*log10(1-0.01);
  >> Rs = -20*log10(0.001);
  >> [N, wp] = cheb1ord(0.3, 0.35, Rp, Rs);
  >> [b, a] = cheby1(N, Rp, wp);
  >> [b11, a11] = iirlp2bp(b, a, 0.3, [0.5 0.7]);
  >> fvtool(b11, a11);
  

  Caution

  Note that since the frequency transformation maps the unit circle onto itself, the specifications of ${\delta }_1$ and ${\delta }_2$ are perserved. However, as the transformation may include frequency shifting and scaling, the width of the transition bands may not be the same as that of the original filter. In addition, the order of the transformed filter may also be larger than that of the original filter.

  In this example, the transformed filter has order $N=32$, which is higher than that of the filter obtained in Example 11. We can also check that the width of transition bands is smaller than that of the filter in Example 11.