1. CHAPTER 7 filter design techniques
7.0 introduction
7.1design of discrete-time IIR filters from continuous-time filters
7.1.1 filter design by impulse invariance
7.1.2 filter design by bilinear transform
not low pass filter design and other method
7.2 design of FIR filters by windowing. 7.3 summary

2. ideal frequency selective filter
7.0 introduction
ideal frequency selective filter
band-stop filter：notch filter；quality factor of band-pass filter=pass-band width/center frequency

3. impulse response of ideal low-pass filter，noncausal，unrealizable
Figure 7.1
impulse response of ideal low-pass filter，noncausal，unrealizable
Specifications for filter design ：given in frequency domain
phase：linear?
magnitude：given by a tolerance scheme
analog or digital，absolute or relative

4. Figure 7.2 absolute specification magnitude response of
equivalent analog system
monotonous descent
passband tolerance
stopband tolerance
passband cutoff frequency
stopband cutoff frequency
3dB cutoff frequency
absolute specification

5. digital specification, finally:
relative specifications：maximum magnitude in passband is normalized to 1, viz. 0dB
maximum attenuation in passband
minimum attenuation in stopband
3dB cutoff frequency：
magnitude response of equivalent analog system:
digital specification, finally:

6. Specifications for bandpass and bandstop filters：
up and down passband cutoff frequency，
up and down stopband cutoff frequency
Design steps：
（1）decide specifications according to application
（2）decide type according to specification：generally , if the phase is required , choose FIR.
（3）approach specifications using causal and stable discrete-time system：
viz. design H(z0) or h[n]，nonuniform
（4）choose a software or hardware realization structure, take effects of limited word length into consideration
H(z) or h[n]

7. 7.1 design of discrete-time IIR filters from continuous-time filters
attention：original analog filter and equivalent analog filter is different, their frequency response is not always the same
7.1.0 introduction of analogy filter
7.1.1filter design by impulse invariance
7.1.2 filter design by bilinear transform

8. 7.1.0 introduction of analog filter
comparison:
1.wave
2.the same order,
increase performance
3.increase design complexity
(A)（C）small aliasing in the impulse invariance design technique

9. magnitude frequency function:
BW design formula: specificationsystem function
magnitude frequency function:
take the specifications into system function and get the results from equation group: OR
confirm the poles of system function:
（in the left half plane）
get system function：

10. design a low pass analogy filter:
EXAMPLE
design a low pass analogy filter:
[N,Wc]=buttord(2000*pi,4000*pi,1,15, 's')
[Bs,As]=butter(N,Wc, 's')
[H,W]=freqs(Bs,As);
plot(W/2/pi,20*(log10(abs(H))))
axis([1000,2000,-16,0])
grid on
OUTPUT:
N = 4
Wc = e+003
Bs = 1.0e+015 *
As = 1.0e+015 *

12. Or： 
[N,Wc]=buttord(2000*pi,4000*pi,1,15, 's')
[z,p,k]=butter(N,Wc, 's' )
Output:
z = Empty matrix: 0-by-1
p = 1.0e+003 * i i i i
k = e+015

13. design a low pass cheby analogy filter:
EXAMPLE
design a low pass cheby analogy filter:
[N,Wc]=cheb1ord(2000*pi,4000*pi,1,15, 's')
[Bs,As]=cheby1(N,1,Wc, 's')
[H,W]=freqs(Bs,As);
plot(W/2/pi,20*(log10(abs(H))))
axis([0,4000,-30,0])
grid on

14. design a low pass cheby analogy filter:
EXAMPLE
design a low pass cheby analogy filter:
[N,Wc]=cheb2ord(2000*pi,4000*pi,1,15, 's')
[Bs,As]=cheby2(N,15,Wc, 's')
[H,W]=freqs(Bs,As);
plot(W/2/pi,20*(log10(abs(H))))
axis([0,4000,-30,0])
grid on

15. design a high pass analogy filter:
EXAMPLE
design a high pass analogy filter:
[N,Wc]=buttord(4000*pi,2000*pi,1,15, 's')
[Bs,As]=butter(N,Wc, 'high', 's')
[H,W]=freqs(Bs,As);
plot(W/2/pi,20*(log10(abs(H))))
axis([0,4000,-16,0])
grid on

16. transformation from analog filter to digital filter:
H(s)H(z)，viz. mapping from S plane to Z plane, must satisfy:
the same frequency response（map imaginary axis to the unit circle）；
causality and stability is preserved（map poles from left half plane to the inside circle）。
imaginary axis the unit circle
poles in left half plane inside the unit circle
S plane  Z plane

17. 7.1.1 filter design by impulse invariance
according to ：
conversion formula：

18. Relationship between poles（causal and stable):
relation between frequencies:
S plane
Z plane

19. relation between frequency response: when aliasing is small, the frequency response is the same.
strongpoint：linear frequency mapping; shortcoming：aliasing in frequency response。
restriction in application：can not used in high-pass and bandstop filter

20. Design steps： （3） about Td: independent of T;
do not influence aliasing;
arbitrary value，generally, take 1（attention（1）and（3）have the same value）.

21. Td=1; Wp=wp/Td; Ws=ws/Td [N,Wc]=buttord(Wp,Ws, ap , as, 's')
EXAMPLE
wp=0.2*pi; ws=0.4*pi
ap=1; as=12
Td=1; Wp=wp/Td; Ws=ws/Td
[N,Wc]=buttord(Wp,Ws, ap , as, 's')
[Bs,As]=butter(N,Wc, 's')
[Bz,Az]=impinvar(Bs,As,1/Td)
[H,W]=freqs(Bs,As);
plot(W/pi,20*(log10(abs(H))), 'r*')
hold on
[H,w]=freqz(Bz,Az);
plot(w/pi,20*(log10(abs(H))))
axis([0.2,0.4,-20,0])
grid

23. 7.1.2 filter design by bilinear transform
design thought：
design formula：

24. relation between frequencies:
left half s-plane  inside the unit circle in the z-plane（causality and stability are preserved）；
imaginary axis of the s-plane the unit circle in z-plane (the same frequency response), one-by-one mapping
relation between frequencies:

25. relation between frequency response:
strongpoint：no aliasing；
shortcoming：nonlinear。
restriction in application：
can not used in differentiator.

26. design steps: (1) (3) about Td: independent of T；
(1) 
(3)
about Td:
independent of T；
arbitrary value，generally, take 1（attention（1）and（3）have the same value）.

27. wp=0.2613*pi; ws=0.4018*pi; ap=0.75; as=20;
EXAMPLE
wp=0.2613*pi; ws=0.4018*pi; ap=0.75; as=20;
Td=1; Ws=2/Td*tan(ws/2); Wp=2/Td*tan(wp/2)
[N,Wc]=buttord(Wp,Ws,ap,as,’s’)
[Bs,As]=butter(N,Wc, ‘s’)
[Bz,Az]=bilinear(Bs,As,1/Td)
[H,W]=freqs(Bs,As);
plot(W/pi,20*(log10(abs(H))),’Rx‘)
hold on
[H,w]=freqz(Bz,Az);
plot(w/pi,20*(log10(abs(H))))
ylabel(‘虚线：模拟滤波器幅度[dB] 实线：数字滤波器幅度[dB]’)
xlabel(‘虚线：模拟角频率[*π弧度/秒] 实线:数字角频率[*π弧度]’)
axis([0.25,0.5,-20,-0.45])
grid

28. OR
wp=0.2613*pi; ws=0.4018*pi; ap=0.75; as=20
[N,wc]=buttord(wp/pi,ws/pi,ap,as)
[Bz,Az]=butter(N,wc)
[H,w]=freqz(Bz,Az);
plot(w/pi,20*(log10(abs(H))))

29. [Bz,Az]=butter(N,Wc, 'high') [H,w]=freqz(Bz,Az);
EXAMPLE
[N,Wc]=buttord(0.6,0.5,ap,as)
[Bz,Az]=butter(N,Wc, 'high')
[H,w]=freqz(Bz,Az);
plot(w/pi,20*(log10(abs(H))))
grid

30. EXAMPLE

31. [N,wc]=buttord([0.45 0.55],[0.4 0.6],3,10) %双线性变换法 [B,A]=butter(N,wc)
[H,w]=freqz(B,A);
plot(w/pi,20*(log10(abs(H))))
ylabel(‘20log|H(ejω)| [dB]’)
xlabel(‘数字角频率[*π弧度]’)
axis([0.4,0.6,-10,0]); grid on
Output:
N = 2
wc =
B =
A =

32. 7.1.3 IIR summary 1.design steps 2. impulse invariance:
frequency axis is linear many-to-one mapping , aliasing in frequency response, inapplicable to high-pass filter.
bilinear transformation：
frequency axis is one-to-one mapping with aberrance, no aliasing in frequency response, inapplicable to differentiator

33. 7.1.4 frequency conversion
Frequency conversion in analog domain：

34. Frequency conversion in digital domain：

35. 7.2 design of FIR filters by windowing
7.2.1 design ideas properties of commonly used windows effect to frequency response design step

36. 7.2.1 design ideas

37. Gibbs phenomenon: frequency response oscillate at wc.
transition width of filer’smainlobe width of window spectrum length and shape of window’s
passband-stopband error of filter’ssidelobe amplitude of window spectrum window’s shape
The smaller mainlobe width, the narrower transition width of filter’s;
The smaller sidelobe amplitude of window spectrum, the smaller relative error of filter’s.
M increases mainlobe width of window spectrum minishes filter oscillate more quickly, transition width minishes, relative amplitude of oscillation is preserved.
In order to improve relative amplitude of oscillation, we need to change the shape of window’s.

38. 7.2.2 properties of commonly used windows

39. A general designation: Blackman window family, except triangular window
Figure 7.21

40. Figure 7.24(a)

41. (a)-(e) attenuation of sidelobe increases, width of mainlobe increases.

42. As B increases, attenuation of sidelobe increases, width of mainlobe increases.
As N increases：
attenuation of sidelobe is preserved,
width of mainlobe decreases.
Figure 7.24(b)(c)

43. Transition width is a little less than mainlobe width
Table 7.1

44. 7.2.3 effect to frequency response
attenuation of window spectrum’s sidelobe attenuation of filter’s stopband
width of window spectrum’s mainlobetransition width of filter’s
check the table for Blackman window;
formulas of Kaiser window:
A：attenuation of stopband, Δω：transition width

45. 7.2.4 design step
1. Write the ideal impulse response：
2.Comfirm the shape of window’s based on attenuation of stopband：
（1）check the table for Blackman window
（2）calculation for Kaiser window

46. (2) calculation for Kaiser window ：
3.Comfirm length of window’s based on transition width（M is even):
(1) check the table for Blackman window: compute the length of window’s based on width of mainlobe:
(2) calculation for Kaiser window ：
4.Cut the ideal impulse response 5.
We can think that the 4 specifications is
transformed into：
The same with IIR, maybe the specification is equivalent analog one.

47. Example 1
Blackman family
(1)
(2)
hamming
(3)
(4)

48. （5）validate the frequency response in MATLAB
h=fir1(26,0.62, 'high',hamming(27))
H=fft(h,512);
plot([0:511]/256,20*log10(abs(H)))
axis([0.5,0.7,-50,0]); grid on

49. h=fir1(24,0.665, 'high',hamming(25)) H=fft(h,512);
（6）correct
h=fir1(24,0.665, 'high',hamming(25))
H=fft(h,512);
plot([0:511]/256,20*log10(abs(H)))
axis([0.5,0.7,-80,0]);
grid on h=

50. Example 2
Kaiser family
Solution1:
(1)
(2)
(3)
(4)

51. （5）validate and correct in MATLAB
h=fir1(18,0.662, 'high', kaiser (19,3.2953)); H=fft(h,512);
k=0:511;
subplot(1,2,1); plot(k/256,20*log10(abs(H))); grid on;
subplot(2,2,2); plot(k/256,20*log10(abs(H)));
axis([0.5,0.55,-50,-0]); grid on
subplot(2,2,4); plot(k/256,20*log10(abs(H)));
axis([0.65,0.75,-5,0]); grid on

53. Example 3
plot the response to signal
h=fir1(18,0.5, 'high',kaiser(19,6));
n=[0:100]; x=10*cos(0.1*pi*n)+3*sin(0.8*pi*n) ;
subplot(2,1,1); plot(n,x)
y=filter(h,1,x) ; subplot(2,1,2); plot(n,y)

55. FIR summary
shape attenuation of sidelobe attenuation of stopband
（the first row in the table）（ the third row in the table）
length width of mainlobe transition width
(the second row) （the second row /2）

56. 7.3 brief introduction of other filter design techniques using matlab
IIR: minimum mean square error in frequency domain: invfreqz( )
FIR: frequency sampling ：fir2( )
FIR: minimum mean square error ：firls( )
FIR: Parks-McCellan/Remez arithmetic：
remez( )

57. 7.4 summary FIR digital filter IIR digital filter
finite-length IR infinite-length IR
no non-zero poles, stable non-zero poles, stable?
realization: convolution /recursion recursion
realize linear phase easily can not realize linear phase in deed
stable maybe instable because of rounding
higher order lower order because of recursion
FFT no fast arithmetic

58. summary 7.1 design of IIR（continuous-time filter）
7.1.1 impulse invariance
7.1.2 bilinear transform
7.2 design of FIR（windowing）
7.2.1 design ideas
7.2.2 properties of commonly used windows
7.2.3 effect to frequency response
7.2.4 design step
7.3 comparison between IIR and FIR

59. requirements：
understand the principles of impulse invariance and bilinear transform and their mapping characteristic, implication;
understand design ideas of windowing;
design various filters using MATLAB.
difficulties：
relationship among prototype analog filter, digital filter and equivalent analog filter;
effects of shape and length of window’s to system characteristic；
why impulse invariance and bilinear transform can not be used in high-pass filter design。

60. exercise and experiment
(a)(b) 7.23
the second experiment 50 52 56 57 59