1. Nonrecursive Digital Filters

2. Digital Filters & Filter Equation
General Equation
- FIR
- Convolution
Frequency response
Transfer function

3. Nonrecursive Filter • Disadvantage : takes computation time
• Advantage : stable (zeros only)
linear phase (no phase distortion)
same phase shift to all frequencies

4. Impulse response of moving average filter
2M+1 coefficients, symmetric to n=0
Smoothness of the signal  correlated to the increment of M
Width of mainlobe  negatively correlated to M
increment of M  narrow band lowpass filter

5. Frequency Response of Moving Average Filter
5-point
(M = 2)
21-point
(M = 10)
No zeros at z=0
since passband around

6. Frequency Response of Moving Average Filter
 = 0  peak value = 1
unwanted side lobe  first side lobe 22% of main lobe
5 terms  4 zeros missing zero at z = 1
21 terms  20 zeros passband contains at  = 0
Zeros lie actually on the unit circle  true nulls in the corresponding frequency
ex)

7. Ideal Lowpass Filter Method
1.0

8. Design of Highpass/Bandpass Filters using Lowpass Filter
(bandwidth : center frequency : )
Lowpass Filter
Replace with
Limit to 2M+1 terms, and start from n=0
Center frequency :
bandwidth :

9. Lowpass Filter Design
Cutoff frequency : Sampling rate :

10. Highpass Filter Design
Cutoff frequency highpass filter

11. Bandpass Filter Design
Cutoff frequency : Sampling rate :
Duration of impulse response :
Center frequency
Bandwidth

12. Frequency Transformation

13. Recursive Digital Filters

14. General Form of Filters
Recursive filter
 powerful : separate control over the numerator and denominator of H(z)
 If the magnitude of the denominator becomes small at the appropriate frequency
 produce sharp response peaks by arranging

15. Example #1 Find the difference equation of Bandpass Filter
(a) Center frequency :  = /2, dB Bandwidth :  /40, Maximum gain : 1
(b) No frequency component at  = 0,  = 
origin
- Assume BC is straight line
- d = 1 - r (r > 0.9)
- 2d = 2 (1-r)
- 2 (1-r) [rad] = /40 = 3.14/40, r = 0.961
- No frequency component at  = 0 and  =  - two zeroes at z = +1 and -1

16. Example #1 ① -3dB band-width : /40 Maximum gain : 26.15 (28.35dB)
in equaiton ①, K = (6.15)-1 =
The corresponding difference equaion is :
y[n+2] y[n] = {x[n+2] - x[n]}
subtracting 2 from each term in brackets
y[n] = y[n-2] {x[n] - x[n-2]}

17. Example #2
Design a band-reject filter which stops 60Hz powerline noise from ECG signal
10Hz cutoff bandwidth at -3dB point
Poles and zeros as in the picture
(solution) - fs = 1.2 kHz
- fmax : 600Hz
- 2 : 1200 = o: 60
- o (60Hz) = 0.1 
y[n+2] y[n+1] y[n] = x[n+2] x[n+1] + x[n]
y[n] = y[n-1] y[n-2] + x[n] x[n-1] + x[n-2]

18. Types of Filters Butterworth Chebyshev – 1st order
Chebyshev – 2nd order
Elliptic

19. Butterworth, Chebyshev, Elliptic Filters
analog
digital
Butterworth
Chebyshev
Elliptic
ripple

20. Example #3 Find the minimum order of Filter Cutoff frequency 1= 0.2
Frequency response of less than 30dB at  = 0.4

21. Bilinear Transformation
H(s) H(z)

22. Bilinear Transformation

23. Impulse-invariant Filters
Another method of deriving a digital filter from an analog filter
A sampled version of that of the reference analog filter

24. Impulse-invariant Filters

25. Impulse-invariant Filters
Transfer function of
analog filter
Impulse-invariant
filter

26. Impulse-invariant Filters
The impulse response of each analog subfilter takes a simple exponential form
For the i-th subfilter
 A zero at the origin of the z-plane
 A polse at

27. Design of Recursive Digital Filters

28. Butterworth LP Analog Filter Design (prototype)
Prototype : when or
frequency responses at N = 1, 2, 3

29. Determination of Poles
When N : odd
When N : even
N = 1 ;
N = 2 ;
N = 3 ; only 3 effective terms

30. Determination of Poles
1st order
2nd order

31. Example
Design a lowpass Butterworth filter : -3dB at 1 rad/sec (prototype filter)
gain of less than 0.1 for the frequency greater than 2 rad/sec
Order of filter

32. Chebyshev LP Analog Filter Design (prototype)or
N : order, : cutoff frequency, r : ripple amplitude ( : ripple parameter)
Order of filter

33. Chebyshev Prototype Denominator Polynomials

34. Example
Maximum passband ripple : 1dB, Cutoff frequency : less than 1.3 rad/sec
Attenuation in stopband : 40dB for greater than 5 rad/sec
ripple parameter
cutoff frequency : -3dB point is half the magnitude
2nd order
Passband characteristic
3rd order

35. Analog Filter Frequency Transformation

36. Example Butterworth bandpass filter Maximum attenuation of 0.2dB for
Minimum attenuation of 50dB for
Prototype equivalent frequency
Filter order
3rd order