# Digital Filter Design

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1 Digital Filter Design Objective - Determination of a realiable transfer function G() approximating a given frequency response specification is an important step in the development of a digital filter If an IIR filter is desired, G() should be a stable real rational function Digital filter design is the process of deriving the transfer function G()

2 Digital Filter Specifications Usually, either the magnitude and/or the phase (delay) response is specified for the design of digital filter for most applications In some situations, the unit sample response or the step response may be specified In most practical applications, the problem of interest is the development of a realiable approximation to a given magnitude response specification

3 Digital Filter Specifications We discuss in this course only the magnitude approximation problem There are four basic types of ideal filters with magnitude responses as shown below H LP (e j ) H HP (e j ) c 0 c H BP (e j ) c 0 c H BS (e j ) 3 c c c c c c c c

4 Digital Filter Specifications 4 As the impulse response corresponding to each of these ideal filters is noncausal and of infinite length, these filters are not realiable In practice, the magnitude response specifications of a digital filter in the passband and in the stopband are given with some acceptable tolerances In addition, a transition band is specified between the passband and stopband

5 Digital Filter Specifications For example, the magnitude response G( e of a digital lowpass filter may be given as indicated below jω ) 5

6 Digital Filter Specifications As indicated in the figure, in the passband, defined by 0 ω ω p, we require that jω G( e ) with an error ± δ p, i.e., δ p G( e jω ) + δ In the stopband, defined by ω s ω π, we jω require that G( e ) 0 with an error δs, i.e., jω G ( e ) δ, ω ω π s s p, ω ω p 6

7 Digital Filter Specifications 7 ω p - passband edge frequency ω s - stopband edge frequency δ p - peak ripple value in the passband δ s - peak ripple value in the stopband jω Since G( e ) is a periodic function of ω, jω and G( e ) of a real-coefficient digital filter is an even function of ω As a result, filter specifications are given only for the frequency range 0 ω π

8 Digital Filter Specifications Specifications are often given in terms of jω loss function A ( ω) 0log0 G( e ) in db Peak passband ripple α log ( δ ) db p 0 0 p Minimum stopband attenuation αs 0log0( δs) db 8

9 Digital Filter Specifications Digital Filter Specifications Magnitude specifications may alternately be given in a normalied form as indicated below 9

10 Digital Filter Specifications Here, the maximum value of the magnitude in the passband is assumed to be unity / + ε - Maximum passband deviation, given by the minimum value of the magnitude in the passband - Maximum stopband magnitude A 0

11 Digital Filter Specifications For the normalied specification, maximum value of the gain function or the minimum value of the loss function is 0 db Maximum passband attenuation - ( ) αmax 0log0 + ε db For δ p <<, it can be shown that α log ( δ ) db max 0 0 p

12 Digital Filter Specifications In practice, passband edge frequency and stopband edge frequency F s are specified in H For digital filter design, normalied bandedge frequencies need to be computed from specifications in H using Ω p πfp ωp πfpt FT FT Ωs πf ω s s πfs T F F T T F p

13 Digital Filter Specifications Example- Let Fp 7 kh, F s 3 kh, and 5 kh F T Then π(7 0 ) ωp 0. 56π π(3 0 ) ωs 0. 4π

14 4 The transfer function H() meeting the frequency response specifications should be a causal transfer function For IIR digital filter design, the IIR transfer function is a real rational function of : H() must be a stable transfer function and must be of lowest order N for reduced computational complexity Selection of Filter Type Selection of Filter Type N M d d d d p p p p H N N M M , ) ( 0 0 L L

15 5 Selection of Filter Type For FIR digital filter design, the FIR transfer function is a polynomial in with real coefficients: H ( ) N h[ n] n 0 For reduced computational complexity, degree N of H() must be as small as possible If a linear phase is desired, the filter coefficients must satisfy the constraint: h[ n] ± h[ N n] n

16 6 Selection of Filter Type Advantages in using an FIR filter - () Can be designed with exact linear phase, () Filter structure always stable with quantied coefficients Disadvantages in using an FIR filter - Order of an FIR filter, in most cases, is considerably higher than the order of an equivalent IIR filter meeting the same specifications, and FIR filter has thus higher computational complexity

17 7 Digital Filter Design: Basic Approaches Most common approach to IIR filter design - () Convert the digital filter specifications into an analog prototype lowpass filter specifications () Determine the analog lowpass filter transfer function H a (s) (3) Transform H a (s) into the desired digital transfer function G()

18 8 Digital Filter Design: Basic Approaches Basic Approaches This approach has been widely used for the following reasons: () Analog approximation techniques are highly advanced () They usually yield closed-form solutions (3) Extensive tables are available for analog filter design (4) Many applications require digital simulation of analog systems

19 9 Digital Filter Design: Basic Approaches An analog transfer function to be denoted as Pa ( s) Ha ( s) D ( s) where the subscript a specifically indicates the analog domain A digital transfer function derived from shall be denoted as P( ) G ( ) D( ) a H a (s)

20 Digital Filter Design: Basic Approaches 0 Basic idea behind the conversion of H a (s) into G() is to apply a mapping from the s-domain to the -domain so that essential properties of the analog frequency response are preserved Thus mapping function should be such that Imaginary ( jω) axis in the s-plane be mapped onto the unit circle of the -plane A stable analog transfer function be mapped into a stable digital transfer function

21 Digital Filter Design: Basic Approaches FIR filter design is based on a direct approximation of the specified magnitude response, with the often added requirement that the phase be linear The design of an FIR filter of order N may be accomplished by finding either the length-(n+) impulse response samples or the (N+) samples of its frequency response H ( e jω ) { h[n] }

22 Digital Filter Design: Basic Approaches Three commonly used approaches to FIR filter design - () Windowed Fourier series approach () Frequency sampling approach (3) Computer-based optimiation methods

23 3 IIR Digital Filter Design: Bilinear Transformation Method Bilinear transformation - s T + Above transformation maps a single point in the s-plane to a unique point in the -plane and vice-versa Relation between G() and H a (s) is then given by G( ) H ( ) a s s T +

24 Bilinear Transformation 4 Digital filter design consists of 3 steps: () Develop the specifications of H a (s) by applying the inverse bilinear transformation to specifications of G() () Design H a (s) (3) Determine G() by applying bilinear transformation to H a (s) As a result, the parameter T has no effect on G() and T is chosen for convenience

25 Bilinear Transformation Inverse bilinear transformation for T is + s s 5 Fors Thus, σ ( + σ ( σ o o o + ) + ) σ σ σ jω o o o o jω jω o o 0 < 0 < > 0 > ( + σ ( σ o o ) ) + Ω + Ω o o

26 Bilinear Transformation Mapping of s-plane into the -plane 6

27 Bilinear Transformation For e with T we have jω jω/ jω/ j e ( e e jω e jω jω/ jω/ j + e e ( e + e jsin( ω/ ) j or cos( ω/ ) Ω tan(ω/ ) jω ω/ ω/ ) ) tan( ω/ ) 7

28 Bilinear Transformation Mapping is highly nonlinear Complete negative imaginary axis in the s- plane from Ω to Ω 0 is mapped into the lower half of the unit circle in the -plane from to Complete positive imaginary axis in the s- plane from Ω 0 to Ω is mapped into the upper half of the unit circle in the -plane from to 8

29 Bilinear Transformation Nonlinear mapping introduces a distortion in the frequency axis called frequency warping Effect of warping shown below 9

30 30 Bilinear Transformation Steps in the design of a digital filter - () Prewarp ( ω p, ωs) to find their analog equivalents ( Ω p, Ω s) () Design the analog filter H a (s) (3) Design the digital filter G() by applying bilinear transformation to H a (s) Transformation can be used only to design digital filters with prescribed magnitude response with piecewise constant values Transformation does not preserve phase response of analog filter

31 IIR Digital Filter Design Using Bilinear Transformation Example- Consider H a ( s) Applying bilinear transformation to the above we get the transfer function of a first-order digital lowpass Butterworth filter s Ωc + Ω c 3 Ω ( ) ( ) ( ) c + G H a s s ( ) + Ω ( + ) + c

32 IIR Digital Filter Design Using Bilinear Transformation Rearranging terms we get where G ( ) α + α Ω α + Ω c c + tan( ω tan( ω c c / / ) ) 3

33 33 IIR Digital Filter Design Using Bilinear Transformation Example- Consider the second-order analog notch transfer function s + Ωo Ha( s) s + B s + Ωo for which H a ( jω o ) 0 Ha( j0) Ha( j ) is called the notch frequency Ω o If Ha( jω ) Ha( jω) / then B Ω is the 3-dB notch bandwidth Ω

34 34 IIR Digital Filter Design Using IIR Digital Filter Design Using Bilinear Transformation Bilinear Transformation Then where ) ( ) ( + s H a s G ) ( ) ( ) ( ) ( ) ( ) ( + Ω + Ω + + Ω + Ω + Ω + Ω B B o o o o o o ) ( α α β β α o o o ω + Ω Ω β cos ) / tan( ) / tan( w w o o B B B B Ω + Ω α

35 IIR Digital Filter Design Using Bilinear Transformation Example- Design a nd-order digital notch filter operating at a sampling rate of 400 H with a notch frequency at 60 H, 3-dB notch bandwidth of 6 H 35 Thus ωo π(60/ 400) 0. 3π B w π(6/ 400) 0. 03π From the above values we get α β

36 Gain, db IIR Digital Filter Design Using Thus Bilinear Transformation G( ) The gain and phase responses are shown below Phase, radians Frequency, H Frequency, H

37 37 IIR Lowpass Digital Filter Design Using Bilinear Transformation Example- Design a lowpass Butterworth digital filter with ωp 0. 5π, ωs 0. 55π, α p 0.5dB, and α s 5 db Thus ε A If G( e j0 ) this implies j0.5π 0log0 G( e ) 0.5 j0.55π 0log0 G( e ) 5

38 38 IIR Lowpass Digital Filter Design Using Bilinear Transformation Prewarping we get Ω tan( ω / ) tan(0.5π/ ) p p Ωs tan( ωs / ) tan(0.55π/ ) The inverse transition ratio is Ωs k Ω p The inverse discrimination ratio is k A ε

39 IIR Lowpass Digital Filter Design Using Bilinear Transformation Thus N log log 0 (/ k) (/ k) We choose N 3 To determine we use Ω c H a( j Ω p ) + ( Ω p / Ω c ) N + ε 39

40 40 IIR Lowpass Digital Filter Design Using Bilinear Transformation We then get Ω.4995( Ω ) c rd-order lowpass Butterworth transfer function for Ω c is H an ( s) ( s + )( s + s + ) Denormaliing to get Ω c we arrive at s H ( s) H a an p

41 IIR Lowpass Digital Filter Design Using Bilinear Transformation Applying bilinear transformation to H a (s) we get the desired digital transfer function G( ) H ( ) a s s + Magnitude and gain responses of G() shown below: 0 Magnitude Gain, db ω/π ω/π

42 Design of IIR Highpass, Bandpass, and Bandstop Digital Filters First Approach - 4 () Prewarp digital frequency specifications of desired digital filter G D () to arrive at frequency specifications of analog filter H D (s) of same type () Convert frequency specifications of H D (s) into that of prototype analog lowpass filter H LP (s) (3) Design analog lowpass filter H LP (s)

43 Design of IIR Highpass, Bandpass, and Bandstop Digital Filters (4) Convert H LP (s) into H D (s) using inverse frequency transformation used in Step (5) Design desired digital filter G D () by applying bilinear transformation to H D (s) 43

44 44 Design of IIR Highpass, Bandpass, and Bandstop Digital Filters Second Approach - () Prewarp digital frequency specifications of desired digital filter G D () to arrive at frequency specifications of analog filter H D (s) of same type () Convert frequency specifications of H D (s) into that of prototype analog lowpass filter H LP (s)

45 Design of IIR Highpass, Bandpass, and Bandstop Digital Filters (3) Design analog lowpass filter H LP (s) (4) Convert H LP (s) into an IIR digital transfer function G LP () using bilinear transformation (5) Transform G LP () into the desired digital transfer function G D () We illustrate the first approach 45

46 IIR Highpass Digital Filter Design 46 Design of a Type Chebyshev IIR digital highpass filter Specifications: Fp 700 H, F s 500 H, α p db, α s 3 db, F T kh Normalied angular bandedge frequencies πfp π 700 ωp 0. 7π FT 000 πf π ω s 500 s 0. 5π F 000 T

47 IIR Highpass Digital Filter Design Prewarping these frequencies we get Ωˆ p tan( ωp / ) Ωˆ tan( ω / ).0 s s For the prototype analog lowpass filter choose Ω p Ω Ω Using Ω p ˆ p we get Ω Ωˆ s Analog lowpass filter specifications:, Ω s.9605, α p db, 3 db α s Ω p 47

48 IIR Highpass Digital Filter Design MATLAB code fragments used for the design [N, Wn] chebord(,.96605,, 3, s ) [B, A] cheby(n,, Wn, s ); [BT, AT] lphp(b, A,.96605); [num, den] bilinear(bt, AT, 0.5); 0-0 Gain, db ω/π

49 49 IIR Bandpass Digital Filter Design Design of a Butterworth IIR digital bandpass filter Specifications: ωp 0. 45π, ωp 0. 65π, ωs 0. 3π, ωs 0. 75π, α p db, α s 40dB Prewarping we get Ωˆ tan( ω / ) Ωˆ Ωˆ Ωˆ p p p tan( ωp / ) tan( ω / ) s s s tan( ωs / ).44356

50 50 IIR Bandpass Digital Filter Design Width of passband B w Ωˆ ˆ p Ω p ˆ Ω ˆ ˆ o Ω pω p Ωˆ ˆ ˆ sω s Ωo We therefore modify Ωˆ s so that Ωˆ s and Ωˆ s exhibit geometric symmetry with respect to Ωˆ o We setωˆ s For the prototype analog lowpass filter we choose Ω p

51 IIR Bandpass Digital Filter Design Ωˆ Using o Ωˆ Ω Ω we get p Ωˆ B Ω s Specifications of prototype analog Butterworth lowpass filter: Ω p, Ω s , α p db, 40 db α s w

52 IIR Bandpass Digital Filter Design MATLAB code fragments used for the design [N, Wn] buttord(,.36767,, 40, s ) [B, A] butter(n, Wn, s ); [BT, AT] lpbp(b, A, , ); [num, den] bilinear(bt, AT, 0.5); 0-0 Gain, db ω/π

53 53 IIR Bandstop Digital Filter Design Design of an elliptic IIR digital bandstop filter Specifications: ωs 0. 45π, ωs 0. 65π, ωp 0. 3π, ωp 0. 75π, α p db, α s 40dB Prewarping we get Ωˆ s , Ωˆ s.63857, Ωˆ , Ω p ˆ p Width of stopband Ωˆ Ωˆ Ωˆ Ωˆ o p Ωˆ Ωˆ p s B w s s Ωˆ s Ωˆ o

54 54 IIR Bandstop Digital Filter Design ˆ p Ωˆ p ˆ p We therefore modify Ω so that and Ω exhibit geometric symmetry with respect to Ωˆ o We setωˆ p For the prototype analog lowpass filter we choose Ω s Ωˆ B Using w Ω Ωs we get ˆ ˆ Ωo Ω Ω p

55 IIR Bandstop Digital Filter Design MATLAB code fragments used for the design [N, Wn] ellipord(0.4346,,, 40, s ); [B, A] ellip(n,, 40, Wn, s ); [BT, AT] lpbs(b, A, , ); [num, den] bilinear(bt, AT, 0.5); 0-0 Gain, db ω/π

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