Discrete-time Symmetric/Antisymmetric FIR Filter Design

Transcription

1 Discrete-time Symmetric/Antisymmetric FIR Filter Design Presenter: Dr. Bingo Wing-Kuen Ling Center for Digital Signal Processing Research, Department of Electronic Engineering, King s College London.

2 Collaborators Department of Electronic Engineering, King s College London Dr. Zoran Cvetković Department of Electronic and Information Engineering, Hong Kong Polytechnic University Dr. Peter Kwong-Shun Tam Department of Mathematics and Statistics, Curtin University of Technology Prof. Kok-Lay Teo

3 Collaborators Department of Mathematics and Statistics, Royal Melbourne Institute of Technology Dr. Yan-Qun Liu School of Mathematical Sciences, Queen Mary, University of London Ms. Charlotte Yuk-Fan Ho 3

4 Outline Introduction Filter Design via Semi-infinite Programming Conclusions References Q&A Session 4

5 Introduction Definition of discrete-time filters Types of discrete-time filters Common discrete-time filters Applications of discrete-time filters Common filter design techniques Challenges in filter design 5

6 Introduction Definition of discrete-time filters Complex exponent signals are eigenfunctions of discrete-time linear time invariant systems, that is y(n)=h(ω)e jωn if x(n)=e jωn for n Z, H(ω) is called the frequency response. A discrete-time filter is a discrete-time linear time invariant system characterized by its frequency response. 6

7 Introduction Types of discrete-time filters Lowpass filters Allow a low frequency band of a signal to pass through and attenuate a high frequency band 7

8 Introduction Types of discrete-time filters Bandpass filters Allow intermediate frequency bands of a signal to pass through and attenuate both low and high frequency bands 8

9 Introduction Types of discrete-time filters Highpass filters Allow a high frequency band of a signal to pass through and attenuate a low frequency band 9

10 Introduction Types of discrete-time filters Band reject filters Allow both low and high frequency bands of a signal to pass through and attenuate intermediate frequency bands 0

11 Introduction Types of discrete-time filters otch filters Allow almost all frequency components to pass through but attenuate particular frequencies

12 Introduction Types of discrete-time filters Oscillators Allow particular frequency components to pass through and attenuate almost all frequency components

13 Introduction Types of discrete-time filters Allpass filters Allow all frequency components to pass through (g) Magnitude response H(ω) Frequency ω 3

14 Introduction Types of discrete-time filters Finite impulse response (FIR) filters The impulse response of the filters has finite time support. ote that FIR filters are strictly bounded input bounded output stable. Infinite impulse response (IIR) filters The impulse response of the filters has infinite time support. ote that rational IIR filters are particular types of IIR filters, but many IIR filters are irrational. For example, sinc filter is irrational IIR filter. However, rational IIR filters are easy to implement. 4

15 Introduction Types of discrete-time filters Linear phase filters The phase response is linear. ote that not all FIR filters are linear phase, but FIR filters can achieve linear phase easily. onlinear phase filters The phase response is nonlinear. ote that not all IIR filters are nonlinear phase, but it is not easy to achieve good linear phase IIR filters. 5

16 Introduction Common discrete-time filters Differentiators H(ω)=jω for ω (-π,π) and π periodic. ote that H(ω) is discontinuous at odd multiples of π. Hilbert transformers H(ω)=sign(ω) for ω (-π,π) and π periodic. ote that H(ω) is discontinuous at integer multiples of π. 6

17 Introduction Applications of discrete-time filters Lowpass and notch filters are widely used in noise reduction applications. Hilbert transformers are widely used in single sideband modulation systems. Differentiators are widely used in measurement systems. Oscillators are widely used as sinusoidal signal generators. etc 7

18 Introduction Common filter design techniques otations: H(e jω ) : H(e j ω ) + δ p - δ p magnitude response of a lowpass discrete-time FIR filter δ p and δ s : maximum passband and stopband δ s ripple magnitudes ω p and ω s : passband and stopband frequencies : filter length passband stopband ω 0 ω p ω s π transition band H d (e j ω ) 8

19 Introduction Common filter design techniques Antisymmetric impulse response For is odd, h(k)= h(--k) for k = 0,,,( 3)/ and h((-)/)=0. The frequency response is H(e jω )=je -j(-)ω/ H 0 (ω) For is even, h(k)= h(--k) for k = 0,,,/-. h[n] The frequency response is H(e jω )=je -j(-)ω/ H 0 (ω) h[n] n center of symmetry 6 7 n 9

20 0 Introduction Common filter design techniques Antisymmetric impulse response = = = is even ]sin [ is odd ]sin [ ) ( k k h k k h H k k o ω ω ω is even is odd ) ( sin sin sin sin T T ω ω ω ω ω L L L L η is even 0 is odd 3 0 h h h h h h T T L L x ( ) x η T o H ) ( ) ( ω ω

21 Introduction Common filter design techniques Symmetric impulse response For is odd, h(k)=h(--k) for k = 0,,,( 3)/. The frequency response is H(e jω )=e -j(-)ω/ H 0 (ω) For is even, h(k)=h(--k) for k = 0,,,/-. h[n] The frequency response is H(e jω )=e -j(-)ω/ H 0 (ω) n h[n] center of symmetry n

22 Introduction Common filter design techniques Symmetric impulse response + = = = is even ]cos [ is odd ]cos [ ) ( k k h k k h h H k k o ω ω ω is even is odd ) ( cos cos cos cos T T ω ω ω ω ω L L L η is even 0 is odd 0 h h h h h h T T L L x ( ) x η T o H ) ( ) ( ω ω

23 Introduction Common filter design techniques Total Weighted Ripple Energy J (x) ( ( ω ) Q W ( ω ) η( ω) η ) B p UB s B p U B s W ( ω ) H T T = x Qx + b x o ( ω ) D ( ω ) + p T d J(x) : total weighted ripple energy B p : {ω: ω ω p }, passband ω dω B s : {ω: ω s ω π}, stopband W(ω) : weighted function, W(ω)>0 D(ω) : desired magnitude response b W ( ω ) D( ω) η( ω) dω p W ( ω) ( D( ω) ) dω B p UB s B p UB s 3

24 Introduction Common filter design techniques Maximum Ripple Magnitude H o ( ω) D( ω) δ [ η( ω) ( ω ] T [ D ( ω) + δ δ D( )] T A( ω ) η ) c δ ( ω) ω A( ω) x + c ( ω) 0 δ δ: the acceptable bound of the maximum ripple magnitude of filters 4

25 Introduction Common filter design techniques Problem (P ) : min x J ( x) T T = x Qx + b x + p x* : optimal solution of problem (P ) δ : minimum value that T F δ : { x : ( η( ω )) x D( ω) δ, ω B p U B s } * ( η( ω) ) T x D( ω) δ ω B U p B s 5

26 Introduction Common filter design techniques Problem (P ) : min x Subject to: J (x) δ g δ ( x, ω) A( ω) x + cδ ( ω) 0 x* : optimal solution of problem (P ) δ : minimum value that T F δ : { x : ( η( ω )) x D( ω) δ, ω B p U B s } ω B U p B s T * ( η( ω )) x D( ω) δ ω Bp U Bs 6

27 Introduction Challenges in filter design Although H approach minimizes the total ripple energy, maximum ripple magnitude may be very large. Although H approach minimizes the maximum ripple magnitude, total ripple energy may be very large. How to tradeoff between the H approach and H approach? 7

28 Filter Design via Semi-infinite Programming Definition of peak constrained least square filter design Computer numerical simulation results of peak constrained least square filter design Open problems in peak constrained least square filter design Properties of peak constrained least square filter design Dual parameterization approach for solving peak constrained least square filter design 8 problem

29 Filter Design via Semi-infinite Programming Definition of peak constrained least square filter design Problem (P) : min x J ( x) T T = x Qx + b x + p subject to g δ ( x, ω) 0 ω B U p B s x* δ : optimal solution of problem (P) δ : the acceptable bound of the maximum ripple magnitude of filters T F : { x : ( η( ω )) x D( ω) δ, ω B p U B s } 9

30 Filter Design via Semi-infinite Programming Computer numerical simulation results of peak constrained least square filter design H design x H infinity design SIP design Ripple response Ripple response Ripple response Frequency ω Frequency ω Frequency ω 30

31 Filter Design via Semi-infinite Programming Open problems in peak constrained least square filter design How to determine the specification for peak constrained least square filter design? In particular, how to determine the value of the acceptable maximum ripple magnitude? 3

32 Filter Design via Semi-infinite Programming Open problems in peak constrained least square filter design ω is a continuous function, so for each frequency, say ω 0, it corresponds to a single constraint. In fact, a continuous function consists of infinite number of discrete frequencies, so the problem is actually an infinite constrained optimization problem. How to guarantee that these infinite number of constraints are satisfied? 3

33 Filter Design via Semi-infinite Programming Properties of peak constrained least square filter design Definition of convex set If x and x are in S, then λx +( λ)x also belongs to S λ [0,]. x x x x (a) convex (b) not convex 33

34 Filter Design via Semi-infinite Programming Properties of Peak constrained least square filter design Definition of convex function: ƒ Let ƒ: S Ε, where S is a nonempty convex set in Ε n. The function ƒ is said to be convex on S if ƒ(λx +( λ)x ) λƒ(x )+( λ)ƒ(x ) for x,x S and λ [0,]. ƒ ƒ x λ x +(- λ)x x x λ x +(- λ)x x x x convex function concave function neither convex nor concave 34

35 Filter Design via Semi-infinite Programming Properties of peak constrained least square filter design Property The feasible set F δ is convex. Let x and x be two distinct elements of F δ, which means that Α(ω)x +c δ (ω) 0and Α(ω)x +c δ (ω) 0. λ [0,], since Α(ω)(λx +(-λ)x )+c δ (ω) 0, this implies that λx +(-λ)x F δ. 35

36 Filter Design via Semi-infinite Programming Properties of peak constrained least square filter design Property The matrix Q is positive definite. T T x Qx = W ( ω) ( η( ω) ) x dωis nonnegative. Suppose that x T B p UB s T Qx=0, which implies that ( η( ω) ) x = 0 ω B. p U B s T In particular, we have [ η ( ω such that ) η( ω) L η( ω )] x = 0 rank( [ η( ω ]) ) η( ω) L η( ω ) =. This implies that x=0. Since x T Qx>0 for x 0, the result follows directly. 36

37 Filter Design via Semi-infinite Properties of peak constrained least square filter design Property 3 Programming The cost function J(x) is strictly convex. T T J ( x) = x Qx + b x + p is twice differentiable with respect to x, and its Hessian matrix is equal to Q which is positive definite. This implies that J(x) is strictly convex. 37

38 Filter Design via Semi-infinite Properties of peak constrained least square filter design Property 4 Programming x* δ is uniquely defined. Let x* a and x* b be optimal solutions of the SIP problem, that is J(x* a )=J(x* b ). Suppose that x* a x* b, since the feasible set F δ is convex and J(x) is strictly convex, this implies that λx* a +( λ)x* b F δ such that J(λx* a +( λ)x* b )<λj(x* a )+( λ)j(x* b )=J(x* a )=J(x* b ). This contradicts to the hypothesis that x* a and x* b are the optimal solutions of the SIP problem because 38 λx* a +( λ)x* b is the optimal solution. Hence, x* a =x* b.

39 Filter Design via Semi-infinite Programming Properties of peak constrained least square filter design Property 5 x* is uniquely defined. Since Q is positive definite, all eigenvalues of Q are positive and Q - exists. Consequently, x* =-Q - b. 39

40 Filter Design via Semi-infinite Programming Properties of peak constrained least square filter design Property 6 x* is uniquely defined. By alternation theorem, x* is uniquely defined. 40

41 Filter Design via Semi-infinite Properties of peak constrained least square filter design Property 7 Programming Suppose that x* x* δ, then * ( η( ω )) T 0 xδ D( ω0) = δ. Bp U B s such that Since x* x* δ, x* F δ and F δ F δ. Otherwise x* F δ implies that J(x* )=J(x* δ ), which contradicts the uniqueness property of the solution. For F δ F δ, J(x* ) J(x* δ ). Otherwise λ (0,) such that J(λx* +(- λ)x* δ )<J(x* δ ) and J(λx* +(-λ)x* δ )<J(x* ), which contradicts the fact that x* and x* δ are the optimal solutions. Hence, J(x* )<J(x* δ ). ω 0 4

42 Filter Design via Semi-infinite Properties of peak constrained least square filter design Property 7 Programming Suppose that (η(ω 0 )) T x* δ -D(ω 0 ) <δ, then λ (0,) and Δx=(-λ)(x* -x* δ ) such that (η(ω 0 )) T (x* δ +Δx)-D(ω 0 ) =δ. Since J(x* δ +Δx)=J(λx* δ +( λ)x* ) and J(x) is strictly convex, we have J(x* δ +Δx)<λJ(x* δ )+( λ)j(x* ). As J(x* )<J(x* δ ), we have J(x* δ +Δx)<J(x* δ ). However, this contradicts to the assumption that x* δ is the optimal solution of the SIP problem. Hence the result follows directly. 4

43 Filter Design via Semi-infinite Programming Properties of peak constrained least square filter design Property 8 Denote x* a and x* b as the solutions of the SIP problems for δ=δ a and δ=δ b, respectively. Denote F δb and F δa as the corresponding feasible sets, respectively. If δ <δ b <δ a <δ, then J(x* )<J(x* a )<J(x* b )<J(x* δ ) and F δ F δb F δa F δ. x F δ implies A( ω) x + D( ω) δ < δb < δ a < δ ω Bp U Bs. This implies that F δ F δb F δa F δ and J(x* ) J(x* a ) J(x* b ) J(x* δ ). Suppose that F δ =F δb, then x* b F δb =F δ. T * ω 0 B p U B s such that ( η( ω0) ) xb D ( ω0) = δb > δ. But this contradicts to the fact that x* b F δ. Hence, x* b F δ and F δ F δb. Since the solution is uniquely defined and J(x) is strictly convex, J(x* b )<J(x* ). Similarly, the result follows directly. 43

44 Filter Design via Semi-infinite Properties of peak constrained least square filter design Property 9 Programming Denote F as a map from the set of the maximum ripple magnitudes to the set of the total ripple energy of the filters. Then F(δ) is convex with respect to δ for δ <δ<δ. Let δ a and δ b be the maximum ripple magnitude such that δ <δ a <δ b <δ. Let x* a and x* b be the solutions of the SIP problems corresponding to δ a and δ b, respectively. Also, let F δa and F δb be the corresponding feasible sets, respectively. Since, x* a F δa and x* b F δb, we have T * ( η( ω) ) xa D( ω) δ T * a ω B p U B and ( η( ω) ) xb D( ω) δ ω B U s b p B s. * 44 Hence λ ( 0,) λa( ω) x a + λd( ω) λδ a and

45 Filter Design via Semi-infinite Properties of peak constrained least square filter design Property 9 Programming * ( λ) A( ω) x b + ( λ) D( ω) ( λ) δ * * It follows that A( ω) λxa + ( λ) xb + D( ω) ( λδ a + λ δ b ) Denote F λδa+( λ)δb as the feasible set corresponding to the maximum ripple magnitude equal to λδ a +( λ)δ b. Then λx* a +( λ)x* b F λδa+( λ)δb. Denote x* λδa+( λ)δb as the solution of the SIP problem corresponding to the maximum ripple magnitude equal to λδ a +( λ)δ b. Then J(x* λδa+( λ)δb ) J(λx* a +( λ)x* b ). Since J(x) is strictly convex, we have J(x* λδa+( λ)δb )<λj(x* a )+( λ)j(x* b ). Hence, F(δ) is convex with respect to δ. b ω B U B [ ] ( ) p s ω B U p B s 45

46 Filter Design via Semi-infinite Programming Properties of peak constrained least square filter design.8 x 0-4 Design using SIP approach Design using H infinity approach Design using H approach.6.4 Total ripple energy Maximum ripple magnitude 46

47 Filter Design via Semi-infinite.8 x 0-4 Programming Properties of peak constrained least square filter design Design using SIP approach Design using H infinity approach Design using H approach.6 H.4 Total ripple energy. 0.8 monotonic decreasing convex 0.6 H SIP 0.4 H SIP H δ 3 δ Maximum ripple magnitude 47 δ min δ

48 Filter Design via Semi-infinite Programming Dual parameterization approach for solving peak constrained least square filter design problem The magnitude response contains finite number of maxima and minima. If the constraints are satisfied in these extrema, then all constraints are satisfied. 48

49 Filter Design via Semi-infinite Programming Dual parameterization approach for solving peak constrained least square filter design problem However, the locations of these extrema are unknown. Hence, we optimize both the filter coefficients and finite number of frequencies so that the cost function is minimized and the constraints are satisfied. 49

50 Conclusions Filters are designed via peak constrained least square approach and the problem can be solved via a dual parameterization approach. The plot of the total ripple energy against the maximum ripple magnitude is monotonic decreasing and convex, this information helps to determine the specifications for filter design. 50

51 References [] Charlotte Yuk-Fan HO, Bingo Wing-Kuen LIG, Yan-Qun LIU, Peter Kwong-Shun TAM and Kok-Lay TEO, Efficient Algorithm for Solving Semi-Infinite Programming Problems and Their Applications to onuniform Filter Bank Designs, IEEE Transactions on Signal Processing, vol. 54, no., pp , 006. [] Charlotte Yuk-Fan HO, Bingo Wing-Kuen LIG, Joshua D. REISS, Yan-Qun LIU, and Kok-Lay TEO, Design of Interpolative Sigma-Delta Modulators via Semi-Infinite Programming, IEEE Transactions on Signal Processing, vol. 54, no. 0, pp , 006. [3] Charlotte Yuk-Fan HO, Bingo Wing-Kuen LIG, Yan-Qun LIU, Peter Kwong-Shun TAM and Kok-Lay TEO, Optimal Design of Magnitude Responses of Rational Infinite Impulse Response Filters, IEEE Transactions on Signal Processing, vol. 54, no. 0, pp , 006. [4] Charlotte Yuk-Fan HO, Bingo Wing-Kuen LIG, Yan-Qun LIU, Peter Kwong-Shun TAM and Kok-Lay TEO, Optimal Design of onuniform FIR Transmultiplexer Using Semi-Infinite Programming, IEEE Transactions on Signal Processing, vol. 53, no. 7, pp , 005. [5] Charlotte Yuk-Fan HO, Bingo Wing-Kuen LIG, Yan-Qun LIU, Peter Kwong-Shun TAM and Kok-Lay TEO, Design of onuniform ear Allpass Complementary FIR Filters via a Semi-Infinite Programming Technique, IEEE Transactions on Signal Processing, vol. 53, no., pp ,

52 Q&A Session Thank you! Let me think Bingo 5