Part 4: IIR Filters Optimization Approach. Tutorial ISCAS 2007

Transcription

1 Part 4: IIR Filters Optimization Approach Tutorial ISCAS 2007 Copyright 2007 Andreas Antoniou Victoria, BC, Canada July 24, 2007 Frame # 1 Slide # 1 A. Antoniou Part4: IIR Filters Optimizatio n Method

2 Introduction IIR filters that would satisfy prescribed specifications can be designed by using a variety of optimization algorithms. Five general steps are involved, as follows: 1. Formulate a suitable objective (cost) function. 2. Deduce the gradient of the objective function and, if necessary, the Hessian matrix. 3. Choose an appropriate weighting function. 4. Select a suitable optimization algorithm. 5. Run the optimization algorithm for increasing filter orders until an order is found that will satisfy the required specifications. Note: The material for this module is taken from Antoniou, Digital Signal Processing: Signals, Systems, and Filters, Chap. 16. Frame # 2 Slide # 2 A. Antoniou Part4: IIR Filters Optimizatio n Method

3 Problem Formulation An Nth-order IIR filter can be represented by a discrete-time transfer function of the form H(z) = H 0 J j=1 a 0j + a 1j z + z 2 b 0j + b 1j z + z 2 where a ij and b ij are real coefficients, J = N/2, and H 0 is a positive multiplier constant. Frame # 3 Slide # 3 A. Antoniou Part4: IIR Filters Optimizatio n Method

4 Problem Formulation An Nth-order IIR filter can be represented by a discrete-time transfer function of the form H(z) = H 0 J j=1 a 0j + a 1j z + z 2 b 0j + b 1j z + z 2 where a ij and b ij are real coefficients, J = N/2, and H 0 is a positive multiplier constant. As presented, H(z) would be of even order; however, an odd-order H(z) can be obtained by letting for one value of j. a 0j = b 0j = 0 Frame # 3 Slide # 4 A. Antoniou Part4: IIR Filters Optimizatio n Method

5 Problem Formulation Cont d The amplitude response of an arbitrary IIR filter can be deduced as M(x,ω)= H(e jωt ) where ω is the frequency and x =[a 01 a 11 b 01 b 11 b 1J H 0 ] T is a column vector with 4J + 1 elements. Frame # 4 Slide # 5 A. Antoniou Part4: IIR Filters Optimizatio n Method

6 Problem Formulation Cont d Straightforward analysis gives the amplitude response as where M(x,ω)= H 0 J j=1 N j (ω) D j (ω) N j (ω) =[1 + a 2 0j + a 2 1j + 2a 1j (1 + a 0j ) cos ωt + 2a 0j cos 2ωT ] 1 2 and D j (ω) =[1 + b 2 0j + b 2 1j + 2b 1j (1 + b 0j ) cos ωt + 2b 0j cos 2ωT ] 1 2 for j = 1, 2,...,J. Frame # 5 Slide # 6 A. Antoniou Part4: IIR Filters Optimizatio n Method

7 Problem Formulation Cont d If M 0 (ω) is the specified amplitude response, the approximation error can be expressed as e(x,ω)= M(x,ω) M 0 (ω) Frame # 6 Slide # 7 A. Antoniou Part4: IIR Filters Optimizatio n Method

8 Problem Formulation Cont d If M 0 (ω) is the specified amplitude response, the approximation error can be expressed as e(x,ω)= M(x,ω) M 0 (ω) By sampling e(x, ω)at frequencies ω 1,ω 2,..., ω K,the column vector can be formed where for i = 1, 2,...,K. E(x) =[e 1 (x) e 2 (x)... e K (x)] T e i (x) = e(x, ω i ) Frame # 6 Slide # 8 A. Antoniou Part4: IIR Filters Optimizatio n Method

9 Problem Formulation Cont d M(x, ω) M 0 (ω) e(x, ω i ) Gain ω 1 ω 2 ω i ω K ω, rad/s Frame # 7 Slide # 9 A. Antoniou Part4: IIR Filters Optimizatio n Method

10 Problem Formulation Cont d An IIR filter can be designed by finding a point x = x such that e i ( x) 0 for i = 1, 2,...,K Frame # 8 Slide # 10 A. Antoniou Part4: IIR Filters Optimizatio n Method

11 Problem Formulation Cont d An IIR filter can be designed by finding a point x = x such that e i ( x) 0 for i = 1, 2,...,K Such a point can be obtained by minimizing the L p norm of E(x), which is defined as [ K ] 1/p (x) = L p (x) = E(x) p = e i (x) p where p is a positive integer. i=1 Frame # 8 Slide # 11 A. Antoniou Part4: IIR Filters Optimizatio n Method

12 Problem Formulation Cont d For filter design, the most important norms are the L 2 and L norms which are defined as [ K ] 1/2 L 2 (x) = e i (x) 2 and where E (x) = i=1 { K } 1/p L (x) = lim e i (x) p p i=1 = E (x) lim p = E (x) max e i (x). 1 i K { K [ i=1 ] p } 1/p e i (x) E (x) Frame # 9 Slide # 12 A. Antoniou Part4: IIR Filters Optimizatio n Method

13 Problem Formulation Cont d In summary, an IIR filter with a amplitude response that approaches a specified amplitude response M 0 (ω) can be designed by solving the optimization problem minimize x (x) Frame # 10 Slide # 13 A. Antoniou Part4: IIR Filters Optimizatio n Method

14 Problem Formulation Cont d In summary, an IIR filter with a amplitude response that approaches a specified amplitude response M 0 (ω) can be designed by solving the optimization problem If minimize x (x) (x) = L 2 (x) a least-squares solution is obtained and if (x) = L (x) the outcome will be a so-called minimax solution. Frame # 10 Slide # 14 A. Antoniou Part4: IIR Filters Optimizatio n Method

15 Gradient of Objective Function For an objective function defined in terms of the L p norm, the gradient of the objective function can be deduced as { K [ ] p } (1/p) 1 K [ ] p 1 e i (x) e i (x) k (x) = e i (x) i=1 E (x ) i=1 E (x ) where with e i (x) =[sgn e i (x)] e i (x) { 1 if ei (x) 0 sgne i (x) = 1 otherwise Frame # 11 Slide # 15 A. Antoniou Part4: IIR Filters Optimizatio n Method

16 Gradient of Objective Function Cont d The elements of e i (x) can be deduced as follows by differentiating the amplitude response: e i (x) a 0l e i (x) a 1l e i (x) b 0l e i (x) b 1l = a 0l + a 1l cos ω i T + cos 2ω i T [N l (ω i )] 2 M(x,ω i ) = a 1l + (1 + a 0l ) cos ω i T [N l (ω i )] 2 M(x,ω i ) = b 0l + b 1l cos ω i T + cos 2ω i T [D l (ω i )] 2 M(x,ω i ) = b 1l + (1 + b 0l ) cos ω i T [D l (ω i )] 2 M(x,ω i ) e i (x) H 0 = 1 H 0 M(x,ω i ) for l = 1, 2,...,J and i = 1, 2,...,K Frame # 12 Slide # 16 A. Antoniou Part4: IIR Filters Optimizatio n Method

17 Optimization Algorithms The optimization problem under consideration can be solved by using a variety of algorithms, such as the steepest-descent or Newton algorithm or one of several quasi-newton algorithms. Frame # 13 Slide # 17 A. Antoniou Part4: IIR Filters Optimizatio n Method

18 Optimization Algorithms The optimization problem under consideration can be solved by using a variety of algorithms, such as the steepest-descent or Newton algorithm or one of several quasi-newton algorithms. Quasi-Newton algorithms offer a number of important advantages as follows: They do not require the Hessian matrix. Can be used with inexact line searches which lead to improved efficiency. Are robust. Offer fast convergence. Frame # 13 Slide # 18 A. Antoniou Part4: IIR Filters Optimizatio n Method

19 Quasi-Newton Algorithms A generic quasi-newton Algorithm is as follows: 1. Input x 0 and ε. SetS 0 = I n and k = 0. Compute g 0. Frame # 14 Slide # 19 A. Antoniou Part4: IIR Filters Optimizatio n Method

20 Quasi-Newton Algorithms A generic quasi-newton Algorithm is as follows: 1. Input x 0 and ε. SetS 0 = I n and k = 0. Compute g Set d k = S k g k and find α k,thevalueofα that minimizes f (x k + α d k ), using a line search. Frame # 14 Slide # 20 A. Antoniou Part4: IIR Filters Optimizatio n Method

21 Quasi-Newton Algorithms A generic quasi-newton Algorithm is as follows: 1. Input x 0 and ε. SetS 0 = I n and k = 0. Compute g Set d k = S k g k and find α k,thevalueofα that minimizes f (x k + α d k ), using a line search. 3. Set δ k = α k d k and x k +1 = x k + δ k, and compute f k +1 = f (x k +1 ). Frame # 14 Slide # 21 A. Antoniou Part4: IIR Filters Optimizatio n Method

22 Quasi-Newton Algorithms A generic quasi-newton Algorithm is as follows: 1. Input x 0 and ε. SetS 0 = I n and k = 0. Compute g Set d k = S k g k and find α k,thevalueofα that minimizes f (x k + α d k ), using a line search. 3. Set δ k = α k d k and x k +1 = x k + δ k, and compute f k +1 = f (x k +1 ). 4. If δ k 2 <ε, then output x = x k +1, f ( x ) = f k +1 and stop. Frame # 14 Slide # 22 A. Antoniou Part4: IIR Filters Optimizatio n Method

23 Quasi-Newton Algorithms A generic quasi-newton Algorithm is as follows: 1. Input x 0 and ε. SetS 0 = I n and k = 0. Compute g Set d k = S k g k and find α k,thevalueofα that minimizes f (x k + α d k ), using a line search. 3. Set δ k = α k d k and x k +1 = x k + δ k, and compute f k +1 = f (x k +1 ). 4. If δ k 2 <ε, then output x = x k +1, f ( x ) = f k +1 and stop. 5. Compute g k +1 and set γ k = g k +1 g k. Frame # 14 Slide # 23 A. Antoniou Part4: IIR Filters Optimizatio n Method

24 Quasi-Newton Algorithms A generic quasi-newton Algorithm is as follows: 1. Input x 0 and ε. SetS 0 = I n and k = 0. Compute g Set d k = S k g k and find α k,thevalueofα that minimizes f (x k + α d k ), using a line search. 3. Set δ k = α k d k and x k +1 = x k + δ k, and compute f k +1 = f (x k +1 ). 4. If δ k 2 <ε, then output x = x k +1, f ( x ) = f k +1 and stop. 5. Compute g k +1 and set γ k = g k +1 g k. 6. Compute S k +1 = S k + C k. Frame # 14 Slide # 24 A. Antoniou Part4: IIR Filters Optimizatio n Method

25 Quasi-Newton Algorithms A generic quasi-newton Algorithm is as follows: 1. Input x 0 and ε. SetS 0 = I n and k = 0. Compute g Set d k = S k g k and find α k,thevalueofα that minimizes f (x k + α d k ), using a line search. 3. Set δ k = α k d k and x k +1 = x k + δ k, and compute f k +1 = f (x k +1 ). 4. If δ k 2 <ε, then output x = x k +1, f ( x ) = f k +1 and stop. 5. Compute g k +1 and set γ k = g k +1 g k. 6. Compute S k +1 = S k + C k. 7. Check S k +1 for positive definiteness and if it is found to be nonpositive definite force it to become positive definite. Frame # 14 Slide # 25 A. Antoniou Part4: IIR Filters Optimizatio n Method

26 Quasi-Newton Algorithms A generic quasi-newton Algorithm is as follows: 1. Input x 0 and ε. SetS 0 = I n and k = 0. Compute g Set d k = S k g k and find α k,thevalueofα that minimizes f (x k + α d k ), using a line search. 3. Set δ k = α k d k and x k +1 = x k + δ k, and compute f k +1 = f (x k +1 ). 4. If δ k 2 <ε, then output x = x k +1, f ( x ) = f k +1 and stop. 5. Compute g k +1 and set γ k = g k +1 g k. 6. Compute S k +1 = S k + C k. 7. Check S k +1 for positive definiteness and if it is found to be nonpositive definite force it to become positive definite. 8. Set k = k + 1 and go to step 2. Frame # 14 Slide # 26 A. Antoniou Part4: IIR Filters Optimizatio n Method

27 Quasi-Newton Algorithms Cont d Notes: x 0 is the initial point. ε is a termination tolerance. I n is the n n identity matrix. d k is the direction vector at the start of the k th iteration. g k is the gradient at the start of the k th iteration. S k is an approximation to the inverse Hessian at the start of the k th iteration. C k is a correction matrix. x is the minimum point and f ( x ) is the minimum value of the objective function. Efficiency can be achieved by using Fletcher s inexact line search in Step 2. Frame # 15 Slide # 27 A. Antoniou Part4: IIR Filters Optimizatio n Method

28 Quasi-Newton Algorithms Cont d Several quasi-newton algorithms are available. They differ from one another in the formula used to update matrix S k +1 in Step 7 of the generic quasi-newton algorithm presented. Frame # 16 Slide # 28 A. Antoniou Part4: IIR Filters Optimizatio n Method

29 Quasi-Newton Algorithms Cont d Several quasi-newton algorithms are available. They differ from one another in the formula used to update matrix S k +1 in Step 7 of the generic quasi-newton algorithm presented. The two most important algorithms of this family are the Davidon-Fletcher-Powell (DFP) algorithm in which S k +1 = S k + δ k δk T γk T δ S k γ k γk T S k k γk T S k γ k and the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm in which ( S k +1 = S k γ k T S ) k γ k δk T γk T δ k γk T δ (δ k γk T S k + S k γ k δk T ) k γk T δ k Frame # 16 Slide # 29 A. Antoniou Part4: IIR Filters Optimizatio n Method

30 Choice between L 2 and L Solutions L 2 solutions are easier to obtain and filters based on these solutions will reject more signal power in stopbands. However, the passband error tends to have large peaks near passband edges, which are undesirable in practice. Frame # 17 Slide # 30 A. Antoniou Part4: IIR Filters Optimizatio n Method

31 Choice between L 2 and L Solutions L 2 solutions are easier to obtain and filters based on these solutions will reject more signal power in stopbands. However, the passband error tends to have large peaks near passband edges, which are undesirable in practice. L solutions are more difficult to obtain because they require the application of minimax algorithms which entail much more computation. However, they offer the advantage that the approximation error tends to be uniformly distributed in passbands, i.e., it tends to be equiripple as in elliptic filters. Frame # 17 Slide # 31 A. Antoniou Part4: IIR Filters Optimizatio n Method

32 Minimax Algorithms Minimax algorithms are essentially sequential algorithms that involve a series of unconstrained optimizations. A representative algorithm of this class is the so-called least-pth algorithm. Another popular minimax algorithm is one proposed by Charalambous (see References). Frame # 18 Slide # 32 A. Antoniou Part4: IIR Filters Optimizatio n Method

33 Least-pth Minimax Algorithm 1. Input x 0 and ε 1.Setk = 1, p = 2,μ= 2, E 0 = Frame # 19 Slide # 33 A. Antoniou Part4: IIR Filters Optimizatio n Method

34 Least-pth Minimax Algorithm 1. Input x 0 and ε 1.Setk = 1, p = 2,μ= 2, E 0 = Initialize frequencies ω 1,ω 2,...,ω K. Frame # 19 Slide # 34 A. Antoniou Part4: IIR Filters Optimizatio n Method

35 Least-pth Minimax Algorithm 1. Input x 0 and ε 1.Setk = 1, p = 2,μ= 2, E 0 = Initialize frequencies ω 1,ω 2,...,ω K. 3. Using x k 1 as initial value, minimize where { k (x) = K [ ] p } 1/p e i (x) E (x) i=1 E (x) E (x) = max 1 i K e i(x) with respect to x, to obtain x k.set E k = E ( x ). Frame # 19 Slide # 35 A. Antoniou Part4: IIR Filters Optimizatio n Method

36 Least-pth Minimax Algorithm 1. Input x 0 and ε 1.Setk = 1, p = 2,μ= 2, E 0 = Initialize frequencies ω 1,ω 2,...,ω K. 3. Using x k 1 as initial value, minimize where { k (x) = K [ ] p } 1/p e i (x) E (x) i=1 E (x) E (x) = max 1 i K e i(x) with respect to x, to obtain x k.set E k = E ( x ). 4. If E k 1 E k <ε 1, then output x k and E k, and stop. Otherwise, set p = μp, k = k + 1 and go to step 3. Frame # 19 Slide # 36 A. Antoniou Part4: IIR Filters Optimizatio n Method

37 Stability Least-squares or minimax algorithms will often yield discrete-time transfer functions with poles outside the unit circle z =1, and such transfer functions represent unstable filters. Frame # 20 Slide # 37 A. Antoniou Part4: IIR Filters Optimizatio n Method

38 Stability Least-squares or minimax algorithms will often yield discrete-time transfer functions with poles outside the unit circle z =1, and such transfer functions represent unstable filters. Fortunately, it is possible to eliminate this problem through a stabilization technique that has been known for a number of years. Frame # 20 Slide # 38 A. Antoniou Part4: IIR Filters Optimizatio n Method

39 Stability Cont d Let us assume that an optimization algorithm has produced a discrete-time transfer function H(z) = N(z) D(z) = N(z) D (z) k i=1 (z p i) with k poles p 1, p 2,..., p k that lie outside the unit circle. Frame # 21 Slide # 39 A. Antoniou Part4: IIR Filters Optimizatio n Method

40 Stability Cont d Let us assume that an optimization algorithm has produced a discrete-time transfer function H(z) = N(z) D(z) = N(z) D (z) k i=1 (z p i) with k poles p 1, p 2,..., p k that lie outside the unit circle. A stable transfer function that yields the same amplitude response can be obtained as where H (z) = H 0 N(z) D (z) k i=1 (z 1/ p i) = H 0 = 1 k i=1 p i N i=0 a i z i 1 + N i=1 b i z i Frame # 21 Slide # 40 A. Antoniou Part4: IIR Filters Optimizatio n Method

41 Prescribed Specifications Optimization algorithms in general tend to yield filters in which the maximum values of the approximation error in the various passbands and stopbands are of the same order. Frame # 22 Slide # 41 A. Antoniou Part4: IIR Filters Optimizatio n Method

42 Prescribed Specifications Optimization algorithms in general tend to yield filters in which the maximum values of the approximation error in the various passbands and stopbands are of the same order. In practice, the prescribed passband ripples and minimum stopband attenuations vary wildly from band to band and from one application to the next, which means that the required maximum values of the passband and stopband errors also vary wildly. Frame # 22 Slide # 42 A. Antoniou Part4: IIR Filters Optimizatio n Method

43 Prescribed Specifications Optimization algorithms in general tend to yield filters in which the maximum values of the approximation error in the various passbands and stopbands are of the same order. In practice, the prescribed passband ripples and minimum stopband attenuations vary wildly from band to band and from one application to the next, which means that the required maximum values of the passband and stopband errors also vary wildly. In order to achieve desired specifications, we need be able to control the relative magnitudes of the maximum passband and stopband errors, and this is achieved through the use of a weighting. Frame # 22 Slide # 43 A. Antoniou Part4: IIR Filters Optimizatio n Method

44 Prescribed Specifications Cont d Weighting essentially involves constructing a modified error function of the form e i (x) = W (ω)e i(x) = W (ω)[m(x,ω i ) M 0 (ω i )] where W (ω) is a scalar weighting function which is used to emphasize or deemphasize the approximation error in selected passbands or stopbands so as to decrease or increase its magnitude in those bands. Frame # 23 Slide # 44 A. Antoniou Part4: IIR Filters Optimizatio n Method

45 Prescribed Specifications Cont d Weighting essentially involves constructing a modified error function of the form e i (x) = W (ω)e i(x) = W (ω)[m(x,ω i ) M 0 (ω i )] where W (ω) is a scalar weighting function which is used to emphasize or deemphasize the approximation error in selected passbands or stopbands so as to decrease or increase its magnitude in those bands. Typically, W (ω) is a piecewise constant function which is assigned a value greater or less than one to decrease or increase the magnitude of the approximation error is a given band. Frame # 23 Slide # 45 A. Antoniou Part4: IIR Filters Optimizatio n Method

46 Prescribed Specifications Cont d Arbitrary filter specifications can be achieved as follows: 1. Choose a suitable weighting function on the basis of the prescribed specifications. Frame # 24 Slide # 46 A. Antoniou Part4: IIR Filters Optimizatio n Method

47 Prescribed Specifications Cont d Arbitrary filter specifications can be achieved as follows: 1. Choose a suitable weighting function on the basis of the prescribed specifications. 2. Design filters for increasing filter orders until a filter order is found that would satisfy the required specifications. Evidently, this is a cut-and-try method and it could entail a considerable amount of computation. Frame # 24 Slide # 47 A. Antoniou Part4: IIR Filters Optimizatio n Method

48 Prescribed Specifications Cont d Assuming a filter with m bands, the weighting function can be deduced from the specified passband ripples and the minimum stopband attenuations as detailed below. Frame # 25 Slide # 48 A. Antoniou Part4: IIR Filters Optimizatio n Method

49 Prescribed Specifications Cont d Assuming a filter with m bands, the weighting function can be deduced from the specified passband ripples and the minimum stopband attenuations as detailed below. Let δ 1, δ 2,..., δ k,..., δ m be the maximum passband and stopband errors imposed by the filter specifications where A p i 1 δ i = A p i + 1 for a passband with ripple A pi db and δ j = A a j for a stopband with a minimum attenuation A aj db. Frame # 25 Slide # 49 A. Antoniou Part4: IIR Filters Optimizatio n Method

50 Prescribed Specifications Cont d Now assume that the weighting function W (ω) is a piecewise-constant function with positive values W 1, W 2,..., W k,..., W m for bands 1, 2,..., k,..., m, respectively. Frame # 26 Slide # 50 A. Antoniou Part4: IIR Filters Optimizatio n Method

51 Prescribed Specifications Cont d Now assume that the weighting function W (ω) is a piecewise-constant function with positive values W 1, W 2,..., W k,..., W m for bands 1, 2,..., k,..., m, respectively. Suitable scaling constants can be assigned as W 1 = δ k, δ 1 W k = 1 W k +1 = W 2 = δ k δ 2,..., W k 1 = δ k δ k 1 δ k δ k +1, W k +2 = δ k δ k +2,..., W m = δ k δ m From example, in a BP filter we could assign W 1 = δ 2 /δ 1, W 2 = 1, W 3 = δ 2 /δ 3 or W 1 = 1, W 2 = δ 1 /δ 2, W 3 = δ 1 /δ 3 or W 1 = δ 3 /δ 1, W 2 = δ 3 /δ 2, W 3 = 1. Frame # 26 Slide # 51 A. Antoniou Part4: IIR Filters Optimizatio n Method

52 Design Examples A variety of design examples can be found in Antoniou, Digital Signal Processing, Chap. 14, and in Antoniou and Lu, Practical Optimization, Chaps. 9 and 16. Frame # 27 Slide # 52 A. Antoniou Part4: IIR Filters Optimizatio n Method

53 Summary A great variety of optimization algorithms can be used for the design of IIR filters. Frame # 28 Slide # 53 A. Antoniou Part4: IIR Filters Optimizatio n Method

54 Summary A great variety of optimization algorithms can be used for the design of IIR filters. Quasi-Newton algorithms work very well. Frame # 28 Slide # 54 A. Antoniou Part4: IIR Filters Optimizatio n Method

55 Summary A great variety of optimization algorithms can be used for the design of IIR filters. Quasi-Newton algorithms work very well. The optimization approach is very flexible in that it can be used to design filters with arbitrary amplitude and/or phase responses. Frame # 28 Slide # 55 A. Antoniou Part4: IIR Filters Optimizatio n Method

56 Summary A great variety of optimization algorithms can be used for the design of IIR filters. Quasi-Newton algorithms work very well. The optimization approach is very flexible in that it can be used to design filters with arbitrary amplitude and/or phase responses. In FIR filters as well as IIR filters based on the bilinear transformation, techniques are availalble that can be used to predict the required filter order to achieve prescribed specifications. Unfortunately, in IIR filters designed by optimization the filter order can only be deduced through a cut-and-try approach. Frame # 28 Slide # 56 A. Antoniou Part4: IIR Filters Optimizatio n Method

57 Summary A great variety of optimization algorithms can be used for the design of IIR filters. Quasi-Newton algorithms work very well. The optimization approach is very flexible in that it can be used to design filters with arbitrary amplitude and/or phase responses. In FIR filters as well as IIR filters based on the bilinear transformation, techniques are availalble that can be used to predict the required filter order to achieve prescribed specifications. Unfortunately, in IIR filters designed by optimization the filter order can only be deduced through a cut-and-try approach. As in any other methodology based on optimization, a large amount of computation is required to complete a design. Frame # 28 Slide # 57 A. Antoniou Part4: IIR Filters Optimizatio n Method

58 References A. Antoniou, Digital Signal Processing: Signals, Systems, and Filters, Chap. 16, McGraw-Hill, A. Antoniou and W.-S. Lu Practical Optimization: Algorithms and Engineering Applications, Chaps. 9 and 16, Springer, C. Charalambous, Acceleration of the least pth algorithm for minimax optimization with engineering applications, Mathematical Programming, vol. 17, pp , A. Antoniou, Improved minimax optimisation algorithms and their application in the design of recursive digital filters, Proc. Inst. Elect. Eng., Part G, vol. 138, pp , Dec Frame # 29 Slide # 58 A. Antoniou Part4: IIR Filters Optimizatio n Method

59 References A. Antoniou and W.-S. Lu, Design of two-dimensional digital filters by using the singular value decomposition, IEEE Trans. Circuits Syst., vol. 34, pp , Oct W.-S. Lu and A. Antoniou, Design of digital filters and filter banks by optimization: A state of the art review, in Proc European Signal Processing Conference, vol. 1, pp , Tampere, Finland, Sept Frame # 30 Slide # 59 A. Antoniou Part4: IIR Filters Optimizatio n Method

60 This slide concludes the presentation. Thank you for your attention. Frame # 31 Slide # 60 A. Antoniou Part4: IIR Filters Optimizatio n Method