Design of CIC of CIC Compensators Compensators With With SPT SPT - PowerPoint PPT Presentation

Design of CIC of CIC Compensators Compensators With With SPT SPT Design Coefficients Based Based o on Interval n Interval Analysis Analysis Coefficients Matija Glavini ć Pecoti ć , Goran Molnar, and Mladen Vu č i ć Siemens CMT d. d. Faculty of Electrical Engineering and Computing, University of Zagreb Opatija, May 2012

Outline Outline o Introduction o CIC compensator with SPT coefficients based on minimax error criterion o Features of proposed CIC compensators o Conclusion 2

Introduction Introduction o Digital down converters usually employ Cascaded- Integrator-Comb (CIC) filter in the first stage o CIC filter o multiplierless structure o significant passband droop 3

Reduction of Passband of Passband Droo Droop p Reduction o Modifying the original CIC structure • sharpening technique • Kwentus et.al. 1997, Stephen et.al. 2004, Dolecek et.al. 2005 o Connecting an additional filter in the cascade with the CIC decimator • CIC compensator • Yeung et.al. 2004, Kim et.al. 2006, Dolecek et.al. 2008, Dolecek 2009, Dolecek et.al. 2010, Molnar et.al. 2011, Vazquez et.al. 2012. 4

CIC Compensator Compensator CIC o FIR filter with linear phase o Multiplierless structure is preferable o Coefficients are expressed as the sum of powers of two (SPT) o Various multiplierless compensators have been proposed • Compensator with three and five coefficients • Compensation over wide and narrow band o Various criteria have been used • Least squares, minimax, maximally flat 5

In this this paper paper... ... In o CIC compensator with SPT coefficients o Minimax error criterion o Optimization based on the interval analysis • Results in global solution! o Recently, it has been used in the design of low-order FIR filters with SPT coefficients o Here, we modify it for the design of CIC compensator 6

Ideal vs. FIR Compensator ompensator Ideal vs. FIR C o The amplitude response of the CIC filter of order N and decimation factor R N   ω   R   sin   1  2    ω = H ( ) CIC ω     R   sin   2     o The amplitude response of the ideal compensator is 1 ω = H ( ) C ω     H CIC R   7

Ideal vs. FIR Compensator ompensator Ideal vs. FIR C o FIR compensator with M coefficients o Compensator with odd number of coefficients is considered 8

Minimax Approximation Approximation with with SPT SPT Coefficients Coefficients Minimax o Objective is to find the optimum SPT coefficients of the compensator which results in the minimax approximation o Such a design is described by [ ] = ε a arg min ( a ) opt a subject to a : is SPT representa ble o The objective function has the form ω   ε = − ω   ( a ) max 1 H CIC H ( , a ) R   ω ≤ ω c 9

Minimax Approximation Approximation with with SPT SPT Coefficients Coefficients Minimax o The problem is solved by using the optimization based on interval analysis o In the paper M. Vucic, G. Molnar, and T. Zgaljic, “Design of FIR filters based on interval analysis,” in Proc. MIPRO, vol. MEET, 2010. the authors deal with the objective function [ ] = ω ω − ω y ( x ) max W ( ) H ( , x ) H ( ) a d ω ∈ Ω o Here, the interval extension of ε ( a ) can be easily obtained by using the extensions of elementary operations 10

Minimax Approximation Approximation with with SPT SPT Coefficients Coefficients Minimax o Vector a contains only right-hand side samples − −   M 1 M 1 = + =   a ( m ) h m ; m 0 , 1 , , K 2 2   o Amplitude response of compensator − ( M 1 ) 2 ∑ ω a = + ω H ( , ) a ( 0 ) 2 a ( m ) cos( m ) = m 1 o SPT coefficient with a given wordlength K K { } − ∑ k ∈ − = b ( m , k ) 1 , 0 , 1 a ( m ) b ( m , k ) 2 = k 1 11

Minimax Approximation Approximation with with SPT SPT Coefficients Coefficients Minimax o Each b ( m , k ) ≠ 0 represents one adder in hardware o Number of terms per each coefficient is limited to a prescribed value P o Structure of compensator o Total number of adders + + ( M 1 )( P 1 ) ≤ − A 1 2 12

Features of of Proposed Proposed Compensators Compensators Features o Three examples are described • Compensators with three coefficients • Wideband compensators • Compensators with the lowest complexity 13

Compensators W With ith T Three hree C Coefficients oefficients Compensators o N =5, R =14 o Narrowband o ω c =0.25 π o Wideband o ω c =0.5 π , 0.6 π o Two compensators • K= 9 • P =2 results in A =5 • P =3 results in A =7 14

Compensators W With ith T Three hree C Coefficients oefficients Compensators o Comparison with other compensators o N =5, R =14 o ω c =0.6 π 15

Wideband Compensators Compensators Wideband o M =5 o N =5, R =32 o ω c =0.5 π o MaxFlat compensator uses 15 adders o Our compensator uses 14 adders 16

Wideband Compensators Compensators Wideband o M =7 o Our compensator uses 19 adders o Compensation error less than 0.02dB 17

Compensators W With ith L Low owest est C Complexity omplexity Compensators o P =1 o N =5, R =32, ω c =0.5 π o Two compensators o M =7 with P =1 o M =3 with the same number of adders o M =7 has a better rolloff characteristic 18

Conclusion n Conclusio o The design of minimax CIC compensators over the SPT coefficient space has been presented o The optimum compensators are obtained using the global optimization based on the interval analysis o It enables the design of the compensators with high number of coefficients and the wideband compensators o Compared with the known SPT compensators, the proposed compensators generally result in a better compensation and a lower complexity 19