Design of Coprime DFT Arrays and Filter Banks

Transcription

1 Design o Coprime DFT Arrays and Filter Banks Chun-Lin Liu clliu@caltechedu P P Vaidyanathan ppvnath@systemscaltechedu Digital Signal Processing Group Electrical Engineering Caliornia Institute o Technology Nov 3, 24 C-L Liu & P P Vaidyanathan Caltech) Coprime DFTFB Nov 3, 24 / 9

2 Introduction Motivation Coprime DFT ilter banks: [] Enhanced degrees o reedom: OMN) based on OM + N) samples Applications in Direction-o-arrival estimation [2], Beamorming [3], and Spectrum estimation [4] Problems: No design guidelines! Interpolated FIR ilter design [5] [7]: IFIR design two ilter designs F i z) = G M z L ) H z) G ) M z L H z) C-L Liu & P P Vaidyanathan Caltech) Coprime DFTFB Nov 3, 24 2 / 9

3 Coprime DFT Filter Bank Design: the Ideal Case Coprime DFT Filter Banks Gz), Hz): FIR Type-I ilters N g, N h : ilter orders N g G z) = gn)z n n= N h H z) = hn)z n n= Gz M ), Hz N ): sparse coeicient ilters G z M) H z N) F l,k z) coprime DFT ilter banks MN-ilters l =,,, N, k =,,, M G ) z M WN l H ) z N WM k DFT ilter banks DFT ilter banks N-ilters M-ilters l =,,, N k =,,, M C-L Liu & P P Vaidyanathan Caltech) Coprime DFTFB Nov 3, 24 3 / 9

4 Coprime DFT Filter Bank Design: the Ideal Case Coprime DFTFBs, the Ideal Case s s s 5 M 5 MN MN 5 MN MN 5 N G e j2π) H e j2π) G e j2πm) H e j2πn) F ), e j2π C-L Liu & P P Vaidyanathan Caltech) Coprime DFTFB Nov 3, 24 4 / 9

5 Coprime DFT Filter Bank Design: the Ideal Case Coprime DFTFBs, the Ideal Case F ) l,k e j2π = G e j2πm e j2πl/n) H e j2πn e j2πk/m) l =, k = l = k = l =, k = 2 l = k = k = 2 C-L Liu & P P Vaidyanathan Caltech) Coprime DFTFB Nov 3, 24 5 / 9

6 Coprime DFT Filter Bank Design: The Practical Case Coprime DFTFBs, the Practical Case s s s 2δ 2δ 2 5 N λ δ 2 λ 5 MN MN 2 M λm δ 2 bumps N λn 5 M G e j2π) H e j2π) G e j2πm) H e j2πn) F ), e j2π C-L Liu & P P Vaidyanathan Caltech) Coprime DFTFB Nov 3, 24 6 / 9

7 Coprime DFT Filter Bank Design: The Practical Case Coprime DFTFBs, Design Parameters Goal: Design gn) and hn) such that F l,k e j2π ) is an approximation o the ideal case Notion o approximation in F l,k e j2π ) : Passband ripples, 2 Stopband ripples 2, 3 Transition band width, 4 Passband edges and stopband edges Idea: Divide the design problem into two sub-problems SPEC o F ) l,k e j2π SPEC o G e j2π) SPEC o H e j2π) the Parks-McClellan ilter design algorithm the Parks-McClellan ilter design algorithm gn) hn) C-L Liu & P P Vaidyanathan Caltech) Coprime DFTFB Nov 3, 24 7 / 9

8 Coprime DFT Filter Bank Design: The Practical Case Design equations Passband ripples and stopband ripples or G e j2π) and H e j2π), δ =, δ 2 = Transition bandwidth, 2 log ) δ δ 2 3 min {MN g, NN h } 2 2 λ determines passband edges and stopband edges C-L Liu & P P Vaidyanathan Caltech) Coprime DFTFB Nov 3, 24 8 / 9

9 How to choose λ? The Role o λ λ =, larger passband edges, with bumps s 2 5 MN MN 2 bumps λ =, smaller passband edges, no bumps F, e j2π ) s 2 λ 5 MN MN 2 no bumps F, e j2π ) C-L Liu & P P Vaidyanathan Caltech) Coprime DFTFB Nov 3, 24 9 / 9

10 How to choose λ? Optimal λ Find the one with maximal passband edge subject to bump-ree constraints λ opt = min λ subject to ) F, e j2π 2, [ λ ] λ), λ), MN MN Relaxation ˆλ = min λ subject to [ λ 5 MN + λ), 5 MN ˆF e j2π ) 2, λ) ˆF e j2π ) might not be realizable in the FIR setting but can be written as some simple closed-orm unctions C-L Liu & P P Vaidyanathan Caltech) Coprime DFTFB Nov 3, 24 / 9 ],

11 How to choose λ? Approximate the transition bands Substitute transition bands o G e j2π) and H e j2π) with simple closed-orm unctions Linear unctions, Q unctions, λ ˆλ li δ 2 δ δ 2 λ ˆλ Q Q ln 4 2 ) Q Q 2, Q Q δ ), Q 2 Q δ 2 ), Q ): inverse Q unctions C-L Liu & P P Vaidyanathan Caltech) Coprime DFTFB Nov 3, 24 / 9

12 How to choose λ? Approximate the transition bands response FIR ilter by irpm Linear unctions Q unctions Normalized requency Figure : A comparison among dierent approximations o the transition band The FIR ilter is designed by the MATLAB unction irpm with speciication p = 8, s = 42, δ =, and δ 2 = The ilter order is 6 C-L Liu & P P Vaidyanathan Caltech) Coprime DFTFB Nov 3, 24 2 / 9

13 How to choose λ? Summary: Coprime DFTFB Design Inputs: M, N, N g, N h,, 2 ) Initialize: δ =, δ 2 = 2 /2 ) ), 2 log δ δ 2 /2 min {MN g, NN h }), λ = ˆλ li = δ 2 )/ δ δ 2 ) or ˆλ Q = Q ln 4 2 ))/Q Q 2 ), Increase λ until stopband ripples or F, e j2π ) are satisied Increase until ripples or G e j2π) and H e j2π) are met Design lowpass ilters gn) and hn) with speciications δ, δ 2, 5/N λm, 5/N + λ)m ) or gn), δ, δ 2, 5/M λn, 5/M + λ)n ) or hn) Output: gn), hn)) C-L Liu & P P Vaidyanathan Caltech) Coprime DFTFB Nov 3, 24 3 / 9

14 Simulation Results Numerical Example M = 8, N = 5, N g =, N h = 6, =, 2 = Deine overall amplitude response A e j2π), A e j2π) = N M l= to measure the spectral coverage k= F ) l,k e j2π, C-L Liu & P P Vaidyanathan Caltech) Coprime DFTFB Nov 3, 24 4 / 9

15 Simulation Results Passband Characteristics e j2π ) λ = λ =λ opt =8477 λ = ˆλ Q =86926 λ = ˆλ li =9699 λ = db Plot o F λ Normalized requency C-L Liu & P P Vaidyanathan Caltech) Coprime DFTFB Nov 3, 24 5 / 9

16 Simulation Results Stopband Characteristics e j2π ) λ λ = λ =λ opt =8477 λ = ˆλ Q =86926 λ = ˆλ li =9699 λ = db Plot o F Normalized requency C-L Liu & P P Vaidyanathan Caltech) Coprime DFTFB Nov 3, 24 6 / 9

17 Simulation Results Spectrum Coverage 2 db Plot o Coprime DFTFB C-L Liu & P P Vaidyanathan Caltech) 4 6 Normalized requency Coprime DFTFB 8 Nov 3, 24 7 / 9

18 Simulation Results Overall Amplitude Responses 2 λ db plot o A e j2π) = λ =λ opt = λ = ˆλ Q =86926 λ = ˆλ li =9699 λ = Normalized requency C-L Liu & P P Vaidyanathan Caltech) Coprime DFTFB Nov 3, 24 8 / 9

19 Reerences Reerences [] P P Vaidyanathan and P Pal, Sparse sensing with co-prime samplers and arrays, IEEE Trans Signal Process, vol 59, no 2, pp , 2 [2] P Pal and P P Vaidyanathan, Coprime sampling and the MUSIC algorithm, in Digital Signal Processing Workshop and IEEE Signal Processing Education Workshop DSP/SPE), 2 IEEE, 2, pp [3] P P Vaidyanathan and C-C Weng, Active beamorming with interpolated FIR iltering, in Proc IEEE Int Symp Circuits and Syst ISCAS), 2, pp [4] B Farhang-Boroujeny, Filter bank spectrum sensing or cognitive radios, IEEE Trans Signal Process, vol 56, no 5, pp 8 8, 28 [5] Y Neuvo, C-Y Dong, and S K Mitra, Interpolated inite impulse response ilters, IEEE Trans Acoust, Speech, Signal Process, vol 32, no 3, pp , 984 [6] T Saramäki, Y Neuvo, and S K Mitra, Design o computationally eicient interpolated FIR ilters, IEEE Trans Circuits Syst, vol 35, no, pp 7 88, 988 [7] P P Vaidyanathan, Multirate Systems And Filter Banks Pearson Prentice Hall, 993 C-L Liu & P P Vaidyanathan Caltech) Coprime DFTFB Nov 3, 24 9 / 9