Coprime DFT Filter Bank Design: Theoretical Bounds and ...systems.caltech.edu/dsp/students/clliu/Files/Co... · Digital Signal Processing Group Electrical Engineering California Institute

Coprime DFT Filter Bank Design:Theoretical Bounds and Guarantees

Chun-Lin [email protected]

P. P. [email protected]

Digital Signal Processing GroupElectrical Engineering

California Institute of Technology

Apr. 23, 2015

C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Apr. 23, 2015 1 / 29

mailto:[email protected]

mailto:[email protected]

Outline

1 Introduction

2 Coprime DFT Filter Bank Design: the Ideal Case

3 Coprime DFT Filter Bank Design: The Practical CaseDesign Method IDesign Method II (Proposed)

4 Theoretical Bump Analysis

5 Conclusion


Introduction

Motivation

Coprime DFT filter banks: [1]Enhanced degrees of freedom: O(MN) based on O(M +N)samples.Applications in Direction-of-arrival estimation [2], Beamforming[3], and Spectrum estimation [4].How to design filter taps?


Introduction

Interpolated FIR filter design [5] [6] [7]:

Design IFIR Fi(z) → design L and two lowpass filters GM(z)and H(z).

.. GM

(zL

). H (z).

Fi (z) = GM

(zL

)H (z)

.

f

.

0

.

Mag

nitu

de

.

f

.

0

.

Mag

nitu

de

.

f

.

0

.

Mag

nitu

deC.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Apr. 23, 2015 4 / 29

Coprime DFT Filter Bank Design: the Ideal Case

Coprime DFT Filter Banks

..

G(z), H(z):FIR Type-I filters

Ng, Nh: filter orders .G (z) =

Ng∑n=0

g(n)z−n

.H (z) =

Nh∑n=0

h(n)z−n

.

G(zM), H(zN):sparse coefficient filters

.

��

��G

(zM

).

��

��H

(zN

).

Fℓ,k (z)

.

��

��G

(zMW ℓ

N

)

.

×

.

��

��H

(zNW k

M

)

.

coprime DFT filter banksMN -filters

ℓ = 0, 1, . . . , N − 1,k = 0, 1, . . . ,M − 1.

.

DFT filter banksN -filters

ℓ = 0, 1, . . . , N − 1.

.

DFT filter banksM -filters

k = 0, 1, . . . ,M − 1.



Coprime DFTFBs, the Ideal Case

.. f.

Mag

nitu

des

.0.5M

.0.5N

.1

.

G(ej2πf

).

H(ej2πf

).

f

.

Mag

nitu

des

.

0.5MN

.

1MN

.

1

.

G(ej2πfM

).

H(ej2πfN

).

f

.

Mag

nitu

des

.

0.5MN

.

1MN

.

1

.

F0,0

(ej2πf

)



Coprime DFTFBs, the Ideal Case..Fℓ,k

(ej2πf

). =. G

(ej2πfMe−j2πℓ/N

). ×. H

(ej2πfNe−j2πk/M

).

f

.

0

.

1

.ℓ = 0

.

Mag

nitu

de

.

f

.

0

.

1

.

ℓ = 1

.

Mag

nitu

de

.

f

.

0

.

1

.k = 0

.

Mag

nitu

de

.

f

.

0

.

1

.

k = 1

.

Mag

nitu

de

.

f

.

0

.

1

.

k = 2

.

Mag

nitu

de

.

f

.

0

.

1

.ℓ = 0, k = 1

.

Mag

nitu

de

.

f

.

0

.

1

.

ℓ = 1, k = 2

.

Mag

nitu

de

.

×

.

×


Coprime DFT Filter Bank Design: The Practical Case

Coprime DFTFBs, the Practical Case

.. f.

Mag

nitu

des

.0.5N

.1− 0.5

M

.1

.

M∆f

.

λM∆f

.

N∆f

.

λN∆f

.

2δ1

.δ2

.

G(ej2πf

).

H(ej2πf

).

f

.

Mag

nitu

des

.

1

.

∆f

.

λ∆f

.2δ1

.

δ2

.

G(ej2πfM

).

H(ej2πfN

).

f

.

Mag

nitu

des

.

0.5MN

.

1MN

.

1

.

∆f

.

λ∆f

.

2∆1

.

∆2

.

bumps

.

F0,0

(ej2πf

)


Coprime DFT Filter Bank Design: The Practical Case

Coprime DFTFB Design Methods

Goal: Design g(n) and h(n) such thatFℓ,k

(ej2πf

)is an approximation of the ideal case.

Notion of approximation in Fℓ,k

(ej2πf

):

1 Passband ripples ∆1,2 Stopband ripples ∆2,3 Transition band width ∆f ,4 Passband edges and stopband edges.

Define an appropriate error measure.


Coprime DFT Filter Bank Design: The Practical Case Design Method I

Design Method I (Main Concept) [8]

Divide the design problem into two sub-problems.

..SPEC ofFℓ,k

(ej2πf

) .

SPEC ofG(ej2πf

).

SPEC ofH

(ej2πf

).

g(n)

.

the Parks-McClellan

.filter design algorithm

.

h(n)

.the Parks-McClellan

.

filter design algorithm


Coprime DFT Filter Bank Design: The Practical Case Design Method I

Design Method I (Design Equations)Passband ripples and stopband ripples for G

(ej2πf

)and

H(ej2πf

),

δ1 = 1−√1−∆1, δ2 =

∆2

2−√1−∆1

.

Transition bandwidth ∆f ,

∆f ≥2 log10

(1

10δ1δ2

)3min {MNg, NNh}

.

Select λ satisfying

λ ≥ λ̂Q ≜ Q1 −√

− ln (4∆2)

Q1 −Q2

,

where Q1 ≜ Q−1 (1− δ1), Q2 ≜ Q−1 (δ2), and Q−1 (· ):inverse Q functions.


Coprime DFT Filter Bank Design: The Practical Case Design Method II (Proposed)

Design Method II (Motivation)Design method I:

Heuristic choice of λ.No control over overall amplitude responses A(ej2πf ).

A(ej2πf

): filter bank coverage to the whole spectrum

A(ej2πf

)=

N−1∑ℓ=0

M−1∑k=0

∣∣Fℓk

(ej2πf

)∣∣,The filter bank satisfying the following criteria is preferred:

1∣∣F00

(ej2πf

)∣∣ is close to unity in the passband.2∣∣F00

(ej2πf

)∣∣ is close to zero in the stopband.3 Overall amplitude responses A

(ej2πf

)is close to unity at all

frequencies.C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Apr. 23, 2015 12 / 29


Design Method II (Problem Formulation)

Optimization Problem

ming(n),h(n)

w1

∥∥∥∣∣F00

(ej2πf

)∣∣f∈[0, 1

2MN )∪(1−1

2MN,1) − 1

∥∥∥p

+ w2

∥∥∥∣∣F00

(ej2πf

)∣∣f∈[ 1

2MN,1− 1

2MN ]

∥∥∥p

+ w3

∥∥A (ej2πf

)− 1

∥∥p,

w1 + w2 + w3 = 1, w1, w2, w3 ≥ 0. Weights among these threefactors.∥· ∥p denotes the p-norm.



Design Method II (Problem Formulation)

Discretized Optimization Problem (P1)By taking Npt uniform samples over f (writing as f), we obtain

mina,b

w1 ∥Jp × [(CMa)⊙ (CNb)− 1]∥p+ w2 ∥Js × [(CMa)⊙ (CNb)]∥p+ w3 ∥P × [(CMa)⊙ (CNb)]− 1∥p ,

where “⊙” indicates the Hadamard product.

Assumptions:g(n) and h(n) are type-I linear phase FIR filters.Stopband ripples (δ2) are much smaller compared to passbandresponses (1± δ1).



Design Method II (Details)a and b:

a =[g (Ng/2) 2g (Ng/2− 1) . . . 2g (0)

]T,

b =[h (Nh/2) 2h (Nh/2− 1) . . . 2h (0)

]T.

CN and CM : Discrete cosine transform matrices.

CM =

cos (2πM [f]1 × 0) . . . cos

(2πM [f]1 ×

Ng

2

)... . . . ...

cos(2πM [f]Npt

× 0)

. . . cos(2πM [f]Npt

× Ng

2

) ,

CN =

cos (2πN [f]1 × 0) . . . cos

(2πN [f]1 ×

Nh

2

)... . . . ...

cos(2πN [f]Npt

× 0)

. . . cos(2πN [f]Npt

× Nh

2

) .



Design Method II (Details)

Jp and Js: selection matrices that choose thepassband/stopband.

Jp =

I Npt2MN

O Npt2MN

×(Npt−

NptMN

) O Npt2MN

O Npt2MN

O Npt2MN

×(Npt−

NptMN

) I Npt2MN

,

Js =[O(

Npt−NptMN

)× Npt

2MN

INpt−

NptMN

O(Npt−

NptMN

)× Npt

2MN

],

P: Generate overall amplitude responses.

P =[I Npt

2MN

I Npt2MN

. . . I Npt2MN

]∈ {0, 1}

Npt2MN

×Npt .

1 all-one column vector.C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Apr. 23, 2015 16 / 29


Design Method II (Solution)(CMa)⊙ (CNb) is a bilinear form of a and b.Alternating minimization to C(a,b) (the cost function in (P1)).Design method I as the initial condition.

..Initialize a and b.(Method I)

. mina C(a,b)for a given b

.

minb C(a,b)for a given a

.

Update b

.Update a



Comparison

Design 1: The example of [8], where M = 8, N = 5, Ng = 100,Nh = 160, ∆1 = 0.01, ∆2 = 0.001, andλ = λ̂Q = 0.86926.

Design 2: M = 8, N = 5, Ng = 100, Nh = 160. Solve (P1) byalternating minimization, where Design 1 above is set asthe initial point. We choose Npt = 2560,w1 = w2 = w3 = 1/3 and p = 1.

Design 3: The same as Design 2 except p = 2.



Passband Response

0 0.0031 0.0063 0.0094 0.0125 0.0156 0.0187 0.0219 0.025

−100

−50

0

Normalized frequency f

dB

Plotof

F00

(

ej2πf)

Design 1Design 2Design 3



Stopband Response

0.375 0.3781 0.3812 0.3844 0.3875 0.3906 0.3937 0.3969 0.4

−100

−50

0


dB

Plotof

F00

(

ej2πf)




Overall Amplitude Response

0 0.0031 0.0063 0.0094 0.0125 0.0156 0.0187 0.0219 0.0250

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


A(

ej2πf)



Theoretical Bump Analysis

Bump Analysis

Definition: (Bumps)A bump in coprime DFTFB results from overlapping between thefinite transition bands of the sparse coefficient filters G

(ej2πfM

)and

H(ej2πfN

).

Bumps are undesired responses.How many bumps are there?Where do these bumps located?What is the level of bumps?



The Number of Bumps

LemmaFor any 0 ≤ ℓ ≤ N − 1, 0 ≤ k ≤ M − 1, there exists a uniquef0 ∈ [0, 1) such that

∣∣Fℓk

(ej2πf

)∣∣ = ∣∣F00

(ej2π(f−f0)

)∣∣.Theorem: (The number of bumps)Fℓk

(ej2πf

)contains exactly two bumps for any 0 ≤ ℓ ≤ N − 1,

0 ≤ k ≤ M − 1.



The Bump Locations

Theorem: (The bump locations)The two bumps of F00

(ej2πf

)are located around f = u/(2MN) and

f = v/(2MN) with

u = 2Mn+ − 1 = 2Nm+ + 1 ̸∈ {−1, 0, 1} ,v = 2Mn− + 1 = 2Nm− − 1 ̸∈ {−1, 0, 1} ,

where m± ∈ {0, 1, . . . ,M − 1}, n± ∈ {0, 1, . . . , N − 1}, andMn± −Nm± = ±1. Also, the amplitude response of F00

(ej2πf

)satisfies∣∣∣F00

(e

jπMN

)∣∣∣ = ∣∣∣F00

(e

−jπMN

)∣∣∣ = ∣∣∣F00

(e

jπuMN

)∣∣∣ = ∣∣∣F00

(e

jπvMN

)∣∣∣ .C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Apr. 23, 2015 24 / 29


The Bump Locations (Illustration)..

f

.∣ ∣ F 0,0

( ej2πf)∣ ∣

.

12MN

.

1MN

.

1

.

u2MN

.

v2MN

Hold true for any coprime DFT filter bank design!



The Bump Level

TheoremAssume the stopband ripples for G

(ej2πf

)and H

(ej2πf

)are ϵ1 and

ϵ2, respectively. The bump level in coprime DFTFB is bounded by

L ≤∣∣∣F00

(e

jπpMN

)∣∣∣ ≤ U,

where

L =1

4

(A(e

jπpMN

)− ϵ

), U =

1

4A(e

jπpMN

),

ϵ = 2 (N − 2) ϵ1 + 2 (M − 2) ϵ2 + (M − 2) (N − 2) ϵ1ϵ2,

p ∈ {±1, u, v}.



The Bump Level (Illustration)..

f

.M

agni

tude

s.

12MN

.

1MN

.

1

.

u2MN

.

v2MN

.

A(ej2πf )

.

Upper bound

.

Lower bound

.

F00(ej2πf )

Hold true for any coprime DFT filter bank design!


Conclusion

Conclusion

Practical coprime DFT filter bank designThe design method in [8] eliminates bumps but neglects overallamplitude responses.Our proposed method provides trade-offs among passbandresponses, stopband responses, and overall amplitude responses.

Theoretical bump analysis (true for any comprime DFT filterbank design)

Exactly two bumps in one filter.Bump locations can be determined from M and N uniquely.Bump levels are approximately 1/4 of overall amplituderesponses.


Conclusion

References[1] P. P. Vaidyanathan and P. Pal, “Sparse sensing with co-prime samplers and arrays,”

IEEE Trans. Signal Process., vol. 59, no. 2, pp. 573–586, 2011.[2] P. Pal and P. P. Vaidyanathan, “Coprime sampling and the MUSIC algorithm,” in Proc.

IEEE Digital Signal Processing Workshop and IEEE Signal Processing EducationWorkshop, 2011, pp. 289–294.

[3] P. P. Vaidyanathan and C.-C. Weng, “Active beamforming with interpolated FIRfiltering,” in Proc. IEEE Int. Symp. Circuits and Syst., 2010, pp. 173–176.

[4] B. Farhang-Boroujeny, “Filter bank spectrum sensing for cognitive radios,” IEEE Trans.Signal Process., vol. 56, no. 5, pp. 1801–1811, 2008.

[5] Y. Neuvo, C.-Y. Dong, and S. K. Mitra, “Interpolated finite impulse response filters,”IEEE Trans. Acoust., Speech, Signal Process., vol. 32, no. 3, pp. 563–570, 1984.

[6] T. Saramäki, Y. Neuvo, and S. K. Mitra, “Design of computationally efficientinterpolated FIR filters,” IEEE Trans. Circuits Syst., vol. 35, no. 1, pp. 70–88, 1988.

[7] P. P. Vaidyanathan, Multirate Systems And Filter Banks. Pearson Prentice Hall, 1993.[8] C.-L. Liu and P. P. Vaidyanathan, “Design of coprime DFT arrays and filter banks,” in

Proc. IEEE Asil. Conf. on Sig., Sys., and Comp., 2014. [Online]. Available:http://systems.caltech.edu/dsp/students/clliu/Files/CoprimeDFTFB.pdf.


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Coprime DFT Filter Bank Design: Theoretical Bounds and ...systems.caltech.edu/dsp/students/clliu/Files/Co... · Digital Signal Processing Group Electrical Engineering California Institute

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