Design of Coprime DFTArrays and Filter Banks

Chun-Lin Liucl.liu@caltech.edu

P. P. Vaidyanathanppvnath@systems.caltech.edu

Digital Signal Processing GroupElectrical Engineering

California Institute of Technology

Nov. 3, 2014

C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Nov. 3, 2014 1 / 19

Introduction

MotivationCoprime 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].Problems: No design guidelines!

Interpolated FIR filter design [5]–[7]: IFIR design → two filterdesigns.

.. 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 Nov. 3, 2014 2 / 19

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.

C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Nov. 3, 2014 3 / 19

Coprime DFT Filter Bank Design: the Ideal Case

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

)

C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Nov. 3, 2014 4 / 19

Coprime DFT Filter Bank Design: the Ideal Case

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

.

×

.

×

C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Nov. 3, 2014 5 / 19

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

)

C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Nov. 3, 2014 6 / 19

Coprime DFT Filter Bank Design: The Practical Case

Coprime DFTFBs, Design ParametersGoal: Design g(n) and h(n) such that Fℓ,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.

Idea: 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

C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Nov. 3, 2014 7 / 19

Coprime DFT Filter Bank Design: The Practical Case

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}

.

λ determines passband edges and stopband edges.

C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Nov. 3, 2014 8 / 19

How to choose λ?

The Role of λ

λ = 0, larger passband edges ☺, with bumps ☹.

.. f.

Mag

nitu

des

.0.5MN

.1

MN

.1

.

∆f

.

2∆1

.∆2

.

bumps

.

F0,0

(ej2πf

)

λ = 1, smaller passband edges ☹, no bumps ☺.

.. f.

Mag

nitu

des

.0.5MN

.1

MN

.1

.

∆f

.

λ∆f

.

2∆1

.∆2

.

no bumps

.

F0,0

(ej2πf

)

C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Nov. 3, 2014 9 / 19

How to choose λ?

Optimal λFind the one with maximal passband edge subject to“bump-free” constraints.

λopt = minλ

λ subject to∣∣F0,0

(ej2πf

)∣∣ ≤ ∆2,

f ∈[0.5

MN+ (1− λ)∆f, 1− 0.5

MN− (1− λ)∆f

],

Relaxation

λ̂ = minλ

λ subject to∣∣∣F̂00

(ej2πf

)∣∣∣ ≤ ∆2,

f ∈[0.5

MN+ (1− λ)∆f, 1− 0.5

MN− (1− λ)∆f

],

F̂00

(ej2πf

)might not be realizable in the FIR setting but can be

written as some simple closed-form functions.C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Nov. 3, 2014 10 / 19

How to choose λ?

Approximate the transition bandsSubstitute transition bands of G

(ej2πf

)and H

(ej2πf

)with

simple closed-form functions.Linear functions,

λ ≥ λ̂li ≜1− δ1 −

√∆2

1− δ1 − δ2.

Q functions,

λ ≥ λ̂Q ≜ Q1 −√

− ln (4∆2)

Q1 −Q2

,

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

C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Nov. 3, 2014 11 / 19

How to choose λ?

Approximate the transition bands

0.011 0.0115 0.012 0.0125 0.013 0.0135 0.014−60

−50

−40

−30

−20

−10

0

Normalized frequency f

Magnituderesponse

FIR filter by firpmLinear functionsQ functions

Figure : A comparison among different approximations of the transitionband. The FIR filter is designed by the MATLAB function firpm withspecification fp = 0.0108, fs = 0.0142, δ1 = 0.01, and δ2 = 0.001. Thefilter order is 160.

C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Nov. 3, 2014 12 / 19

How to choose λ?

Summary: Coprime DFTFB DesignInputs: (M,N,Ng, Nh,∆1,∆2)

Initialize:δ1 = 1−

√1−∆1, δ2 = ∆2/(2−

√1−∆1),

∆f ≥ 2 log1 0(

110δ1δ2

)/(2min {MNg, NNh}),

λ = λ̂li = (1− δ1 −√∆2)/(1− δ1 − δ2)

or λ̂Q = (Q1 −√

− ln (4∆2))/(Q1 −Q2),Increase λ until stopband ripples for F0,0

(ej2πf

)are satisfied.

Increase ∆f until ripples for G(ej2πf

)and H

(ej2πf

)are met.

Design lowpass filters g(n) and h(n) with specifications(δ1, δ2, 0.5/N − λM∆f, 0.5/N + (1− λ)M∆f) for g(n),(δ1, δ2, 0.5/M − λN∆f, 0.5/M + (1− λ)N∆f) for h(n).

Output: (g(n), h(n))

C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Nov. 3, 2014 13 / 19

Simulation Results

Numerical Example

M = 8, N = 5,Ng = 100, Nh = 160,∆1 = 0.01,∆2 = 0.001.Define overall amplitude response A

(ej2πf

),

A(ej2πf

)=

N−1∑ℓ=0

M−1∑k=0

∣∣Fℓ,k

(ej2πf

)∣∣,to measure the spectral coverage.

C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Nov. 3, 2014 14 / 19

Simulation Results

Passband Characteristics

..

0 0.0031 0.0063 0.0094 0.0125 0.0156 0.0187 0.0219 0.025−140

−120

−100

−80

−60

−40

−20

0

20

Normalized frequency f

dB

Plotof

F00

(

ej2πf)

λ = 0λ = λopt = 0.8477

λ = λ̂Q = 0.86926

λ = λ̂li = 0.96919λ = 1

. λ

C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Nov. 3, 2014 15 / 19

Simulation Results

Stopband Characteristics

..

0.375 0.3781 0.3812 0.3844 0.3875 0.3906 0.3937 0.3969 0.4−140

−120

−100

−80

−60

−40

−20

0

20

Normalized frequency f

dB

Plotof

F00

(

ej2πf)

λ = 0λ = λopt = 0.8477

λ = λ̂Q = 0.86926

λ = λ̂li = 0.96919λ = 1

.

λ

C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Nov. 3, 2014 16 / 19

Simulation Results

Spectrum Coverage

..

0 0.2 0.4 0.6 0.8 1−140

−120

−100

−80

−60

−40

−20

0

20

Normalized frequency f

dB

Plotof

CoprimeDFTFB

C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Nov. 3, 2014 17 / 19

Simulation Results

Overall Amplitude Responses

..

0 0.0031 0.0063 0.0094 0.0125 0.0156 0.0187 0.0219 0.025−100

−80

−60

−40

−20

0

20

Normalized frequency f

dB

plotof

A(

ej2πf)

λ = 0λ = λopt = 0.8477

λ = λ̂Q = 0.86926

λ = λ̂li = 0.96919λ = 1

.

λ

C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Nov. 3, 2014 18 / 19

References

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,” inDigital Signal Processing Workshop and IEEE Signal Processing Education Workshop(DSP/SPE), 2011 IEEE, 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. (ISCAS), 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.

C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Nov. 3, 2014 19 / 19