Design of Coprime DFT Arrays and Filter...
Transcript of Design of Coprime DFT Arrays and Filter...
Design of Coprime DFTArrays and Filter Banks
Chun-Lin [email protected]
P. P. [email protected]
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