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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
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
C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Apr. 23, 2015 2 / 29
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?
C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Apr. 23, 2015 3 / 29
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.
C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Apr. 23, 2015 5 / 29
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 Apr. 23, 2015 6 / 29
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 Apr. 23, 2015 7 / 29
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 Apr. 23, 2015 8 / 29
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.
C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Apr. 23, 2015 9 / 29
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
C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Apr. 23, 2015 10 / 29
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.
C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Apr. 23, 2015 11 / 29
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
Coprime DFT Filter Bank Design: The Practical Case Design Method II (Proposed)
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.
C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Apr. 23, 2015 13 / 29
Coprime DFT Filter Bank Design: The Practical Case Design Method II (Proposed)
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).
C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Apr. 23, 2015 14 / 29
Coprime DFT Filter Bank Design: The Practical Case Design Method II (Proposed)
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
) .
C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Apr. 23, 2015 15 / 29
Coprime DFT Filter Bank Design: The Practical Case Design Method II (Proposed)
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
Coprime DFT Filter Bank Design: The Practical Case Design Method II (Proposed)
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
C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Apr. 23, 2015 17 / 29
Coprime DFT Filter Bank Design: The Practical Case Design Method II (Proposed)
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.
C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Apr. 23, 2015 18 / 29
Coprime DFT Filter Bank Design: The Practical Case Design Method II (Proposed)
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
C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Apr. 23, 2015 19 / 29
Coprime DFT Filter Bank Design: The Practical Case Design Method II (Proposed)
Stopband Response
0.375 0.3781 0.3812 0.3844 0.3875 0.3906 0.3937 0.3969 0.4
−100
−50
0
Normalized frequency f
dB
Plotof
F00
(
ej2πf)
Design 1Design 2Design 3
C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Apr. 23, 2015 20 / 29
Coprime DFT Filter Bank Design: The Practical Case Design Method II (Proposed)
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
Normalized frequency f
A(
ej2πf)
Design 1Design 2Design 3
C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Apr. 23, 2015 21 / 29
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?
C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Apr. 23, 2015 22 / 29
Theoretical Bump Analysis
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.
C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Apr. 23, 2015 23 / 29
Theoretical Bump Analysis
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
Theoretical Bump Analysis
The Bump Locations (Illustration)..
f
.∣ ∣ F 0,0
( ej2πf)∣ ∣
.
12MN
.
1MN
.
1
.
u2MN
.
v2MN
Hold true for any coprime DFT filter bank design!
C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Apr. 23, 2015 25 / 29
Theoretical Bump Analysis
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}.
C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Apr. 23, 2015 26 / 29
Theoretical Bump Analysis
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!
C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Apr. 23, 2015 27 / 29
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.
C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Apr. 23, 2015 28 / 29
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.
C.-L. Liu & P. P. Vaidyanathan (Caltech) Coprime DFTFB Apr. 23, 2015 29 / 29