Single Sensor Estimation of Radio Activity via Blind Block- - - PowerPoint PPT Presentation

Single Sensor Estimation of Radio Activity via Blind Block- Partitioned Tensor Decomposition

Christopher Mueller-Smith

chris.muellersmith@rutgers.edu

Problem Formulation

Sensor

Problem Formulation

Transmissions are non-orthogonal in time and frequency

Problem Formulation

Transmissions are non-orthogonal in time and frequency

Problem Formulation

Signal 1 Signal 2 Signal 3

Problem Formulation

Sensor OFDM PSK FHSS ? ? ?

Problem Formulation

Sensor

Separate Receivers: OFDM PSK FHSS

• ?-

OFDM PSK FHSS ? ? ?

Problem Formulation

Sensor

Separate Receivers: OFDM PSK FHSS

• ?-

OFDM PSK FHSS ? ? ?

System Model

• M transmitters
• Linear/Linear Approx. Modulation:
• Packetized (non-continuous in-time transmissions)
• Overlapping in time and frequency
• Received at sensor:

xm(t) =

Rm

X

r=1 ∞

X

k=−∞

am,r,kpm,r(t − kTm,r) y(t) =

M

X

m=1

hm(t) ∗ xm(t) + n(t)

Preprocessing

Frequency Time 1 1 1 1 2 2 2 3 3 3 4 4 4

[1] G. Ivkovic, P. Spasojevic, and I.Seskar,“Mean shift based segmentation for time frequency analysis of packet based radio signals,” in 2010 Conference Record of the Forty Fourth Asilomar Conference on Signals, Systems and Computers, Nov 2010, pp. 1526–1530.

Group statistically similar receptions together into Icl groups.

Trispectrum Slice

Sx

04(f, v) =Sx 4 (f, v, −v)

=E{|X(f)X(v)|2} − |E{X(f)X∗(v)}|2 − |E{X(f)X(v)}|2 − E{|X(f)|2}E{|X(v)|2} Estimate anti-diagonal slice of the trispectrum for each group:

Trispectrum Slice Components

S04(f, v, i) =

M

X

m=1

|Hm(f)|2|Hm(v)|2Sxm

04 (f, v)cim + Sn 04(f, v)

Sxm

04 (f, v) = Rm

X

r=1

Samr

04 (f, v)

Tmr |Pmr(f)|2|Pmr(v)|2 Samr

04 (f, v)

Tmr = cum4(amrk, amrk, a∗

mrk, a∗ mrk)

Tmr = kmr Icl trispectrum slices:

Pulse shape

Symbol sequence trispectrum

Trispectrum Slice Components

S04(f, v, i) =

M

X

m=1

|Hm(f)|2|Hm(v)|2Sxm

04 (f, v)cim + Sn 04(f, v) Channel Response Signal trispectrum Noise trispectrum

Sxm

04 (f, v) = Rm

X

r=1

Samr

04 (f, v)

Tmr |Pmr(f)|2|Pmr(v)|2 Samr

04 (f, v)

Tmr = cum4(amrk, amrk, a∗

mrk, a∗ mrk)

Tmr = kmr Icl trispectrum slices:

Signal activity Pulse shape

Symbol sequence trispectrum

Trispectrum Slice Components

S04(f, v, i) =

M

X

m=1

|Hm(f)|2|Hm(v)|2Sxm

04 (f, v)cim + Sn 04(f, v) Channel Response Signal trispectrum Noise trispectrum

Sxm

04 (f, v) = Rm

X

r=1

Samr

04 (f, v)

Tmr |Pmr(f)|2|Pmr(v)|2 Samr

04 (f, v)

Tmr = cum4(amrk, amrk, a∗

mrk, a∗ mrk)

Tmr = kmr Icl trispectrum slices:

Signal activity Pulse shape

Symbol sequence trispectrum

Trispectrum Slice Components

S04(f, v, i) =

M

X

m=1

|Hm(f)|2|Hm(v)|2Sxm

04 (f, v)cim + Sn 04(f, v) Channel Response Signal trispectrum Noise trispectrum

Sxm

04 (f, v) = Rm

X

r=1

Samr

04 (f, v)

Tmr |Pmr(f)|2|Pmr(v)|2 Samr

04 (f, v)

Tmr = cum4(amrk, amrk, a∗

mrk, a∗ mrk)

Tmr = kmr Icl trispectrum slices:

Signal activity Pulse shape

Symbol sequence trispectrum

Discretize Trispectrum

Icl Nfb Nfb

Y

yjni = Z j∆f

(j−1)∆f

Z n∆f

(n−1)∆f

Sy

04(f, v, i) d

f dv =

M

X

m=1

cim

Rm

X

r=1

krmfjrmfnrm

fnrm = Z n∆f

(n−1)∆f

|Hrm(f)|2|Prm(f)|2 d f

CP Tensor Model

[3]

• C. Mueller-Smith and P. Spasojevic, “Single Sensor Blind Time-Frequency Activity Estimation of a Mixture of Radio Signals via CP Tensor Decomposition,” presented at the Military

Communications Conference (MILCOM), 2014 IEEE, 2014, pp. 617–622.

• Explored fitting tensor to Canonical Decomposition/

Parallel Factors (CP) Tensor Model in [3]

• Good performance for linear modulations
• Poor performance for non-linear modulations

(decomposition not unique) X = JU, V, WK =

R

X

r=1

ur vr wr

Rank-(Rm,Rm,1) Block Partitioned Tensor A = ⇥ A1 | A2 | . . . | AM ⇤ B = ⇥ B1 | B2 | . . . | BM ⇤ C = ⇥ c1 c2 . . . cM ⇤

T =

M

X

m=1

(Am · BT

m) cm

The 3-way trispectrum array can be modeled as a Rank-(Rm,Rm,1) Block Partitioned Tensor (BPT): tjni =

M

X

m=1

cim

Rm

X

r=1

ajrmbnrm

[2]

• L. De Lathauwer, “Decompositions of a Higher-Order Tensor in Block Terms—Part II: Definitions and Uniqueness,” SIAM. J. Matrix Anal. & Appl., vol. 30, no. 3, pp. 1033–1066, Jan. 2008.

Rank-(Rm,Rm,1) Block Partitioned Tensor

tjni =

M

X

m=1

cim

Rm

X

r=1

ajrmbnrm yjni =

M

X

m=1

cim

Rm

X

r=1

krmfjrmfnrm

General BPT: Trispectrum BPT:

F = ⇥F1 | F2 | . . . | FM ⇤ C = ⇥c1 c2 . . . cM ⇤ k = ⇥ kT

1 | kT 2 | . . . | kT M

⇤T

BPT Formulation for Trispectrum Slice

Y =

M

X

m=1

Fm · diag(km) · FT

m cm

Note in our case we have repeated factor matrix e.g. A=B.

BPT Formulation for Trispectrum Slice

Y =

M

X

m=1

Fm · diag(km) · FT

m cm

Current BPT Decomposition Algorithms

• Alternating Least Squares (ALS)
• Non-blind (partitioning known)
• A≠B - ALS not intended for

decompositions with repeated factor matrices

Reformulate BPT

Reformulate BPT to look like a CANDECOMP/PARAFAC (CP) tensor model:

Y = JF, F, CP · diag(k)K = JF, F, CWK

X = JU, V, WK =

R

X

r=1

ur vr wr

Reformulate BPT

Reformulate BPT to look like a CANDECOMP/PARAFAC (CP) tensor model:

Y = JF, F, CP · diag(k)K = JF, F, CWK

P =           1R1 0R−R1 0R1 1R2 0R−R1−R2 . . . ... . . . 0Pm

i=1 Ri−R1

1Rm 0R−Pm

i=1 Ri

. . . ... . . . 0PM

i=1 Ri−R1

1RM          

Reformulate BPT

Reformulate BPT to look like a CANDECOMP/PARAFAC (CP) tensor model:

Y = JF, F, CP · diag(k)K = JF, F, CWK

R-sparse R-sparse P =           1R1 0R−R1 0R1 1R2 0R−R1−R2 . . . ... . . . 0Pm

i=1 Ri−R1

1Rm 0R−Pm

i=1 Ri

. . . ... . . . 0PM

i=1 Ri−R1

1RM          

• Under certain conditions the BPT decomposition of Y is

unique (up to permutation and scaling of columns)

• Output of algorithms is
• F: estimates of signals’ power spectra
• C: estimates of signals’ on/off activity (in time)
• W: estimate of signal grouping

Estimation Strategy

[F, C, W] = arg min

F,C,W

1 2kY JF, F, CWKk2 + λkWk1

Simulation

−1 −0.5 0.5 1 −140 −120 −100 −80 −60 −40 −20 Power Spectrum Frequency (radians) Power (dB)

• 3 overlapping signal

groups

• Each signal group

experiences different Rayleigh fading channels

• Varying SNR

Preliminary Results

5 10 15 20 25 30 35 40 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Probability of Correct Activity Sequence Estimation SNR (dB) Probability of correct C Nfb=32, M=3, Rm={3,2,1} Nfb=64, M=3, Rm={3,2,2} 5 10 15 20 25 30 35 40 −13 −12 −11 −10 −9 −8 −7 −6 −5 Normalized Mean Square Error of F SNR (dB) NMSE (dB) Nfb=32, M=3, Rm={3,2,1} Nfb=64, M=3, Rm={3,2,2}

[3]

• C. Mueller-Smith and P. Spasojevic, “Single Sensor Blind Time-Frequency Activity Estimation of a Mixture of Radio Signals via CP Tensor Decomposition,” presented at the Military

Communications Conference (MILCOM), 2014 IEEE, 2014, pp. 617–622.

Estimate of PSD

−10 −8 −6 −4 −2 2 4 6 8 10 −20 −15 −10 −5 5 10 15 Power Spectrum Frequency Power (dB) PSD m=1 PSD m=2

• Est. PSD m=1
• Est. PSD m=2

Estimate of PSD

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 −20 −18 −16 −14 −12 −10 −8 −6 −4 −2 Power Spectrum Frequency (π radians) Power (dB)

• How to choose appropriate L1 regularization coefficient.
• Alternative algorithms that avoid tuning of L1 regularization

coefficient.

• Alternative algorithms that do not require estimating

appropriate values for M and R.

• Characterization of algorithm performance in varying

conditions (SNR, signal orthogonality in freq. & time).

• System implementation in software defined radio.

Continuing Work

Thank You