The document proposes a sub-Nyquist rate spectrum sensing method for wideband cognitive radio using a multicoset sampler and nonlinear least squares (NLLS) estimation. It models the wideband signal sampling process using a multicoset sampler to reduce the sampling rate. It then estimates the active channel set using an NLLS method on the sample correlation matrix. Simulations show the NLLS method provides higher detection probability and lower false alarm rates than conventional energy detection, while allowing sampling below the Nyquist rate.

1. An NLLS Based Sub-Nyquist Rate Spectrum Sensing for
Wideband Cognitive Radio
M. R. Avendi, K. Haghighi, A. Panahi, M. Viberg
Department of Signal and Systems
Chalmers University of Thechnology
May 2011
M. R. Avendi, K. Haghighi, A. Panahi, M. Viberg Wideband Spectrum Sensing May 2011 1 / 21

2. Outline
Introduction
Problem Statement
Proposed Model
Comparison and Simulation
Summary
M. R. Avendi, K. Haghighi, A. Panahi, M. Viberg Wideband Spectrum Sensing May 2011 2 / 21

3. Introduction
Spectrum Sensing
M. R. Avendi, K. Haghighi, A. Panahi, M. Viberg Wideband Spectrum Sensing May 2011 3 / 21

4. Introduction
Spectrum Sensing
Narrowband
Energy Detection (ED), ...
M. R. Avendi, K. Haghighi, A. Panahi, M. Viberg Wideband Spectrum Sensing May 2011 3 / 21

5. Introduction
Spectrum Sensing
Narrowband
Energy Detection (ED), ...
Wideband
Challenge: High Sample Rate ADC
M. R. Avendi, K. Haghighi, A. Panahi, M. Viberg Wideband Spectrum Sensing May 2011 3 / 21

6. Problem Statement
Signal
Complex signal x(t)
Fourier X(f ), f ∈ [0, Bmax ]
Nyquist rate: Bmax = L × B
frequency[MHz]
Spectrum
0 Bmax
index L = {0, 1, ..., L − 1}
M. R. Avendi, K. Haghighi, A. Panahi, M. Viberg Wideband Spectrum Sensing May 2011 4 / 21

7. Problem Statement Cont.
Active channel set b = [b1, b2, ..., bN ]
Example: b = [8, 16, 17, 18, 29, 30]
frequency[MHz]
Spectrum
0 8 16 24 32
Given B, Bmax, Ωmax = Nmax
L and x(t)
Find b and N ?
at fsample < Bmax
M. R. Avendi, K. Haghighi, A. Panahi, M. Viberg Wideband Spectrum Sensing May 2011 5 / 21

8. Proposed Model
LL
xi (m)x(t) Delay
xdi 1
M Σxd x∗
d
ˆR ˆb
y(f )
Multicoset Sampler
Sample Correlation matrix
NLLS Estimator
favg = αBmax
M. R. Avendi, K. Haghighi, A. Panahi, M. Viberg Wideband Spectrum Sensing May 2011 6 / 21

9. Multicoset Sampler
Non-uniform sampling: xi (m) = x[(mL + ci )/Bmax ]; m ∈ Z
0 5 10 15 20 25 30 35 40
−3
−2
−1
0
1
2
3
time
x(t)
M. R. Avendi, K. Haghighi, A. Panahi, M. Viberg Wideband Spectrum Sensing May 2011 7 / 21

10. Multicoset Sampler
Sampling frequency: favg = p
L Bmax
Landau’s lower bound: Nmax < p ≪ L
Random sample pattern: ci ∈ L
x1(m)
x(t) x2(m)
xp(m)
t = (mL + c1)/Bmax
t = (mL + cp)/Bmax
M. R. Avendi, K. Haghighi, A. Panahi, M. Viberg Wideband Spectrum Sensing May 2011 8 / 21

11. Recall Model
LL
xi (m)x(t) Delay
xdi 1
M Σxd x∗
d
ˆR ˆb
y(f )
Multicoset Sampler
Sample Correlation matrix
NLLS Estimator
favg = αBmax
M. R. Avendi, K. Haghighi, A. Panahi, M. Viberg Wideband Spectrum Sensing May 2011 9 / 21

12. Configuration
Upsampling: factor L
Low pass filtering: [0, B]
Delaying: with ci samples
L
xi (m)
Delay
xci
, y(f )
M. R. Avendi, K. Haghighi, A. Panahi, M. Viberg Wideband Spectrum Sensing May 2011 10 / 21

13. Frequency domain Model
Matrix form:
y(f ) = A(b)x(f ) + n(f ), f ∈ [0, B]
M. R. Avendi, K. Haghighi, A. Panahi, M. Viberg Wideband Spectrum Sensing May 2011 11 / 21

14. Frequency domain Model
Matrix form:
y(f ) = A(b)x(f ) + n(f ), f ∈ [0, B]
y(f ): Known vector of DFT of configured sequences
x(f ): Unknown vector of signal spectrum in the active channels
n(f ): Gaussian complex noise, N(0, σ2I)
A(b)(i, k) = B exp j2πci bk
L
M. R. Avendi, K. Haghighi, A. Panahi, M. Viberg Wideband Spectrum Sensing May 2011 11 / 21

15. Recall Model
LL
xi (m)x(t) Delay
xdi 1
M Σxd x∗
d
ˆR ˆb
y(f )
Multicoset Sampler
Sample Correlation matrix
NLLS Estimator
favg = αBmax
M. R. Avendi, K. Haghighi, A. Panahi, M. Viberg Wideband Spectrum Sensing May 2011 12 / 21

16. Correlation Matrix
True matrix: R = E[y(f )y∗(f )]
Estimated in time domain using Parseval’s identity
ˆR =
B
0
y(f )y∗
(f )df =
+∞
m=−∞
xci
[m]x∗
ci
[m]
Reduce complexity, downsampling xdi (m) = xci [mL]
ˆR =
1
M
M
m=1
xd (m)x∗
d (m)
Lxci
xdi 1
M Σxd x∗
d
ˆR
M. R. Avendi, K. Haghighi, A. Panahi, M. Viberg Wideband Spectrum Sensing May 2011 13 / 21

17. NLLS Based Method
Recall model y(f ) = A(b)x(f ) + n(f ) ⇒ b ?
Minimizing the least square error J(b) = tr{(Ip − A(b)A†(b))ˆR}
Detection threshold
M. R. Avendi, K. Haghighi, A. Panahi, M. Viberg Wideband Spectrum Sensing May 2011 14 / 21

18. NLLS Based Method
Recall model y(f ) = A(b)x(f ) + n(f ) ⇒ b ?
Minimizing the least square error J(b) = tr{(Ip − A(b)A†(b))ˆR}
Detection threshold
Jmin = σ2
(p − N)
M. R. Avendi, K. Haghighi, A. Panahi, M. Viberg Wideband Spectrum Sensing May 2011 14 / 21

19. NLLS method
Sequential Forward NLLS Algorithm
Typical Example: p = 10, N = 6, σ2 = 1
1 2 3 4 5 6
4
6
8
10
12
14
16
18
J(bi )
LSE
i
Jmin
(p − i)σ2
M. R. Avendi, K. Haghighi, A. Panahi, M. Viberg Wideband Spectrum Sensing May 2011 15 / 21

20. Comparison and Simulation
Signal: Bmax = 320MHz, B = 10MHz, Ωmax = 0.25
Multicoset sampler: L = 32, p = 10, M = 64
favg = p
L Bmax = 100MHz!!
0 80 160 240 320
frequency[MHz]
Spectrum
M. R. Avendi, K. Haghighi, A. Panahi, M. Viberg Wideband Spectrum Sensing May 2011 16 / 21

21. Energy Detection Model
Conventional ED model
x(t) x(nT)
Uniform Sampler
fs = Bmax
Filter Bank
1
M |.|2
1
M |.|2
≷1
0 η
≷1
0 η
H0
H0
H1
H1
M. R. Avendi, K. Haghighi, A. Panahi, M. Viberg Wideband Spectrum Sensing May 2011 17 / 21

22. Numerical Results
Probability of detection
−5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
α=0.3, NLLS
α=0.5, NLLS
ED
MUSIC
Pd
SNR, [dB]
M. R. Avendi, K. Haghighi, A. Panahi, M. Viberg Wideband Spectrum Sensing May 2011 18 / 21

23. Numerical Results
Probability of false alarm
−5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10
0
0.005
0.01
0.015
0.02
0.025
α=0.3, NLLS
α=0.5, NLLS
ED
MUSIC
Pf
SNR, [dB]
M. R. Avendi, K. Haghighi, A. Panahi, M. Viberg Wideband Spectrum Sensing May 2011 19 / 21

24. Summary
Wideband Spectrum Sensing
MulticosetSampler
NLLS method
Comparison
M. R. Avendi, K. Haghighi, A. Panahi, M. Viberg Wideband Spectrum Sensing May 2011 20 / 21

25. Thank you for your attention
M. R. Avendi, K. Haghighi, A. Panahi, M. Viberg Wideband Spectrum Sensing May 2011 21 / 21