Uniform Linear Array based Spectrum Sensing
from sub-Nyquist Samples
Or Yair, Shahar Stein
Supervised by Deborah Cohen and Prof. Yonina C. Eldar
December 7, 2015
Signal Model
• Multiband sparse signal
• Each transmission 𝑠𝑖 corresponds to a carrier frequency, 𝑓𝑖
• Each transmission 𝑠𝑖 is narrow band of maximum bandwidth 𝐵
• All transmissions are assumed to have identical and known angle of arrival 𝜃 ≠
90𝑜
2
Goal: Perfect blind signal reconstruction from subNyquist samples
Proposed Algorithm
• We suggest a ULA based system
• Each sensor of the array, followed by one branch of the MWC with the same
periodic function
• Each sampler is of rate 𝐵
• From the samples we can form a classic DOA
equation 𝒙 = 𝑨𝒘 and use known techniques
to obtain 𝒘
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We show that the a sufficient condition is 𝑴 + 𝟏
samplers
System Description
• The received signal at the 𝑛’th sensor:
𝑢𝑛 𝑡 ≈
𝑀
𝑖=1 𝑠𝑖
𝑡 𝑒 𝑗2𝜋𝑓𝑖 (𝑡+𝜏𝑛)
𝑆1 (𝑓 − 𝑓1 )
𝑆3 (𝑓 − 𝑓3 )
0
𝑆2 (𝑓 − 𝑓2 )
𝐵
• The mixed signal after multiplying with periodic function:
𝑌𝑛 𝑓 =
∞
𝑙=−∞ 𝑐𝑙
𝑀
𝑖=1 𝑆𝑖
0
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𝑓 − 𝑓𝑖 − 𝑙𝑓𝑝 𝑒 𝑗2𝜋𝑓𝑖 𝜏𝑛
System Description
• The filtered signal at the baseband:
𝑌𝑛 𝑓 =
𝑀
𝑆𝑖
𝑖=1
𝐿
𝑓 𝑒 𝑗2𝜋𝑓𝑖 𝜏𝑛 ,
0
𝑆𝑖 𝑓 =
𝑐𝑙 𝑆𝑖 𝑓 − 𝑓𝑖 − 𝑙𝑓𝑝
𝑙=−𝐿0
0
• The sampled signal:
𝑀
𝑋𝑛 𝑒 𝑗2𝜋𝑓𝑇𝑠 =
𝑊𝑖 𝑒 𝑗2𝜋𝑓𝑇𝑠 𝑒 𝑗2𝜋𝑓𝑖 𝜏𝑛
𝑖=1
𝑤𝑖 𝑘 = 𝑠𝑖 (𝑘𝑇𝑠 )
The unknown frequencies are held at the relative
accumulated phase
5
Sampling Scheme
• Modified MWC sampling chain.
• All sensors use the same periodic function with period 𝑓𝑝
• Single sensor output:
Sufficient condition on 𝒇𝒔 , 𝒇𝒑 :
𝒇𝒔 ≥ 𝒇𝒑 ≥ 𝑩
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Sampling Scheme
• Our measurements
The sampling scheme is simpler than the MWC, since
the same sequence can be used in all cannels
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Basic Equation
• Similar to classic DOA equation
• Source Signal vector is scaled and cyclic-shifted
0
0
0
0
𝑿 𝑒 𝑗2𝜋𝑓𝑇𝑠
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|
𝒂 𝑓1
|
|
𝒂 𝑓2
|
𝑨𝑁×𝑀
|
𝒂 𝑓3
|
0
0
0
𝑾 𝑒 𝑗2𝜋𝑓𝑇𝑠
Goal: estimate 𝒇 and 𝒘
Reconstruction Steps
1. Estimate all frequencies 𝑓𝑖 .
2. Reconstruct the steering matrix 𝑨 𝒇 .
3. Calculate 𝒘 = 𝑨† 𝒙
4. Uniquely recover 𝒔 from 𝒘.
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Theorem
• For multiband signal (as presented),
• If the following conditions hold:
• Minimal number of sensors: 𝑀 + 1
• 𝑓𝑝 = 𝑓𝑠 > 𝐵
𝑐
• 𝑑<
𝑓𝑁𝑌𝑄
∞
2𝜋𝑙𝑓𝑝 𝑡
𝑐
𝑒
𝑙=−∞ 𝑙
• p 𝑡 =
• Then: 𝑓𝑖 , 𝑠𝑖 𝑓
, 𝑐𝑙 ≠ 0, ∀𝑙: 𝑙𝑓𝑝 ≤
𝑓𝑁𝑌𝑄
2
can be perfectly reconstructed
Achievable sampling rate: 𝑴 + 𝟏 𝑩
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Simulations
• For the ULA based system 2 reconstruction methods were tested:
• ESPRIT – analytic method based on SVD
• MMV – CS method based on OMP algorithm
• The performance were compared against the MWC
• Both systems used the same amount of samplers
• At the ULA based system – the number of sensors
• At the MWC – the number of branches
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Simulations
• Performance against different number of sensors
• For the MWC system: the amount of branches
12
SNR
10dB
Number of
Snapshots
400
Number of
Signals
3
𝑩
𝟓𝟎𝑴𝑯𝒛
𝒇𝑵𝒀𝑸
𝟏𝟎𝑮𝑯𝒛
𝒇𝒔
𝟓𝟎𝑴𝑯𝒛
Simulations
• Performance against different SNR
Number of
Sensors
10
Number of
Snapshots
400
Number of
Signals
3
𝑩
𝟓𝟎𝑴𝑯𝒛
𝒇𝑵𝒀𝑸
𝟏𝟎𝑮𝑯𝒛
𝒇𝒔
𝟓𝟎𝑴𝑯𝒛
⇓
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Sampling Rate:
𝟓𝟎𝟎𝑴𝑯𝒛
Summary
• We suggest an ULA based system for the spectrum sensing
problem
• In each sensor of the array, we sample using one branch of the
MWC with the same periodic function
• We relate the unknown parameter and signal to the sub-Nyquist
samples
• We show that a sufficient condition for perfect recovery are 𝑴 + 𝟏
sensors, each sampling at rate 𝑩
• We perform parameter estimation out of the DOA equation
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Questions?
• Thank you for listening
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