Technion - Israel institute of technology
department of Electrical Engineering
Asaf Barel
Eli Ovits
Supervisor: Debby Cohen
June 2013
High speed digital systems laboratory
Project Motivation
 Communication Signals are wideband with very high
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Nyquist rate
Communication Signals are Sparse, therefore
subnyquist sampling is possible
Possible application: Cognitive Radio
Current system suffers from low noise robustness
Project goal: implementing algorithm for cyclic
detection with high noise robustness
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Background: Sub-Nyquist Sampling
MWC system
Background: Sub-Nyquist Sampling
Digital Processing
System Output
 Full signal reconstruction, or support recovery using
Energy Detection
 The problem: Noise is enhanced by Aliasing
Energy Detection: simulation
Signal: 𝑁 = 6, 𝐵 = 19.5𝑒6, 𝑡𝑦𝑝𝑒 = 𝑞𝑝𝑠𝑘, 𝑓𝑚𝑎𝑥 = 2.46𝑒9
SNR = 10 dB
SNR = -10 dB
 Original support:
 Original support:
8 72 90 162 180 244
24 35 117 135 217 228
 Reconstructed support:
 Reconstructed support:
90 180 244 21 200 241
162 72 8 231 52 11
24 87 107 217 232 168
228 165 145 35 20 84
Original support is contained!
Original support is not contained!
Cyclostationary Signals
 Wide sense Cyclostationary signal: mean and
autocorrelation are periodic with 𝑇0
𝜇𝑥 𝑡 + 𝑇0 = 𝜇𝑥 𝑡
𝑅𝑥 𝑡 + 𝑇0 , 𝜏 = 𝑅𝑥 𝑡, 𝜏
Cyclostationary Signals
 The Autocorrelation can be expanded in a fourier
series:
𝑅𝑥𝛼
𝑅𝑥 𝑡, 𝜏 =
𝛼
𝑅𝑥𝛼
1
𝜏 =
𝑇
𝜏
𝑒 𝑗2𝜋𝛼𝑡
𝑛
,𝛼 =
𝑇0
∞
𝑅𝑥 𝑡, 𝜏 𝑒 −𝑗2𝜋𝛼𝑡 𝑑𝑡
−∞
Cyclostationary Signals
 Specral Correlation Function (SCF):
∞
𝑆𝑥𝛼 𝑓 =
𝑅𝑥𝛼 𝜏 𝑒 −𝑗2𝜋𝑓𝜏 𝑑𝜏
−∞
[Gardner, 1994]
Cyclostationary Signals
 The Cyclic Autocorrelation function can also be
viewed as cross correlation between frequency
modulations of the signal:
𝑅𝑥𝛼 𝜏 = 𝑥 𝑡 𝑒 −𝑗𝜋𝛼𝑡 𝑥 𝑡 − 𝜏 𝑒 𝑗𝜋𝛼
[Gardner, 1994]
𝑡−𝜏
Cyclic Detection
 Signal Model: Sparse, Cyclostationary signal. No
correlation between different bands.
 The goal: blind detection
 Support Recovery: instead of simple energy detection,
we will use our samples to reconstruct the SCF, and
then recover the signal’s support.
SCF Reconstruction
 Using the latter definition for cyclic Autocorrelation,
we can get Autocorrelation from a signal:
𝑅𝑥 = 𝑬 𝑥 𝑓 𝑥 𝐻 𝑓
For a Stationary Signal
For a Cyclostationary Signal
𝑅𝑥 =
𝑅𝑥 =
𝛼=0
SCF Reconstruction – Mathematical derivation
𝑧 𝑓 = 𝐴𝑥 𝑓
𝑅𝑧 = 𝑬 𝑧 𝑓 𝑧 𝐻 𝑓
= 𝐴𝑅𝑥 𝐴𝐻
𝑟𝑧 = 𝑣𝑒𝑐𝑡𝑜𝑟 𝑅𝑧
𝑟𝑧 = 𝐴∗ ⊗ 𝐴 𝑣𝑒𝑐𝑡𝑜𝑟 𝑅𝑥
Discarding zero elements from 𝑅𝑥 :
𝑟𝑧 = 𝐴∗ ⊗ 𝐴 B 𝑟𝑥 = Φ𝑟𝑥
Algorithm Pseudo Code
Pseudo Code
Further Objectives
 MATLAB implementation of the Algorithm
 Simulation of the new system, including Comparison
to the Energy Detection system (Receiver operating
characteristic (ROC) in different SNR scenarios )
 Comparison to Cyclic detection at Nyquist rate (mean
square error )
Gantt Chart
5.6.13
Adaptation of exisiting algorithm to the cyclic
case
Implementing MATLAB code for SCF
reconstruction
Adding signal detecion from the SCF
Simulations and comparison
Optional: Implementing cyclic detection in
Hardware simulating enviroument
15.6.13
25.6.13
5.7.13
15.7.13
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10
10
7
14
25.7.13