Mid Semester Presentation - High Speed Digital Systems Lab
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Transcript Mid Semester Presentation - High Speed Digital Systems Lab
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
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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25.7.13