Transcript Slide 1

APPLICATION OF A WAVELET-BASED
RECEIVER FOR THE COHERENT DETECTION
OF FSK SIGNALS
Dr. Robert Barsanti, Charles Lehman
SSST March 2008, University of New Orleans
Overview
•
•
•
•
•
•
Introduction
FSK Signals
Wavelet Domain Filtering
Wavelet Domain Correlation Receiver
Simulations and Results
Summary
FSK Signals
In binary frequency shift keying modulation, the binary information is transmitted
using signals at two different frequencies. These signals can be represented as
sm (t )  A cos(2 f m t ) . (1)
The symbol A represents the signal amplitude, and T is the bit duration. It
is easy to show that the bit energy is given by
A2
E 
.
2T
( 2)
FSK Signal
This figure shows an example of a binary FSK signal. Notice that
the transmitted signal case a constant envelope and abrupt phase
changes at the beginning of each signal interval.
FSK Probability of Bit Error
•
Assuming the received signal at the input to the correlator is corrupted with
Gaussian noise of zero mean and variance No/2. The probability of bit error can be
computed to be [5]
 Eb
Pe  Q
 No





(3)
where
Q( x) 
•
1
 
2
x
e
Z2
2
dz .
(4)
It can be seen that the probability does not depend on the detailed signal
and noise characteristics, but only upon the signal to noise ratio per bit
(SNR) [6].
The Classical Cross Correlation Receiver for
two transmitted signals
T
X
 ( )d
0
ri
Detector
(choose
largest)
So
T
X
 ( )d
0
Sample at t = T
S1
Output
Symbol
Noise Removal
• Separate the signal from the noise
Signal
Noisy
Signal
TRANSFORMATION
Noise
Wavelet Filtering
Wavelet Based Filtering
S1 Signal + Noise
THREE STEP DENOISING
DWT of S1
10
0
-2
1. PERFORM DWT
5
2. THRESHOLD COEFFICIENTS
0
3. PERFORM INVERSE DWT
-4
-6
-8
-5
0
0.5
1
-10
0
S2 Denoised Signal
0
4
-2
2
-4
0
-6
-2
-8
0
0.5
1
DWT of S2
6
-4
0.5
1
-10
0
0.5
1
Wavelet Filtering of an FSK Signal
DWT of a noise free FSK signal.
DWT of noisy FSK signal.
Wavelet Domain Correlation
(1) transform prototype signal into the wavelet domain and pre-stored DWT coefficients
(2) transform received signal into the wavelet domain via the DWT,
(3) apply a non-linear threshold to the DWT coefficients (to remove noise),
(4) correlate the noise free DWT coefficients of the signal, and the pre-stored
DWT coefficients of the prototype signal.
The Wavelet Domain Correlation (WDC) Receiver
ri
So
S1
DWT
DWT
DWT
Wavelet
De-noise
Bank of
CrossCorrelation
Receivers
Detector
(choose
largest)
Output
Symbol
Simulation
• FSK signals were generated using 128 samples per
symbol
• Monte Carlo Runs at each SNR with different
instance of AWGN
• 10 SNR’s between -6 and +10 dB
• Symmlet 8 wavelet & soft threshold
• Threshold set to σ/10.
• Only 32 coefficients retained
Bit Error Rate Curve for FSK
Bit Error Rate Curves for MFSK
Results
(1) Both the WDC and classical TDC provided similar results.
(2) The WDC provides improvement in processing speed since only
32 vice 128 coefficients were used in the correlations.
Summary
(1) Receiver for FSK signals in the presence of AWGN.
(2) Uses the cross- correlation between DWT coefficients
(3) Procedure is enhanced by using standard wavelet noise removal techniques
(4) Simulations of the performance of the proposed algorithm were presented.
Wavelets
H aar W avelet
0.2
0.1
D 4 Wavelet
Some S8 Symmlets at Various Scales and Locations
0.2
9
0
8
0
7
-0.2
-0.1
6
-0.4
0.5
0.5
C 3 Coiflet
S8 Symmlet
Scale j
-0.2
5
4
0.2
0.2
0.1
0.1
3
0
0
2
-0.1
-0.1
1
-0.2
-0.2
0.5
0.5
0
0
0.2
0.4
0.6
time index k
0.8
1
1. Can be defined by a wavelet function (Morlet & Mexican hat)
2. Can be defined by an FIR Filter Function (Haar, D4, S8)
EFFECTIVENESS OF WAVELET ANALYSIS
• Wavelets are adjustable and adaptable by virtue of
large number of possible wavelet bases.
• DWT well suited to digital implementation. ~O (N)
• Ideally suited for analysis non-stationary signals [
Strang, 1996]
• Has been shown to be a viable denoising technique
for transients [Donoho, 1995]
• Has been shown to be a viable detection technique
for transients [Carter, 1994]
• Has been shown to be a viable TDOA technique for
transients [Wu, 1997]
Wavelet Implementation
Response
LPF
HP
Filter
Details
LP
Filter
Averages
HPF
X(n)
Frequency
F/2
Pair of Half Band Quadrature Mirror Filters (QMF)
[Vetterli, 1995]
Signal Reconstruction
-----Analysis Section------LPF
2
> >
-----Synthesis Section-----2
LPF
x(n)
+
HPF
2
> >
2
x(n)
HPF
Two Channel Perfect Reconstruction QMF Bank
Analysis + Synthesis = LTI system
Wavelet Implementation [Mallat, 1989]
LP
2
LP
HP
HP
2
LP
2
LPLPLP
J=4
HP
2
LPLPHP
J=3
2
LPHP
J=2
2
HP
J=1
J
2
samples
LP
HP
LPHP
LPLPLP
LPLPHP
Symmlet Wavelet vs. Time and Frequency
Calculating a Threshold
Let the DWT coefficient be a series of noisy observations y(n)
then the following parameter estimation problem exists:
y(n) = f(n) +s z(n),
z ~N(0,1)
and
n = 1,2,….
s = noise std.
s is estimated from the data by analysis of the coefficients
at the lowest scale.
s = E/0.6475
where E is the absolute median deviation
[Kenny]
Thresholding Techniques
* Hard Thresholding (keep or kill)
C j ,k
 (C j ,k )  
0
H
| C j ,k |  T 
 ,
| C j ,k |  T 
* Soft Thresholding (reduce all by Threshold)

sign(C j ,k )[| C j ,k | T ]
 (C j ,k )  

0
S
| C j ,k |  T 

 ,
| C j ,k |  T 

The Threshold Value is determined as a multiple
of the noise standard deviation,
eg., T = ms where typically 2< m <5
Hard vs. Soft Thresholds