1. PERFORMANCE OF A WAVELET-BASED RECEIVER FOR BPSK AND QPSK SIGNALS IN ADDITIVE WHITE GAUSSIAN NOISE CHANNELS Dr. Robert Barsanti, Timothy Smith, Robert Lee SSST March 2007

2. Overview Introduction BPSK and QPSK Signals Wavelet Domain Filtering
Wavelet Domain Correlation Receiver
Simulations and Results
Summary

3. BPSK and QPSK Signals
In carrier phase modulation, the transmitted information is contained in the
phase of the carrier waveform. Since the carrier phase is limited to the
range 0 ≤ θ ≤ 2π, the carrier phases used to transmit the digital information
are given by
For binary phase modulation M = 2, For QPSK M = 4. The general representation of the phase carrier modulated waveform is given by
The symbol A represents the signal amplitude, and gT(t) is the signal wave-shape.
In the case of a rectangular pulse shape gT(t) is defined as

4. Phase Modulated Signal
and the transmitted waveform on the symbol interval 0 ≤ t ≤ T is represented as
Where E is the energy transmitted per symbol

5. The Classical Cross Correlation Receiver for two transmitted signals
Detector
(choose largest) X
Output
Symbol ri So X
Sample at t = T S1

6. Noise Removal Separate the signal from the noise Signal Noisy Signal
TRANSFORMATION
Noisy Signal
Signal
Noise

7. Wavelet Filtering

8. Wavelet Based Filtering
THREE STEP DENOISING
1. PERFORM DWT
2. THRESHOLD COEFFICIENTS
3. PERFORM INVERSE DWT

9. Wavelet Domain Correlation
transform prototype signal into the wavelet domain and pre-stored DWT coefficients
transform received signal into the wavelet domain via the DWT,
apply a non-linear threshold to the DWT coefficients (to remove noise),
correlate the noise free DWT coefficients of the signal, and the pre-stored
DWT coefficients of the prototype signal.

10. The Wavelet Domain Correlation (WDC) Receiver
Bank of
Cross-Correlation
Receivers
Detector
(choose largest)
Wavelet
De-noise
r i
DWT
Output Symbol So
DWT S1
DWT

11. Simulation
BPSK and QPSK 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 16 coefficients retained

12. Bit Error Rate Curves for BPSK

13. Bit Error Rate Curves for QPSK

14. Results Both the WDC and classical TDC provided similar results.
The WDC provides improvement in processing speed since only
16 vice 128 coefficients were used in the correlations.

15. Summary Receiver for BPSK and QPSK signals in the presence of AWGN.
Uses the cross- correlation between DWT coefficients
Procedure is enhanced by using standard wavelet noise removal techniques
Simulations of the performance of the proposed algorithm were presented.

17. Wavelets
Some S8 Symmlets at Various Scales and Locations 9 8 7 6 5
Scale j 4 3 2 1
0.2
0.4
0.6
0.8 1
time index k
1. Can be defined by a wavelet function (Morlet & Mexican hat)
2. Can be defined by an FIR Filter Function (Haar, D4, S8)

18. 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]

19. Wavelet Implementation
Response
LPF HPF HP
Filter
Details
X(n) LP
Filter
Averages
Frequency
F/2
Pair of Half Band Quadrature Mirror Filters (QMF)
[Vetterli, 1995]

20. Signal Reconstruction
Two Channel Perfect Reconstruction QMF Bank
Analysis + Synthesis = LTI system

21. Wavelet Implementation [Mallat, 1989]LP 2 LP 2 LP 2
LPLPLP
J = 4 HP 2
LPLPHP
J = 3 HP 2
LPHP
J = 2 HP 2 HP
J = 1
2J samples LP HP
LPHP
LPLPLP
LPLPHP

22. Symmlet Wavelet vs. Time and Frequency

23. 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), n = 1,2,….
z ~N(0,1) and s = noise std.
s is estimated from the data by analysis of the coefficients
at the lowest scale.
s = E/ where E is the absolute median deviation
[Kenny]

24. Thresholding Techniques
* Hard Thresholding (keep or kill)
* Soft Thresholding (reduce all by Threshold)
The Threshold Value is determined as a multiple
of the noise standard deviation,
eg., T = ms where typically 2< m <5

25. Hard vs. Soft Thresholds