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

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

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

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

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

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

Wavelet Filtering

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

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.

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

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

Bit Error Rate Curves for BPSK

Bit Error Rate Curves for QPSK

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.

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.

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)

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

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

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

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), 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/0.6475 where E is the absolute median deviation [Kenny]

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

Hard vs. Soft Thresholds