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 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

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] 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 Detector ( choose largest ) So S1 Sample at t = T riri X X Output Symbol

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

Wavelet Filtering

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

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 Output Symbol Detector ( choose largest ) So S1 r i Bank of Cross- Correlation Receivers Wavelet De-noise DWT

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 1. Can be defined by a wavelet function (Morlet & Mexican hat) 2. Can be defined by an FIR Filter Function (Haar, D4, S8) Some S8 Symmlets at Various Scales and Locations time index k Scale j

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

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

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

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  = noise std.  is estimated from the data by analysis of the coefficients at the lowest scale.  = E/ 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 = m  where typically 2< m <5

Hard vs. Soft Thresholds