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APPLICATION OF A WAVELET-BASED RECEIVER FOR THE COHERENT DETECTION OF FSK SIGNALS Dr. Robert Barsanti, Charles LehmanSSST March 2008, University of New Orleans

Overview DETECTION OF FSK SIGNALS

- Introduction
- FSK Signals
- Wavelet Domain Filtering
- Wavelet Domain Correlation Receiver
- Simulations and Results
- Summary

FSK Signals DETECTION OF 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 DETECTION OF FSK SIGNALS

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 DETECTION OF FSK SIGNALS

- 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)

X

Output

Symbol

ri

So

X

Sample at t = T

S1

Wavelet Filtering signals

Wavelet Based Filtering signals

THREE STEP DENOISING

1. PERFORM DWT

2. THRESHOLD COEFFICIENTS

3. PERFORM INVERSE DWT

Wavelet Domain Correlation signals

- 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 signals

Bank of

Cross-Correlation

Receivers

Detector

(choose largest)

Wavelet

De-noise

r i

DWT

Output Symbol

So

DWT

S1

DWT

Simulation signals

- 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 signals

Bit Error Rate Curves for MFSK signals

Results signals

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

Summary signals

- Receiver for FSK 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 signals

Some S8 Symmlets at Various Scales and Locations

9

8

7

6

5

Scale j

4

3

2

1

0

0

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 signals

- 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 signals

Response

LPF HPF

HP

Filter

Details

X(n)

LP

Filter

Frequency

Averages

F/2

Pair of Half Band Quadrature Mirror Filters (QMF)

[Vetterli, 1995]

Signal Reconstruction signals

Two Channel Perfect Reconstruction QMF Bank

Analysis + Synthesis = LTI system

Wavelet Implementation [Mallat, 1989] signals

2

LPLPLP

J = 4

2

2

LP

LP

LP

2

HP

LPLPHP

J = 3

2

LPHP

J = 2

HP

2

HP

HP

J = 1

2J samples

LP

HP

LPHP

LPLPHP

LPLPLP

Calculating a Threshold signals

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 signals

* 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 signals

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