Experimental Study

Chapter 7 EXPERIMENTAL STUDY : Chapter 7 EXPERIMENTAL STUDY Statistical Signal Processing June 11, 2008 Abibullaev Berdakh Microsystems Lab. (305-1) Dept. of Electronic Engineering, Graduate School, Yeungnam University, KOREA (Tel : +82-53-810-3924; Fax : +82-53-810-4742 E-mail : berdakho@ynu.ac.kr)

Outline : Outline Introduction Generalized detector in brief. Experimental Conditions The Experimental investigations SIGNAL – TO – NOISE RATIO = 15.92dB SIGNAL – TO – NOISE RATIO = 0.96 dB Conclusions

Introduction : Introduction Signal detection, determination, and estimation of the signal parameters are carried out in this experiment. using detectors of two types : The generalized detector The correlation detector Signal detection is implemented under conditions of two input power SNR : Signal –to – noise is 15.92 dB (clear signal detection by correlation detector ) Signal –to – noise is 0.96 dB (failure to detect signal by correlation detector) The experimental study is carried out for the following cases. Case 1: the effective bandwidth of the PF is the same value as the effective bandwidth of the AF. Case 2: the effective bandwidth of the AF is one order of value less than the effective bandwidth of the PF.

Generalized detector : First, we have to satisfy the condition Generalized detector

Experimental conditions : - The noise is imitated by a random uncorrelated number generator obeying the Gaussian distribution law with zero mean and finite variance. - The signal is a zero phase-manipulated sequence of recurrent code with the number of the first digits equal to 5 and period N . (N = 31). - The following process is observed at the outputs of the detectors: - for the correlation detector: - for the generalized detector: Experimental conditions where is the process at the output of the PF.

Slide 6 : Experimental conditions (cont.) In the experimental investigations the location of the signal on the time axis is determined in the following manner: - is the instant reckoned from and is set equal to zero Given the model signal coincides with the structure of the actual signal [0,T]. the energy of the model signal for the generalized detector must be varied from zero to a definite fixed value that exceeds the energy of the signal. The energy of the model signal is varied by changing the amplitude of the model signal (the element of the tracking system). - To provide sufficient decision statistics at the outputs of the correlation and generalized detectors the sample size is set to six hundred at each instant (each discrete reading).

Experimental conditions (cont.) : Experimental conditions (cont.) The process at the outputs of the correlation and generalized detectors is observed in the following coordinate systems: the optimal detector: by the indicator where t - is the discrete time; - the generalized detector: by the indicators and simultaneously.

Experimental conditions (cont.) : Experimental conditions (cont.) The experimental study is carried out for the following cases. Case 1: the effective bandwidth of the PF is the same value as the effective bandwidth of the AF. The generalized detector is shown in Fig. 7.l. Case 2: the effective bandwidth of the AF is one order of value less than the effective bandwidth of the PF. The generalized detector is shown in Fig. 7.2 (the filter Fl is added to the AF). The effective bandwidth of the AF may be less than the effective bandwidth of the PF to suit statements discussed in [13,17,24,25,37,38]

Correlation detector SNR = 15.92dB : Correlation detector SNR = 15.92dB - Given a yes signal in the input stochastic process The noise component of the correlation detector, is formed at the output of the PF, and has zero mean and variance equal approximately to 1.47. The decision statistics at the 30th reading, indicate a yes signal in the input stochastic process a yes signal the term is the average energy of the signal when

Generalized Detector : Indicator f1[Z(out),t] : Generalized Detector : Indicator f1[Z(out),t] is characterized by zero mean and variance equal approximately to 0.64 (case 1, in Fig. 7.4) 0.33 (case 2, Fig. 7.5) is observed at the output of the generalized detector for ; the model signal generator is switched off ; The variance of the background noise at the output of the generalized detector is much less than that of the noise component at the output of the correlation detector SNR = 15.92dB a no signal in the input stochastic process. The decision statistics of the background noise a no signal

Generalized Detector : Indicator : Generalized Detector : Indicator The variance of the noise component at the output of the correlation detector is determined by with an increase in the energy of the model signal the variance of the noise component at the output of the correlation detector increases. The variance of the background noise at the output of the generalized detector is independent of both the energy of the model signal and the average energy of the signal. The decision statistics of the background noise at the output of the generalized detector are shown in Figs. 7.4 (case 1) and 7.5 (case 2). Noise component: CORRELATION DETECTOR Noise component: GENERALIZED DETECTOR SNR = 15.92dB a no signal

Generalized Detector (cont.) : Generalized Detector (cont.) The decision statistics at the output of the generalized detector, which are obtained under the conditions are shown in Figs. 7.6 (case 1) and 7.7 ( case 2). These conditions imply that the model signal generator is switched off and a yes signal exists in the input stochastic process. The decision statistics, which are observed at the output of the generalized detector under these conditions, are determined by The average energy of the signal (the first term in Eq. (7.12)) and the random component of the autocorrelation channel of the generalized detector, which is caused by the interaction between the signal and noise formed at the output of the PF (the second term in Eq. (7.12)), are added with a minus sign to the background noise at the output of the generalized detector. SNR = 15.92dB a yes signal

Generalized Detector (cont.) : Generalized Detector (cont.) Comparing Figs. 7.4 and 7.6 (case 1) and Figs. 7.5 and 7.7 (case 2), respectively, one can readily see how the decision statistics at the output of the generalized detector vary given a yes signal in the input stochastic process. SNR = 15.92dB a yes signal a no signal

Signal structure definition : study case : Signal structure definition : study case Case 1: Noncoincidence in time between the model signal and the signal Case 2: Coincidence in time between the model signal and the signal model signal generator is switched on to define the signal structure and parameters, a yes signal in the input stochastic process. The decision statistics are observed at the output of the generalized detector for : SNR = 15.92dB a yes signal

The case 1 : : The case 1 : Let’s observe those sections where: 1) the time axis where signal influence is present (readings 0-60) 2) the time axis where signal influence is absent (readings 61-68). The decision statistics at the output of the generalized detector, which are analyzed within the limits of the time interval where a no signal in the input stochastic process, are determined by The noise component of the correlation channel of the generalized detector is added to the background noise at the output of the generalized detector; The variance of these decision statistics is increased in comparison with that of the background noise at the output of the generalized detector owing to an increase in the degree of uncertainty. SNR = 15.92dB a yes signal

The case 1 : : The case 1 : The identification process suggests the following definition of signal structure: a signal with an expected structure or some another structure is detected. The decision statistics : are added to the background noise at the output of the generalized detector within the limits of the time interval given a yes signal in the input stochastic process. Notice the drastic increase in the variance of the decision statistics at the output of the generalized detector due to the noncorrelation between the signal and the model signal. SNR = 15.92dB (7.15) a yes signal

The case 2: : The case 2: The decision statistics at the output of the generalized detector are given by Fig. 7.9 Generalized detector : Correlation and Autocorrelation channels. Case 2 Fig. 7.8 Generalized detector : Correlation and Autocorrelation channels. Case 1 SNR = 15.92dB a yes signal Consider the case at reading 30.

Generalized Detector : Indicator : Generalized Detector : Indicator The necessity of the indicator is due to the following: for the correlation detector there is no need to satisfy the condition The amplitude of the model signal (the reference voltage) is chosen from the condition where k is the coefficient of proportionality, which can be, in general, more or less than 1. For the generalized detector the compensation between the noise component of the correlation channel and the random component of the autocorrelation channel in the statistical sense is realized only if SNR = 15.92dB

The case 1 : : The case 1 : Fig. 7.10 Noncoincidence in Time between Model Signal and Signal. Case 1 Fig. 7.11 Noncoincidence in Time between Model Signal and Signal. Case 2 - the decision statistics at the output of the generalized detector are displaced downward by the value relative to zero axis, when the model signal generator is switched off; SNR = 15.92dB

The case 2: : The case 2: The tracking window of the correlation channel of the generalized detector coincides in time with the signal within the limits of the time interval [0,T] on the time axis. The variation of the mean of the decision statistics at the output of the generalized detector The variance of the decision statistics at the output of the generalized detector is given by SNR = 15.92dB

Conclusions (SNR= 15.92dB) : Conclusions (SNR= 15.92dB) The signal component exceeds the background noise at the output of the generalized detector if the model signal generator is switched off, - The downward displacement of the decision statistics at the output of the generalized detector is caused by the minus sign of the autocorrelation channel of the generalized detector When the tracking window and the signal on the time axis do not coincide in time in time interval [0,T], the decision statistics at the output of the generalized detector are displaced downward by that part of the signal energy that is equivalent to the degree of coincidence in time between the tracking window of the correlation channel of the generalized detector and the signal on the time axis (see Figs. 7.10 [case 1] and 7.11 [case 2], respectively). - When the tracking window coincides completely in time with the signal in interval [0,T], on the time axis, the decision statistics at the output of the generalized detector are displaced downward by the average energy of the signal (see Figs. 7.12 [case 1] and 7.13 [case 2], respectively) - For incomplete coincidence in time between the tracking window of the correlation channel of the generalized detector and the signal on the time axis, the variance of the decision statistics at the output of the generalized detector increases relative to a symmetry axis, which is defined by the energy of the signal with an increase in the amplitude of the model signal. In this case the energy of the signal is equivalent to the degree of coincidence in time between the tracking window of the correlation channel of the generalized detector and the signal within the limits of the time interval [0,T] on the time axis. . SNR = 15.92dB

Signal – to - Noise ratio = 0.96 dB : Signal – to - Noise ratio = 0.96 dB The decision statistics at the output of the correlation detector if a no signal exists in the input stochastic process are shown in Fig. 7.14. and the case a yes signal in Fig. 7.15 The power signal-to-noise ratio at the output of the correlation detector is equal to 0.75 dB. - the decision statistics at the output of the correlation detector are the same both for a no and a yes signal in the input stochastic process. The detection of signals during the use of the correlation detector at the input power signal-to-noise ratio 0.96dB lies in the region of the failure to detect a signal for the correlation detector. CORRELATION DETECTOR a no signal a yes signal

Generalized Detector : Indicator : Generalized Detector : Indicator if a yes signal exists in the input stochastic process, and the model signal generator is switched off The background noise a yes signal a yes signal a no signal a no signal

The case 1 : : The case 1 : The decision statistics at the output of the generalized detector are shown in Figs. 7.18 (case 1) for and a no signal in the input stochastic process within the limits of the time interval [0,T] (readings 61-68), a no signal

The case 2: : The case 2: The signal and the model signal appear to be correlated for . - the decision statistics at the output of the generalized detector are determined by Eq. (7.16), The signal is clearly detected at the reading 30.

Generalized Detector : Indicator : Generalized Detector : Indicator The case : We can see that the decision statistics at the output of the generalized detector are shifted downward by the value - and relative to the zero axis when the model signal generator is switched off,

The case 2: : The case 2: - With an increase in the amplitude of the model signal, the variance of the decision statistics at the output of the generalized detector decreases. - this variance reaches its minimum and is equal to that of the background noise at the output of the generalized detector.

Conclusion – weak signals : Conclusion – weak signals The signal is not detected using the correlation detector when the input power signal-to-noise ratio is equal to 0.96 dB. The signal is clearly detected during the use of the generalized detector for the input power signal-to-noise ratio 0.96 dB. The generalized detector is more informative during signal detection and estimation of the signal parameters than the correlation detector. The coincidence in time between the model signal and signal within the limits of the time interval [0,T] allows us to define the position of the signal on the time axis. The minimum of the variance of the decision statistics at the output of the generalized detector at the reading 30 is equal to that of the background noise at the output of the generalized detector. The energy parameters of the signal are determined with the help of this fact.

Slide 29 : Thank you for your attention!