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Fundamentals of electronic equipment computer aided design

1. Detection theory and signal detection

Signal is a carrier of new information for the observer. Presence of it in the input process causes statically characteristics change of this process. It can be change of average value (shift parameter), dispersion change (scale parameter), change of a correlation function (frequency spectrum of a power), change of phase distribution (time delay) etc. in general case it is possible to say that the presence of a signal leads to the change of multidirectional distribution of signal mixtures and interferences probabilities.

Detection of signals at a time of interference action is one of the fundamental tasks in theory signal processing. The mixture of signal and interferences on the input of the received device I general case is a random process. In case of signal absence it is a process with definite statically characteristics (values of probability distribution, correlation function). In most cases random process is known to be stationary and ergodic.

In digital systems of signal processing random processes are discretized in time, that means that analogue-digitizer (analogue-digital converter) registries the realization process values through definite intervals of discretization Дt.

Results of registrations make the sequence of random values

This sequence can be written in such a way:

and be called a sample.

A sample of a random process is multidetectional random value and is characterized by multidetectional distribution of probability function

The result of digital experiment over random process X (t) is a registration of series n coordinate values of its definite realization

which is called random process sample realization.

The statement of task of signal detection in such that relatively to a sample realization two hypothesis are advanced: hypothesis H₀ - realization of sample contains only interference; alternative hypothesis H₁ - realization of sample contains interference and signal.

During making decision about choice between H₀ and H₁two types of errors are possible. First of all there could be errors connected with correct zero hypothesis deviation H_0. Such error is called an error of the first type. Its probability is defined by the letter б. Secondly, it is possible to recognize alternative H₁ incorrect when it will happen. It will be the error of the second order. Its probability is defined by letter в.

In theory of signal detection the error of the first order is called false alert, and corresponding probability б - probability of false alert. The situation of the missing signal corresponds to the error of the second order. The probability of signal missing в is directly connected with the probability of correct signal detection D.

2. Detectors

Classification of signal detector

A detector is a device that recovers information of interest contained in a modulated wave. The term dates from the early days of radio when all transmissions were in Morse code, and it was only necessary to detect the presence (or absence) of a radio wave using a device such as a coherer without necessarily making it audible.

During the study of course fundamentals of electronic equipment computer aided design we observe three types of detectors such as parametric, adaptive and nonparametric detectors, which have differences in their work.

a) Parametric detector

For the parametric detector:

б - Var, в - Var - is no stability meanings

A parametric detector contains a coupled film sensor stripe and an excitation conductor which, electrically insulated from each other, cross at right angles. A sensor stripe consists of a pair of super positioned stripes which are separated by a thin nonmagnetic layer. A high frequency excitation current in the conductor produces an inductive output signal in the sensor stripe. The waveform of this periodic signal is responsive to domains proximate to the sensor/conductor crossover area.

A parametric detector comprising a coupled film sensor strip, and excitation means interacting over a limited area with said stripe wherein the passage of a periodically varying current in said excitation means produces a similarly periodic inductive output signal from said stripe having a waveform responsive to the present of a domain proximal to said limited area.

b) Adaptive Detector

For the adaptive detector:

б - is constant, в - Var - no stability meanings

3. Let's define the detection abilities of different detector's types

signal detector information

We must choose the detector which shows the smallest probability of errors with our given data.

According to my number in the list of our group (N=10), I have such values:

m - number of ranks (for non-parametric detector)

C1 - loss from a false alert

m = 5

C1 = 0.1

R= C1б + (1-C1)в

Text of a program:

Parametrical

DIM x(200), y(200)

ps = 1: s = 5: v = 2: ai = 0: bi = 0: mx = 2: my = 10: ax = 200: m = 5: c1 = 0.1:

FOR i = 1 TO 200

x(i) = SQR (-2 * ps * LOG(RND)) * COS (6.28 * RND)

IF i > 100 THEN x(i) = x(i) + s

IF x(i) > v THEN y(i) = 1 ELSE y(i) = 0

NEXT i

FOR i = 1 TO 200

IF i < 100 AND y(i) = 1 THEN ai = ai + 1 / 100

IF i > 100 AND y(i) < 1 THEN bi = bi + 1 / 100

NEXT i

R = c1 * ai + (1 - c1) * bi

PRINT ai, bi, R

Adaptive

DIM x(200), y(200)

ps = 1: s = 5: v = 2: ai = 0: bi = 0: mx = 2: my = 10: ax = 200: m = 5: c1 = 0.1:

ymax = -100: ymin = 100: k = 1.8

FOR i = 1 TO m

y1 (i) = SQR (-2 * ps * LOG(RND)) * COS (6.28 * RND)

IF ymax < y1 (i) THEN ymax = y1 (i)

IF ymin > y1 (i) THEN ymin = y1 (i)

NEXT i

sm = (ymax - ymin) / 6

v = sm * k

FOR i = 1 TO 200

x(i) = SQR (-2 * ps * LOG(RND)) * COS (6.28 * RND)

IF i > 100 THEN x(i) = x(i) + s

IF x(i) > v THEN y(i) = 1 ELSE y(i) = 0

NEXT i

FOR i = 1 TO 200

IF i < 100 AND y(i) = 1 THEN ai = ai + 1 / 100

IF i > 100 AND y(i) < 1 THEN bi = bi + 1 / 100

NEXT i

R = c1 * ai + (1 - c1) * bi

PRINT ai, bi, R

For my variant

Our input signal looks like

When у=1

When у=3

When у=5

For parametrical adder

When у=1

When у=3

When у=5

For adaptive adder

When у=1

When у=3

When у=5

Our adder:

Conclusion

signal detector information

In our work we investigate three type of detectors: parametric and adaptive for all we calculated probability of first and second kind error and risk. Due to my data, i make the conclusion, that the most optimal parametrical detector, because it have the smallest number of R - risk. Of course it have great probability of the first kind error, but it is most optimal choice between our detectors.

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