Communication Theory. I. Frigyes 2009-10/II. http://docs.mht.bme.hu/~frigyes/hirkelm hirkelm01bEnglish. Topics. (0. Math. Introduction: Stochastic processes, Complex envelope) 1. Basics of decision and estimation theory 2. Transmission of digital signels over analog channels: noise effects

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Communication Theory

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Stochastic processes • Also called random waveforms. • 3 different meanings: • As a function of ξnumber of realizations: a series of infinite number of random variables ordered in time • As a function of time t:a member of a time-function family of irregular variation • As a function of ξ and t: one member of a family of time functions drawn at random

Stochastic processes: how to characerize them? • According to the third definition • And with some probability distribution. • As the number of random variables is infinite: with their joint distribution (or density) • (not only infinite but continuum cardinality) • Taking these into account:

Stochastic processes: how to characerize them? • (Say: density) • First prob. density of x(t) • second: joint t1,t2 • nth:n-fold joint • The stochastic process is completly characterized, if there is a rule to compose density of any order (even for n→). • (We’ll see processes depending on 2 parameters)

Stochastic processes: how to characerize them? • Comment: although precisely the process (function of t and ξ) and one sample function (function of t belonging to say ξ16)are distinguished we’ll not always make this distinction.

45o 45o Continuing the example: In two distinct time-slots In the same time-slot

Stochastic processes: the Gaussian process • A stoch. proc. is Gaussianif its n-th density is that of an n-dimensional vector random variable • m is the expected value vector, K the covariance matrix. • nth density can be produced if are given • are given

Stochastic processes: the Gaussian process • An interesting property of Gaussian processes (more precisely: of Gaussian variables): • These can be realizations of one process at different times

Stochastic processes: stationary processes • A process is stationary if it does not change (much) as time is passing • E.g. the semirandom binary signal is (almost) like that • Phone: to transmit 300-3400 Hz sufficient (always, for everybody). (What could we do if this didn’t hold?) • etc.

Stochastic processes: stationary processes • Precise definitions: what is almost unchanged: • A process is stationary (in the strict sense) if for the distribution function of any order and any at any time and time difference • Is stationary in order n if the first n distributions are stationary • E.g.: the seen example is first order stationary • In general: if stationary in order n also in any order <n

Stochastic processes: stationary processes • Comment: to prove strict sense stationarity is difficult • But: if a Gaussian process is second order stationary (i.e. in this case: if K(t1,t2) does not change if time is shifted) it is strict sense (i.e. any order) stationary. As: if we know K(t1,t2)nth density can be computed (any n)

Stochastic processes: stationarity in wide sense • Wide sense stationary: if the correlation function is unchanged if time is shifted (to be defined) • A few definitions:. • a process is called a Hilbert-process if • (That means: instantaneous power is finite.)

Stochastic processes: wide sense stationary processes • (Auto)correlation function of a Hilbert-process: • The process is wide sense stationary if • the expected value is time-invariant and • R depends only on τ=t2-t1 for any time and any τ.

e T Stochastic processes: once again on binary transmission • The semi-random binary transmission can be transformed in random by introducing a dummy random variable e distributed uniformly in (0,1) • like x:

-T T τ Stochastic processes: once again on binary transmission • I.e. :

Stochastic processes: other type of stationarity • Given two processes, x and y, these are jointly stationary, if their joint distributions are alle invariant on any τtime shift. • Thus a complex process is stationary in the strict sense if x and y are jointly stationary. • A process is periodic (or ciklostat.) if distributions are invariant to kT time shift

Stochastic processes: comment on complex processes • Appropriate definition of correlation for these: • A complex process is stationary in the wide sense if both real and imaginary parts are wide sense stationary and they are that jointly as well

Stochastic processes: continuity • There are various definitions • Mean square continuity • That is valid if the correlation is continuous

Stochastic processes: stochastic integral • x(t) be a stoch. proc. Maybe that Rieman integral exists for all realizations: • Then s is a random variable (RV). But if not, we can define an RV converging (e.g. mean square) to the integral-approximate sum:

Stochastic processes: time average • Integral is needed – among others –to define time average • Time average of a process is its DC component; • time average of its square is the mean power • definition:

Stochastic processes: time average • In general this is a random variable. It would be nice if this were the statistical average. This is really the case if • Similarly we can define

Stochastic processes: time average • This is in general also a RV. But equal to the correlation if • If these equalities hold the process is called ergodic • The process is mean square ergodic if

x(t) y(t) FILTERh(t) Spectral density and linear transformation • As known in time functions output function is convolution • h(t): impulse response

Spectral density and linear transformation • Comment.: h(t<0)≡ 0; (why?); and: h(t) = F-1[H(ω)] • It is plausible: the same for stochastic processes • Based on that it can be shown : • (And also )

x(t) y(t) FILTERh(t) H(ω) Sy(ω) (its integral is negative) Spectral density and linear transformation • FurtherS(ω) ≥ 0 (all frequ.) • For: if not, there is a domain where S(ω) <0 (ω1, ω2) Sx(ω)

H(ω) ω Spectral density and linear transformation • S(ω) is the spectral density (in rad/sec).As:

Modulated signals – the complex envelope • In previous studies we’ve seen that in radio, optical transmission • one parameter is influenced (e.g. made proportional) • of a sinusoidal carrier • by the modulating signal . • A general modulated signal:

Modulated signals – the complex envelope • Here d(t) and/or (t) carries theinformation – e.g. are in linear relationship with the modulating signal • An other description method (quadrature form): • d, , a and q are real time functions – deterministic or realizations of a stoch. proc.

Modulated signals – the complex envelope • Here a+jq is the complex envelope. Question: when, how to apply. • To beguine with: Fourier transform of a real function is conjugate symmetric: • But if so: X(ω>0)describes the signal completly: knowing that we can form theω<0 partand, retransform.

↓„Hilbert” filter Modulated signals – the complex envelope • Thus instead of X(ω) we can take that: • By the way: • The relevant time function:

Modulated signals – the complex envelope • We can write: • The shown inverse Fourier transform is 1/t. • So • Imaginary part is the so-callerd Hilbert-transzform of x(t)

Modulated signals – the complex envelope • Now introduced function is the analyticfunction assigned to x(t) (as it is an analytic function of the z=t+ju complexvariable). • An analytic function can be assigned to any (baseband or modulated) function; relationship between the time function and the analytic function is

Modulated signals – the complex envelope • It is applicable to modulated signals: analytic signal of cosωct is ejωct. Similarly that of sinωct is jejωct. So if quadrature components of the modulated signal a(t), q(t) are • band limited and • their band limiting frequency is < ωc/2π (narrow band signal) • then NB. Modulation is a linearoperation in a,q: frequencydisplacement.

Modulated signals – the complex envelope • Thus complexenvelope determines uniquely the modulated signals. In the time domain • Comment: according to its name can be complex. (X(ω) is not conjugate symmetric around ωc.) • Comment 2: if the bandwidt B>fc, is not analytic, its real part does not define the modulated signal.) • Comment 3: a és q can be independent signals (QAM) or can be related (FM or PM).

X(ω) X(ω) X˚(ω) X̃(ω) ω Modulated signals – the complex envelope • In frequency domain? On analytic signal we saw.