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1、1,Chapter 2 Signals,2.1 Classification of Signals2.1.1 Deterministic signals and random signalsWhat is deterministic signal?What is random signal?2.1.2 Energy signals and power signalsSignal power: Let R = 1, then P = V2/R = I2R = V2 = I2Signal energy：Let S represent V or I，if S varies with time，the
2、n S can be rewritten as s(t), Hence, the signal energy E = s2(t)dtEnergy signal satisfies Average power: , then P = 0 for energy signal.For power signal: P 0, i.e., power signal has infinite duration.Energy signal has finite energy, but its average power equals 0.Power signal has finite average powe
3、r, but its energy equals infinity.,2,2.2 Characteristics of deterministic signals,2.2.1 Characteristics in frequency domainFrequency spectrum of power signal: let s(t) be a periodic power signal, its period is T0, then we havewhere 0 = 2 / T0 = 2f0 C(jn0) is a complex function, C(jn0) = |Cn|ejnwhere
4、 |Cn| amplitude of the component with frequency nf0 n phase of the component with frequency nf0Fourier series of signal s(t):,3,【Example 2.1】 Find the spectrum of a periodic rectangular wave. Solution: Assume the period of a periodic rectangular wave is T , the width is , and the amplitude isV, then
5、Its frequency spectrum is,4,Frequency spectrum figure,5,【Example 2.2】Find the frequency spectrum of a sinusoidal wave after full-wave rectification.Solution：Assume the expression of the signal isIts frequency spectrum:The Fourier series of the signal is:,6,Frequency spectral density of energy signal
6、sLet an energy signal be s(t), then its frequency spectral density isThe inverse Fourier transform of S() is the original signal:【Example 2.3】Find the frequency spectral density of a rectangular pulse.Solution: Let the expression of the rectangular pulse beThen its frequency spectral density is its
7、Fourier transform:,7,【Example 2.4】Find the waveform and the frequency spectral density of a sample function.Solution: The definition of the sample function isthe frequency spectral density Sa(t) is:From the above equation, we see that Sa() is a gate function.【Example 2.5】Find the unit impulse functi
8、on and its frequency spectral density. Solution: Unit impulse function is usually called d function d(t). Its definition isThe frequency spectral density of (t):,8,d(t) and its frequency spectral density:Physical meaning of function:It is a pulse with infinite height, infinitesimal width, and unit a
9、rea.Sa(t) has the following property:When k , amplitude , and the zero-spacing of the waveform 0，Hence,9,Characterisitics of (t) (t) is an even function:(t) is the derivative of unit step function:Difference between frequency spectral density S(f) of energy signal and frequency spectrum of periodic
10、power signal:S(f ) continuous spectrum； C(jn0) discreteUnit of S(f ): V/Hz； Unit of C(jn0): VAmplitude of S(f ) at a frequency point infinitesimal,10,【Example 2.6】Find the frequency spectral density of a cosinusoidal wave with infinite length. Solution: Let the expression of a cosinusoidal wave be f
11、 (t) = cos0t, then according to eq. (2.2-10), F() can be written asReferencing eq.(2.2-19), the above equation can be written as:Introducing (t), the concept of frequency spectral density can be generalized to power signal.,11,Energy spectral densityLet the energy of an energy signal s(t) be E, then
12、 the energy of the signal is decided byIf its frequency spectral density is S(f ), then from Parsevals theorem we havewhere |S(f )|2 is called energy spectral density. The above equation can be rewritten as：where G(f )|S(f)|2 （J / Hz） is energy spectral density.Property of G(f ): Since s(t) is a rea
13、l function, |S(f )|2 is an even function,12,Power spectral densityLet the truncated signal of s(t) is sT(t)，-T/2 t T/2, thenTo define the power spectral density of the signal as:obtain the signal power:,13,2.2.2 Characteristics in time domainAutocorrelation functionDefinition of the autocorrelation
14、function for energy signal:Definition of the autocorrelation function for power signal: Characteristics:R() is only dependent on , but independent of t.When = 0, R() of energy signal equals the energy of the signal, and R() of power signal equals the average power of the signal.,14,Cross-correlation
15、 functionDefinition of the cross-correlation function for energy signal:Definition of the cross-correlation function for power signal:Characteristics:1. R12() is dependent on , and independent of t.2. Proof: Let x = t + ，then,15,2.3 Characteristics of random signals,2.3.1 Probability distribution of
16、 random variableConcept of random variable: If the random outcome of a trial A is expressed by X, then we call X a random variable, and let its value be x. For example, the number of calls received within a given period of time at the telephone exchange is a random variable.Distribution function of
17、random variableDefinition: FX(x) = P(X x) Characteristics: P(a X b) + P(X a) = P(X b),P(a X b) = P(X b) P(X a)， P(a X b) = FX(b) FX(a),16,Distribution function of discrete random variable:Let the values of X be: x1 x2 xi xn，their probabilities are respectively p1, p2, , pi, , pn, thenP (X x1) = 0,P(
18、X xn) = 1 P(X xi) = P(X = x1) + P(X = x2) + + P(X = xi), Characteristics: FX(- ) = 0 FX(+) = 1 If x1 x2, then FX(x1) FX(x2) - monotonic increasing function.,17,Distribution function of continuous random variable:When x is continuous, from the definition of distribution function FX(x) = P(X x) we kno
19、w that FX(x) is a continuous monotonic increasing function.,18,2.3.2 Probability density of random variableProbability density of continuous random variable pX (x)Definition of pX (x):Meaning of pX (x):pX (x) is the derivative of FX (x), and is the slope of the curve of FX (x) P(a X b) can be found
20、from pX (x):Characteristics of pX (x):,pX(x) 0,19,Probability density of discrete random variableDistribution function of discrete random variable can be written as: where pi probability of x = xi u(x) unit step functionFinding the derivatives of the two sides of the above equation, we obtain its pr
21、obability density:Characteristics:When x xi , px (x) = 0 When x = xi , px (x) = ,20,2.4 Examples of frequently used random variables,Random variable with normal distributionDefinition: Probability densitywhere 0, a = const.Probability density curve:,21,Random variable with uniform distributionDefini
22、tion: probability densitywhere a, b are constants.Probability density curve:,22,Random variable with Rayleigh distributionDefinition: Probability densitywhere a 0, and is a constant.Probability density curve:,23,2.5 Numerical characteristics of random variable,2.5.1 Mathematical expectationDefinitio
23、n: for continuouse random variableCharacteristics: If X and Y are independent of each other, and E(X) and E(Y) exist, then,24,2.5.2 VarianceDefinition: where Variance can be rewritten as:Proof:For discrete variable:For continuous variable:Characteristics:D( C ) = 0 D(X+C)=D(X)，D(CX)=C2D(X) D(X+Y)=D(
24、X)+D(Y)D(X1 + X2 + + Xn)=D(X1) + D(X2) + + D(Xn),25,2.5.3 MomentDefinition: the k-th moment of a random variable X isk-th origin moment is the moment when a = 0:k-th central moment is the moment when :Characteristics: The first origin moment is the mathematical expectation:The second central moment
25、is the variance:,26,2.6 Random process,2.6.1 Basic concept of random processX(A, t) ensumble consisting of all possible “realizations” of an event AX(Ai , t) a realization of event A, it is a determined time functionX(A, tk) value of the function at the given time tkDenote for short: X(A, t) X(t) X(
26、Ai , t) Xi (t),27,Example: receiver noiseNumerical characteristics of random process:Statistical mean:Variance:Autocorrelation function:,28,2.6.2 Stationary random processDefinition of stationary random process:A random process whose statistical characteristics is independent of the time origin is c
27、alled a stationary random process. (or, strict stationary random process) Definition of generalized stationary random process: The random process whose mean, variance and autocorrelation function are independent of the time originCharacteristics of generalized stationary random process: A strict sta
28、tionary random process must be a generalized stationary random process; but a generalized stationary random process is not always a strict stationary random process.,29,2.6.3 ErgodicitySignificance of ergodicityA realization of a stationary random process can go through all states of the process.Cha
29、racteristic of ergodicity: time average may be replaed by statistical mean. For example,Statistical mean of ergodic process mX：Autocorrelation function of ergodic process RX():If a random process has ergodicity, then it must be a strict stationary random process. However, a strict stationary random
30、process is not always ergodic.,30,Ergodicity of stationary communication system If the signal and the noise are both ergodic, thenFirst origin moment mX = EX(t) D. C. component of signalSquare of first origin moment mX 2 power of normalized D.C. component of signalSecond origin moment E X 2( t ) nor
31、malized average power of signalSquare root of second origin moment E X 2(t)1/2 root mean square of signal current or voltageSecond central moment X2 normalized average power fo A. C. component of signalIf mX = mX 2 = 0, thenX2 = E X 2( t ) ；Standard deviation X root mean square of A. C. component of
32、 signal If mX = 0, then X is root mean square of signal,31,2.6.4 Autocorrelation function and power spectral density of stationary random processCharacteristics of autocorrelation function,32,Characteristics of power spectral density Review: power spectral density of deterministic signalSimilarly, p
33、ower spectral density of stationary random process equals:Average power:,33,Relationship between autocorrelation function & power spectral densityFrom where, Let =t t，k =t + t, then the above equation can be reduced to Hence,34,The above equation demonstrates that PX(f ) and R( ) are a pair of Fouri
34、er transform.Characteristics of PX(f ):PX(f ) 0, and PX(f ) is a real function. PX(f ) PX(-f ), i.e., PX(f ) is an even function 【Example 2.7】Let a binary signal be x(t) as shown in the figure. Its amplitude is +a or a; and the number k of its sign changes in time interval T obeys Poisson distributi
35、on: where, is average number of sign changes of amplitude in unit time.Find its autocorrelation function R() and power spectral density P(f).,35,Solution: It can be seen from the above figure, x(t)x(t-) has only two possible values: a2 or -a2. Hence, equation can be reduced to R() = a2 occurrence pr
36、obability of a2 + (-a2) occurrence probability of (-a2)where, the occurrence probability can be calculated according to Poisson distribution P(k).If the number of sign changes of x(t) is even in second, then +a2 occurs; if the number of sign changes of x(t) is odd in second, then -a2 occurs。Therefor
37、eUse instead of T in Poisson distribution, then obtain,36,The in Poisson distribution is a time interval, so it should be non negative. Hence, when the value of is negative, the above equation should be rewritten asCombining the above two equations, finally obtain:The P( f ) can be obtained from the
38、 Fourier transform of the R( ): Curves of P( f ) and R():,37,【Example 2.8】Assume the power spectral density P( f ) of a random process is shown as in the figure. Find its autocorrelation function R().Solution:P( f ) is known, where,Curve of autocorrelation function:,38,【Example 2.9】Find the autocorr
39、elation function and the power spectral density of white noise. Solution: White noise has uniform power spectral density Pn( f ):Pn( f ) n0/2where, n0 - single side power spectral density（W/Hz） The autocorrelation function of white noise can be obtained from its power spectral density: As can be see
40、n the samples of white noise at any two adjacent instants (i.e., 0) are uncorrelated. Average power of white noise: The above equation shows that the average power of white noise is infinity.,39,Power spectral density & autocorrelation function of band-limited white noisePower spectral density of ba
41、ndlimited white noise:If the bandwidth of a white noise in limited in the interval (-fH, fH), then we have Pn(f) = n0 / 2,-fH f fH= 0,elseits autocorrelation functin is:Curves:,40,2.7 Gaussian process,DefinitionOne dimensional probability density of Gauss process:where, a = EX(t) - mean 2 = EX(t) -
42、a2 - variance - standard deviationGauss process is a stationary process, hence its probability density pX (x, t1) is independent of t1i.e., pX (x, t1) pX (x)Curve of pX (x):,41,Strict definition of Gaussian process: Joint probability density of arbitrary dimension satisfying the following condition:
43、where, ak - mathematical expetaction of xk k - standard deviation of xk |B| - determinant of the normalized covariance matrix:|B|jk - algebraic cofactor of bjk in |B| bjk - normalized covariance function, i.e.,42,Characteristics of n-dimensional Gauss processpX (x1, x2, , xn; t1, t2, , tn) is decide
44、d only by ai , i , and bjk of every random variables, so it is a generalized stationary random process.If x1, x2, , xn are uncorrelated one another, then when j k, bjk = 0. Now,i.e., the n-dimensional joint probability density equals the product of each one dimensional probability density.If the cro
45、ss-correlation function of two random variables equals 0, then they are uncorrelated to each other; if the two dimensional joint probability density of two random variables is equal to the product of the one dimensional probability densities, then it is independent of each other. Two uncorrelated ra
46、ndom variables are not always independent of each other; and two independent random variables are certainly uncorrelated.Random variables of Gaussian process are uncorrelated and independent of one another.,43,Characteristics of probability density of normal distributionp(x) is symmetrical to x = a，
47、i.e.,p(x) is monotonically increasing in (-, a), and monotonically decreasing in (a, ), and reaches its max. at a. The maximum value is When x - or x + , p(x) 0. If a = 0, = 1，then the distribution is called standard normal distribution:,44,Normal distribution functionThe integral of normal probabil
48、ity density function is defined as normal distribution function. It can be expressed as:where, (x) - probability integral function:This integral is difficult to calculate. Usually table-lookup method is used instead of calculation.,45,Normal distribution expressed by error functionDefinition of erro
49、r function:Definition of Complementary error function: Expression of normal distribution,46,频率近似为fc,2.8 Narrow band random process,2.8.1 Basic concept of narrow band random processWhat does it mean narrow band? Assume the bandwidth of a random process is f , the central frequency is fc . If f fc , t
50、hen the random process is called a narrow band random process.Waveform and expression of narrow band random processWaveform and spectrum:,47,Expressionwhere, aX(t) random envelope of narrow band random processX(t) random phase of narrow band random process 0 angular frequency of sinusoidal wave The
