Introduction The cumulative distribution function(CDF) for the Gaussian distribution is: X-m F(x)=fexp X-m k=Q()=(1/2)e √2πo 2c Where the Q function is defined by Q(=) exp(-)dn T And the error function(erf) defined as erT ex(2) √2兀 And the complementary error function(erfc)defined as 2r exp(-n)aa And e()=1erf(2) ere()=2-2Q√2z) or erf(=)=2Q(√2z)-1
11 Introduction • The cumulative distribution function (CDF) for the Gaussian distribution is: ) 2σ - ) = (1/ 2) ( σ - ] = ( 2σ ( - ) exp[- 2πσ 1 ( ) = ∫-∞ 2 2 x x x mx x erfc x m dz Q z m F x • Where the Q function is defined by: ) λ 2 λ exp(- 2π 1 ( ) = 2 ∞ Q z ∫ d z • And the error function (erf) defined as: exp(-λ ) λ 2π 1 ( ) = ∫ ∞ 2 erfc z d z • And the complementary error function (erfc) defined as: exp(-λ ) λ 2π 1 ( ) = z 2 ∫0 erf z d • And ( ) = 2 2 2 or ( ) = 2 2 -1 ( ) = 1 erfc z - Q( z) erf z Q( z) erfc z -erf(z)
6.2 Power Spectral Density (definition) The definition of the psd for the case of deterministic waveform is Eq (2-66) w(f o(=lim T→∞ T Definition: The power spectral density (PSD)for a random process x(t) is given by ()=1im(m( (6-42) → where Xr()=,x(t)e2mat(6-43)
12 6.2 Power Spectral Density (definition) • The definition of the PSD for the case of deterministic waveform is Eq.(2-66): (2 - 66) ( ) ( ) = lim 2 →∞ ω T W f f T P • Definition: The power spectral density (PSD) for a random process x(t) is given by: ) (6 - 42) [ ( ) ] ( ) = lim ( 2 →∞ T X f f T T Px • where ( ) ∫ ( ) (6 - 43) T/2 T/2 2π － ＝ － X f x t e dt j f t T
6.2 Power Spectral Density (Wiener-Khintchine Theorem) When x(t) is a wide-sense stationary process, the Psd can be obtained from the Fourier transform of the autocorrelation function ()=[R(=R2()et6-44 Converse R()=y|m;()=r()e/2nd(6-45) Provided that r(t) becomes sufficiently small for large values of t so that τR3(r)|t<(6-46) This theorem is also valid for a nonstationary process, provided that we replace r(t) by <r(t, t+t)>. ° Proof:( notebook p) 13
13 6.2 Power Spectral Density (Wiener-Khintchine Theorem) • When x(t) is a wide-sense stationary process, the PSD can be obtained from the Fourier transform of the autocorrelation function: ( ) = [ (τ)] = ∫ (τ) τ (6 - 44) ∞ -∞ - 2π τ ω f R R e d j f P F x x • Conversely, (τ) = [ ( )] = ∫ ( ) (6 - 45) ∞ -∞ 1 2π τ R f f e df j f x Px Px －F • Provided that R(τ) becomes sufficiently small for large values of τ, so that ∫| τ (τ) | τ < ∞ (6 - 46) ∞ -∞ Rx d • This theorem is also valid for a nonstationary process, provided that we replace R(τ) by < R(t,t+τ) >. • Proof: (notebook p)
6.2 Power Spectral Density (Wiener-Khintchine Theorem There are two different methods that may be used to e evaluate the PSd of a random process I direct method Pr()=lim xr()] 2 using the indirect method by evaluating the fourier → transform ofr(τ), where r(τ) has to obtained first Properties of the psd ·(1) PU is always real ·(2)(>=0; e.(3)When x() is real, PG-f=r); (4)When x(t)is wide-sense stationary, e()df-P=X-R,()6 (5)2(0)=」R3()dr(6-55
14 6.2 Power Spectral Density (Wiener-Khintchine Theorem) • There are two different methods that may be used to evaluate the PSD of a random process: ) (6 - 42) [ ( ) ] 1 ( ) = lim ( 2 →∞ T X f f T T Px direct method • 2 using the indirect method by evaluating the Fourier transform of Rx (τ) , where Rx (τ) has to obtained first • Properties of the PSD: • (1) Px (f) is always real; • (2) Px (f)>=0; • (3) When x(t) is real, Px (-f)= Px (f); • (4) When x(t) is wide-sense stationary, ( ) P x (0) (6 - 54) ∞ 2 ∫-∞ x df Rx P f ＝ = = (0) ∫ ( ) (6 - 55) ∞ -∞R d Px = x (5)
6.2 Power Spectral Density Example 6-3: (notebook p (b) signaling Pulse Shape x(f) eP,0)-T (inTo) (e) Power Spectral Density of a Polar signal Figure c-5 Random polar signal and its PSD
15 6.2 Power Spectral Density • Example 6-3:(notebook p)