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i i i in Software Development European Article Number 13 in Software i i i




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i i i using software toincoporate ean/ucc-13 with asp.net web,windows application ASP.NET i i Trante Software ean13+5 rBook 2003/11/18 16:12 page 513 #531. Section 13.3. Random Process Models Input PSD S ~~ ( f ) = A2 ( f f 0 ) xx Output PSD S ~~ ( f ) = A2 Sh h ( f f 0 ) ~~ yy Figure 13.11 Spectral spreading in a time-varying system. rate treati Software EAN-13 ng the system as time invariant and increasing the sampling rate to accommodate the excess bandwidth, which is equal to the bandwidth of the random process modeling the time-varying nature of the system. A more general version of the time-varying model treats h( , t) as a stationary random process in t with an autocorrelation function Ree ( 1 , 2 , ) = E h ( 1 , t)h( 2 , t + ) hh (13.54).

The most co mmonly used model for h( , t) is a zero mean stationary Gaussian process that leads to a Rayleigh probability density function for h( , t) . In this model, it is usually assumed that h( 1 , t) and h( 2 , t) are uncorrelated for 1 = 2 . In other words: Ree ( 1 , 2 , ) = E h ( 1 , t)h( 2 , t + ) hh = Ree ( 1 , ) ( 1 2 ) hh (13.

55). For this ca ean13+2 for None se, the autocorrelation of the output of the system can be obtained from Ryy ( ) = E {y (t)y(t + )} ee Substituting for y(t) yields Ryy ( ) = ee. (13.56). h ( 1 , t)x (t 1 ) d 1 h( 2 , t + )x(t + 2 ) d 2 (13.57). i i i i i Trante Software EAN-13 rBook 2003/11/18 16:12 page 514 #532. Modeling and Simulation of Time-Varying Systems 13 . Interchanging orders of expectation and integration results in Ryy ( ) = ee E h ( 1 , t)h( 2 , t + )x (t 1 )x(t + 2 ) d 1 d 2 (13.58). Assuming x(t) and h(t) independent, which is certainly reasonable, yields Ryy ( ) = ee E h ( 1 , Software ean13 t)h( 2 , t + ) (13.59). E {x (t 1 )x(t + 2 )} d 1 d 2 Recognizing Software EAN-13 that the two expectations are autocorrelation functions and invoking (13.55) provides the simpli cation. Ryy ( ) = ee Ree ( 1 , ) ( 1 2 )Rxx ( + 1 2 ) d 1 d 2 ee hh (13.60). Performing the integration on 2 using the sifting property gives Ryy ( ) = Rxx ( ) ee ee which can be expressed by Ree ( 1 , ) d 1 hh (13.61). Ryy ( ) = Rxx ( )Ree ( ) ee ee hh where Ree ( ) = hh (13.62). Ree ( 1 , ) d 1 hh (13.63). The power s Software EAN13 pectral density of the output can be obtained by taking the Fourier transform of (13.62), which leads to the convolution Syy (f ) = Sxx (f ) ee ee See (f ) hh (13.64).

where , a s always, denotes convolution. Note that See (f ) is averaged power hh spectral density and is the Fourier transform of the averaged autocorrelation function de ned in (13.63).

Note that the output power spectral density is a convolution of the input power spectral density and the averaged power spectral density of the random process that models the time variations. In contrast, the power spectral density relationship for an LTIV system is given by Syy (f ) = Sxx (f ) . H(f ). ee ee (13.65). Once again, when the input is a tone, an LTIV system produces an output tone at the same frequency, whereas the output of an LTV system could be shifted and spread in frequency.. i i i i i Trante rBook 2003/11/18 16:12 page 515 #533. Section 13.4. Simulation Models for LTV Systems Simulation Models for LTV Systems Given a description of an L TV system in the form of its impulse response h( , t), a simulation model can be derived using the sampling theorem assuming that the input to the channel is bandlimited [1]. We start from the input-output relationship given by the convolution integral. y(t) =. h( , t)x(t ) d and use the sampling theore EAN/UCC-13 for None m to represent the input in terms of its sampled values. From the sampling theorem we know that a lowpass signal w( ) bandlimited to B Hz can be represented in terms of its sampled values as. w( ) =. n= . w(nT ). sin(2 B( nT )) 2 B( n ean13+5 for None T ). (13.66). where 1/T is the sampling r EAN/UCC-13 for None ate, which is set equal to the Nyquist rate 2B. The minimum sampling rate of 2B is chosen to minimize the computational burden of the simulation model. Using the representation given above, with x(t ) = w( ), we can replace x(t ) in the convolution integral by.

x(t ) =. n= . x(t nT ). sin(2 B( nT )) 2 B( n Software EAN13 T ). (13.67). which leads to y( ) =. h( , t). n= . x(t nT ). sin(2 B( nT )) 2 B( n T ) sin(2 B( nT )) 2 B( nT ). d d (13.68). n= . x(t nT ). h( , t).
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