Praise for Fundamentals of WiMAX

96 Chapter 3 • The Challenge of Broadband Wireless Channels3.5.2.1 Time CorrelationIn the time domain, the channel h( τ = 0, t)can intuitively be thought of as consisting of approximatelyone new sample from a Rayleigh distribution every T cseconds, with the values inbetween interpolated. But, it will be useful to be more rigorous and accurate in our descriptionof the fading envelope. As discussed in Section 3.4, the autocorrelation function A ( ∆ t ) tdescribes how the channel is correlated in time. Similarly, its frequency-domain Doppler powerspectrum ρ t( ∆f) provides a band-limited description of the same correlation, since it is simplythe Fourier transform of A ( ∆ t ) . In other words, the power-spectral density of the channelth( τ = 0, t)should be ρ t( ∆f). Since uncorrelated random variables have a flat power spectrum, asequence of independent complex Gaussian random numbers can be multiplied by the desiredDoppler power spectrum ρ t( ∆f); then, by taking the inverse fast fourier transform, a correlatednarrowband sample signal h( τ = 0, t)can be generated. The signal will have a time correlationdefined by ρ t( ∆f) and be Rayleigh, owing to the Gaussian random samples in frequency.For the specific case of uniform scattering [16], it can been shown that the Doppler powerspectrum becomes⎧ Pr1f fDπ fρt( f )= 4 , | ∆ | ≤⎪ ∆ 2∆ ⎨ fD1 − ( )f⎪D⎪⎩0, ∆f> fD. (3.45)A plot of this realization of ρ t( ∆f) is shown in Figure 3.15. It is well known that the inverseFourier transform of this function is the 0th order Bessel function of the first kind, which is oftenused to model the time autocorrelation function, A ( δ t ) , and hence predict the time-correlationcproperties of narrowband fading signals. A specific example of how to generate a Rayleigh fadingsignal envelope with a desired Doppler f D, and hence channel coherence time T , isc≈ f−1Dprovided in Matlab (see Sidebar 3.4).3.5.2.2 Frequency CorrelationSimilarly to time correlation, a simple intuitive notion of fading in frequency is that the channelin the frequency domain, H( f, t =0) , can be thought of as consisting of approximately one newrandom sample every B cHz, with the values in between interpolated. The Rayleigh fadingmodel assumes that the received quadrature signals in time are complex Gaussian. Similar to thedevelopment in the previous section where by complex Gaussian values in the frequency domaincan be converted to a correlated Rayleigh envelope in the time domain, complex Gaussian valuesin the time domain can likewise be converted to a correlated Rayleigh frequency envelope| H( f)|.The correlation function that maps from uncorrelated time-domain ( τ domain) random variablesto a correlated frequency response is the multipath intensity profile, A τ( ∆τ). This makessense: Just as ρ t( ∆f) describes the channel time correlation in the frequency domain, A τ( ∆τ)describes the channel frequency correlation in the time domain. Note that in one familiar special