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3.5.1 Correlation Models for Frequency Selective Fading

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3.5.1 Correlation Models for Frequency Selective Fading Environment Spread Flat Rural 0.5 µs Urban 5 µs Hilly 20 µs Mall 0.3 µs Indoors 0.1 µs Table 3.1: Typical delay spreads for various environments. If W > 1 , then the fading is said to be frequency selective, and h(t; τ) can no longer be replaced by a τds multiplication by the flat fading process E(t) as in (3.12). The goal of this section is to study how the channel model should be modified to account for frequency selectivity. 3.5.1 CORRELATION MODELS FOR FREQUENCY SELECTIVE FADING Recall that time-varying impulse response is given by h(t; τ) = β ejφn(t) δ(τ τ ) (3.88) n − n n X where the phase process φn(t) evolves as: φn(t) 'n + 2πνnt ≈ (3.89) = 'n + 2πνmaxt cos θn with the phases ' being i.i.d. Unif [ π; π] (see Assumption 3.2 and (3.14)). We can rewrite (3.88) as f ng − h(t; τ) = β ej'n ej2πνnt δ(τ τ ) (3.90) n − n n X Before we develop a stochastic model for h(t; τ), we introduce two additional equivalent represensentations of the time-varying channel [Bel63]. Definition 3.11. For a channel with impulse response h(t; τ), the time-varying transfer function is given by j2πfτ H(t; f) = h(t; τ)e− dτ Z (3.91) j'n j2π(νnt fτn) = βne e − n X and the delay-Doppler spreading function is given by j2πνt C(ν; τ) = h(t; τ)e− dt : Z (3.92) = β ej'n δ(ν ν ) δ(τ τ ) n − n − n n X c V. V. Veeravalli, 2007 44 Statistical Characterization of H(t; f) In characterizing the channel stochastically, it is easiest to deal with the time-varying transfer function H(t; f) since it is written as a sum of complex exponentials (rather than impulses). Since H is a function of two variables, it must be modelled as a random field. The contribution of the n-th path to H(t; f) can be written as j'n j2π(νnt fτn) ηn(t; f) = βne e − (3.93) Analogous to the result we derived earlier in the context of frequency flat fading (Result 3.2), ηn(t; f) can easily be seen to be a zero-mean proper complex field. The autocorrelation function (ACF) of the field H(t; f) can be derived in a manner similar to the way in which we derived the ACF of the flat fading process in (3.20). In particular, we have ? 2 j2πνn(t1 t2) j2πτn(f1 f2) E [H(t ; f )H (t ; f )] β e − e− − : (3.94) 1 1 2 2 ≈ n n X Thus H(t; f) is approximately stationary in both t and f, i.e., it is approximately a homogeneous random field. The homogeneous autocorrelation function can then be defined as ? RH (ξ; ζ) = E [H(t + ξ; f + ζ)H (t; f)] 2 j2πνnξ j2πτnζ (3.95) β e e− : ≈ n n X Using the fact that νn = νmax cos θn, we can rewrite (3.95) as 2 j2πνmaxξ cos θn j2πτnζ RH (ξ; ζ) = βn e e− (3.96) n X We can express the autocorrelation function RH (ξ; ζ) in terms a quantity, which is defined below, that is analogous to the Doppler power spectrum of (3.25): Definition 3.12. The delay-Doppler scattering function is given by Ψ(ν; τ) = β2 δ(ν ν ) δ(τ τ ) : (3.97) n − n − n n X The scattering function Ψ(ν; τ) describes the distribution of power as a function of the Doppler frequency and delay. The discrete joint density of (3.97) can be approximated by continuous density, by considering the paths to be forming a continuum. As mentioned earlier, the continuous model is representative of diffuse scattering. The support of the scattering function is restricted to the rectangular region [ ν ; ν ] [0; τ ]. − max max × ds Furthermore, the Doppler power spectrum Ψ(ν) is the marginal of Ψ(ν; τ), i.e., τds Ψ(ν) = Ψ(ν; τ)dτ : (3.98) Z0 Based on (3.97), we can rewrite (3.96) as νmax τds j2πνξ j2πτζ RH (ξ; ζ) = Ψ(ν; τ) e e− dτdν (3.99) ν 0 Z− max Z c V. V. Veeravalli, 2007 45 We may also express the autocorrelation function RH (ξ; ζ) in terms of a power gain density. To this end, we generalize the angular gain density γ(θ) to a joint density that describes the allocation of power to both angle of arrival and delay. Definition 3.13. The joint angle-delay gain density γ(θ; τ) is given by γ(θ; τ) = β2 δ(θ θ ) δ(τ τ ) : (3.100) n − n − n n X Obviously, there is a one-one relationship between the scattering function, Ψ(ν; τ), and the angle-delay gain density γ(θ; τ). A straightforward change of variables argument yields γ(θ; τ) = νmax sin θ Ψ(νmax cos θ; τ) : (3.101) Based on (3.100), we can rewrite (3.96) as π τds j2πνmaxξ cos θ j2πτζ RH (ξ; ζ) = γ(θ; τ) e e− dτdθ (3.102) π 0 Z− Z WSSUS Model The zero-mean proper complex field model for H(t; f) implies that the time-varying transfer function, h(t; τ), and the delay-Doppler spreading function, C(ν; τ), are both also zero-mean proper complex fields. The autocorrelation function of h(t; τ) is easily shown to be ? 2 j2πνn(t1 t2) E [h(t ; τ )h (t ; τ )] = β e − δ(τ τ ) δ(τ τ ) (3.103) 1 1 2 2 n 1 − n 1 − 2 " n # X Note that h(t; τ) is not a homogeneous random field since it is not stationary in the delay variable τ. However, it is indeed stationary in the time variable and uncorrelated in the delay variable. The continuous path general- ization of this model for h(t; τ) is referred to as the Wide Sense Stationary Uncorrelated Scattering (WSSUS) model, and was first studied in great detail by Bello [Bel63]. For the continuous path generalization, we can rewrite autocorrelation function in terms of the delay-Doppler scattering function: νmax τds ? j2πνξ E [h(t + ξ; τ1)h (t; τ2)] = Ψ(ν; τ) e δ(τ1 τ) dτdθ δ(τ1 τ2) (3.104) ν 0 − − Z− max Z The autocorrelation function of the delay-Doppler spreading function, C(ν; τ), is given by E [C(ν ; τ )C?(ν ; τ )] = β2 δ(ν ν ) δ(τ τ ) δ(τ τ ) δ(ν ν ) : (3.105) 1 1 2 2 n 1 − n 1 − n 1 − 2 1 − 2 " n # X Thus, C(ν; τ) is not stationary in ν or τ, but it is uncorrelated in both variables. Note that E [C(ν ; τ )C?(ν ; τ )] = Ψ(ν ; τ ) δ(τ τ ) δ(ν ν ) : (3.106) 1 1 2 2 1 1 1 − 2 1 − 2 c V. V. Veeravalli, 2007 46 3.5.2 RAYLEIGH AND RICEAN FADING As we did in the study of frequency flat fading, we first characterize the distribution of the time-varying transfer function for the situation where the scattering is purely diffuse, i.e., there is no specular component. The Central Limit Theorem (Result B.3) can be applied to obtain: Result 3.10. For purely diffuse scattering, H(t; f) is well-modelled as a zero-mean, proper complex Gaussian random field. As a consequence, for fixed t, f, the magnitude H(t; f) has a Rayleigh distribution and the phase \H(t; f) j j is uniformly distributed on [ π; π]. The corresponding model for h(t; τ) is called the Gaussian WSSUS (or − GWSSUS) model. If there is a specular component (with possibly multiple paths) in addition to the diffuse paths, then we can write the time-varying transfer function as Ns j'n j2π(νnt τnf) H(t; f) = βn e e − + β~ Hˇ (t; f) (3.107) n=1 X where Hˇ (t; f) is a zero-mean, proper complex Gaussian, homogeneous field, and N β~2 = 1 β2 : (3.108) − n n=XNs+1 Thus, H(t; f) is a zero-mean, proper complex, homogeneous field, but is non-Gaussian. However, conditioned on '1; : : : ; 'Ns , H(t; f) is a proper complex, non-homogeneous field, with non-zero mean. Also, just as in the case of the flat fading process E(t), for fixed t, f, the distribution of the envelope H(t; f) , conditioned on j j '1; : : : ; 'Ns , is Ricean with Rice factor given in (3.71). If there is only one specular path, the unconditional distribution of H(t; f) is also Ricean. j j The time-varying impulse response and delay-Doppler spreading functions get modified in a similar fashion: Ns h(t; τ) = β ej'n ej2πνntδ(τ τ ) + β~ hˇ(t; τ) (3.109) n − n nX=1 where hˇ(t; τ) is a zero-mean, GWSSUS field. Ns C(ν; τ) = β ej'n δ(ν ν ) δ(τ τ ) + β~ Cˇ(ν; τ) (3.110) n − n − n n=1 X where Cˇ(ν; τ) is a zero-mean Gaussian field that is uncorrelated in ν and τ. c V. V. Veeravalli, 2007 47 Chapter 4 Sampled Delay-Doppler Channel Representations 4.1 INTRODUCTION In this chapter we develop sampled representations for the time-varying channel h(t; τ). The motivation for such representations is that channel modeling independent of the signal space is irrelevant from a communica- tion viewpoint — an effective channel representation commensurate with signal space characteristics is what is important.
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