118 Channel measurements 5.6.3 Analysis of the impact on angular parameter estimation Ho) c(c( (5.36) )"ac()pfiu)tjexpiun c(0)c(0) the zimuth of arrival of the path.It is n the analytical expression of p(e)for over multiple observation snapshots.It can be shown that Ep()]can be calculated as Eir()=Bexpfju)c(o"c(+ajexpijne(c(△e+(O"n(门 c(c(9 -ecp("c+jpne("e" (5.37 Ea1=0+cap2 jopijnico"e4oaoe创+aon门 (5.38) We use p()and p()to represent the second and the third terms in the right hand side of (5.38),i.e. po=cocopjxpjuie(0"c△oe门 (5.39) a=a(o)"n(门 (5.40) It can be shown that p()can be calculated as )-eocoFEpno") he2.cwra-时 -三s.pE-时 P2∑l.(0)c.(@)P
118 Channel measurements 5.6.3 Analysis of the impact on angular parameter estimation The conventional Bartlett beamforming technique is usually used to estimate the angular parameters of propagation paths. Here the impact of phase noise is analyzed for the estimates obtained by using the Bartlett beamforming technique. The angular power spectrum computed using the Bartlett beamformer Bartlett (1948) can be written as p(θ) = c(θ) Hy(t) c(θ)Hc(θ) 2 , (5.36) where [·] H denotes the Hermitian operation. Inserting the phase-noise-impaired received signal (5.35) in (5.36), we obtain p(θ) = c(θ) H[αc(θ ′ ) ⊙ [exp{jµ} + j exp{jµ}∆φ(t)] + n(t)] c(θ)Hc(θ) 2 , = α exp{jµ}c(θ) Hc(θ ′ ) + αj exp{jµ}c(θ) H[c(θ ′ ) ⊙ ∆φ(t)] + c(θ) Hn(t) c(θ)Hc(θ) 2 , where θ ′ represents the true azimuth of arrival of the path. It is non-trivial to get the analytical expression of p(θ) for instantaneous scenarios. Thus, we derive the expectation of p(θ) for the case where p(θ) is computed by averaging over multiple observation snapshots. It can be shown that E[p(θ)] can be calculated as E[p(θ)] = E α exp{jµ}c(θ) Hc(θ ′ ) + αj exp{jµ}c(θ) H[c(θ ′ ) ⊙ ∆φ(t)] + c(θ) Hn(t) c(θ)Hc(θ) 2 , = 1 |c(θ)Hc(θ)| 2 E α exp{jµ}c(θ) Hc(θ ′ ) + αj exp{jµ}c(θ) H[c(θ ′ ) ⊙ ∆φ(t)] + c(θ) Hn(t) 2 , = 1 |c(θ)Hc(θ)| 2 E α exp{jµ}c(θ) Hc(θ ′ ) 2 + αj exp{jµ}c(θ) H[c(θ ′ ) ⊙ ∆φ(t)] 2 + c(θ) Hn(t) 2 . (5.37) Notice that the components in E[p(θ)] have their expectation equal to zero. These components are dropped in (5.37). We may further write (5.37) as E[p(θ)] = ¯p(θ) + 1 |c(θ)Hc(θ)| 2 E αj exp{jµ}c(θ) H[c(θ ′ ) ⊙ ∆φ(t)] 2 + c(θ) Hn(t) 2 . (5.38) We use p ′ (θ) and pn(θ) to represent the second and the third terms in the right hand side of (5.38), i.e. p ′ (θ) = 1 |c(θ)Hc(θ)| 2 E αj exp{jµ}c(θ) H[c(θ ′ ) ⊙ ∆φ(t)] 2 . (5.39) pn(θ) = 1 |c(θ)Hc(θ)| 2 E c(θ) Hn(t) 2 . (5.40) It can be shown that p ′ (θ) can be calculated as p ′ (θ) = 1 |c(θ)Hc(θ)| 2 E αj exp{jµ}c(θ) H[c(θ ′ ) ⊙ ∆φ(t)] 2 = 1 |c(θ)Hc(θ)| 2 |α| 2E X M m=1 |cm(θ)cm(θ ′ )| 2 (φ1(t) − µ) 2 = 1 |c(θ)Hc(θ)| 2 |α| 2 X M m=1 |cm(θ) ∗ cm(θ ′ )| 2E (φ1(t) − µ) 2 = 1 |c(θ)Hc(θ)| 2 |α| 2σ 2 X M m=1 |cm(θ) ∗ cm(θ ′ )| 2
Channel measurements 119 It isevident thatp()is proportional to the inner product ofc()and(),ie. p()=laPo2cm(0)"cm(@)2 where c(o Similarly,p(can be calculated to be ()-Ic()cl 1 2 =0丽 Thus,by talking multiple independent snapshots,the angular power spectrum computed by using the Bartlett beamformer is written as 2 =laP((0"c (5.41) 、0rs如E一 5.6.4 Cramer-Rao lower bound derivation The Cramer-Rao lower boun vides the ex is un-baise In this section we noise present in nformation Matrix r=元()2 (5.42) where st0)=ac(⊙p() (5.43) s(t:0)=ac(0)exp(ju)+jexp((0)(t) (5.44) For the simplicity of derivation,we assume that the steering vectorsc()of the array can be written as c(0)=[exp(j2/Ad(n-1)cos(0)}:n =1....,N]. (5.45) Thus,the first order derivative of s()can be written as )sa(0)eptpas/M(-)o(p). (5.46)
Channel measurements 119 It is evident that p ′ (θ) is proportional to the inner product of c(θ) and c(θ ′ ), i.e. p ′ (θ) = |α| 2σ 2 |c˜m(θ) H cm(θ ′ )| 2 , where c˜(θ) = c(θ) c(θ)Hc(θ) . Similarly, pn(θ) can be calculated to be pn(θ) = 1 |c(θ)Hc(θ)| 2 |c(θ)| 2σ 2 n = σ 2 n kc(θ)k 2 . Thus, by talking multiple independent snapshots, the angular power spectrum computed by using the Bartlett beamformer is written as E[p(θ)] = ¯p(θ) + |α| 2σ 2 |c˜m(θ) H cm(θ ′ )| 2 + σ 2 n kc(θ)k 2 = |α| 2 (1 + σ 2 )|c˜m(θ) H cm(θ ′ )| 2 + σ 2 n kc(θ)k 2 . (5.41) From (5.41), it can be observed that angular power spectrum can still be used to find the estimate of θ ′ when the phase noise is i.i.d. across the subchannels. Furthermore, we also observe that when the norm of the steering vector kc(θ)k 2 is much less than the noise variance σ 2 n , the power spectrum computed may exhibit local maximum. The latter observation is consistent with the conjecture that by using highly directional antenna array, the estimates of θ ′ can be significantly erroneous, and ghost paths may appear at θ where the steering vectors c(θ) have small norms. 5.6.4 Cramer-Rao lower bound derivation The Cramer-Rao lower bound provides the expectation of the RMSEE of a parameter when the estimator is un-baised. In this section we show the derivation of the Cramer-Rao lower bound for estimation of θ with phase noise present in the received signal. The Cramer-Rao bound can be calculated by taking the inverse of the Fisher Information Matrix (FIM). In our case, the FIM reduces to a scalar, which can be calculated as F(θ) = 1 No R Z ∂s(t; θ) ∂θ H ∂s(t; θ) ∂θ dt (5.42) where s(t; θ) = αc(θ) ⊙ p(t) (5.43) represents the signal components distorted by the phase noise. As illustrated in the previous section, signal s(t; θ) can be approximated by applying the Taylor-series expansion as s(t; θ) = αc(θ) exp{jµ} + j exp{jµ}αc(θ) ⊙ ∆φ(t) (5.44) For the simplicity of derivation, we assume that the steering vectors c(θ) of the array can be written as c(θ) = [exp{j2π/λd(n − 1) cos(θ)}; n = 1, . . . , N]. (5.45) Thus, the first order derivative of s(t; θ) can be written as ∂s(θ) ∂θ = [αj 2π λ d(n − 1) sin(θ) exp{j2π/λd(n − 1) cos(θ)} exp{jφn}; n = 1, . . . , N]. (5.46)