152 Deterministic channel parameter estimation The correlogram spectral estimation method is c(o)- (6.2) k=(-N-1) where hatr()represents the autocorrelation function of u(n). 1.For a given direction of arrival (DoA)0,the filter passed undistorted signals. 2.The filter attenuates all the other DoAs different from. hould the cmdu tha)h ond to the c s”of the spatial filte c8) h=c(0)We(0) (6.3) Thus,when this optimal weights are applied to the spatial filter,we obtain for the output signal of the spatial filter: Eflu(t))=h"Rh =hE(y(t)y(t)"] _c(0)Eiy(t)(t))c(0) (6.4) c()Hc(6) This equation shows that when the DoA of the impinging signal isonly when the weights of the fiter satisfy (6.3). c(0)Rc(e) pB()= (6.5) c(0Hc(0) whereis the covariance matrix of the received signal,which can be calculated according to R=E(y(t)u(t)) (6.6) nshebembgmhmhehenhngerecetedsgalsnom likelihood e thod Inmthe following an example of using the boinmethod the dband MIMO underOSou-builin ofu were not used in the Rx during the measurements.Thus,totally50 1000 spatial subchannels are measured in one measurement cycle. polarization and horizontal polarization.In this example,both vertical and horiz nt al polarization array responses are ctra ct on the 152 Deterministic channel parameter estimation The correlogram spectral estimation method is pˆc(ω) = N X−1 k=(−N−1) rˆ(k)e −jωk (6.2) where hatr(k) represents the autocorrelation function of y(n). Beamforming method is based on the assumption that the array response, which is also called steering vector c(θ) is know. The beamforming method is proposed for designing a spatial filter, which satisfies the following two conditions: 1. For a given direction of arrival (DoA) θ, the filter passed undistorted signals. 2. The filter attenuates all the other DoAs different from θ. These two conditions correspond to the constraint that the weights h of the multiple spatial “taps” of the spatial filter should satisfy the condition that min h h Hh subject to h Hc(θ) = 1 ??. It can be shown that in such a case, h = c(θ) c(θ)Hc(θ) . (6.3) Thus, when this optimal weights are applied to the spatial filter, we obtain for the output signal of the spatial filter: E{|y(t)| 2 } = h HRh = h HE{y(t)y(t) H} = c(θ) HE{y(t)y(t) H}c(θ) c(θ)Hc(θ) (6.4) This equation shows that when the DoA of the impinging signal is θ, only when the weights of the filter satisfy (6.3), the output signal has the maximum power. Thus, when the DoA of the impinging signal is unknown, we can vary the weights of the filter by letting θ being selected within a certain range. Therefore, a pseudo-power spectrum p(θ) with respect to θ can be computed. By selecting the specific value of θ which leads to the peak of p(θ), the estimate of the DoA is obtained. Based on this rationale, the power spectrum computed can be expressed as pB(θ) = c(θ) HRcb (θ) c(θ)Hc(θ) , (6.5) where Rb is the covariance matrix of the received signal, which can be calculated according to Rb = E{y(t)y(t) H} = 1 N X N n=1 yn(t)yn(t) H (6.6) discussing about the rank... It is worth mentioning that the maximum likelihood estimation method when assuming the received signal is normal distributed has the same expression as the beamforming method in the single-path scenario. In the following an example of using the beamforming method to estimate the direction of arrival is illustrated. In this example, measurement data is used which was collected in a campaign jointly conducted by Elektrobit and Technology University of Vienna in 2005 using the wideband MIMO sounder -PROPSound- in a building of Oulu University. Figure 6.1 (a) depicts the photograph of the 50-element antenna array used in the Tx and Rx during the measurements. Figure 6.1 (b) illustrates the indices of the antennas in the array. Notice that 18 antennas from No. 19 to 36 were not used in the Rx during the measurements. Thus, totally 50 × 32 = 1600 spatial subchannels are measured in one measurement cycle. Fig. 6.3 depicts the power spectrum estimated by using the beamforming method. The signals emitted by 50 Tx antennas and received from the first Rx antenna are used. The responses of the antennas are separated into vertical polarization and horizontal polarization. In this example, both vertical and horizontal polarization array responses are considered. It can be observed from Fig. 6.3 that the power spectra are not exactly the same. The spectrum estimated by using the vertical polarization has more fluctuations than that observed when horizontal polarization is considered. Furthermore, it can be seen that the power spectrum estimated by using the horizontal polarization has higher spectral height than that computed by using the vertical polarization. It is obvious that the array response may have significant impact on the estimated power spectrum