Vehicular Technologies Increasing Connectivity Part 11 pptx

Correlation Coefficients of Received Signal I and Q Components 291 ρ ( Δ x ) ∆ fσ f m T s 0.5 0 0 0.5 1 1.5 2 1.5 1 1 0.5 0 -0.5 (a) Bird’s-eye view (theory) 0.5 ρ ( ∆x) 0.5 0 0 0.5 1 1.5 2 1.5 1 1 0.5 0 -0.5 ∆ fσ f m T s (b) Bird’s-eye view (simulation) f m T s Δ fσ 0 0.3 0.2 0.1 0.1 0.3 0.2 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 ρ = 0.1 (c) Locus of theory (fine) & simulation(solid) Fig. 7. Correlation coefficient in a domain with time and frequency axes (Type 1: exponential distribution , K = -∞ dB) ρ ( Δ x) ρ ( Δ x) Vehicular Technologies: Increasing Connectivity 292 correlation for K= -∞ dB is slightly lower than the theoretical one. We suppose that the difference is due to different conditions between the theoretical and simulation models for the delay profile, such as the number of waves N and the counted effective amplitude in h i . Therefore, we repeated the simulation, changing from N=10 and effective amplitude of -20 dB to N>100 and effective amplitude of less than -50 dB. As a result, the correlation became close to the theoretical one. The above results show that the autocorrelation is independent of the K factor, while the frequency correlation depends on the K factor. The correlation coefficient in a domain with time and frequency axes for NLOS is shown in Fig. 7. Figure 7(a) is a bird’s-eye view of ρ ( Δ x) obtained from (13), which makes it easy to comprehend the overall ρ ( Δ x). We can see that ρ ( Δ x) has a peak ρ (0)=1 at the origin Δ x = 0 and that ρ ( Δ x) becomes smaller with increasing Δ x or as f’ m T s and Δ f σ become larger. Furthermore, ρ ( Δ x) decreases in a fluctuating manner on the time axis, but decreases monotonically on the frequency axis. The high area of ρ ( Δ x) needed to allocate the pilot signal in coherent detection, such as ρ ( Δ x) >0.8, is very small in the domain, whereas the low area needed to design the diversity antenna, such as ρ ( Δ x) <0.5, is spread out widely. Figure 7(b) also shows a bird’s-eye view of the simulated correlation; Figs. 7(a) and (b) both exhibit almost the same features. Figure 7(c) shows the loci of the theoretical and simulated correlations in a small area up to 0.3 on both axes, with contour lines of ρ ( Δ x). It is easy to compare them. The theoretical value on the time axis agrees well with the simulated one, but the theoretical value on the frequency axis is slightly higher than the simulated one. The reason for this is a different model for the delay profile, as described earlier. We can see from Fig. 7(c) that the theoretical value agrees roughly with the simulated one. Furthermore, we note that in Fig. 7(c), the locus of the correlation coefficient ρ ( Δ x) seems to be an ellipse with its major axis on the time axis and its origin Δ x=0 in the domain. ii) Type 2: Delay profile with random distribution Figure 8 shows the correlation coefficient for a delay profile with random distributions of K=-∞ and 5 dB. The values plotted by symbols of × and ▲ in Figs. 8(a) and (b) were simulated in a similar way to Fig. 6, and the fine and broken lines show the theoretical value of ρ ( Δ x) obtained from (16) by putting Δ f =0 for autocorrelation or ρ ( Δ x)=J 0 (2 π f’ m T s ) and by putting f’ m T s =0 for frequency correlation or f k f k πΔ τ πΔ τ + + max max sin(2 ) 2 1 . As shown in Fig. 8(a), the simulated autocorrelations for both K=-∞ and 5 [dB] have almost the same features and agree well with the theoretical values, so the correlation seems to be independent of the K factor. On the other hand, the simulated frequency correlations for K=-∞ and 5 dB in Fig. 8(b) have different features that also depend on the K factor. The simulated and theoretical values agree well. The correlation coefficient in a domain with time and frequency axes for NLOS is shown in Fig. 9 by a similar method to that for Fig. 7. Figure 9(a) is a bird’s-eye view of the theoretical value of ρ ( Δ x) obtained from (16). We can also see that ρ ( Δ x) has a peak ρ (0)=1 at the origin Δ x = 0, and ρ ( Δ x) becomes smaller with increasing Δ x. However, in this case, ρ ( Δ x) decreases in a fluctuating manner not only on the time axis, but also on the frequency axis. The high area of ρ ( Δ x) in the domain, such as ρ ( Δ x) >0.8, is larger than that for an exponential distribution; the low area, such as ρ ( Δ x) <0.5, is spread out widely. Correlation Coefficients of Received Signal I and Q Components 293 Figure 9(b) is the simulated correlation, which exhibits similar features to the theoretical one in Fig. 9(a). Figure 9(c) shows the loci of the theoretical and simulated correlations. The theoretical value in the domain agrees well with the simulated ones. For a delay profile with random distribution, the locus of correlation coefficient ρ ( Δ x) is also an ellipse in the domain. -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 00.511.52 Theory (K = - ∞, 5 dB) K = - ∞ dB K = 5 dB (a) Autocorrelation -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 00.511.52 K = - ∞ dB Theory (K = 5 dB) K = 5 dB Theory (K = - ∞ dB) (b) Frequency correlation Fig. 8. Correlation coefficient (Type 2 : random distribution) ρ ( Δ x) f m T s ρ ( Δ x) Δ f τ max Vehicular Technologies: Increasing Connectivity 294 ρ ( Δ x ) Δ f τ max f m T s 0.5 0 0 0.5 1 1.5 2 1.5 1 1 0.5 0 -0.5 (a) Bird’s-eye view (theory) ρ ( Δ x ) Δ f τ max f m T s 0.5 0 0 0.5 1 1.5 2 1.5 1 1 0.5 0 -0.5 (b) Bird’s-eye view (simulation) f m T s Δ f τ max 0 0.3 0.2 0.1 0.1 0.3 0.2 0.9 0.8 0.7 0.6 0.5 0.4 0.3 ρ = 0.2 (c) Locus of theory (fine) & simulation (solid) Fig. 9. Correlation coefficient in a domain with time and frequency axes (Type 2: random distribution , K = -∞ dB) ρ ( Δ x) ρ ( Δ x) Correlation Coefficients of Received Signal I and Q Components 295 5. Conclusion An analysis model having a domain with time, frequency, and space axes was prepared to study the correlation coefficients of the I and Q components in a mobile channel, which are needed in order to allocate a pilot signal with M-ary QAM detection, such as in an OFDM system, and to compose antennas in the MIMO technique. For a multipath environment, the general correlation coefficient formula was derived on the basis of a delay profile with and without a directive wave, and as examples, the formulas for delay profiles with exponential and random distributions were then derived. The formulas exhibit some interesting features: the autocorrelation on the time axis is independent of the K factor and is expressed by J 0 (2 π f’ m T s ), but the frequency correlation depends on the K factor and delay profile type. The locus of a fixed value correlation is an ellipse in the domain with time and frequency axes. The correlations were also shown in the domain using bird’s-eye views for easy comprehension. Furthermore, computer simulation was performed to verify the derived formulas and the theoretical and simulated values agree well. Therefore, it is possible to estimate logically the pilot signal allocation of M-ary QAM in OFDM and antenna construction for MIMO in the domain. 6. Appendix 6.1 Appendix A derivation of (8) Under the conditions of the propagation model in 2-A and assuming that N is a large number, <I(0)I( Δ x)> is analyzed as follows. It is separated into terms of the same ith and other ith arriving waves, as shown in (17). NN iciii cimsii ii N icii cimsii i IIx h f h f f fT hf fffT ' 00 2' 0 (0) ( ) [ cos(2 )][ cos{2 (( ) cos ) }] cos(2 )cos{2 (( ) cos ) } ΔπτφπΔτξφ πτ φ π Δτ ξ φ == = 〈〉=〈 + +− +〉 =〈 + + − + ∑∑ ∑ NN ij ci i c j ms j j ij hh f f f f T πτ φ π Δτ ξ φ == + ++−+〉 ∑∑ '' ' 00 cos(2 )cos{2 (( ) cos ) } , (17) NN ij where i j on == ≠ ∑ ∑ '' 00 Furthermore, the second term in (17) vanishes because the values of τ i , ξ i , and φ i for the ith wave and τ j , ξ j , and φ j for the jth wave are independent of each other and are also random values, and the sum of the products of cos θ i and cos θ j then becomes zero. With this in mind and considering the small values of Δ f τ i and f m ’T s , the first term of (17) is modified as (18) to (20) by using, for example, a transforming trigonometric function, τ i , ξ i , and φ i with random values, and an odd function of sine. In this procedure, the second terms in (18) and (19) also vanish, so we finally get (20) as the result for <I(0)I( Δ x)>. N iimsi cimsii i Eq h f fT f f fT π Δτ ξ π Δ τ ξ φ = =〈 − + + − + 〉 ∑ 2' ' 0 1 . (17) [cos(2 ( cos )) cos{2 ((2 ) cos ) 2 }] 2 (18) Vehicular Technologies: Increasing Connectivity 296 N i i ms i i ms i i hf fT f fT πΔ τ π ξ πΔ τ π ξ = =〈 − − − 〉 ∑ 2' ' 0 1 [cos(2 )cos( 2 cos ) sin(2 )sin( 2 cos )] 2 (19) N iimsi i hf fT π Δτ π ξ = =〈 − 〉 ∑ 2' 0 1 cos(2 )cos( 2 cos ) 2 (20) 6.2 Appendix B derivation of (12) In the first term A 1 in (11), we change A 1 to (21) because the directive wave’s amplitude h 0 is usually much larger than that of the nondirective one and because τ 0 =0. Moreover, the amplitude h i of the ith arriving wave depends on excess delay time τ i according to (10). So by substituting (10) for h i , we can rewrite (21) as (22). N ii i Ahh f πΔ τ = =〈 + ∑ 22 10 1 1 [cos(2)] 2 (21) () N iav i i hh f 22 0 1 11 [ exp ] cos(2 ) 22 τ τπΔτ = = 〈〉+〈 − 〉 ∑ (22) We try to calculate by replacing the ensemble average of the second term in (22) by integration with respect to τ i assuming a large N. As a result, we get (23) assuming τ min is close to 0 and τ max is large. av h Ah fd τ τ τ τπΔττ =〈 〉+ − ∫ max min 2 2 10 1 exp( 2 / )cos(2 ) 22 av av h h f τ τπΔ =〈 〉+ + 2 2 0 22 1/4 21(/2)(2) (23) Furthermore, we need to adjust h in (23) to a normalized power of 1 when N and τ max are large, so we get () NN iiav ii av hh h d max min 22 11 2 11 [exp ] 22 exp( 2 / ) 2 τ τ ττ τ ττ == =− =− ∑∑ ∫ av h 2 1 1. 4 τ = = (24) Consequently, the numerator of the second term in (23) should be 1. Furthermore, considering h 2 0 1 2 , we get (25). av Ak f τπΔ =+ + 1 22 1 1( /2)(2 ) (25) Correlation Coefficients of Received Signal I and Q Components 297 6.3 Appendix C derivation of (15) We calculate the ensemble average of A 2 in (14) in a similar manner to that for (23) by integrating with the provability density function 1/ τ max of τ . Considering the independence of h i and τ i and the power of the directive and nondirective waves, or k and 1, we get (26). N ii i Ah h f πΔ τ = =+ ∑ 22 20 1 11 cos(2 ) 22 NN ii ii kh f π Δτ == = +〈 〉〈 〉 ∑∑ 2 11 1 cos(2 2 kfd τ τ π Δτ τ τ =+ ∫ max min max 1 cos(2 ) f k f πΔ τ πΔ τ =+ max max sin(2 ) 2 (26) 7. References [1] ITU Circular Letter 5/LCCE/2, Radiocommunication Bureau, 7 March 2008. [2] Richard van Nee and Ramjee Prasad, OFDM FOR WIRELESS MULTIMEDIA COMMUNICATIONS, Artech House, 1999. [3] T. Hwang, C. Yang, G. Wu, S. Li, and G. Y. Li, OFDM and Its Wireless Applications: A Survey, IEEE Trans. Veh. Technol., Vol. 58, No. 4, pp. 1673-1694, May 2009. [4] G. J. Foschini and M. J. Gans, On limits of wireless communications in fading environments when using multiple antennas, Wireless Personal Commun. Vol. 6, pp. 311-335, 1998. [5] D. Shiu, G. Foschini, M. J. Gans, and J. Kahn, Fading Correlation and Its Effect on the Capacity of Multielement Antenna Systems, IEEE Trans. Commun., Vol. 48, No. 3, pp. 502-513, March 2000. [6] Andreas F. Molisch, Martin Steinbauer, Martin Toeltsch, Ernst Bonek, and Reiner S. Thoma, Capacity of MIMO Systems Based on Measured Wireless Channel, IEEE JSAC, Vol. 20, No. 3, pp. 561-569, April 2002. [7] H. Nishimoto, Y. Ogawa, T. Nishimura, and T. Ohgana, Measurement-Based Performance Evaluation of MIMO Spatial Multiplexing in Multipath-Rich Indoor Environment, IEEE Trans. Antennas and Propag., Vol. 55, No. 12, pp. 3677-3689, Dec. 2007. [8] Seiichi Sampei and Terumi Sunaga, Rayleigh Fading Compensation for QAM in Land Mobile Radio Communications, IEEE Trans. Veh. Technol., Vol. 42, No. 2, pp. 137- 147, May 1993. [9] R. H. Clarke, A statistical theory of mobile-radio reception, Bell Syst. Tech. J., Vol. 47, No. 6, pp. 957-1000, 1968. [10] William C. Jakes, MICROWAVE MOBILE COMMUNICATIONS, John Wiley & Sons, Inc., 1974. Vehicular Technologies: Increasing Connectivity 298 [11] S. Kozono, T. Tsuruhara, and M. Sakamoto, Base Station Polarization Diversity Reception for Mobile Radio, IEEE Trans. Veh. Technol., Vol. VT33, No. 4, pp. 301- 306, Nov. 1984. [12] H. Nakabayashi and S. Kozono, Theoretical Analysis of Frequency-Correlation Coefficient for Received Signal Level in Mobile Communications, IEEE Trans. Veh. Technol., Vol. 51, No. 4, pp. 729-737, July 2002. [13] S. Kozono, K. Ookubo, and K. Yoshida, Study of Correlation Coefficients of Complex Envelope and Phase in a Domain with Time and Frequency Axes in Narrowband Multipath Channel, in 69 th IEEE Veh. Technol. Conf., Barcelona, Spain, April 2009. Jenq-Tay Yuan 1 and Tzu-Chao Lin 2 Graduate Institute of Applied Science and Engineering, Fu Jen Catholic University Taipei 24205, Taiwan, R.O.C. 1. Introduction Adaptive channel equalization without a training sequence is known as blind equalization [1]-[11]. Consider a complex baseband model with a channel impulse response of c n .The channel input, additive white Gaussian noise, and equalizer input are denoted by s n , w n and u n , respectively, as shown in Fig. 1. The transmitted data symbols, s n , are assumed to consist of stationary independently and identically distributed (i.i.d.) complex non-Gaussian random variables. The channel i s possibly a non-minimum phase linear time-invariant filter. The equalizer input, u n = s n ∗ c n + w n is then sent to a tap-delay-line blind equalizer, f n , intended to equalize the distortion caused by inter-symbol interference (ISI) without a training signal, where ∗ denotes the convolution operation. The output of the blind equalizer, y n = f  n ∗ u n = s n ∗ h n + f  n ∗ w n , can be used to recover the transmitted data symbols, s n ,where denotes complex conjugation and h n = f  n ∗ c n denotes the impulse response of the combined channel-equalizer system whose parameter vector can be written as the time-varying vector h n =[h n (1), h n (2), ] T with M arbitrarily located non-zero components at a particular instant, n, during the blind equalization process, where M = 1,2, 3, . . For example, if M = 3andI M = {1, 2, 5} is any M-element subset of the integers, then h n =[h n (1), h n (2),0,0,h n (5),0 0] T is a representative value of h n . Fig. 1. A complex basedband-equivalent model. The constant modulus algorithm (CMA) is one of the most widely used blind equalization algorithms [1]-[3]. CMA is known to be phase-independent and one way to deal with its phase ambiguity is through the use of a carrier phase rotator to produce the correct constellation orientation, which increases the c omplexity of the implementation of the receiver. Moreover, for high-order quadrature amplitude modulation (QAM) signal constellations (especially for cross constellations such as 128-QAM, in which the corner points containing significant phase information are not available), both the large adaptation noise and the increased sensitivity Multimodulus Blind Equalization Algorithm Using Oblong QAM Constellations for Fast Carrier Phase Recovery 17 to phase jitter may make the phase rotator spin due to the crowded signal constellations [10]-[13]. Wesolowski [4], [5] Oh and Chin [6], and Yang, Werner and Dumont [7], proposed the multimodulus algorithm (MMA), whose cost function is given by J MMA = E{[y 2 R,n − R 2,R ] 2 }+ E{[ y 2 I,n − R 2,I ] 2 } (1) where y R,n and y I,n are the real and imaginary parts of the equalizer output, respectively, while R 2,R and R 2,I are given by R 2,R = E[s 4 R,n ]/E[s 2 R,n ] and R 2,I = E[s 4 I,n ]/E[s 2 I,n ],inwhich s R,n and s I,n denote the real and imaginary parts of s n , respectively. Decomposing the cost function of the MMA into real and imaginary parts thus allows both the modulus and the phase of the equalizer output to be considered; therefore, joint blind equalization and carrier phase recovery may be simultaneously accomplished, eliminating the need for an adaptive phase rotator to perform separate constellation phase recovery in steady-state operation. The tap-weight vector of the MMA, f n , is updated according to the stochastic gradient descent (SGD) to obtain the blind equalizer output y n = f H n u n f n+1 = f n −μ ·∇J MMA = f n −μ · ∂J MMA ∂f  n = f n −μ ·e  n ·u n (2) where u n =[u n+l , ,u n−l ] T and e n = e R,n + je I,n in which e R,n = y R,n ·[y 2 R,n − R 2,R ], e I,n = y I,n ·[y 2 I,n − R 2,I ] and L = 2l + 1 is the tap length of the equalizer. The analysis in [9], which concerns only the square constellations, indicates that the MMA can remove inter-symbol interference (ISI) and simultaneously correct the phase error. However, when the transmitted symbols are drawn from a Q AM constellation having an odd number of bits per symbol (N = 2 2i+1 , i = 2,3, ),theN-points constellations can be arranged into an oblong constellation [14], [15] so l ong as E [s 2 R,n ] = E[s 2 I,n ] is satisfied. For example, Fig. 2 illustrates a 128-QAM arranged by oblong (8 ×16)-QAM with the required average energy of 82. The conventional 128-cross QAM constellations with the required average energy of 82 can be obtained from a square constellation of 12 × 12 = 144 points by removing the four outer points in each corner as illustrated in Fig. 3 [7]. Notably, the distance between two adjacent message points in the oblong constellations illustrated in Fig. 2 has been modified to be 1.759 instead of 2 as in the conventional cross 128-QAM, so that the average energies required by both cross and oblong constellations are almost identical. In this chapter, the oblong constellation illustrated in Fig. 2 is used as an example to demonstrate that the MMA using asymmetric oblong QAM constellations with an odd number of bits per symbol may significantly outperform its cross counterpart in the recovery of the carrier phase introduced by channels, without requiring additional average transmitted power. We use the te rm asymmetric because the oblong QAM is not quadrantally symmetric, i.e., E [s 2 R,n ] = E[s 2 I,n ],and as a c onsequence E [s 2 n ] = 0. Although reducing the distance between adjacent message points in the proposed oblong constellation in Fig. 2 may increase the steady-state symbol-error rate (SER) or mean-squared error (MSE) o f the adaptive equalizer, this chapter is concerned with the unique feature of fast carrier phase recovery associated with the MMA using oblong constellations during blind equalization p r ocess owing to its non-identical nature of the real and imaginary parts of the source statistics. 2. Analysis of MMA using oblong constellations This section presents an analysis of the MMA using oblong QAM constellations from the perspective of its stationary points. Our analytical results demonstrate that the four saddle 300 Vehicular Technologies: Increasing Connectivity [...]... I 4 ] + E [ s4 ] + [3( M − 1)] · ( E2 [ s2 ] + E2 [ s2 ]) E[sR I R I R2,R · E [ s2 ] + R2,I · E [ s2 ] I R E [ s4 ] + E [ s4 ] + [3( M − 1)] · ( E2 [ s2 ] + E2 [ s2 ]) R I R I (11) (12) 304 Vehicular Technologies: Increasing Connectivity The following form for the four stationary points along θ (k) = 0, π/2, π, 3π/2 for each of the M non-zero components of hn (for oblong constellations only) can thus... by (1) reduces to J MMA = 2E {[ y2 − R2,R ]2 }, R,n J MMA = 2E {[ y2 − R2,R ]} which essentially considers only the real part of the equalizer R,n output, when the symmetric constellations are such that R2,R = R2,I , E [ s4 ] = E [ s4 ] and R I 312 Vehicular Technologies: Increasing Connectivity Fig 7 Ensemble-averaged SER performance using the Brazil Ensemble C for SNR = 30dB and phase rotation 40◦... the PAPR problem The objectives of this chapter are twofold Firstly, we provide an overview of the PAPR reduction methods already published in the literature for WPM systems As part of the 316 Vehicular Technologies: Increasing Connectivity overview, we discuss the merits and limitations associated with the existing PAPR reduction schemes Secondly, we formulate the design criteria for a set of PAPR minimizing... entirely by sin 2θ (k) Hence, the non-zero component of hn with maximum modulus based on the SGD in (2) can be readily shown to move toward the stationary points along θ (k) = 0 and 308 Vehicular Technologies: Increasing Connectivity θ (k) = 180◦ , respectively, when −90◦ < θ (k) < 90◦ and 90◦ < θ (k) < 270◦ , as shown in Fig 6(a), provided that C > 0 We consider the strictest condition when cos 2θ (l... and slopecross = r2 ( M) + r2 ( M) + − A = r2 ( M) + r2 ( M) + − r ( M) r ( M) J MMA [ ×√ , ×√ ] − J MMA [r± ( M ), 0] 2 ( [ r ×√M) 2 2 ( − r± ( M )]2 + [ r ×√M) − 0]2 2 (17) (18) 310 Vehicular Technologies: Increasing Connectivity = B r2 ( M) + r2 ( M) − × ± √ 2 · r± ( M) · r× ( M) 4 where A = (−1/4) · E [ s4 ] · [r4 ( M ) − r4 ( M )] + (3/4) · k s σs · [r4 ( M ) − r4 ( M )] − [ R2,R + R2,I ] · + −... of (3), (1/4) · i 4 E [ s2 ] and k s = E [| sn |4 ] /σs is the source kurtosis The I,n 4 (i )}, which is related to the fourth-order statistics (or ∑i h E [ s2 ] + R,n { E [ s4 ] · 302 Vehicular Technologies: Increasing Connectivity Fig 3 cross constellations for 128-QAM sources fourth-power phase estimator) in [12], [13], [16]-[18] containing the phase information and is absent from the CMA cost function,... 327-329 [18] R Lopez-Valcarce (2004) "Cost minimization interpretation of fourth power phase estimator and links to multimodulus algorithm," Electron Lett., vol 40, no 4, Feb 2004 314 Vehicular Technologies: Increasing Connectivity [19] J P LeBlanc, I Fijalkow, and C R Johnson, Jr (1998) "CMA fractionally spaced equalizers: stationary points and stability under iid and temporally correlated sources,"... adopts the oblong QAM based on the SGD, the MMA cost can be considered in terms of h(k) with maximum modulus alone by substituting the approximations ∑k ∑ l =k r2 (k)r2 (l ) ∼ 0, = 306 Vehicular Technologies: Increasing Connectivity = = ∑ l r4 (l ) ∼ r4 (k), and ∑l r2 (l ) ∼ r2 (k) (i.e., the sum of magnitude square of the remaining ( M − 1) non-zero components of hn are very small once the MMA begins... the successive application of time-reversed QMF pairs (i.e., h(n) and g(n) ) For a system with 2 M sub-channels, we require M levels of QMF pairs at the transmitter, wherein the mth 318 Vehicular Technologies: Increasing Connectivity level consists of 2 m − 1 QMF pairs Generally, the successive application of the M levels of QMF pairs yields the output y(n) of the WPM transmitter It should be noted that... method In reducing the processing time and computational complexity, it is shown in (Baro & Ilow, 2007b) that the multi-pass tree pruning method provides good PAPR reduction capability 320 Vehicular Technologies: Increasing Connectivity when the number of iterations (or passes) is set to 5 Furthermore, a computationally simpler version of the multi-pass tree pruning algorithm based on setting a PAPR threshold . MICROWAVE MOBILE COMMUNICATIONS, John Wiley & Sons, Inc., 1974. Vehicular Technologies: Increasing Connectivity 298 [11] S. Kozono, T. Tsuruhara, and M. Sakamoto, Base Station Polarization. (Type 2 : random distribution) ρ ( Δ x) f m T s ρ ( Δ x) Δ f τ max Vehicular Technologies: Increasing Connectivity 294 ρ ( Δ x ) Δ f τ max f m T s 0.5 0 0 0.5 1 1.5. 〉 ∑ 2' ' 0 1 . (17) [cos(2 ( cos )) cos{2 ((2 ) cos ) 2 }] 2 (18) Vehicular Technologies: Increasing Connectivity 296 N i i ms i i ms i i hf fT f fT πΔ τ π ξ πΔ τ π ξ = =〈 −