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Antennas for WLAN (WiFi) Applications        253

                  Eq. 7.13 if the electric fields for both polarizations over the entire three-
                  dimensional space are known.

                                                      *
                                               *
                                            |S S   + S S  | 2
                              ρ =              11 12  21  22                  (7.12)
                                                         |
                                                                     2
                                                              2
                                     (
                                  1 (  − |S  | 2  +|S  | ))( 1  − (|S  | + |S  | ))
                                                  2
                                        11      21         22      12
                                                                2
                                              v       v
                                               ∫∫  F i ( θ φ , ) F j ( θ φ , ) dΩ
                                                           
                                                    •
                                                           
                                   ρ   =   4 π                                (7.13)
                                    e,ij   v     2      v     2
                                         ∫∫  F i ( θ φ , ) dΩ ∫∫  F ( , )  dΩ
                                                            φ
                                                          θ
                                                   d
                                                         j
                                         4 π         4 π
                    From Eq. 7.12, the effects of mutual coupling and impedance matching
                  on the correlation can be studied directly. Figure 7.3 shows that as the
                  return loss (|S |) increases, the mutual coupling S has to be reduced
                                11
                                                                  21
                  in order to achieve low correlation. The mutual coupling requirement
                  can be significantly relaxed when the antenna is well-matched. For
                  instance, when the return loss is –10 dB, the mutual coupling can be
                  around –4 dB and yet achieve a correlation of 0.7, assuming that the
                  S  and S  are both in phase.
                    11
                           21
                    However, if they are not in phase, the correlation will be sensitive
                  to the phase difference if the mutual coupling (|S |) is too high, as
                                                                  21
                  shown in Figure 7.4. The effects of the phase difference on the correla-
                  tion becomes insignificant when |S 21 | < –10 dB. Hence, if the mutual
                  coupling and return loss are kept lower than –10 dB, a very low correla-
                  tion of less than 0.1 can be achieved.
                     1.0
                               21
                          S 11 , S  Phase = 0°
                     0.9      S = −3 dB
                                11
                     0.8       −6 dB
                     0.7       −10 dB
                               −14 dB
                     0.6
                    r  0.5
                     0.4
                     0.3
                     0.2
                     0.1
                     0.0
                       −20 −18 −16 −14 −12 −10  −8  −6  −4  −2  0
                                         S , dB
                                           21
                  Figure 7.3  Effects of |S 11 | and |S 21 | on correlation
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