Tripling the capacity of wireless communications using electromagnetic polarization.pdf
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1、. Triplingthecapacityofwireless communicationsusing electromagneticpolarization Michael R. Andrews*, Partha P. Mitra* techniques such as polariza- tion diversity already take advantage of this3. Recent work47has shown that environments with scattering, such as urban areas or indoors, also possess in
2、dependent spatial channels that can be usedtoenhancecapacitygreatly.Ineithercase,therelevantsignal processingtechniquescomeundertheheadingofmultiple-input/ multiple-outputcommunications,becausemultipleantennaeare required to access the polarization or spatial channels. Here we show that, in a scatte
3、ring environment, an extra factor of three in channel capacity can be obtained, relative to the conventional limit using dual-polarized radio signals. The extra capacity arises because there are six distinguishable electric and magnetic states of polarization at a given point, rather than two as is
4、usually assumed. In Fig. 1, we show why the presence of a scattering environment violates our intuitive notion of there being only two polarization degrees of freedom for electromagnetic radiation. This situation arises because in free space, radiated electric and magnetic fi elds are constrainedto
5、be perpendicular to one another and to the direction of propagation2 . Thus, once the direction of propagation is fi xed, only two degrees of freedom remain which are typically referred to aseitherhorizontalorvertical(linear)polarizations.Inthepresence of a refl ecting surface, however, multiple pat
6、hs are possible between thetwopoints.Althoughthewavepropagatingalongthedirectpath cannot have an electric fi eld component parallel to that path, the wave propagating along the refl ected path can contribute such a component to the fi eld at the receiver (it is effectively a longitudinal component w
7、hen referred to the line-of-sight). Thus, by using appropriate transmit antennae we can propagate waves which cause independent fl uctuations in all three electric fi eld compo- nents. The electromagnetic polarization is no longer constrained to be perpendicular to the line-of-sight8. The presence o
8、f a single refl ecting surface allows the use of three channels of electric-fi eld polarization for wireless communication. In order to make our statements more precise, we introduce H, a 6 3 6 matrix relating the electric (E) and magnetic (B) fi elds measured at a point r, owing to idealized oscill
9、ating electric (p) and magnetic (m) dipole moments, produced by transmitting antennae at point r9 E?r? cB?r? ? ? 2H?k;r2r9? cp m ? ?1? where in free space and in the far-fi eld, H is given by a compact version of the standard formulas that describe the fi elds radiated by oscillating electric and ma
10、gnetic dipoles2(in SI units): H0?k;r? ? jkj3 e0c eik?r 4pk?r J2? r?J? r? 2J? r?J2? r? ? ?2? Here k is the wave vector (jkj ? q=c ? 2p=l) and 1=e0c 377Q is the impedance of free space (c is the speed of light). J(r ) is a 3 3 3 matrix given by Jij? r? ? Skeikj r k , defi ned so that J? r?p ? r 3 p (w
11、ith r ? r=jrj). The communication channel associated with equation (1) includes additive noise measured by the receiver. For simplicity, we make the canonical assumption that its components are uncor- related gaussian white noise with equal variance, and that commu- nication takes place over a suffi
12、 ciently narrow bandwidth that H has negligible frequency dependence (fl at fading). For a time- independent H, the rate at which information can be transferred letters to nature 316NATURE|VOL 409|18 JANUARY 2001| Mirror E0 E0 E1E2 E2 I I TransmitterReceiver Figure 1 Communicating with electric fi e
13、lds in the presence of a mirror. In the two- dimensionalplaneofthefi gure,orthogonalcurrentdipolesatthetransmitter,IandI9(left), controltwodegreesofelectricfi eldfreedomatthereceiver,E0andE2(right).Thedirection perpendicular to the plane contributes a third degree of freedom (E0, E1, and E2are trans
14、verse fi elds on the three paths shown). Normally, due to transverse propagation of electromagnetic waves, there would be no longitudinal electric fi eld component at the receiver. Thus, as is drawn, without the mirror the current dipole labelled I would not produce electric fi elds at the receiver.
15、 However, the alternative propagation path shown causes the component E2to appear at the receiver, which has non-zero longitudinal projection when referred to the non-refl ected path. 10 10 30 50 70 Normalized eigenvalues (dB) 210100 Distance along line (wavelengths) TR Figure 2 Eigenvaluesof HHin a
16、 two-mirror idealized scatteringenvironment. Data were obtainedbysimulatingthereceiver(R)atavariabledistancefromthetransmitter(T)along the line indicated by the inset. The line joins the points ?x;y;z? ?9:9;7:7;10:5?l and (15.1, 109.8, 8.1)l, while the mirrors are in the planes x ? 0 and z ? 0. Note
17、 that rank?H? ? 6 (that is, there are no eigenvalues equal to zero), indicating a threefold capacity increase over what is possible in free space with no scattering. Eigenvalues are normalized to those that would be obtained in free space: ?jkj2=4pe0jrjc?2. (At large distances the refl ections are g
18、lancing, which leads to widely spaced eigenvalues). See equations (1) and (2). 2001 Macmillan Magazines Ltd between an n-antennae transmitter/receiver pair (n ? 6 in equa- tions (1) and (2) above) is characterized by the quantity M?H? ? log2det?I ? ?r=n?HH?3? whereM(H),inbitspersecond perhertz,isthe
19、mutualinformation between the transmitter and receiver when the transmitted signals are uncorrelated white gaussian stochastic processes with equal power. We use units such that the noise components have unit variance,allowingustodenotebothtotalpowerandsignal-to-noise ratio by the same symbol r. The
20、 signifi cance of M(H) is that for a fl uctuating H, under appropriate conditions, the capacity C of the channel is given by the expectation of M(H) taken over the probability distribution of H. The conditions are that H is known by the receiver but not by the transmitter (as would be the case if tr
21、ansmitted pilotsignalshelpthereceivercalculate H).We notethat ifthechannelisknowntothetransmitter,thecapacitywillbehigher and is given by the water fi llingsolution where power is unevenly divided among transmitting antennae6. Fluctuations in H may be caused by a fl uctuating environment or by movem
22、ent of the antennae. Clearly, the capacity depends on the rank of H. It follows by inspection that at large signal-to-noise ratio r, M(H) (and therefore C)tends tothe valuemlog2r,wherem ? rank?H?. Wenotethat the largerlimitistakenforfi xedH.ItiseasytoverifythattherankofH0 (see equation (2) is two, w
23、hich corresponds to our intuitivenotion of there being only two polarization degrees of freedom in free space. We now formally defi ne the number of polarization channels as rank(H). Our basic observation here may be summarized by the statement that it is possible to have rank?H? ? 6 in an environ-
24、ment with scattering, with a concomitant increase in the capacityof the wireless communication channel. We note that six independent signals have to be transmitted in order to take advantage of this increased capacity. A simple geometry which shows that we would expect rank?H? ? 6 in scattering envi
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