Satellite multiple beam antenna (MBA) began to develop in the 1970s to 1980s. Single Feed per Beam (SFB) technology was adopted in the early stage. Later, multiple feeds per beam (MFB) technology was employed. Compared with phased-array antenna, feed beam of MBA is more narrow, and stronger. So the MBA has been widely used in satellite mobile communication systems in 1990s.

According to the structure, MBA can be divided to three categories: phased array antenna, reflector antenna and lens antenna. In this paper, we use reflector antenna to create multi-beam model.

As is shown in Figure 1, the distance of ground receiver to center of component beam is different, which makes the signal strength varies.

Figure 2 shows the space relationship among satellite, user u and component beam center n. h is the height of satellite orbit. R is radius of the earth.

According to Figure 2, suppose that is the angle of ground receiver u to satellite and component beam center n to satellite, can be given by

where d_Sn and d_Su are the distance between satellite and component beam center n and user u respectively. is the distance between sub-satellite point and user u.

Suppose that is the normalized far-field radiation pattern of component beam, the adoption of different radiation pattern results different antenna gain, when the direction angle is the same. represents the antenna gain of the n^th component beam. The can be written as

where can be calculated by (1), G_max,n is the maximum gain of the n^th component beam.

In satellite communication, a signal is fading during propagation. Those fading may cause by environment and various attenuation facts, for instance cloud attenuation, rain attenuation and ionospheric scintillation. Taking into account this paper research mainly discusses spatial correlation, since attenuation facts like cloud attenuation and ionospheric scintillation are stay the same for a long time, we focus on the influence of ground environment near receiver, namely the multipath which arise from scattering, reflection, refraction or diffraction of the radiated energy off scatterers that lie in the environment.

For satellite communication, typically the height of the ground receiver is lower than nearby scatterers. The received signals in the both horizontal and

vertical plane are distributed. While establishing the channel model, we need to consider the height of scatterers, so the traditional two-dimensional geometric channel model no longer applies. Assuming that scatterers are distributed in a right cylinder which has ground receiver as it bottom center, a three-dimension geometry channel model is shown in Figure 3. A large picture of ground receiver in the model is shown in Figure 4.

For a downlink of satellite communication system, we assume that p and q are the number of satellite antennas, l and m are the number of ground antennas,. N_r and N_r are the maximum number of transmitting antenna and receiving antenna. The received signal is given by

where, H(t) is channel impulse response, n(t) is noise. represents convolution.

The received signal includes direct-path and multipath component in satellite communication system. So the channel impulse response is given by

where H_loS(t) is impulse response of direct-path, H_NLoS(t) is impulse response of multipath. H_loS(t) and H_NLoS(t) are both N_T × N_R matrix. K is Rician Index, which describes ratio of direct-path power and multipath power (unit is dB). The value of K depends on angel of elevation from ground receiver to satellite, carrier frequency and environment around receiver. In this paper, since the interval between ground antennas is much small than shadow fading in different states, we

can approximate that received signal shallow fading of two antennas are the same.

In the model shown in Figure 3 and Figure 4, R_m and R_l are the m^th and the l^th antenna in ground receiver respectively, T_p and T_q are the p^th antenna and the q^th antenna in satellite respectively. x − y plane is parallel to horizontal surface, and is perpendicular to OO_R. O_R is the midpoint of two antennas R_m and R_l ground projection line. O_S is the point of intersection of x axis and y axis and also the midpoint of x − y plane projection line of two antennas T_p and T_q. O_T is the center of transmitting antenna array. H_S,max is the maximum height of scatterers distribution. R_S,max is the maximum range of scatterer distribution (the maximum radius). and represent projection of T_p and T_q in x − y plane respectively. is the angle between projection of transmitting antennas array and x axis. is the angle of receiving antennas array R_m R_l and x axis. is the angle of R_m R_l and x − y plane. is the angle between segment of receiving antenna arrays and satellite and x-y plane, namely the elevation. d_T and d_R is the distance of adjacent antenna on satellite and ground respectively. is the central angle between sub-satellite point and receiver antenna. To facilitate the analysis, d_a→b represent distance of a and b. the distance of and O_R is. And the distance of and O_T is .

According to the model, the impulse responses of sub channel from T_p to R_l can be written as

Since the large number of hypothetical scatterers in the model, let S_n be the n^th scatterer. Let R_S,max and H_S,max represent the radius and height of cylinder in the model, namely, the maximum distance and height of scatterers distribution. R_S(n) is the distance between O_R and S_n projection in x-y plane,. H_S(n) is the height of S_n,. Without considering the height of ground antennas, the n^th scatterer elevation in scattering path can be approximately shown as. and respectively describe the leaving angle and arriving angle of the n^th path, which is influenced by S_n. In the model, , , R_S(n), H_S(n) and are all random variables.

In multi-beam channels of mobile satellite system, the impulse responses of sub _channel from T_p to R_l can be written in terms of (5). Considering the influence from component beam on to received signal, the direct-path component of the channel response can be written as

And multipath component of the channel response can be written as

For the large number of scatterers, according to Central Limit Theorem, is a deterministic process. is a complex Gaussian process with a low-pass nonzero mean, and its envelop consistent with Rayleigh distribution.

In (6), is gain of satellite antenna p. is angle that ground antenna l deviates from beam center p. is gain of ground antenna l. Generally, ground antenna is omnidirectional antenna, so.

In (7), is the n^th multipath component random phase. is carrier frequency. A (n) is the amplitude of multipath component which is influenced by the n^th scatterer. K_pl represents sub-channel Rician Index from T_p to R_l,.

In the early stage of channel research, multi-antenna technology will get a huge channel capacity upgrade in the independent and identically distributed (iid) Rayleigh channel [5] [6], under the assumption that the interval between the antennas is large enough and the angle of arrival (AOA) angular spread is sufficiently large. However, in an actual mobile communication system, the angular spread of the received signal AOA is not large enough under the affection of the interval among antennas and scatterers distribution around. It results that the fading of received signal among different antennas and the signal received among the same antenna from different transmitting antenna are not statistically independent. So the correlation exists. This correlation is called by spatial correlation, its definition is written as

where p and q are two transmitting antennas. l and m are two receiving antennas. h_pl(t) represents channel response of antenna p to l. In the same way, h_qm(t) represents channel response of antenna q to m. E[•] is expectation. ρ_pl,qm is correlation coefficient between received signal of antenna l and m.

Since and are both zero-mean random process (),and are deterministic process, can be written as

According to the model shown in Figure 3. The spatial correlation coefficient of direct-path component is

where, which is described the distance of transmitting antenna p and q. is the distance of receiving antenna m and l.

Also, the spatial correlation coefficient of multipath component is

Since the large number of scatterers around ground receiver antennas (a larger number N), R_S, H_S and α_R can replace the discrete variables R_S(n), H_S(n) and α_R,n as an approximation. (11) can change into integral form

R_S, which is the distance of scatterer and ground receiver, can be described by hyperbolic distribution [4]. The height of scatterer H_S, which PDF depends on environment around receiver, is normally described by normal distribution [7], Lon-normal distribution [8] etc. The normal distribution is used for modeling in this paper. In ground wireless communication system, the arrival angle of multipath signal is described by evenly distribution [9] or Laplace distribution. But in mobile satellite communication system, Laplace distribution is more suitable under testing in urban area [10], which is affected by the number and the distribution of scatterers around ground receiver.

When the arrival angle of received signal in x-y plane α_R is Laplace distribution, the spatial correlation coefficient of multipath component in the received signal is shown as

where, and are given by (14)