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Influence of transmitter motion on received signal parameters – Analysis of the Doppler effect
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Wave Motion 45 (2008) 178–190 www.elsevier.com/locate/wavemoti
Influence of transmitter motion on received signal parameters – Analysis of the Doppler effect Jo´zef Rafa, Cezary Zio´łkowski
*
Military University of Technology, Gen. Sylwestra Kaliskiego 2 Str., 00-908 Warsaw, Poland Received 17 July 2006; received in revised form 26 April 2007; accepted 17 May 2007 Available online 26 May 2007
Abstract The result of the theoretical analysis of the electric field strength generated by a moving signal source is presented in this paper. The wave equation with source function taking a signal source motion into consideration is the basis of the analysis problem. To solve of the wave equation we have applied integral transformations i.e., the Laplace transform (with respect to normalized time variable) and the Fourier transform (with respect to space variables). The solution form of the analysis problem has been constructed as following: at first, we use the Cagniard-de Hoop’s method to calculate the inverse Fourier transform and next we apply inverse Laplace transform. As a numerical example, the influence of the velocity and the space location of a transmitter motion trajectory on the Doppler frequency is presented. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Propagation of electromagnetic wave in free space; Doppler effect; Cagniard-de Hoop’s method
1. Introduction The characteristic of the mobile communications is continual change of the transmitters and receivers location. Signal source motion results in changes of the received signal parameters – especially its frequency. This phenomena is called the Doppler effect. The solution of the Maxwell equations (taking into consideration stationary space conditions of the system transmitter–receiver) is the basis of the analytic description of the Doppler effect in the electrodynamics. In classic approach to this problem, the formula describing the electric field strength for a far-field region is applied to description of the signal source motion effect. The electric field strength in the case of the half wavelength dipole as an antenna is given by (cf. [1–3])
EðtÞ ¼ if0 l0 *
I 0 cosððp=2Þ cos hÞ ið2pf0 tbrÞ e sin h br
Corresponding author. Tel.: +48 22 683 96 19; fax: +48 22 683 90 38. E-mail address: [email protected] (C. Zio´łkowski).
0165-2125/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.wavemoti.2007.05.003
ð1Þ
J. Rafa, C. Zio´łkowski / Wave Motion 45 (2008) 178–190
179
where f0 is the signal frequency, b ¼ 2p ¼ 2p fc0 is the wave number (k – wavelength, c – speed of light), l0 is the k magnetic permeability of free-space, h is the elevation angle, I0 is the amplitude of the antenna current and r denotes distance (in general) between transmitter and receiver. Therefore the phase angle U(t) is following UðtÞ ¼ 2pf0 t br
ð2Þ
whereas when the transmitter is covering a distance Dr, the phase of the received signal can be represented as UðtÞ ¼ 2pf0 t bðr DrÞ
ð3Þ
(‘‘+’’ the signal source moves away the receiver, ‘‘–’’ the signal source moves towards the receiver). In the case of the constant velocity v, the displacement of the signal source amounts to Dr = v Æ t. Thus, the offset of a signal frequency is described by the following expression (cf. [4]) 1 d d 1 v v f ðtÞ ¼ UðtÞ ¼ f0 t br t ¼ f0 ð4Þ 2p dt dt 2p k k If the receiving site is situated at an angle u with respect to the motion direction of the signal source, then (cf. [5,6]) v f ðtÞ ¼ f0 cos u ð5Þ k The frequency fD ¼ kv cos u is called the Doppler frequency. Using the above equations we can obtain the analytic description of a frequency offset as a function of a transmitter–receiver location in space. The approximate expression describing the frequency offset with respect to space co-ordinates is presented in [7]. As you can see the analytic description of the Doppler effect is based on the solution of the Maxwell equations taking stationary space conditions of the system transmitter–receiver into consideration. In our paper we shall present the way to solve the Maxwell equations in the case of a signal source motion with a constant velocity. In this case the electric far-field strength is obtained by following methodology. Basing on the Maxwell equations, taking into account a signal source motion we get the vector wave equation of the electric field strength. To solve this wave equation we apply integral transformations i.e., the Laplace transform (with respect to normalized time variable) and the Fourier transform (with respect to space variables). Final solution form of the analysis problem we construct as follows: first, we use the Cagniard-de Hoop’s method to calculate the inverse Fourier transform and then we apply inverse Laplace transform. The analytic form of the obtained solution makes possible assessment of a received signal parameters change as a result of a moving transmitter. In particular, it concerns the frequency offset of a received signal. Numerical example of the frequency offset as a function of the transmitter location and motion parameters in some air-to-ground mobile communication system is presented in our paper. 2. Statement of the problem The propagation of electromagnetic wave in free space is examined in our paper. Basing on the Faraday and the Ampere equations as well as on the property of a vector field double rotation, we get a wave equation describing vector of the electric field strength E(x, t) = (Ex(x, t), Ey(x, t), Ez(x, t)) in the following form 2 1 o2 o o2 o2 o Eðx; tÞ Eðx; tÞ þ Eðx; tÞ þ Eðx; tÞ ¼ l0 i0 ðx; tÞ ð6Þ c2 ot2 ot ox2 oy 2 oz2 where i0(x, t) = (ix(x, t), iy(x, t), iz(x, t),) is the current density vector (distribution of current in antenna – source of the electromagnetic field) and x = (x, y, z) is the three-dimensional space co-ordinates. It is easy to see that the calculation of the electric field strength E(x, t) requires solving the system of the wave equations. In order to simplify the analysis we shall concentrate on the linear antenna system, i.e., we assume that the current density vector has the following form i0(x, t) = (0, 0, iz(x, t) = i0(z, t)d(x)d(y)), where d(Æ) is the Dirac distribution. Additionally, we assume the stationary conditions of the current propagation in the antenna, i.e., we pass over the transient state. Therefore the distribution of the current density amplitude
180
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along antenna is invariable in time. Hence, we can write i0(z, t) = I(z)i0(t), where I(z) denotes the distribution of the current density amplitude along the antenna and i0(t) describes a change of the current density in time. In view of the above assumptions the wave equation has the following form 2 1 o2 o o2 o2 o Eðx; tÞ Eðx; tÞ þ 2 Eðx; tÞ þ 2 Eðx; tÞ ¼ l0 iz ðx; tÞ ð7Þ c2 ot2 ot ox2 oy oz where E(x,t) = Ez(x,t). In our paper we shall concentrate on the assessment of a signal source motion influence on the received signal parameters. Taking the above assumptions into consideration, we will examine a signal source motion along x-axis with velocity ~ v ¼ ðvx ¼ v; 0; 0Þ. Space structure of the analysed problem is shown in Fig. 1. In this case we have o o iz ðx; tÞ ¼ ðIðzÞ i0 ðtÞdðx vtÞdðyÞÞ ot ot where oto ðÞ is the differential operator in the distribution sense. Thus the analytic description of the analysed problem is given by following equation 2 1 o2 o o2 o2 o Eðx; tÞ Eðx; tÞ þ Eðx; tÞ þ Eðx; tÞ ¼ l0 IðzÞ ði0 ðtÞ dðx vtÞÞ dðyÞ c2 ot2 ot ox2 oy 2 oz2
ð8Þ
ð9Þ
The aim of our paper is to construct the solution E(x, t) (the electric field strength) of the above equation as well as analysis of property its parameters, especially the frequency. The solution of Eq. (9) will be carried out in two steps. At first, we will find the fundamental solution of this equation. Next, the solution of the analysing problem will be obtained by the convolution of the fundamental solution with the right hand side of Eq. (9) i.e., the function describing space distribution of a current density. 3. Construction of the fundamental solution In the first step we shall find the fundamental solution Eðx; tÞ of Eq. (9). For a start, we apply a new system variables, namely x0 = (x0, x1, x2, x3) where x0 = ct, x1 = x vt, x2 = y, x3 = z and we use following denotations Eðx; tÞ ¼ Eðx; y; z; tÞ ¼ V ðx0 ; x1 ; x2 ; x3 Þ ¼ V ðx0 Þ In a new system of variables the found fundamental solution V(x0) satisfies the following equation
Fig. 1. Space structure of the analysed problem.
J. Rafa, C. Zio´łkowski / Wave Motion 45 (2008) 178–190 2 o2 o2 o2 o2 2 o V ðx Þ 2k V ðx Þ ð1 k Þ V ðx Þ V ðx Þ V ðx0 Þ ¼ dðx0 Þdðx1 Þdðx2 Þdðx3 Þ 0 0 0 0 ox0 ox1 ox20 ox21 ox22 ox23
181
ð10Þ
where k ¼ vc is the ratio between the signal source velocity and the speed of electromagnetic wave. 3.1. Fundamental solution of the Laplace transform and the Fourier transform We calculate the solution of Eq. (10) by means of the integral transformations. We apply the Laplace transform in relation to the variable x0 as well as Fourier transform in relation to the x2 and x3. Let V ðn0 Þ and Vb ðnÞ denote the Laplace and the Fourier transform of the V(x0) and V ðn0 Þ function, respectively, i.e., Z 1 V ðn0 Þ ¼ V ðn0 ; x1 ; x2 ; x3 Þ ¼ LfV ðx0 Þg ¼ V ðx0 ; x1 ; x2 ; x3 ; Þen0 x0 dx0 ð11Þ 0
where n0 = (n0, x1, x2, x3) is a multidimensional variable and n0 denotes a complex parameter of the Laplace transform. Now we apply the Fourier transform Z Z 1 Z 1 b iðn2 ;x2 Þ V ðnÞ ¼ F fV ðn0 Þg ¼ V ðn0 Þe dx2 ¼ V ðn0 ; x1 ; x1 ; x1 Þeiðn2 x2 þn3 x3 Þ dx2 dx3 ð12Þ 1
RR
1
where n = (n0, x1, n2, n3) is a multidimensional variable, n2 = (n2, n3) denotes a parameter of the two-dimensional Fourier transform and x2 = (x2, x3). Using the Laplace and the Fourier integration transforms (cf. [8]) with reference to Eq. (10), we obtain the following expressions – the Laplace integral transform (with respect to the x0 variable) n20 V ðn0 Þ 2kn0
o o2 o2 o2 V ðn0 Þ ð1 k 2 Þ 2 V ðn0 Þ 2 V ðn0 Þ 2 V ðn0 Þ ¼ dðx1 Þdðx2 Þdðx3 Þ ox1 ox1 ox2 ox3
ð13Þ
– the Fourier integral transform (with respect to x2 and x3 variables) n20 Vb ðnÞ 2kn0
d b d2 V ðnÞ ð1 k 2 Þ 2 Vb ðnÞ þ n22 Vb ðnÞ þ n23 Vb ðnÞ ¼ dðx1 Þ dx1 dx1
ð14Þ
2 which is the consequence of F fdðxi Þg ¼ 1 and F foxo 2 V ðn0 ; x1 ; x2 Þg ¼ n2i Vb ðnÞ, (i = 2, 3). i After the appropriate arrangement of Eq. (14) elements we obtain an ordinary differential equation second order
d2 b 2k d b ðn20 þ n22 þ n23 Þ b 1 V ðnÞ þ V ðnÞ V ðnÞ ¼ n dðx1 Þ 0 2 2 2 dx1 ð1 k Þ dx1 ð1 k Þ ð1 k 2 Þ
ð15Þ
The above equation is a linear differential equation with constant coefficients. Thus its a solution has the following form (cf. [9,10]) k x
H ðx1 Þe 1 1 þ H ðx1 Þe Vb ðnÞ ¼ ð1 k 2 Þðk2 k1 Þ
k2 x1
ð16Þ
where H(x1) is the Heaviside function and the k1, k2 are roots of characteristic multinomial of Eq. (15) that is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kn n20 þð1k 2 Þðn22 þn23 Þ k1;2 ¼ 0 for k < 1. 1k 2 The differential operator given by expression (15) is a self-adjoint operator. In this case, basing on property of the Green function ([11]) we can determine boundary conditions of the solution (16) as follows
J. Rafa, C. Zio´łkowski / Wave Motion 45 (2008) 178–190
182
lim Vb ðnÞ lim Vb ðnÞ ¼ 0
x1 !0þ
ð17Þ
x1 !0
d Vb d Vb ðnÞ lim ðnÞ ¼ 1 x1 !0 dx1 x1 !0 dx1 lim Vb ðnÞ ¼ 0
ð18Þ
limþ
ð19Þ
jx1 j!þ1
Taking into account the above expressions we calculate the integration constants of the solution (16). 3.2. Fundamental solution of the analysis problem In order to obtain the fundamental solution of Eq. (9) we make use of the expression (16) as well as of inverse Fourier and Laplace transforms. The inverse Fourier transform is calculated by applying the Cagniard-de Hoop’s method (cf. [12]). Example of practical usage of this method (for analysis of a microstrip problem) was presented in [13]. The form of the inverse Fourier transform (with respect to variables n2 and n3) is expressed as follows Z Z 1 Z 1 1 1 1 b b iðn2 ;x2 Þ V ðn0 Þ ¼ F n2 f V ðnÞg ¼ 2 V ðnÞe dn2 ¼ 2 Vb ðn0 ; x1 ; n2 ; n3 Þeiðn2 x2 þn3 x3 Þ dn2 dn3 ð20Þ 4p RR 4p 1 1 Taking it into account we obtain pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n20 þð1k 2 Þðn22 þn23 Þ Z 1 Z 1 H ðx1 Þ exp x1 1k 2 1 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi V ðn0 Þ ¼ eiðn2 x2 þn3 x3 Þ dn2 dn3 2 4p2 1 1 2 2 2 n0 þ 1 k n2 þ n3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kn0 þ n20 þð1k 2 Þðn22 þn23 Þ Z 1 Z 1 H ðx1 Þ exp x1 1k 2 1 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi þ 2 eiðn2 x2 þn3 x3 Þ dn2 dn3 4p 1 1 2 2 2 2 n0 þ ð1 k Þðn2 þ n3 Þ
kn0
ð21Þ
In order to calculate the above integrals we apply the de Hoop’s transform in the following form (cf. [9,10]) n2 x2 þ n3 x3 ¼ n0 qðx2 Þp
and
n2 x3 þ n3 x2 ¼ n0 qðx2 Þq
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where p and q are new variables of the Fourier transform and qðx2 Þ ¼ x22 þ x23 . After a simple transformation we obtain n2 ¼ ðpx2 qx3 Þ
n0 qðx2 Þ
and
n3 ¼ ðpx3 þ qx2 Þ
n0 qðx2 Þ
Hence we can write qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 kn0 n0 þ ð1 k Þðn2 þ n3 Þ ¼ n0 k 1 þ ð1 k 2 Þðp2 þ q2 Þ Simultaneously, we see that the Jakobian of this transform has the form on on n0 n0 2 2 n20 n20 op oq x2 qðx2 Þ x3 qðx2 Þ 2 2 ¼ x þ x ¼ n20 jJ j ¼ on on ¼ 2 2 3 2 3 3 x3 n0 x2 n0 q ðx2 Þ q ðx2 Þ op oq qðx2 Þ qðx2 Þ
ð22Þ
ð23Þ
ð24Þ
ð25Þ
Taking the above relationships into consideration the Laplace transform of the fundamental solution is described as follows qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 2 2 Z 1 Z 1 exp n0 1k2 kx1 þ jx1 j 1 þ ð1 k Þðp þ q Þ ipqðx2 Þ n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V ðn0 Þ ¼ 02 dpdq ð26Þ 4p 1 1 1 þ ð1 k 2 Þðp2 þ q2 Þ
J. Rafa, C. Zio´łkowski / Wave Motion 45 (2008) 178–190
Let g be a new parameter with the form qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 g¼ kx þ jx j 1 þ ð1 k 2 Þðp2 þ q2 Þ ipqðx2 Þ 1 1 1 k2
183
ð27Þ
In this way we can use the concept of the Cagniard contour C defined in our case by the following expression (cf. [14]) C : pðgÞ ¼ p1 ðgÞ þ ip2 ðgÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ffi 1þq2 ð1k Þ k where g P 1k , and p1(g) is the real component of the Cagniard contour 2 x1 þ R1 ðx1 ; x2 Þ 1k 2 ffi! pffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 jx1 j 1 k 2 k 1 þ q2 ð1 k 2 Þ 2 x1 R1 ðx1 ; x2 Þ p1 ðgÞ ¼ 2 g 2 R1 ðx1 ; x2 Þ 1 k2 ð1 k 2 Þ
ð28Þ
p2(g) is the imaginary component of the Cagniard contour qðx2 Þð1 k 2 Þ k g p2 ðgÞ ¼ x1 R21 ðx1 ; x2 Þ 1 k2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Where R1 ðx1 ; x2 Þ ¼ x21 þ q2 ðx2 Þð1 k 2 Þ. Using the Cagniard’s method we replace the integral along real axis p by the integral over a suitably chosen contour C (cf. Fig. 2). The integrals over way C+ and C vanish in view of the Jordan Lemma, and also, the residual term and the contribution coming from the integrand branch point located over the contour C (see Fig. 2) are equal zero. Taking the above notice into consideration the double integral (26) can be expressed as follows Z 1 Z 1 n en0 g rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi dgdq V ðn0 Þ ¼ 02 ð29Þ 2 4p 1 g1 1þq2 ð1k 2 Þ 2 k g 1k2 x1 R1 ðx1 ; x2 Þ ð1k2 Þ2 where g1 is the lower limit of the integration with respect to variable g and equals qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ q2 ð1 k 2 Þ k g1 ¼ x þ R ðx ; x Þ 1 1 1 2 1 k2 1 k2 After the iteration of the above integrations Eq. (29) obtain
Fig. 2. Cagniard contour C and mutual location of the vertex point and the integrand branch point.
J. Rafa, C. Zio´łkowski / Wave Motion 45 (2008) 178–190
184
V ðn0 Þ ¼
n0 4p2
Z
1
en0 g
g0
Z
q0
q0
dq r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi dg g
k 1k 2
2
x1
R21 ðx1 ;x2 Þ ð1k 2 Þ
q2 þ
ð30Þ
1 1k 2
where ±q0 are the integration limits with respect to variable q and equal vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 u k u g 1k ð1 k 2 Þ 2 x1 1 q0 ¼ t R21 ðx1 ; x2 Þ ð1 k 2 Þ2 and g0 is the lower limit of the integration with respect to variable g and equals g0 ðx1 ; x2 Þ ¼
k R1 ðx1 ; x2 Þ x1 þ 1 k2 ð1 k 2 Þ
Hence, the Laplace transform of the fundamental solution takes the following form V ðn0 Þ ¼
en0 g0 ðx1 ;x2 Þ 4pR1 ðx1 ; x2 Þ
ð31Þ
Thus, taking the translation theorem (cf. [8]) into consideration the fundamental solution of the analysis Eq. (10) expresses the following dependence V ðx0 Þ ¼
1 dðx0 g0 ðx1 ; x2 ÞÞ 4pR1 ðx1 ; x2 Þ
So, the fundamental solution Eðx; tÞ for Eq. (9) is given by formula 1 g ðx; tÞ Eðx; tÞ ¼ d t 0 4pcR0 ðx; tÞ c
ð32Þ
ð33Þ
where R0 is a reduced distance from a signal source to a receiver at the moment t qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R0 ðx; tÞ ¼ ðx vtÞ2 þ ð1 k 2 Þðy 2 þ z2 Þ x, y, z are location co-ordinate of a receiver with reference to the established co-ordinate system and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 kðx vtÞ þ ðx vtÞ þ ð1 k 2 Þðy 2 þ z2 Þ . g0 ðx; tÞ ¼ 1k 2 4. The solution form of the analysis problem The final solution of the analysis problem we obtain as a convolution of the fundamental solution with the right hand side of Eq. (9), i.e., with the function describing space distribution of a current density. Therefore the solution of Eq. (9) is described as follows Eðx; tÞ ¼ Eðx; tÞ l0 IðzÞ
o ði0 ðtÞ dðx vtÞÞ dðyÞ ot
ð34Þ
In our paper we concentrate on one elementary case of a signal source. We assume that the antenna is supplied by a harmonic signal. So, in each point of the antenna the change in time of a current density is described as follows i0 ðtÞ ¼ eix0 t . In this case, changing the variables ((x, y, z, t) ! (x0, x1, x2, x3)), we get x0 o o ox0 o ox1 o ixc0 x0 iz ðx; tÞ ¼ þ e dðx1 Þdðx2 ÞIðx3 Þ iz ðx0 Þ ¼ ix0 ei c x0 dðx1 Þdðx2 ÞIðx3 Þ v ot ox0 ot ox1 ot ox1 o ¼ ix0 hðx0 Þ v hðx0 Þ ð35Þ ox1 x
where hðx0 Þ ¼ ei c x0 dðx1 Þdðx2 ÞIðx3 Þ. Taking the above expression into consideration we can rewrite the formula (34) in the following form
J. Rafa, C. Zio´łkowski / Wave Motion 45 (2008) 178–190
o Aðx0 Þ ¼ l0 V ðx0 Þ ix0 hðx0 Þ v hðx0 Þ ox1
¼ il0 x0 ðV ðx0 Þ hðx0 ÞÞ þ l0 v
185
o ðV ðx0 Þ hðx0 ÞÞ ox1
ð36Þ
where A(x0) = E(x,t). Thus, in order to obtain the solution of the analysed problem we calculate the convolution V(x0) * h(x0) which equals Z x0 0 1 1 V ðx0 Þ hðx0 Þ ¼ dðx0 g0 x00 Þei c x0 dðx01 Þdðx02 ÞIðx03 Þdx00 dx01 dx02 dx03 0 0 0 4p X R1 ðx1 x1 ; x2 x2 ; x3 x3 Þ x0 Z 1 ei c ðx0 g0 Þ Iðx0 Þdx0 ¼ ð37Þ 4p R R1 ðx1 ; x2 ; x3 x03 Þ 3 3 where X ¼ R3 ð0; x0 Þ. Differentiating the above expression with respect to x1 we get Z x0 o 1 x0 x1 x0 ðV ðx0 Þ hðx0 ÞÞ ¼ ei c ðx0 g0 Þ i þi 2 2 0 ox1 4p R ð1 k Þc R1 ðx1 ; x2 ; x3 x3 Þ ð1 k 2 Þc k x1 þ Iðx03 Þdx03 R1 ðx1 ; x2 ; x3 x03 Þ R31 ðx1 ; x2 ; x3 x03 Þ Putting (37) and (38) into (36) and changing the variables ((x0, x1, x2, x3) ! (x, y, z, t)), we obtain Z g ðx;tÞ 0 l 1 x vt x vt Eðx; tÞ ¼ 0 þv 3 þ ikx1 2 ix1 eix0 t c Ið1Þd1 R0 ðx; y; z 1; tÞ 4p R R0 ðx; y; z 1; tÞ R0 ðx; y; z 1; tÞ g ðx; tÞ P0 for t 0 c
ð38Þ
ð39Þ
x0 where x1 ¼ 1k 2 Now, we calculate the solution of the analysed problem for the far zone, i.e., bR0(x, t) 1 for each t P 0. It is a practical case occurring in mobile communications. In this case we make use of analogous space simplifications as for example in classic cases of the Hertz dipole (cf. [1,2]). So, we can write sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2z1 þ 12 z1 2 2 2 ð40Þ ffi R0 ðx; tÞ ð1 k 2 Þ ðx vtÞ þ ð1 k Þðy 2 þ ðz 1Þ Þ ¼ R0 ðx; tÞ 1 ð1 k 2 Þ 2 R0 ðx; tÞ R0 ðx; tÞ
Furthermore we assume that an antenna system is a half-wavelength dipole. Then the distribution of a current density amplitude along the antenna has the following form (cf. [1,2]) ( I 0 cos bz for jzj 6 2l IðzÞ ¼ ð41Þ 0 for jzj > 2l ðl length of the dipoleÞ Applying the above classic approaches, analogous as for example in [1,2] or [3] the practical value of the electric field strength E(x, t), for the far zone is expressed as follows l0 1 x vt p x1 I 0 þk 2 Eðx; tÞ ¼ eix1 tib1 kxib1 R0 ðx;tÞþi2 R0 ðx; tÞ 4p R0 ðx; tÞ Z l=2 z 1 1 k ið1k 2 Þb1 R ðx;tÞ 0 ð42Þ e cosðð1 k 2 Þb1 1Þd1 for t P R0 ðx; tÞ þ x c c l=2 2p where b1 ¼ xc1 ¼ kð1k 2 Þ Taking into account l/2 = k/4 for the half-wavelength dipole and after simple calculations as for example in [1], we obtain
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186
Z
l=2
e
ið1k
2
z 1 Þb1 R ðx;tÞ 0
cosðð1 k 2 Þb1 1d1 ¼ 2
l=2
Basing on the following expressions Z 0 ¼
cos
p z 2 R0 ðx;tÞ
2 ð1 k 2 Þb1 1 R2 zðx;tÞ
qffiffiffiffi l0 e0
ð43Þ
0
and c ¼
1 pffiffiffiffiffiffi , l0 e0
we can write l0c = Z0, where Z0 is a wave imped-
ance of the propagation medium and e0 denotes a permittivity. So, the analytic form of the electric far-field strength generated by moving transmitter is p z Z0 1 x vt cos 2 R0 ðx;tÞ ib1 kxib1 R0 ðx;tÞþip ix1 t 2e Eðx; tÞ ¼ I0 e ð44Þ þk 2 2 R0 ðx; tÞ R0 ðx; tÞ 2pð1 k 2 Þ 1 R2 zðx;tÞ 0
But E(x,t) is an analytic signal, hence we can write Eðx; tÞ ¼ E0 ðx; tÞeiUðx;tÞ
ð45Þ
where E0(x, t) = |E(x, t)| denotes the signal amplitude and U(x, t) = arg E(x, t) is the phase angle of a signal. Thus, it is easy to see that the amplitude E0(x, t) and the phase angle U(x, t) of the signal E(x, t) are described as follows p z Z0 1 x vt cos 2 R0 ðx;tÞ þk 2 E0 ðx; tÞ ¼ ð46Þ I0 2 R0 ðx; tÞ R0 ðx; tÞ 2pð1 k 2 Þ 1 R2 zðx;tÞ 0
and Uðx; tÞ ¼ x1 t b1 kx b1 R0 ðx; tÞ
p 2
ð47Þ
Hence, the instantaneous frequency f(x, t) is expressed as follows f ðx; tÞ ¼
1 d 1 f0 d Uðx; tÞ ¼ R0 ðx; tÞ f 2 0 2 2p dt dt cð1 k Þ 1k
ð48Þ
But d d R0 ðx; tÞ ¼ dt dt
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vðx vtÞ ðx vtÞ2 þ ð1 k 2 Þðy 2 þ z2 Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx vtÞ2 þ ð1 k 2 Þðy 2 þ z2 Þ
therefore the instantaneous frequency f(x, t) has the form 1 k x vt qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f0 f ðx; tÞ ¼ f þ 2 0 2 1k 1k ðx vtÞ2 þ ð1 k 2 Þðy 2 þ z2 Þ So, the Doppler frequency is given by the following formula 0 fD ðx; tÞ ¼ f ðx; tÞ f0 ¼
ð49Þ
ð50Þ
1
k B x vt C k þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAf0 2@ 2 1k 2 2 2 ðx vtÞ þ ð1 k Þðy þ z Þ
ð51Þ
Hence, we notice that the value of the Doppler frequency fD(x, t) depends not only on a signal source velocity and a carrier frequency but also space location of a motion trajectory. The Doppler frequency fD(x, t) is linear function of the frequency carrier, whereas the dependence on velocity and space co-ordinates has a more complex character. 5. Numerical example of the Doppler frequency calculation The influence of the velocity and the transmitter–receiver space location on value of the Doppler frequency is presented in this part of our paper. We concentrate on the example of the air-to-ground mobile communi-
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187
Fig. 3. Space structure model for numerical calculation.
Fig. 4. Doppler frequency fD versus distance vt and height y of the aircraft flight.
cations where the change of the transmitter–receiver space location concerns big altitudes. The numerical calculations were conducted for a line-of-sight region. In this case we considered the space co-ordinates y and z as a constant parameter of the motion. Space structure model for analysis of the motion influence on the Doppler frequency is shown in Fig. 3. We assumed that the first space co-ordinate of the receiver is x = 5000 m and the signal frequency is f0 = 900 MHz. Fig. 4 shows the three-dimensional plot of the Doppler frequency fD versus vt and y where vt denotes the value of the transmitter–receiver distance, and y is the height of the aircraft flight. Fig. 5 shows the Doppler frequency versus distance vt for some heights of the aircraft flight. In this case the curves in Fig. 5 are suitable intersections of the three-dimensional plot shown in Fig. 4. The influence of an aircraft velocity on the Doppler frequency offset is presented in Figs. 6 and 7. Fig. 6 shows the three-dimensional plot of the Doppler frequency fD versus distance vt and velocity v. Fig. 7 shows the Doppler frequency fD versus distance vt for some velocities of the aircraft flight. In this case the curves in Fig. 7 are suitable intersections of the three-dimensional plot shown in Fig. 6. In Figs. 4–7 we present the results when aircraft flies exactly above the site of the receiver location i.e. orthogonal project of the flight trajectory to the ground area includes the point of a receiver location. Next two figures present numerical calculation results for the case when a trajectory of the aircraft flight goes near a receiver location site. The influence of a distance between the receiver location point and the aircraft trajec-
188
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Fig. 5. Doppler frequency fD versus distance vt for some heights of the aircraft flight.
Fig. 6. Doppler frequency fD versus distance vt and velocity v of the aircraft flight.
Fig. 7. Doppler frequency fD versus distance vt for some velocities of the aircraft flight.
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189
Fig. 8. Doppler frequency fD versus distance vt and distance z between a receiver location site and aircraft flight trajectory.
Fig. 9. Doppler frequency fD versus distance vt for some distances between a receiver location site and aircraft flight trajectory.
tory trace on the ground area is presented in Figs. 8 and 9. Fig. 8 shows the three-dimensional plot of the Doppler frequency fD versus distance vt and space co-ordinate z. Fig. 9 shows the Doppler frequency fD versus distance vt for some values z. In this case the curves in Fig. 9 are suitable intersections of the three-dimensional plot shown in Fig. 8. 6. Summary and conclusions The detailed mathematical analysis of a transmitter motion influence on received signal parameters especially the Doppler frequency has been presented in this paper. The calculation of the dependence describing the electric field strength generated by moving signal source has been the main aim of our analysis. On the basis of the obtained formula we can carry out an assessment of a received signal frequency offset in time and space. Namely following three categories of the parameters influence the Doppler frequency value: space parameter (co-ordinates of a receiver location site), motion parameter (velocity of a signal source), signal parameter (signal frequency). In the characteristics presented in Section 5 we can distinguish three fundamental ranges: the range of a maximum value, the range of dynamic changes, the range of a minimum value. The influence of individual parameter on running of the Doppler frequency characteristics is diverse. Namely, the mutual location of a signal source-receiver influences the slope of the Doppler characteristic in the range of dynamic changes. The increase of a distance from a transmitter motion trajectory to a receiver location site
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190
decreases a slope of the frequency offset plot. However motion and signal parameters determine minimum and maximum value of the Doppler frequency, i.e., the increase of the velocity and carrier frequency values causes increases of the frequency offset range. One-dimensional distribution of a current density in antenna has been considered in our paper but presented methodology has a universal character. In a case of two- or three-dimensional distribution the analysed problem requires solving system equations by using presented methodology with reference to each equation. In this case, taking the analytic form of an analysis signal into consideration, the resultant solution is sum of the component solutions. In order to describe the Doppler effect occurring in mobile communications we put simplified formula into practice. This approximation is based on solution of the Maxwell equations taking stationary space conditions of the system transmitter–receiver into consideration. In this case the Doppler frequency can be estimated by the expressions (4) or (5). In most mobile communications we observe relation c v, i.e., k 1. Thus, in this case estimation mistake is negligible. But applying the formula (51) to describe the Doppler effect we can investigate influence of the space, motion and signal parameters on a received signal frequency offset. Furthermore, in mobile communications where objects move with high velocity (for example in satellite communications, radio-astronomy) the exact formula (51) describing the Doppler frequency is required. Finally, we want to notice that the solving methodology of the signal source motion problem, described in our paper has an universal character. This method presented here can be applied to all branches of the classic physics. For example very interesting results for this problem are expected in the acoustics where parameter k achieves difference value with reference to 1. Acknowledgement The authors would like to express thanks to Mrs. B. Niklewicz and Mr. J.M. Kelner for preparing final edition form of this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
C.A. Balanis, Antenna Theory: Analysis and Design, Wiley, New York, 1997. M. Zahn, Electromagnetic Field Theory: A Problem Solving Approach, Wiley, New York, 1979. T.S. Rappaport, Wireless Communications – Principles and Practice, IEEE Press, New York, 1996. M. Kayton, W.R. Fried, Avionics Navigation Systems, Wiley, New York, 1969. F.B. Berger, The Design of Airborne Doppler Velocity Measuring Systems, IRE Trans. Aeron. Navig. Elect., vol. ANE-4 (1957) 157– 175. F.B. Berger, The Nature of Doppler Velocity Measurement, IRE Trans. Aeron. Navig. Elect., vol. ANE-4 (1957) 104–112. R.K. Hawkins, P.J. Farris-Manning, J.R. Gibson, K.P. Singh, Calibration of the CCRS Airborne Scatterometers, IEEE Trans. Antennas Propagat. 38 (2004) 903–918. A.D. Poularakis, The Transforms and Applications Handbook, CRC Press, Boca Raton, Florida, 2000. D.G. Duffy, Transform Methods for Solving Partial Differential Equations, CRC Press, Boca Raton, Florida, 1994. A.T. de Hoop, Handbook of Radiation and Scattering of Waves: Acoustic Waves in Fluids, Elastic Waves in Solids, Electromagnetic Waves, Academic Press, San Diego, 1995. B. Friedman, Principles and Techniques of Applied Mathematics, Wiley, New York, 1956. A.T. de Hoop, A modification of Cagniard’s method for solving seismic pulse problems, Appl. Sci. Res. B8 (1960) 349–356. M.-Y. Xia, Ch. H. Chan, Y. Xu, W.Ch. Chew, Time-domain Green’s functions for mictrostrip structures Using Cagniarda-deHoop method, IEEE Trans. Antennas Propagat. 52 (2004) 1578–1584. L. Cagniard, Reflection and Refraction of Progressive Seismic Waves, Mc Grow-Hill, New York, 1962.
Wave Motion 45 (2008) 178–190 www.elsevier.com/locate/wavemoti
Influence of transmitter motion on received signal parameters – Analysis of the Doppler effect Jo´zef Rafa, Cezary Zio´łkowski
*
Military University of Technology, Gen. Sylwestra Kaliskiego 2 Str., 00-908 Warsaw, Poland Received 17 July 2006; received in revised form 26 April 2007; accepted 17 May 2007 Available online 26 May 2007
Abstract The result of the theoretical analysis of the electric field strength generated by a moving signal source is presented in this paper. The wave equation with source function taking a signal source motion into consideration is the basis of the analysis problem. To solve of the wave equation we have applied integral transformations i.e., the Laplace transform (with respect to normalized time variable) and the Fourier transform (with respect to space variables). The solution form of the analysis problem has been constructed as following: at first, we use the Cagniard-de Hoop’s method to calculate the inverse Fourier transform and next we apply inverse Laplace transform. As a numerical example, the influence of the velocity and the space location of a transmitter motion trajectory on the Doppler frequency is presented. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Propagation of electromagnetic wave in free space; Doppler effect; Cagniard-de Hoop’s method
1. Introduction The characteristic of the mobile communications is continual change of the transmitters and receivers location. Signal source motion results in changes of the received signal parameters – especially its frequency. This phenomena is called the Doppler effect. The solution of the Maxwell equations (taking into consideration stationary space conditions of the system transmitter–receiver) is the basis of the analytic description of the Doppler effect in the electrodynamics. In classic approach to this problem, the formula describing the electric field strength for a far-field region is applied to description of the signal source motion effect. The electric field strength in the case of the half wavelength dipole as an antenna is given by (cf. [1–3])
EðtÞ ¼ if0 l0 *
I 0 cosððp=2Þ cos hÞ ið2pf0 tbrÞ e sin h br
Corresponding author. Tel.: +48 22 683 96 19; fax: +48 22 683 90 38. E-mail address: [email protected] (C. Zio´łkowski).
0165-2125/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.wavemoti.2007.05.003
ð1Þ
J. Rafa, C. Zio´łkowski / Wave Motion 45 (2008) 178–190
179
where f0 is the signal frequency, b ¼ 2p ¼ 2p fc0 is the wave number (k – wavelength, c – speed of light), l0 is the k magnetic permeability of free-space, h is the elevation angle, I0 is the amplitude of the antenna current and r denotes distance (in general) between transmitter and receiver. Therefore the phase angle U(t) is following UðtÞ ¼ 2pf0 t br
ð2Þ
whereas when the transmitter is covering a distance Dr, the phase of the received signal can be represented as UðtÞ ¼ 2pf0 t bðr DrÞ
ð3Þ
(‘‘+’’ the signal source moves away the receiver, ‘‘–’’ the signal source moves towards the receiver). In the case of the constant velocity v, the displacement of the signal source amounts to Dr = v Æ t. Thus, the offset of a signal frequency is described by the following expression (cf. [4]) 1 d d 1 v v f ðtÞ ¼ UðtÞ ¼ f0 t br t ¼ f0 ð4Þ 2p dt dt 2p k k If the receiving site is situated at an angle u with respect to the motion direction of the signal source, then (cf. [5,6]) v f ðtÞ ¼ f0 cos u ð5Þ k The frequency fD ¼ kv cos u is called the Doppler frequency. Using the above equations we can obtain the analytic description of a frequency offset as a function of a transmitter–receiver location in space. The approximate expression describing the frequency offset with respect to space co-ordinates is presented in [7]. As you can see the analytic description of the Doppler effect is based on the solution of the Maxwell equations taking stationary space conditions of the system transmitter–receiver into consideration. In our paper we shall present the way to solve the Maxwell equations in the case of a signal source motion with a constant velocity. In this case the electric far-field strength is obtained by following methodology. Basing on the Maxwell equations, taking into account a signal source motion we get the vector wave equation of the electric field strength. To solve this wave equation we apply integral transformations i.e., the Laplace transform (with respect to normalized time variable) and the Fourier transform (with respect to space variables). Final solution form of the analysis problem we construct as follows: first, we use the Cagniard-de Hoop’s method to calculate the inverse Fourier transform and then we apply inverse Laplace transform. The analytic form of the obtained solution makes possible assessment of a received signal parameters change as a result of a moving transmitter. In particular, it concerns the frequency offset of a received signal. Numerical example of the frequency offset as a function of the transmitter location and motion parameters in some air-to-ground mobile communication system is presented in our paper. 2. Statement of the problem The propagation of electromagnetic wave in free space is examined in our paper. Basing on the Faraday and the Ampere equations as well as on the property of a vector field double rotation, we get a wave equation describing vector of the electric field strength E(x, t) = (Ex(x, t), Ey(x, t), Ez(x, t)) in the following form 2 1 o2 o o2 o2 o Eðx; tÞ Eðx; tÞ þ Eðx; tÞ þ Eðx; tÞ ¼ l0 i0 ðx; tÞ ð6Þ c2 ot2 ot ox2 oy 2 oz2 where i0(x, t) = (ix(x, t), iy(x, t), iz(x, t),) is the current density vector (distribution of current in antenna – source of the electromagnetic field) and x = (x, y, z) is the three-dimensional space co-ordinates. It is easy to see that the calculation of the electric field strength E(x, t) requires solving the system of the wave equations. In order to simplify the analysis we shall concentrate on the linear antenna system, i.e., we assume that the current density vector has the following form i0(x, t) = (0, 0, iz(x, t) = i0(z, t)d(x)d(y)), where d(Æ) is the Dirac distribution. Additionally, we assume the stationary conditions of the current propagation in the antenna, i.e., we pass over the transient state. Therefore the distribution of the current density amplitude
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along antenna is invariable in time. Hence, we can write i0(z, t) = I(z)i0(t), where I(z) denotes the distribution of the current density amplitude along the antenna and i0(t) describes a change of the current density in time. In view of the above assumptions the wave equation has the following form 2 1 o2 o o2 o2 o Eðx; tÞ Eðx; tÞ þ 2 Eðx; tÞ þ 2 Eðx; tÞ ¼ l0 iz ðx; tÞ ð7Þ c2 ot2 ot ox2 oy oz where E(x,t) = Ez(x,t). In our paper we shall concentrate on the assessment of a signal source motion influence on the received signal parameters. Taking the above assumptions into consideration, we will examine a signal source motion along x-axis with velocity ~ v ¼ ðvx ¼ v; 0; 0Þ. Space structure of the analysed problem is shown in Fig. 1. In this case we have o o iz ðx; tÞ ¼ ðIðzÞ i0 ðtÞdðx vtÞdðyÞÞ ot ot where oto ðÞ is the differential operator in the distribution sense. Thus the analytic description of the analysed problem is given by following equation 2 1 o2 o o2 o2 o Eðx; tÞ Eðx; tÞ þ Eðx; tÞ þ Eðx; tÞ ¼ l0 IðzÞ ði0 ðtÞ dðx vtÞÞ dðyÞ c2 ot2 ot ox2 oy 2 oz2
ð8Þ
ð9Þ
The aim of our paper is to construct the solution E(x, t) (the electric field strength) of the above equation as well as analysis of property its parameters, especially the frequency. The solution of Eq. (9) will be carried out in two steps. At first, we will find the fundamental solution of this equation. Next, the solution of the analysing problem will be obtained by the convolution of the fundamental solution with the right hand side of Eq. (9) i.e., the function describing space distribution of a current density. 3. Construction of the fundamental solution In the first step we shall find the fundamental solution Eðx; tÞ of Eq. (9). For a start, we apply a new system variables, namely x0 = (x0, x1, x2, x3) where x0 = ct, x1 = x vt, x2 = y, x3 = z and we use following denotations Eðx; tÞ ¼ Eðx; y; z; tÞ ¼ V ðx0 ; x1 ; x2 ; x3 Þ ¼ V ðx0 Þ In a new system of variables the found fundamental solution V(x0) satisfies the following equation
Fig. 1. Space structure of the analysed problem.
J. Rafa, C. Zio´łkowski / Wave Motion 45 (2008) 178–190 2 o2 o2 o2 o2 2 o V ðx Þ 2k V ðx Þ ð1 k Þ V ðx Þ V ðx Þ V ðx0 Þ ¼ dðx0 Þdðx1 Þdðx2 Þdðx3 Þ 0 0 0 0 ox0 ox1 ox20 ox21 ox22 ox23
181
ð10Þ
where k ¼ vc is the ratio between the signal source velocity and the speed of electromagnetic wave. 3.1. Fundamental solution of the Laplace transform and the Fourier transform We calculate the solution of Eq. (10) by means of the integral transformations. We apply the Laplace transform in relation to the variable x0 as well as Fourier transform in relation to the x2 and x3. Let V ðn0 Þ and Vb ðnÞ denote the Laplace and the Fourier transform of the V(x0) and V ðn0 Þ function, respectively, i.e., Z 1 V ðn0 Þ ¼ V ðn0 ; x1 ; x2 ; x3 Þ ¼ LfV ðx0 Þg ¼ V ðx0 ; x1 ; x2 ; x3 ; Þen0 x0 dx0 ð11Þ 0
where n0 = (n0, x1, x2, x3) is a multidimensional variable and n0 denotes a complex parameter of the Laplace transform. Now we apply the Fourier transform Z Z 1 Z 1 b iðn2 ;x2 Þ V ðnÞ ¼ F fV ðn0 Þg ¼ V ðn0 Þe dx2 ¼ V ðn0 ; x1 ; x1 ; x1 Þeiðn2 x2 þn3 x3 Þ dx2 dx3 ð12Þ 1
RR
1
where n = (n0, x1, n2, n3) is a multidimensional variable, n2 = (n2, n3) denotes a parameter of the two-dimensional Fourier transform and x2 = (x2, x3). Using the Laplace and the Fourier integration transforms (cf. [8]) with reference to Eq. (10), we obtain the following expressions – the Laplace integral transform (with respect to the x0 variable) n20 V ðn0 Þ 2kn0
o o2 o2 o2 V ðn0 Þ ð1 k 2 Þ 2 V ðn0 Þ 2 V ðn0 Þ 2 V ðn0 Þ ¼ dðx1 Þdðx2 Þdðx3 Þ ox1 ox1 ox2 ox3
ð13Þ
– the Fourier integral transform (with respect to x2 and x3 variables) n20 Vb ðnÞ 2kn0
d b d2 V ðnÞ ð1 k 2 Þ 2 Vb ðnÞ þ n22 Vb ðnÞ þ n23 Vb ðnÞ ¼ dðx1 Þ dx1 dx1
ð14Þ
2 which is the consequence of F fdðxi Þg ¼ 1 and F foxo 2 V ðn0 ; x1 ; x2 Þg ¼ n2i Vb ðnÞ, (i = 2, 3). i After the appropriate arrangement of Eq. (14) elements we obtain an ordinary differential equation second order
d2 b 2k d b ðn20 þ n22 þ n23 Þ b 1 V ðnÞ þ V ðnÞ V ðnÞ ¼ n dðx1 Þ 0 2 2 2 dx1 ð1 k Þ dx1 ð1 k Þ ð1 k 2 Þ
ð15Þ
The above equation is a linear differential equation with constant coefficients. Thus its a solution has the following form (cf. [9,10]) k x
H ðx1 Þe 1 1 þ H ðx1 Þe Vb ðnÞ ¼ ð1 k 2 Þðk2 k1 Þ
k2 x1
ð16Þ
where H(x1) is the Heaviside function and the k1, k2 are roots of characteristic multinomial of Eq. (15) that is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kn n20 þð1k 2 Þðn22 þn23 Þ k1;2 ¼ 0 for k < 1. 1k 2 The differential operator given by expression (15) is a self-adjoint operator. In this case, basing on property of the Green function ([11]) we can determine boundary conditions of the solution (16) as follows
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182
lim Vb ðnÞ lim Vb ðnÞ ¼ 0
x1 !0þ
ð17Þ
x1 !0
d Vb d Vb ðnÞ lim ðnÞ ¼ 1 x1 !0 dx1 x1 !0 dx1 lim Vb ðnÞ ¼ 0
ð18Þ
limþ
ð19Þ
jx1 j!þ1
Taking into account the above expressions we calculate the integration constants of the solution (16). 3.2. Fundamental solution of the analysis problem In order to obtain the fundamental solution of Eq. (9) we make use of the expression (16) as well as of inverse Fourier and Laplace transforms. The inverse Fourier transform is calculated by applying the Cagniard-de Hoop’s method (cf. [12]). Example of practical usage of this method (for analysis of a microstrip problem) was presented in [13]. The form of the inverse Fourier transform (with respect to variables n2 and n3) is expressed as follows Z Z 1 Z 1 1 1 1 b b iðn2 ;x2 Þ V ðn0 Þ ¼ F n2 f V ðnÞg ¼ 2 V ðnÞe dn2 ¼ 2 Vb ðn0 ; x1 ; n2 ; n3 Þeiðn2 x2 þn3 x3 Þ dn2 dn3 ð20Þ 4p RR 4p 1 1 Taking it into account we obtain pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n20 þð1k 2 Þðn22 þn23 Þ Z 1 Z 1 H ðx1 Þ exp x1 1k 2 1 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi V ðn0 Þ ¼ eiðn2 x2 þn3 x3 Þ dn2 dn3 2 4p2 1 1 2 2 2 n0 þ 1 k n2 þ n3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kn0 þ n20 þð1k 2 Þðn22 þn23 Þ Z 1 Z 1 H ðx1 Þ exp x1 1k 2 1 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi þ 2 eiðn2 x2 þn3 x3 Þ dn2 dn3 4p 1 1 2 2 2 2 n0 þ ð1 k Þðn2 þ n3 Þ
kn0
ð21Þ
In order to calculate the above integrals we apply the de Hoop’s transform in the following form (cf. [9,10]) n2 x2 þ n3 x3 ¼ n0 qðx2 Þp
and
n2 x3 þ n3 x2 ¼ n0 qðx2 Þq
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where p and q are new variables of the Fourier transform and qðx2 Þ ¼ x22 þ x23 . After a simple transformation we obtain n2 ¼ ðpx2 qx3 Þ
n0 qðx2 Þ
and
n3 ¼ ðpx3 þ qx2 Þ
n0 qðx2 Þ
Hence we can write qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 kn0 n0 þ ð1 k Þðn2 þ n3 Þ ¼ n0 k 1 þ ð1 k 2 Þðp2 þ q2 Þ Simultaneously, we see that the Jakobian of this transform has the form on on n0 n0 2 2 n20 n20 op oq x2 qðx2 Þ x3 qðx2 Þ 2 2 ¼ x þ x ¼ n20 jJ j ¼ on on ¼ 2 2 3 2 3 3 x3 n0 x2 n0 q ðx2 Þ q ðx2 Þ op oq qðx2 Þ qðx2 Þ
ð22Þ
ð23Þ
ð24Þ
ð25Þ
Taking the above relationships into consideration the Laplace transform of the fundamental solution is described as follows qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 2 2 Z 1 Z 1 exp n0 1k2 kx1 þ jx1 j 1 þ ð1 k Þðp þ q Þ ipqðx2 Þ n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V ðn0 Þ ¼ 02 dpdq ð26Þ 4p 1 1 1 þ ð1 k 2 Þðp2 þ q2 Þ
J. Rafa, C. Zio´łkowski / Wave Motion 45 (2008) 178–190
Let g be a new parameter with the form qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 g¼ kx þ jx j 1 þ ð1 k 2 Þðp2 þ q2 Þ ipqðx2 Þ 1 1 1 k2
183
ð27Þ
In this way we can use the concept of the Cagniard contour C defined in our case by the following expression (cf. [14]) C : pðgÞ ¼ p1 ðgÞ þ ip2 ðgÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ffi 1þq2 ð1k Þ k where g P 1k , and p1(g) is the real component of the Cagniard contour 2 x1 þ R1 ðx1 ; x2 Þ 1k 2 ffi! pffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 jx1 j 1 k 2 k 1 þ q2 ð1 k 2 Þ 2 x1 R1 ðx1 ; x2 Þ p1 ðgÞ ¼ 2 g 2 R1 ðx1 ; x2 Þ 1 k2 ð1 k 2 Þ
ð28Þ
p2(g) is the imaginary component of the Cagniard contour qðx2 Þð1 k 2 Þ k g p2 ðgÞ ¼ x1 R21 ðx1 ; x2 Þ 1 k2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Where R1 ðx1 ; x2 Þ ¼ x21 þ q2 ðx2 Þð1 k 2 Þ. Using the Cagniard’s method we replace the integral along real axis p by the integral over a suitably chosen contour C (cf. Fig. 2). The integrals over way C+ and C vanish in view of the Jordan Lemma, and also, the residual term and the contribution coming from the integrand branch point located over the contour C (see Fig. 2) are equal zero. Taking the above notice into consideration the double integral (26) can be expressed as follows Z 1 Z 1 n en0 g rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi dgdq V ðn0 Þ ¼ 02 ð29Þ 2 4p 1 g1 1þq2 ð1k 2 Þ 2 k g 1k2 x1 R1 ðx1 ; x2 Þ ð1k2 Þ2 where g1 is the lower limit of the integration with respect to variable g and equals qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ q2 ð1 k 2 Þ k g1 ¼ x þ R ðx ; x Þ 1 1 1 2 1 k2 1 k2 After the iteration of the above integrations Eq. (29) obtain
Fig. 2. Cagniard contour C and mutual location of the vertex point and the integrand branch point.
J. Rafa, C. Zio´łkowski / Wave Motion 45 (2008) 178–190
184
V ðn0 Þ ¼
n0 4p2
Z
1
en0 g
g0
Z
q0
q0
dq r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi dg g
k 1k 2
2
x1
R21 ðx1 ;x2 Þ ð1k 2 Þ
q2 þ
ð30Þ
1 1k 2
where ±q0 are the integration limits with respect to variable q and equal vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 u k u g 1k ð1 k 2 Þ 2 x1 1 q0 ¼ t R21 ðx1 ; x2 Þ ð1 k 2 Þ2 and g0 is the lower limit of the integration with respect to variable g and equals g0 ðx1 ; x2 Þ ¼
k R1 ðx1 ; x2 Þ x1 þ 1 k2 ð1 k 2 Þ
Hence, the Laplace transform of the fundamental solution takes the following form V ðn0 Þ ¼
en0 g0 ðx1 ;x2 Þ 4pR1 ðx1 ; x2 Þ
ð31Þ
Thus, taking the translation theorem (cf. [8]) into consideration the fundamental solution of the analysis Eq. (10) expresses the following dependence V ðx0 Þ ¼
1 dðx0 g0 ðx1 ; x2 ÞÞ 4pR1 ðx1 ; x2 Þ
So, the fundamental solution Eðx; tÞ for Eq. (9) is given by formula 1 g ðx; tÞ Eðx; tÞ ¼ d t 0 4pcR0 ðx; tÞ c
ð32Þ
ð33Þ
where R0 is a reduced distance from a signal source to a receiver at the moment t qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R0 ðx; tÞ ¼ ðx vtÞ2 þ ð1 k 2 Þðy 2 þ z2 Þ x, y, z are location co-ordinate of a receiver with reference to the established co-ordinate system and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 kðx vtÞ þ ðx vtÞ þ ð1 k 2 Þðy 2 þ z2 Þ . g0 ðx; tÞ ¼ 1k 2 4. The solution form of the analysis problem The final solution of the analysis problem we obtain as a convolution of the fundamental solution with the right hand side of Eq. (9), i.e., with the function describing space distribution of a current density. Therefore the solution of Eq. (9) is described as follows Eðx; tÞ ¼ Eðx; tÞ l0 IðzÞ
o ði0 ðtÞ dðx vtÞÞ dðyÞ ot
ð34Þ
In our paper we concentrate on one elementary case of a signal source. We assume that the antenna is supplied by a harmonic signal. So, in each point of the antenna the change in time of a current density is described as follows i0 ðtÞ ¼ eix0 t . In this case, changing the variables ((x, y, z, t) ! (x0, x1, x2, x3)), we get x0 o o ox0 o ox1 o ixc0 x0 iz ðx; tÞ ¼ þ e dðx1 Þdðx2 ÞIðx3 Þ iz ðx0 Þ ¼ ix0 ei c x0 dðx1 Þdðx2 ÞIðx3 Þ v ot ox0 ot ox1 ot ox1 o ¼ ix0 hðx0 Þ v hðx0 Þ ð35Þ ox1 x
where hðx0 Þ ¼ ei c x0 dðx1 Þdðx2 ÞIðx3 Þ. Taking the above expression into consideration we can rewrite the formula (34) in the following form
J. Rafa, C. Zio´łkowski / Wave Motion 45 (2008) 178–190
o Aðx0 Þ ¼ l0 V ðx0 Þ ix0 hðx0 Þ v hðx0 Þ ox1
¼ il0 x0 ðV ðx0 Þ hðx0 ÞÞ þ l0 v
185
o ðV ðx0 Þ hðx0 ÞÞ ox1
ð36Þ
where A(x0) = E(x,t). Thus, in order to obtain the solution of the analysed problem we calculate the convolution V(x0) * h(x0) which equals Z x0 0 1 1 V ðx0 Þ hðx0 Þ ¼ dðx0 g0 x00 Þei c x0 dðx01 Þdðx02 ÞIðx03 Þdx00 dx01 dx02 dx03 0 0 0 4p X R1 ðx1 x1 ; x2 x2 ; x3 x3 Þ x0 Z 1 ei c ðx0 g0 Þ Iðx0 Þdx0 ¼ ð37Þ 4p R R1 ðx1 ; x2 ; x3 x03 Þ 3 3 where X ¼ R3 ð0; x0 Þ. Differentiating the above expression with respect to x1 we get Z x0 o 1 x0 x1 x0 ðV ðx0 Þ hðx0 ÞÞ ¼ ei c ðx0 g0 Þ i þi 2 2 0 ox1 4p R ð1 k Þc R1 ðx1 ; x2 ; x3 x3 Þ ð1 k 2 Þc k x1 þ Iðx03 Þdx03 R1 ðx1 ; x2 ; x3 x03 Þ R31 ðx1 ; x2 ; x3 x03 Þ Putting (37) and (38) into (36) and changing the variables ((x0, x1, x2, x3) ! (x, y, z, t)), we obtain Z g ðx;tÞ 0 l 1 x vt x vt Eðx; tÞ ¼ 0 þv 3 þ ikx1 2 ix1 eix0 t c Ið1Þd1 R0 ðx; y; z 1; tÞ 4p R R0 ðx; y; z 1; tÞ R0 ðx; y; z 1; tÞ g ðx; tÞ P0 for t 0 c
ð38Þ
ð39Þ
x0 where x1 ¼ 1k 2 Now, we calculate the solution of the analysed problem for the far zone, i.e., bR0(x, t) 1 for each t P 0. It is a practical case occurring in mobile communications. In this case we make use of analogous space simplifications as for example in classic cases of the Hertz dipole (cf. [1,2]). So, we can write sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2z1 þ 12 z1 2 2 2 ð40Þ ffi R0 ðx; tÞ ð1 k 2 Þ ðx vtÞ þ ð1 k Þðy 2 þ ðz 1Þ Þ ¼ R0 ðx; tÞ 1 ð1 k 2 Þ 2 R0 ðx; tÞ R0 ðx; tÞ
Furthermore we assume that an antenna system is a half-wavelength dipole. Then the distribution of a current density amplitude along the antenna has the following form (cf. [1,2]) ( I 0 cos bz for jzj 6 2l IðzÞ ¼ ð41Þ 0 for jzj > 2l ðl length of the dipoleÞ Applying the above classic approaches, analogous as for example in [1,2] or [3] the practical value of the electric field strength E(x, t), for the far zone is expressed as follows l0 1 x vt p x1 I 0 þk 2 Eðx; tÞ ¼ eix1 tib1 kxib1 R0 ðx;tÞþi2 R0 ðx; tÞ 4p R0 ðx; tÞ Z l=2 z 1 1 k ið1k 2 Þb1 R ðx;tÞ 0 ð42Þ e cosðð1 k 2 Þb1 1Þd1 for t P R0 ðx; tÞ þ x c c l=2 2p where b1 ¼ xc1 ¼ kð1k 2 Þ Taking into account l/2 = k/4 for the half-wavelength dipole and after simple calculations as for example in [1], we obtain
J. Rafa, C. Zio´łkowski / Wave Motion 45 (2008) 178–190
186
Z
l=2
e
ið1k
2
z 1 Þb1 R ðx;tÞ 0
cosðð1 k 2 Þb1 1d1 ¼ 2
l=2
Basing on the following expressions Z 0 ¼
cos
p z 2 R0 ðx;tÞ
2 ð1 k 2 Þb1 1 R2 zðx;tÞ
qffiffiffiffi l0 e0
ð43Þ
0
and c ¼
1 pffiffiffiffiffiffi , l0 e0
we can write l0c = Z0, where Z0 is a wave imped-
ance of the propagation medium and e0 denotes a permittivity. So, the analytic form of the electric far-field strength generated by moving transmitter is p z Z0 1 x vt cos 2 R0 ðx;tÞ ib1 kxib1 R0 ðx;tÞþip ix1 t 2e Eðx; tÞ ¼ I0 e ð44Þ þk 2 2 R0 ðx; tÞ R0 ðx; tÞ 2pð1 k 2 Þ 1 R2 zðx;tÞ 0
But E(x,t) is an analytic signal, hence we can write Eðx; tÞ ¼ E0 ðx; tÞeiUðx;tÞ
ð45Þ
where E0(x, t) = |E(x, t)| denotes the signal amplitude and U(x, t) = arg E(x, t) is the phase angle of a signal. Thus, it is easy to see that the amplitude E0(x, t) and the phase angle U(x, t) of the signal E(x, t) are described as follows p z Z0 1 x vt cos 2 R0 ðx;tÞ þk 2 E0 ðx; tÞ ¼ ð46Þ I0 2 R0 ðx; tÞ R0 ðx; tÞ 2pð1 k 2 Þ 1 R2 zðx;tÞ 0
and Uðx; tÞ ¼ x1 t b1 kx b1 R0 ðx; tÞ
p 2
ð47Þ
Hence, the instantaneous frequency f(x, t) is expressed as follows f ðx; tÞ ¼
1 d 1 f0 d Uðx; tÞ ¼ R0 ðx; tÞ f 2 0 2 2p dt dt cð1 k Þ 1k
ð48Þ
But d d R0 ðx; tÞ ¼ dt dt
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vðx vtÞ ðx vtÞ2 þ ð1 k 2 Þðy 2 þ z2 Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx vtÞ2 þ ð1 k 2 Þðy 2 þ z2 Þ
therefore the instantaneous frequency f(x, t) has the form 1 k x vt qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f0 f ðx; tÞ ¼ f þ 2 0 2 1k 1k ðx vtÞ2 þ ð1 k 2 Þðy 2 þ z2 Þ So, the Doppler frequency is given by the following formula 0 fD ðx; tÞ ¼ f ðx; tÞ f0 ¼
ð49Þ
ð50Þ
1
k B x vt C k þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAf0 2@ 2 1k 2 2 2 ðx vtÞ þ ð1 k Þðy þ z Þ
ð51Þ
Hence, we notice that the value of the Doppler frequency fD(x, t) depends not only on a signal source velocity and a carrier frequency but also space location of a motion trajectory. The Doppler frequency fD(x, t) is linear function of the frequency carrier, whereas the dependence on velocity and space co-ordinates has a more complex character. 5. Numerical example of the Doppler frequency calculation The influence of the velocity and the transmitter–receiver space location on value of the Doppler frequency is presented in this part of our paper. We concentrate on the example of the air-to-ground mobile communi-
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187
Fig. 3. Space structure model for numerical calculation.
Fig. 4. Doppler frequency fD versus distance vt and height y of the aircraft flight.
cations where the change of the transmitter–receiver space location concerns big altitudes. The numerical calculations were conducted for a line-of-sight region. In this case we considered the space co-ordinates y and z as a constant parameter of the motion. Space structure model for analysis of the motion influence on the Doppler frequency is shown in Fig. 3. We assumed that the first space co-ordinate of the receiver is x = 5000 m and the signal frequency is f0 = 900 MHz. Fig. 4 shows the three-dimensional plot of the Doppler frequency fD versus vt and y where vt denotes the value of the transmitter–receiver distance, and y is the height of the aircraft flight. Fig. 5 shows the Doppler frequency versus distance vt for some heights of the aircraft flight. In this case the curves in Fig. 5 are suitable intersections of the three-dimensional plot shown in Fig. 4. The influence of an aircraft velocity on the Doppler frequency offset is presented in Figs. 6 and 7. Fig. 6 shows the three-dimensional plot of the Doppler frequency fD versus distance vt and velocity v. Fig. 7 shows the Doppler frequency fD versus distance vt for some velocities of the aircraft flight. In this case the curves in Fig. 7 are suitable intersections of the three-dimensional plot shown in Fig. 6. In Figs. 4–7 we present the results when aircraft flies exactly above the site of the receiver location i.e. orthogonal project of the flight trajectory to the ground area includes the point of a receiver location. Next two figures present numerical calculation results for the case when a trajectory of the aircraft flight goes near a receiver location site. The influence of a distance between the receiver location point and the aircraft trajec-
188
J. Rafa, C. Zio´łkowski / Wave Motion 45 (2008) 178–190
Fig. 5. Doppler frequency fD versus distance vt for some heights of the aircraft flight.
Fig. 6. Doppler frequency fD versus distance vt and velocity v of the aircraft flight.
Fig. 7. Doppler frequency fD versus distance vt for some velocities of the aircraft flight.
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189
Fig. 8. Doppler frequency fD versus distance vt and distance z between a receiver location site and aircraft flight trajectory.
Fig. 9. Doppler frequency fD versus distance vt for some distances between a receiver location site and aircraft flight trajectory.
tory trace on the ground area is presented in Figs. 8 and 9. Fig. 8 shows the three-dimensional plot of the Doppler frequency fD versus distance vt and space co-ordinate z. Fig. 9 shows the Doppler frequency fD versus distance vt for some values z. In this case the curves in Fig. 9 are suitable intersections of the three-dimensional plot shown in Fig. 8. 6. Summary and conclusions The detailed mathematical analysis of a transmitter motion influence on received signal parameters especially the Doppler frequency has been presented in this paper. The calculation of the dependence describing the electric field strength generated by moving signal source has been the main aim of our analysis. On the basis of the obtained formula we can carry out an assessment of a received signal frequency offset in time and space. Namely following three categories of the parameters influence the Doppler frequency value: space parameter (co-ordinates of a receiver location site), motion parameter (velocity of a signal source), signal parameter (signal frequency). In the characteristics presented in Section 5 we can distinguish three fundamental ranges: the range of a maximum value, the range of dynamic changes, the range of a minimum value. The influence of individual parameter on running of the Doppler frequency characteristics is diverse. Namely, the mutual location of a signal source-receiver influences the slope of the Doppler characteristic in the range of dynamic changes. The increase of a distance from a transmitter motion trajectory to a receiver location site
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190
decreases a slope of the frequency offset plot. However motion and signal parameters determine minimum and maximum value of the Doppler frequency, i.e., the increase of the velocity and carrier frequency values causes increases of the frequency offset range. One-dimensional distribution of a current density in antenna has been considered in our paper but presented methodology has a universal character. In a case of two- or three-dimensional distribution the analysed problem requires solving system equations by using presented methodology with reference to each equation. In this case, taking the analytic form of an analysis signal into consideration, the resultant solution is sum of the component solutions. In order to describe the Doppler effect occurring in mobile communications we put simplified formula into practice. This approximation is based on solution of the Maxwell equations taking stationary space conditions of the system transmitter–receiver into consideration. In this case the Doppler frequency can be estimated by the expressions (4) or (5). In most mobile communications we observe relation c v, i.e., k 1. Thus, in this case estimation mistake is negligible. But applying the formula (51) to describe the Doppler effect we can investigate influence of the space, motion and signal parameters on a received signal frequency offset. Furthermore, in mobile communications where objects move with high velocity (for example in satellite communications, radio-astronomy) the exact formula (51) describing the Doppler frequency is required. Finally, we want to notice that the solving methodology of the signal source motion problem, described in our paper has an universal character. This method presented here can be applied to all branches of the classic physics. For example very interesting results for this problem are expected in the acoustics where parameter k achieves difference value with reference to 1. Acknowledgement The authors would like to express thanks to Mrs. B. Niklewicz and Mr. J.M. Kelner for preparing final edition form of this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
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