Influence of Terrain on Multipath Propagation of Fm Signal

Journal of ELECTRICAL ENGINEERING, VOL. 56, NO. 5-6, 2005, 113–120 INFLUENCE OF TERRAIN ON MULTIPATH PROPAGATION OF FM SIGNAL Ján Klima ∗ — Marián Moˇzucha ∗∗ FM signal propagating through the troposphere interacts with the terrain as a reflecting plane. These reflected signals are not included into calculations and predictions of the field strength. The reason is simple — it is too difficult for processing, and demanding high quality data. For recent capabilities of computers and resolution level of digital terrain models this is not an irresolvable problem. In the report the authors show a practical demonstration how to include the reflected FM signals into the field strength calculations as well as the comparison with standard methods of field strength predictions. K e y w o r d s: digital terrain model (DTM), reflections, diffraction, multipath propagation 1 INTRODUCTION Here GRF represents a set of points of the terrain, z is the height in a point located in coordinates [x,y], A digital terrain model, as a discrete field of repre- KN F M is the curvature of the terrain as it is seen from sentative and positionally expressed points of the terrain the transmitter, FF M is the form, γF M is the slope, AF M and its approximation equations securing its continuity is the aspect, and δexp F M is irradiation — all from the [8] can serve, besides many other tasks, as a tool for lo- direction of the transmitter. cating, testing and calculating the reflecting planes for We will come from the known fact that the received propagating FM signals. When calculating the reflected field strength (E) of the radio signal propagated above a FM signals we cannot distinguish all, only statistically vertically dissected terrain can oscillate by ∆E = ±6 dB important directions expressed on the chosen resolution level. That is why measuring the field strength of FM sig- due to the reflected signals (Fig. 1). nal is a measurement of an integral quantity — it means The existence of reflection planes σI [1] between the of all possible inducted streams on the receiving antenna. transmitter Tx and receiver Rx is taken without testing An antenna constructed in the form of antenna field re- reflections in 3D dimensional space. The main reasons ceiving FM signals from different directions and summing are: them separately does not exist but we can find these di- – there is no exact procedure how to calculate statisti- rections iteratively. cally important reflected signals, In radiocommunication literature there are attempts – it is assumed that these reflections can be neglected to sum up direct and reflected waves [2], mainly those (though practice shows the opposite), reflected from determined shapes of obstacles — build- – it is difficult to describe the shapes and other parame- ings. Terrain with its irregular shape was not dealt with ters of all important obstacles on a huge area (though in these calculations. We will proceed from the assump- it is not a problem to get statistical values). tion that thanks to the additional field strength of FM In some special cases, the terrain with its configuration signal reflected from the terrain — not only on the line and reflection parameters can become a massive reflect- between the transmitter and receiver — we can get bet- ter predicted results. We will also assume that we can ing plane, this is why we are to predict with some sta- express that shape of objects of majority of obstacles on tistical error. If we considered a valley with dimensions the terrain with statistical methods, so we will get most responding to the 3D dimensions of the first Fresnel el- of reflected planes. As for the terrain, most calculation lipsoid (Fig. 2), then we could get an extreme increase of methods count with reflections only on a vertical pro- the field strength [10]. file though it does not correspond with reality. To take all reflections into accounts we can use effective values of morphometric parameters of the terrain such as elevation, 2 INPUT CONDITIONS AND PRESUMPTIONS form (convex, concave or linear), slope, aspect and angle of irradiation: 1. We have a terrain GRF (1) defined as a continuous topographic plane with continuous changes of its mor- GRF = z, KN F M ,FF M , γF M , AF M , (δexp)F M . (1) phometric parameters, in the form of a digital terrain ∗ Department of Physics, Faculty of Natural Sciences, Universitu of Matej Bel, Tajovského 74, 974 01 BanskáBystrica, ∗∗ E-mail: [email protected]; Trans World Radio–Slovakia, Banˇselova 17, 821 04 Bratislava 2, Slovakia, E-mail: [email protected] ISSN 1335-3632 c 2005 FEI STU 114 J. Klima — M. Moˇzucha: INFLUENCE OF TERRAIN ON MULTIPATH PROPAGATION OF FM SIGNAL Tx Tx Rx a) Rx Tx Rx s 1 s s 2 3 b) d c) Fig. 1. Reflection planes σ . Fig. 2. Valley with dimension of the first Fresnel zone a) side view, b) upper view, c) profile view. T Tx x R N x L O A1 d LN d LN d A3 0 d exp Rx A2 Fig. 3. Multipath of FM signal from transmitter Tx to receiver Fig. 4. Equal size of the angle of incident δexp and angle of Rx . reflection δ0 in the same plane. model (DTM) defined by approximation equation 3 GEOMETRIC EXPRESSION z = f(x,y) . (2) The impact angle δexp is equal to the angle of reflection δ0 , so the reflected signal will be in the same plane, (Fig. 4). 2. We have a known and definite set of objects (for- The normal N~ of the terrain in point Ai[xi,yi,zi] is est, water planes, urban objects etc) creating the land expressed as: cover with its height and geometric (morphometric) parameters. N~ = {Nx, Ny, Nz} , (3) 3. We will work with radio FM signal from the frequency band h87.5, 108i (MHz). where 4. Important interactions between the terrain and FM signal are mathematically described by geometric and Nx = |N~ | cos αN = sin γN · cos AN , wave optics. N = |N~ | cos β = sin γ · sin A , 5. Only important and steady reflections will be taken y N N N into account. The neglected reflections are expressed Nz = |N~ | cos γN = cos γN by means of statistics (included in the mean values of the field strength). where cos αN , cos βN , cos γN are direction angles (Fig. 5) 6. Calculation of the field strength as a sum of direct which are expressed through relations and reflected signals in huge areas (Fig. 3) is still in- sufficiently examined, hence there is a problem how to −zx −zy cos αN = , cos βN = , verify the outputs. 2 2 2 2 zx + zy + 1 zx + zy + 1 7. DTM is created by means of interpolation, smoothing q q 1 and various generalization methods — a lot of infor- cos γN = . 2 2 mation is lost due to these processes. The angle of re- zx + zy + 1 flection can be rapidly changing in a real terrain, while q there is a smooth terrain in DTM. where zx and zy are partial derivatives of function (2). 8. We do not know the reflection ratio of the terrain and The unit vector |N~ | = 1 is expressed by equation its land cover, hence it would be best to find them 2 2 2 iteratively. cos αN + cos βN + cos γN = 1 . (4) Journal of ELECTRICAL ENGINEERING 56, NO. 5–6, 2005 115 In the sense of Fig. 4, let us define the plateau σLN where ~ ~ zt both vectors L and N are placed. We want to find the g L Ni angle between these vectors. We will use the dot product a b g za L L N of the two vectors L~ and N~ [8]: yt b AL N a ~ ~ ◦ xt N ya L · N = cos δLN = sin δexp δLN + δexp = 90 . (7) t xa n In scalar form L~ · N~ = N L + N L + N L Fig. 5. Unit vectors N~ and L~ . x x y y z z = cos αN cos αL + cos βN cos βL + cos γN cos γL = −zx −zy sin γL ·cos AL + sin γL ·sin AL 2 2 2 2 Tx g L n zx + zy + 1 zx + zy + 1 q q 1 d 0 g R a) FM x + cos γL . (8) d exp 2 2 zx + zy + 1 Ai q Tx g L n From (7) and (8) we can get d exp Rx b) g FM d 0 (−zx∆xat) + (−zy∆yat) + ∆zat = sin δexp . (9) A 2 2 2 2 2 i zx + zy + 1 ∆xat + ∆yat + ∆zat q p Fig. 6. Reflections. From relation (8) we can work with angles AL and γL , FM signal represented by the electromagnetic wave and γF M , AF M , γ0, A0 , in sense of Fig. 6. interacting with the terrain in a given point Ai[xi,yi,zi] can be expressed in a similar way: ◦ ◦ ◦ ◦ ◦ γN < 45 , γF M < 45 , γ0 > 0 , AL = h0 , 360 ), L~ = {Lx, Ly, Lz} , (5) ◦ ◦ ◦ ◦ (10) AF M = h0 , 360 ) , A0 = h0 , 360 ) where ◦ ◦ where AF M = 180 −AL +AN = 180 −AL +arctg zy/zx , Lx = cos αL = sin γL · cosAL , AL is the direction of the FM signal towards the south Ly = cos βL = sin γL · sin AL, (Fig. 5): Lz = cos γL = cos γL , ~ 2 2 2 2 2 for |L| = 1 Lx + Ly + Lz = 1 . (6) γF M = γN cos AF M = arctg zx + zy cos AF M , q ya − yt Direction angles cos αL , cos βL , cos γL are expressed in A = arctg , L x − x a similar way, see Fig.

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