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whether the mobile receiver is moving toward or away from the transmitter. In fact, from Equation (1.11), if the mobile moves toward the direction of arrival of the signal with radiated frequency fc, then the received frequency is increased, that is the apparent frequency is fc fD . When the mobile moves away from the direction of arrival of the signal then the received frequency is decreased, that is the apparent frequency is fc fD . The maximum Doppler shift is fDmax v=l, which, in our futher description will simply be denoted as fm . There are many probability distribution functions that can be used to describe the fast fading effects, such as, Rayleigh, Suzuki, Rician, Gamma, Gamma Gamma, and so on. Because the Rician distribution is very general [1 4,10], as it includes both line-of-sight (LOS) together with scattering and diffraction with non-LOS, we brie y describe it in the following paragraph. To estimate the contribution of each signal component, at the receiver, due to the dominant (or LOS) and the secondary (or multipath), the Rician parameter K is usually introduced, as a ratio between these components [1 4,10], that is, K LOS Component power Multipath Component power 1:12
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The Rician PDF distribution of the signal strength or voltage envelope r can be de ned as [1 4,10]: & 2 '   r r A2 Ar I0 2 ; PDF r 2 exp 2s2 s s for A > 0; r ! 0 1:13
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where A denotes the peak strength or voltage of the dominant component envelope, s is the standard deviation of signal envelope, and I0 is the modi ed Bessel function of the rst kind and zero-order. According to de nition (1.12), we can now rewrite the parameter K, which was de ned above as the ratio between the dominant and the multipath component power. It is given by K A2 2s2 1:14
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Using (1.14), we can rewrite (1.13) as a function of K only, [1 3,10]: PDF x & '  r p  r r2 exp 2 exp K I0 2K 2s s2 s 1:15
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For K 0, exp K 1 and I0 0 1, that is, the worst case of the fading channel. The Rayleigh PDF, when there is no LOS signal and is equal to: PDF x & ' r r2 exp 2 2s s2 1:16
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3.5 3.0 2.5 PDF 2.0 1.5 1.0 0.5 0 30 25 20 15 10 5 0 5 10 Strength/rms, dB K=4 K = 0, Rayleigh
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FIGURE 1.4. Rician PDF distribution versus ratio of signal to rms.
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Conversely, in a situation of good clearance between two terminals with no multipath components, that is, when K ! 1, the Rician fading approaches a Gaussian one yielding a Dirac-delta shaped PDF described by formula (1.9) (see Fig. 1.4). We will use these de nitions in 5 for the link budget design inside a terrestrial radio communication system. 1.4.3. Signal Presentation in Wireless Communication Channels To understand how to describe mathematically multipath fading in communication channels, we need to understand what kinds of signals we deal with in each channel. Narrowband (CW) Signals. First of all, we consider a continuous wave CW or narrowband signals. A voice-modulated CW signal occupies a very narrow bandwidth surrounding the carrier frequency fc of the radio frequency (RF) signal (e.g., the carrier), which can be expressed as: x t A t cos 2pfc t j t 1:17
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where A t is the signal envelope (i.e., slowly-varied amplitude) and j t is its signal phase. For example, for a modulated 1 GHz carrier signal by a wire signal of bandwidth f 2fm 8 KHz, the fractional bandwidth is very narrow, that is, 8 103 Hz=1 109 Hz 8 10 6 or 8 10 4 %. Since all information in the signal is contained within the phase and envelope-time variations, an alternative form of a bandpass signal x t is introduced [1,2,6 10]: y t A t expfjj t g 1:18
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