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the time varying discrete-time impulse response can be expressed as: h t; t ( N 1 X
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If the channel impulse response is assumed to be time invariant, or is at least stationary over a short-time interval or over a small-scale displacement of the receiver/transmitter, then the impulse response (1.28) reduces to h t; t
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where yi 2pfc ti j t . If so, the received power delay pro le for a wideband or pulsed signal averaged over a small area can be presented simply as a sum of the powers of the individual multipath components, where each component has a random amplitude and phase at any time, that is,  Ppulse  *
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The received power of the wideband or pulse signal does not uctuate signi cantly when the subscriber moves within a local area, because in practice, the amplitudes of the individual multipath components do not change widely in a local area of service. Comparison between small-scale presentations of the average power of the narrowband (CW) and wideband (pulse) signals that is, (1.27) and (1.30), shows that when hAi Aj i 0 or/and hcos ji jj i 0, the average power for CW signal and that for pulse are equivalent. This can occur when either the path amplitudes are uncorrelated, that is, each multipath component is independent after multiple re ections, diffractions, and scattering from obstructions surrounding both the receiver and the transmitter or the base station and the subscriber antenna. It can also occur when multipath phases are independently and uniformly distributed over the range of b0; 2pc. This property is correct for UHF/X-waveband when the multipath components traverse differential radio paths having hundreds of wavelengths [6 10]. 1.4.4. Parameters of the Multipath Communication Channel So the question that is remains to be answered which kind of fading occurs in a given wireless channel. Time Dispersion Parameters. First some important parameters for a wideband (pulse) signal passing through a wireless channel, can be determined, for a certain threshold level X (in dB) of the channel under consideration, from the signal
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power delay pro le, such as mean excess delay, rms delay spread and excess delay spread. The mean excess delay is the rst moment of the power delay pro le of the pulse signal and is de ned as:
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The rms delay spread is the square root of the second central moment of the power delay pro le and is de ned as q st ht2 i hti2 where  2 t
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These delays are measured relative to the rst detectable signal arriving at the receiver at t0 0. We must note that these parameters are de ned from a single power delay pro le, which was obtained after temporal or local (small-scale) spatial averaging of measured impulse response of the channel [1 3,7 10]. Coherence Bandwidth. The power delay pro le in the time domain and the power spectral response in the frequency domain are related through the Fourier transform. Hence, to describe a multipath channel in full, both the delay spread parameters in the time domain, and the coherence bandwidth in the frequency domain are used. As mentioned earlier the coherence bandwidth is the statistical measure of the frequency range over which the channel is considered at. In other words, this is a frequency range over which two frequency signals are strongly amplitude correlated. This parameter, actually, describes the time dispersive nature of the channel in a small-scale (local) area. Depending on the degree of amplitude correlation of two frequency separated signals, there are different de nitions for this parameter. The rst de nition is the coherence bandwidth, Bc , which describes a bandwidth over which the frequency correlation function is above 0.9 or 90%, and it is given by: Bc % 0:02s 1 t 1:34