# Effects of Fading Channels on OFDM

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3 4) Velocity of objects within the channel: Objects in the communication environment sometimes are moving. If that movement is of a speed that is more than that of the receiver, small-scale fading happens otherwise we can ignore those movements. III. CHANNEL MODELING Time variant impulse response of the channel is used to model it. Motion of the receiver is the variable that affects time changing. Equation (3) is calculates the received signal. y d, t = t x τ h d, t τ d(t) (3) Where h (d, t) is the impulse response of the channel and x (t) is the signal. If d = vt is the position of the receiver and v is a constant, it is possible to replace d in equation (3) with vt as in equation (4). t y vt, t = x τ h vt, t τ d(t) (4) h b t, τ = N 1 a i (t, τ)e j 2πf cτ i t +φ i t,τ i=0 δ(τ τ i t ) (8) Channel impulse response can be found using (5). Where, a i (t, τ) are the amplitudes and τ i (t) are the delays whereas θ i t, τ = 2πf c τ i t + φ i t, τ represent phase shift in the ith component. The impulse response can be found using equation (9) for time invariant channel where every multipath components of that cannel have delay. N 1 h b τ = a i (t, τ)e jθ i i=0 t,τ δ(τ τ i t ) (9) 3.1 Clarke s Model Clarke s Model relies on scattering to discover the statistical characteristics of the channel. Beside the assumption of fixed transmission antenna that is vertically polarized, N normalized plane waves of the antenna with a random carrier phases angels of arrival are assumed while amplitude remains unchanged. When the receiver is moving, m th wave that has arriving angle α m with respect to the x-axis. Doppler shift is given by equation (10). f m = v λ cosα m (10) Where, λ is the wavelength of incident wave. Every received wave has a different carrier frequency with small shift from the center frequency. The power spectral density of the output is given by equation 11. It is clear from that equation that the PDF is zero when f f c > f m. With center frequency f c, Spectrum is limited to ±f m and all the received waves has different carrier frequencies that are shifted. Assuming a vertical λ 4 (12). A[q α G + G( ) S f = (11) f n 1 ( f f c fm )2 1 antenna, G α = 1.5 and p α = over 0 to 2π 180, then S f becomes as in equation S f = 1.5 πf m 1 ( f f c fm )2 (12) Figure 3 Rayleigh fading implementation 118 P a g e

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