Chapter: Wireless Propagation Channels
One can define 'narrowband' transmission also in the time domain, considering the interarrival
times of multipath reflections and the time scale of variations in the signal caused by modulation:
A signal sees a narrowband channel if the bit duration is sufficiently larger than the interarrival time of reflected waves. In such case, the intersymbol interference is small.
Formally the coherence bandwidth is the bandwidth for which the auto co-variance of the signal amplitudes at two extreme frequencies reduces from 1 to 0.5. For a Rayleigh-fading WSSUS channel with an exponential delay profile, one finds
Bc = 1/(2 p Trms)where Trms is the rms delay spread. This results follows from the derivation of correlation of the fading at two different frequencies.
Correlation of Fading
From the delay profile, one can compute the correlation of the fading
at different carrier frequencies.
Delay Profile <- -> E [H(f1) H*(f2) ]After some algebraic operations, this can be used to express the auto-correlation and auto-covariance of the amplitude, versus frequency separation f1 - f2.
Figure: Auto-Covariance of the received amplitude of two carriers transmitted with certain frequency offset.
The normalized co-variance of the amplitudes R1 and R2 of two carriers, one at f1 and another at f2 is
E R1 R2 - E R1 E R2 C = ---------------------- s(R1) s(R2)That is, we normalized for unity standard deviations sigma of R1 and R2. One finds, for an exponential delay profile
1 C = -------------------------------- 1 + 4 p2 (f1 - f2)2 Trms2