JPL's Wireless Communication Reference WebsiteChapter: Wireless Propagation Channels

In a multipath channel, the relative delay and amplitude of reflected waves are random variables. The correlation between the signal amplitude at two frequencies is denoted as E H(f_{1}) H^{*}(f_{2}).Its derivation is an important step towards the definition of the coherence bandwidth. The delay profile describes the expected energy received in a particular bin of delay times. Here, our analysis diverts from the original derivation, as for instance covered in textbooks by Jakes. By modeling a multipath channel and it delay profile, not as a continuoustime channel but by a timediscrete channel impulse response, sampled at interval T_{s}, the analysis can be shortened. The generalized timecontinuous situation is then covered by taking the limit for T_{s} decreases to zero.
The coefficients h_{m}, i.e., the channel impulse response taps are complex Gaussian random variables. The variance of each tap is determined by the delay profile. For an exponential delay profile with rms delay spread T_{rms}, an appropriate choice is
E h_{m} h^{*}_{m} = [1  exp{T_{s}/T_{rms}}] exp{m T_{s}/T_{rms}}
Here the constant [1  exp{T_{s}/T_{rms}}] ensures that the local mean power (found as the sum of E h_{m} h^{*}_{m} over all m) equals unity.Using the definition of the (continuous time) Fourier Transform, the frequency transfer function for this particular channel is
The correlation between the transfer function at frequencies f_{1} and f_{2} is
To solve these sums analytically, we substitute q = k  m, so
In the summing over q, only the term with q = 0 is nonzero, so
Inserting the sampled exponential power delay profile, we arrive at
Interpreting the sum as a geometric series gives
Next we will denote the difference between the two carrier frequencies as D_{f} = f_{1}  f_{2}. Taking the limit for the sampling resolution T_{s} of the delay profile reducing to zero,
or
E H(f_{1}) H^{*}(f_{2}) = [I_{1} + j Q_{1}] [I_{2}  j Q_{2}] = I_{1} I_{2} + Q_{1} Q_{2} + j[I_{2} Q_{1} I_{1} Q_{2} ].
Further, I_{1} I_{2} = Q_{1} Q_{2} and I_{2} Q_{1} = I_{1} Q_{2}. This allows us to split the above expression for E H(f_{1}) H^{*}(f_{2}) into a correlation matrix G for I_{1}, Q_{1}, I_{2} and Q_{2}.
G= 0.5 [  1  0  c_{1}  c_{2}  ]  
0  1  c_{2}  c_{1}  
c_{1}  c_{2}  1  0  
c_{2}  c_{1}  0  1 
c_{1} = 1 / [1 +(2p D_{f}T_{rms})^{2}]
and
c_{2} = 2p D_{f}T_{rms} / [1 +(2p D_{f}T_{rms})^{2}]
Note the factor of 0.5, which reflects that half the power is in the I component, and the other half is in the Q component.
E (r_{1}  E r_{1}) (r_{2}  E r_{2}) = 1 / [1 +(2p D_{f}T_{rms})^{2}]