Logo der Uni Stuttgart
Channel Evaluation using Autocorrelation

The autocorrelation function (ACF) can be useful in evaluating channel behavior. It reveals how self-similar the impulse and frequency response are. The shape of the ACF can be used to determine the coherence time and coherence bandwidth [2]. The results are normalized in such a way that the value of the ACF will always be between -1 and 1. A value of one implies that signal is the same, at $\tilde{t} = 0$ or $\tilde{f} = 0$ its value will always be equal to one.

Temporal ACF
Theoretically the ACF is described by the following function $$ R_{aa,i}(\tilde{t}) = \dfrac{1}{K+1} \left[\mathrm{J_0}\left(2\pi f_{\mathrm{D,max}}\tilde{t}\right) + K \exp\left(\mathrm{j}2\pi f_{\mathrm{D,max}} \cos\left(\theta_0\right) \cdot \tilde{t} \right)\right] $$ where $K = 0 \quad \text{if } i \neq 0$ [3]. The function $\mathrm{J_0}(x)$ is the Bessel function of the first kind and zeroth order.
The visual appearance of the temporal ACF will differ greatly depending on the amount of symbols $N_\mathrm{Symbol}$. By increasing $N_\mathrm{Symbol}$ the temporal ACF will be calculated for a greater span of time. The temporal ACF is directly computed in the time domain using $h(t, 0)$.

Spectral ACF
The spectral ACF is quite a bit simpler. It can be described using [6] $$ R_{HH}(t,f) = \sum_{i}P_{\tau,i}\cdot \exp\left[-\mathrm{j}2\pi \tau_i \cdot f\right]. $$ The power $P_{\tau,i}$ is dependent on $w_i$ and $K$. The spectral ACF is directly computed in the frequency domain using $H(t, f)$.

[2] T. S. Rappaport, Wireless Communications: Principles and Practice. Prentice Hall PTR, 2002.
[3] N. C. Beaulieu, Chengshan Xiao, and Yahong Rosa Zheng, “Novel Sum-of-Sinusoids Simulation Models for Rayleigh and Rician fading Channels,” IEEE Transactions on Wireless Communications, vol. 5, no. 12, pp. 3667–3679, 2006.
[6] R. J. C. Bultitude, “Estimating frequency correlation functions from propagation measurements on fading radio channels: a critical review”, IEEE Journal on Selected Areas in Communications, vol. 20, no. 6, pp. 1133–1143, 2002.