Delay statistics are a measure of the signal echos caused by multipath. Delay statistics are important for estimating bit error rates and for equalizer design. The ITS radio channel impulse response measurement system can be used to quantify delay statistics. The starting point are the measured received impulses or power delay profiles (PDPs). The resulting delay statistics are usually based on averaged power delay profiles (APDPs)

,

where t_i is the i-th time step (sampling point), and N is the number of PDPs used to form the APDP. The use of APDPs significantly reduces the contribution of noise to the delay statistics. To avoid including noise peaks in the delay statistics, an APDP should have an interval of discrimination (ID) sufficient to insure that noise is not calculated into the fading statistics. Referring to the figure below, the interval of discrimination (ID) is defined as the difference in power levels between the peak of the intended signal (impulse) and the peak noise. Delay statistics are typically based on those portions of APDP greater than the ID + 3 dB, here referred to as the corrected APDP.

A sketch of an APDP showing the delay statistic parameters.

Three delay measures are typically considered: maximum delay, mean delay, and RMS delay spread. The maximum delay is defined as the time delay between the first and last signals of the corrected APDP (see figure above). The mean delay (d) is the time-weighted average, or first moment, of the corrected APDP normalized by the average signal power,

,

where t_k is the time delay (in seconds) relative to the start of the corrected APDP (i.e. t = 0), P is the signal power (W), and N is the index of the final corrected APDP signal point considered. The RMS delay spread (S) measures the standard deviation of the delay spread of each corrected APDP about its mean delay (d). It is the second central moment of the corrected APDP given by

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The figures below show an example of a measured PDP (1920 MHz) and the cumulative probability distribution for delay statistics gathered in a suburban environment (near to the ITS boulder Labs location).

A measured power delay profile (1920 MHz) showing significant multipath.
(Click image above to enlarge.)

1920 MHz delay statistics for a suburban environment.