Estimating the RMS Delay Spread
from the Frequency-Domain Level Crossing Rate
Contributed by Klaus Witrisal
The root-mean-square (RMS) delay spread is probably the most important
single measure for the delay time extent of a multipath radio channel.
Since the impulse response (IR) and the transfer function (TF) of a channel
are related by the Fourier transform, it is intuitively understandable
that the TF's magnitude shows more fades per bandwidth, the longer the
As shown by Witrisal et al. , , there exists a well-defined
relationship between the so-called level crossing rate in the frequency-domain
(LCRf) and the RMS delay spread (Trms), written as
LCRf = Trms * f(K, r', u).
As seen from this equation, Trms and the LCRf
are proportional, where the proportionality factor is a function of
For the LCRf at the RMS amplitude value of the channel transfer
function, the factor f(K,r' = 1,u = 0) can be approximated by
the Ricean K-factor K
the threshold value at which the LCRf is determined, r' (r'
is normlized to the RMS amplitude value of the TF)
and the channel model, expressed by u. (This influence is very small, therefore
it can be neglected .)
Application to Channel Measurements
Equation (1) allows for the estimation of a complete set of wide-band channel
parameters (average received power, Ricean K-factor, and RMS delay spread)
from rather simple swept-frequency power measurements of the channel. (Note
that the Fourier transform cannot be used to calculate an IR from a measured
power response, due to the lack of phase information). The following measurement
procedure is suggested:
An observation bandwidth of 10/Trms (equivalent to the observation
of approx. 20 level crossings) allows for estimating Trms at
an accuracy in the order of 10 %. Higher bandwidths can enhance the accuracy.
Alternatively, multiple measurements from within a small local area can
be combined to increase the observation bandwidth without modifying the
Measure the narrowband power (or magnitude) response of the channel as
a function of frequency. (A continuous wave frequency generator and a spectrum
analyzer can be used to conduct such measurements).
Calculate the average received power and the Ricean K-factor from the measured
Count the number of level crossings at a specific threshold, preferably
at the RMS amplitude.
Use equation (1) for estimating Trms.
Other issues that should be considered when applying this method for
channel investigations are:
Influence of the sampling interval of the channel's frequency response:
The sampling interval must be selected according to the sampling theorem,
otherwise some fades may be missed when counting the level crossings.
Influence of measurement equipment noise: Noise may introduce additional
level crossings, which would lead to overestimation of Trms.
The sampling interval mentioned above should be as large as possible to
minimize this effect
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