Estimating the RMS Delay Spread
from the FrequencyDomain Level Crossing Rate
Contributed by Klaus Witrisal
The rootmeansquare (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
IR.
As shown by Witrisal et al. [2], [3], there exists a welldefined
relationship between the socalled level crossing rate in the frequencydomain
(LCR_{f}) and the RMS delay spread (T_{rms}), written as
LCR_{f} = T_{rms }* f(K, r', u).
(1)
As seen from this equation, T_{rms} and the LCR_{f}
are proportional, where the proportionality factor is a function of

the Ricean Kfactor K

the threshold value at which the LCR_{f} 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 [2].)
For the LCR_{f} at the RMS amplitude value of the channel transfer
function, the factor f(K,r' = 1,u = 0) can be approximated by
.
Application to Channel Measurements
Equation (1) allows for the estimation of a complete set of wideband channel
parameters (average received power, Ricean Kfactor, and RMS delay spread)
from rather simple sweptfrequency 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:

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 Kfactor from the measured
power response.

Count the number of level crossings at a specific threshold, preferably
at the RMS amplitude.

Use equation (1) for estimating T_{rms}.
An observation bandwidth of 10/T_{rms} (equivalent to the observation
of approx. 20 level crossings) allows for estimating T_{rms} 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
measurement equipment.
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 T_{rms}.
The sampling interval mentioned above should be as large as possible to
minimize this effect
Wireless Communication ©
Klaus
Witrisal
Centre for Wireless Personal Communications (CEWPC)
TUDelft, IRCTR
Mekelweg 4
NL2628 CD Delft
The Netherlands
Phone: +31(0)152787371
Fax: +31(0)152784046
email: K.Witrisal@ITS.TUDelft.NL