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The Power Delay Profile gives the distribution of signal power

received over a multipath channel as a function of propagation

delays. It is obtained as the spatial average of the complex baseband

channel impulse response as

As discussed in the previous post, it can also be derived from

scattering function as given below

where denotes Doppler Frequency, denotes multipath

propagation delay, denotes the scattering function.

In a Power Delay Profile plot, the signal power of each multipath is

plotted against their respective propagation delays. A sample power

Power Delay Profile Mathuranathan July 9, 2014 Channel Modelling No Comment

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delay profile plot, shown below, indicates how a transmitted pulse

gets received at the receiver with different signal strength as it travels

through a multipath channel with different propagation delays (

and .

Power Delay Profile is usually supplied as a table of values obtained

from empirical data and it serves as a guidance to system design.

Nevertheless, it is not an accurate representation of the real

environment in which the mobile is destined to operate at. For

example, the 3GPP spec specifies the power delay profile for various

environments as follows.

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Typical values for power delay profile listed in a 3GPP spec

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Maximum Excess Delay Definitionand applications

With Power Delay Profile, one can classify a multipath channel into

frequency selective or frequency non-selective category. The derived

parameter, namely, Maximum Excess Delay together with the

symbol time of each transmitted symbol, can be used to classify the

channel into frequency selective or non-selective channel.

Power Delay Profile can be used to estimated the average power of a

multipath channel, measured from the first signal that strikes the

receiver to the last signal whose power level is above certain

threshold. This threshold is chosen based on receiver design

specification and is dependent on receiver sensitivity and noise floor

at the receiver.

Maximum Excess Delay, also called Maximum Delay Spread, denoted

as , is the relative time difference between the first signal

component arriving at the receiver to the last component whose

power level is above some threshold. Maximum Delay Spread

and the symbol time peroid can be used to classify a channel

into frequency selective or non-selective category. This classification

can also be done using Coherence Bandwidth (a derived parameter

from Spaced Frequency Correlation Function which in turn is the

frequency domain representation of power delay profile).

Maximum Excess Delay is also an important parameter in mobile

positioning algorithm. The accuracy of such algorithm depends on

how well the Maximum Excess Delay parameter conforms with

measurement results from actual environment.

When a mobile channel is modeled as a FIR filter (tapped delay line

implementation), as in CODIT channel model[1], the number of taps

of the FIR filter is determined by the product of maximum excess

delay and the system sampling rate.

The Cyclic Prefix in a OFDM system is typically determined by the

maximum excess delay or by the RMS delay spread of that

environment [2].

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Classification of Channel signalspreading in Time domain:

A channel is classified as Frequency Selective, if the maximum excess

delay is greater than the symbol time period, i.e, . This

introduces Inter Symbol Interference (ISI) into the signal that is being

transmitted, thereby distorting it. This occurs since the signal

components (whose power are above the threshold or the maximum

excess delay) due to multipath extend beyond the symbol time. ISI

can be mitigated at the receiver by an equalizer.

In a frequency selective channel , the channel output can be

expressed as the convolution of input signal and the channel impulse

response plus some noise.

On the other hand, if the maximum excess delay is less than the

symbol time period, i.e, , the channel is classified as

frequency non-selective or flat channel. Here, all the scattered

signal components (whose power are above the specified threshold

or the maximum excess delay) due to the multipath, arrive at the

receiver within the symbol time. This will not introduce any ISI, but

the received signal is distorted due to inherent channel effects like

SNR condition. Equalizers in the receiver are not needed. A time

varying non-frequency selective channel is obtained by assuming

that the impulse response . Thus the output of the

channel can be expressed as

Note, that the output of the channel can be expressed simply as

product of time varying channel response and the input signal. If the

channel impulse response is a deterministic constant, i.e, time in-

varying, then the non-frequency selective channel is expressed as

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15/7/2557 Power Delay Profile | GaussianWaves

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follows by assuming

This is the simplest situation that can occur. In addition to that, if the

noise in the above equation is white Gaussian noise, the channel is

called Additive White Gaussian Noise (AWGN) channel.

Characterization of FrequencySelective Channels:

Average delay and the RMS delay spread are two most important

parameters that characterize a frequency selective channel. They are

derived from Power Delay Profile.

Average delay:

Simply the statistical mean of the delay that a signal undergoes when

transmitted over a multipath channel. For a frequency selective

WSSUS channel, the average delay is equal to the first moment of the

power delay profile . For a discrete channel it is calculated as

If the given PDP values are continuous in terms of time delays (as

given in the sample plot below), replace the summation with

integral and integrate it with respect to .

RMS delay spread:

RMS delay spread is equal to the second central moment of power

delay profile . It is similar to the standard deviation of a

statistical distribution. For a discrete channel it is given by

where,

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If the given PDP values are continuous in terms of time delays (as

given in the sample plot below), replace the summation with integral

and integrate it with respect to .

The ratio of RMS delay spread and symbol time duration quantifies

the strength of Inter Symbol Interference. This ratio determines the

complexity of the equalizer required at the receiver. Typically, when

the symbol time period is greater than 10 times the RMS delay

spread, no ISI equalizer is needed in the receiver.

A sample power delay profile is plotted below. Average delay, RMS

delay spread, and the maximum excess delay ( given a threshold of

100 dBm ) are all marked.

References:

[1] Andermo, P.G.; Larsson, G., Code division testbed, CODIT, Universal

Personal Communications, 1993. Personal Communications:

Gateway to the 21st Century. Conference Record., 2nd International

Conference on , vol.1, no., pp.397,401 vol.1, 12-15 Oct 1993

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Power Delay Profile with Mean delay, RMS delay spread,

Maximum Excess Delay

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[2] Huseyin Arslan, Cognitive Radio, Software Dened Radio, and Adaptive

Wireless Systems, pp. 238, 2007, Dordrecht, Netherlands, Springer.

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autocorrelation function, average delay, AWGN, channel modeling, CoherenceBandwidth, Coherence Time, Doppler Power Spectrum, Fading, flat fading, frequencynon-selective, Frequency selective, Maximum excess delay, mean delay, multipath,Power Delay Profile (PDP), RMS delay spread, scattering function, spaced-frequencycorrelation function, spaced-time correlation function

About Mathuranathan

Mathuranathan Viswanathan - Founder and Author @

gaussianwaves.com which has garnered worldwide readership.

He is a masters in communication engineering and has 7 years

of technical expertise in channel modeling and has worked in

various technologies ranging from read channel design for

hard drives, GSM/EDGE/GPRS, OFDM, MIMO, 3GPP PHY layer

and DSL. He also specializes in tutoring on various subjects like

signal processing, random process, digital communication

etc..,He can be contacted at [email protected]

View all posts by Mathuranathan

15/7/2557 Power Delay Profile | GaussianWaves

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