Power Delay Profile

Power delay profile gives the signal power received on each multipath as a function of the propagation delays of the respective multipaths.

Power delay profile (PDP)

A multipath channel can be characterized in multiple ways for deterministic modeling and power delay profile (PDP) is one such measure. In a typical PDP plot, the signal power on each multipath is plotted against their respective propagation delays.

In a typical PDP plot, the signal power () of each multipath is plotted against their respective propagation delays (). A sample power delay profile plot, shown in Figure 1, indicates how a transmitted pulse gets received at the receiver with different signal strengths as it travels through a multipath channel with different propagation delays. PDP 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.

The PDP, when expressed as an intensity function , gives the signal intensity received over a multipath channel as a function of propagation delays. The PDP plots, like the one shown in Figure 1, can be obtained as the spatial average of the complex channel impulse response as

RMS delay spread and mean delay

The RMS delay spread and mean delay are two most important parameters that characterize a frequency selective channel. They are derived from power delay profile. The delay spread of a multipath channel at any time instant, is a measure of duration of time over which most of the symbol energy from the transmitter arrives the receiver.

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In the wide-sense stationary uncorrelated scattering (WSSUS) channel models [1], the delays of received waves arriving at a receive antenna are treated as uncorrelated. Therefore, for the WSSUS model, the underlying complex process is assumed as zero-mean Gaussian random proces and hence the RMS value calculated from the normalized PDP corresponds to standard deviation of PDP distribution.

For continuous PDP (as in Figure 2), the RMS delay spread () can be calculated as

where, the mean delay  is given by

For discrete PDP (as in Figure 1), the RMS delay spread () can be calculated as 

where, is the power of the path, is the delay of the path and the mean delay is given by

Knowledge of the delay spread is essential in system design for determining the trade-off between the symbol rate of the system and the complexity of the equalizers at the receiver. The ratio of RMS delay spread () and symbol time duration () quantifies the strength of intersymbol interference (ISI). Typically, when the symbol time period is greater than 10 times the RMS delay spread, no ISI equalizer is needed in the receiver. The RMS delay spread obtained from the PDP must be compared with the symbol duration to arrive at this conclusion.

Frequency selective and non-selective channels

With the power delay profile, one can classify a multipath channel into frequency selective or frequency non-selective category. The derived parameter, namely, the 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.

PDP can be used to estimate 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 period () 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 [2], 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 [3].

A channel is classified as frequency selective, if the maximum excess delay is greater than the symbol time period, i.e, . This introduces intersymbol interference into the signal that is being transmitted, thereby distorting it. This occurs since the signal components (whose powers are above either a threshold or the maximum excess delay), due to multipath, extend beyond the symbol time. Intersymbol interference 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 zero-mean channel. Here, all the scattered signal components (whose powers are above either a 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 the impulse response . Thus the output of the channel can be expressed as

Therefore, for a frequency non-selective channel, the channel output 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 invariant, then the non-frequency selective channel is expressed as 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.

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References:

[1] P. A. Bello, Characterization of randomly time-variant linear channels, IEEE Trans. Comm. Syst., vol. 11, no. 4, pp. 360–393, Dec. 1963.↗
[2] Andermo, P.G and 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.↗
[3] Huseyin Arslan, Cognitive Radio, Software Defined Radio, and Adaptive Wireless Systems, pp. 238, 2007, Dordrecht, Netherlands, Springer.↗

Topics in this chapter

Small-scale Models for Multipath Effects ● Introduction ● Statistical characteristics of multipath channels  □ Mutipath channel models  □ Scattering function  □ Power delay profile  □ Doppler power spectrum  □ Classification of small-scale fading ● Rayleigh and Rice processes  □ Probability density function of amplitude  □ Probability density function of frequency ● Modeling frequency flat channel ● Modeling frequency selective channel  □ Method of equal distances (MED) to model specified power delay profiles  □ Simulating a frequency selective channel using TDL model

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