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## THEORY

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 TFmagnitude shows more fades per bandwidth, the longer the IR.

Because of multipath reflections, the channel impulse response of a wireless channel looks likes a series of pulses.

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Figure is an example of impulse response and frequency transfer function of a multipath channel.

We can define the local-mean average received power with excess delay within the interval (T, T + dt). This gives the "delay profile" of the channel.  The delay profile determines to what extent the channel fading at two different frequencies f1 and f2 are correlated.

In order to compare different multipath channels and to develop some general guidelines for wireless systems, parameters which grossly quantify the multipath channel are used. The mean excess delay, rms delay spread and excess delay spread (X dB) are multipath channel parameters that can be determined from a power delay profile. The mean excess delay is the first moment of the power delay profile and is defined to be

Where P (τ) is the power measured at time τ.

The rms delay spread is the square root of the second central moment of the power delay profile and is defined to be

These delays are measured relative to the first detectable signal to at the receiver at t0=0. The above equations do not rely on the absolute power level of P(t), but only the relative amplitudes of the multipath components within P(t).Typical values of rms delay spread are on the order of microseconds in outdoor mobile radio channels and on the order of  nanoseconds in indoor radio channels.

### Maximum Excess Delay

The maximum excess delay (X dB) of power delay profile is defined to be the time delay during which multipath energy falls to X dB below the maximum. In other words, the maximum excess delay is defined as tx - t0.where t0 is the first arriving signal and tx is the maximum delay at which a multipath component is within X dB of the strongest arriving multipath signal (which does not necessarily arrive at t0).The figure below illustrates the computation of the maximum excess delay for multipath components within 10 dB of maximum. The maximum excess delay (X dB) defines the temporal extent of the multipath that is above a particular threshold. The value of tx –is sometimes called the excess spread of a power delay profile, but in all cases must be specified with a threshold that relates the multipath noise floor to the maximum received multipath component.

### Coherence Bandwidth

Coherence bandwidth is a statistical measure of the range of frequencies over which the channel can be considered “flat” (i.e. a channel which passes all spectral components with approximately equal gain and linear phase). In other words, coherence bandwidth is the range of frequencies over which two frequency components have a strong potential for amplitude correlation. Two sinusoids with frequency separation greater than Bc are affected quite differently by the channel. If the coherence bandwidth is defined as the bandwidth over which the frequency correlation function is above 0.9, then the coherence bandwidth is approximately

Where the denominator sigma corresponds to the rms delay spread. If the definition is relaxed so that the frequency correlation function is above 0.5, then the coherence bandwidth is approximately

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