Tag: power delay profile

Model a Frequency Selective Multipath Fading channel

A brief intro to modeling a frequency selective fading channel using tapped delay line (TDL) filters. Rayleigh & Rician frequency-selective fading channel models explained.

Tapped delay line filters

Tapped-delay line filters (FIR filters) are best to simulate multiple echoes originating from same source. Hence they can be used to model multipath scenarios. Tapped-Delay-Line (TDL) filters with number taps can be used to simulate a multipath frequency selective fading channel. Frequency selective channels are characterized by time varying nature of the channel. For simulating a frequency selective channel, it is mandatory to have N > 1. In contrast, if N = 1, it simulates a zero-mean fading channel where all the multipath signals arrive at the receiver at the same time.

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Let be the associated path attenuation corresponding to the received power and propagation delay of the th path. In continuous time, the complex path attenuation is given by

The complex channel response is given by

In the equation above, the attenuation and path delay vary with time. This simulates a time-variant complex channel.

As a special case, in the absence any movements or other changes in the transmission channel, the channel can remain fairly time invariant (fixed channel with respect to instantaneous time ) even though the multipath is present. Thus the time-invariant complex channel becomes

Usually, the pair is described as a Power Delay Profile (PDP) plot. A sample power delay profile plot for a fixed, discrete, three ray model with its corresponding implementation using a tapped-delay line filter is shown in the following figure

3-ray multipath time-invariant channel and its equivalent TDL implementation (Frequency selective fading channel)
Figure 1: 3-ray multipath time-invariant channel and its equivalent TDL implementation (path attenuations and
propagation delays are fixed)

Choose underlying distribution:

The next level of modeling involves, introduction of randomness in the above mentioned model there by rendering the channel response time variant. If the path attenuations are typically drawn from a complex Gaussian random variable, then at any given time , the absolute value of the impulse response is

● Rayleigh distributed – if the mean of the distribution
● Rician distributed – if the mean of the distribution

Respectively, these two scenarios model the presence or absence of a Line of Sight (LOS) path between the transmitter and the receiver. The first propagation delay has no effect on the model behavior and hence it can be removed.

Similarly, the propagation delays can also be randomized, resulting in a more realistic but extremely complex model to implement. Furthermore, the power-delay-profile specifications with arbitrary time delays, warrant non-uniformly spaced tapped-delay-line filters, that are not suitable for practical simulation. For ease of implementation, the given PDP model with arbitrary time delays can be converted to tractable uniformly spaced statistical model by a combination of interpolation/approximation/uniform-sampling of the given power-delay-profile.

Real-life modelling:

Usually continuous domain equations for modeling multipath are specified in standards like COST-207 model in GSM specification. Such continuous time power-delay-profile models can be simulated using discrete-time Tapped Delay Line (TDL) filter with number of taps with variable tap gains. Given the order , the problem boils down to determining the discrete tap spacing and the path gains , in such a way that the simulated channel closely follows the specified multipath PDP characteristics. A survey of method to find a solution for this problem can be found in [2].

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References:

[1] Julius O. Smith III, Physical Audio Signal Processing, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.↗
[2] M. Paetzold, A. Szczepanski, N. Youssef, Methods for Modeling of Specified and Measured Multipath Power-Delay Profiles, IEEE Trans. on Vehicular Techn., vol.51, no.5, pp.978-988, Sep.2002.↗

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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Significance of RMS (Root Mean Square) value

Root Mean Square (RMS) value is the most important parameter that signifies the size of a signal.

Defining the term “size”:

In signal processing, a signal is viewed as a function of time. The term “size of a signal” is used to represent “strength of the signal”. It is crucial to know the “size” of a signal used in a certain application. For example, we may be interested to know the amount of electricity needed to power a LCD monitor as opposed to a CRT monitor. Both of these applications are different and have different tolerances. Thus the amount of electricity driving these devices will also be different.

A given signal’s size can be measured in many ways. Some of them are,

► Total energy
► Square root of total energy
► Integral absolute value
► Maximum or peak absolute value
► Root Mean Square (RMS) value
► Average Absolute (AA) value

Parseval’s theorem

The Parseval’s theorem expresses the energy of a signal in time-domain in terms of the average energy in its frequency components.

Suppose if the x[n] is a sequence of complex numbers of length N : xn={x0,x1,…,xN-1}, its N-point discrete Fourier transform (DFT): Xk={X0,X1,…,XN-1} is given by

The inverse discrete Fourier transform is given by

Suppose if x[n] and y[n] are two such sequences that follows the above definitions, the Parseval’s theorem is written as

where, indicates conjugate operation.

Deriving Parseval’s theorem

Energy content

Given a discrete-time sequence length N : xn={x0,x1,…,xN-1}, according to Parseval’s theorem, the energy content of the signal in the time-domain is equivalent to the average of the energy contained in its frequency components.

If the samples x[n] and X[k] are real-valued, then

Mean Square value

Mean square value is the arithmetic mean of squares of a given set of numbers. For a complex-valued signal set represented as discrete sampled values – , the mean square xMS value is given as

Applying Parseval’s theorem, the mean square value can also be computed using frequency domain components X[k]

RMS value

RMS value of a signal is calculated as the square root of average of squared value of the signal. For a complex-valued signal set represented as discrete sampled values – , the mean square xRMS value is given as

Applying Parseval’s theorem, the root mean square value can also be computed using frequency domain components X[k]

Implementing in Matlab:

Following Matlab code demonstrates the calculation of RMS value for a random sequence using time-domain and frequency domain approach. Figure 1, depicts the simulation results for RMS values for some well-known waveforms.

N=100; %length of the signal
x=randn(1,N); %a random signal to test 
X=fft(x); %Frequency domain representation of the signal 

RMS1 = sqrt(mean(x.*conj(x))) %RMS value from time domain samples 
RMS2 = sqrt(sum(X.*conj(X))/length(x)^2) %RMS value from frequency domain representation

%Result: RMS1 = 0.9814, RMS2 = 0.9814

%Matlab has inbuilt 'rms' function, it can also be used.
Figure 1: RMS values of some well known signals

Significance of RMS value

► One of the most important parameter that is used to describe the strength of an Alternating Current (AC).

► RMS value of an AC voltage/current is equivalent to the DC voltage/current that produces the same heating effect when applied across an identical resistor. Hence, it is also a measure of energy content in a given signal.

► In statistics, for any zero-mean random stationary signal, the RMS value is same as the standard deviation of the signal. Example : Delay spread of a multipath channel is often calculated as the RMS value of the Power Delay Profile (PDP)

► When two uncorrelated (or orthogonal ) signals are added together, such as noise from two independent sources, the RMS value of their sum is equal to the square-root of sum of the square of their individual RMS values.

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See also

Basics of – Power and Energy of a signal
Calculation of power of a signal and verifying it through Matlab.

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