Tag: Fading

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.

A typical discrete power delay profile plot for a multipath channel with 3 paths
Figure 1: A typical discrete power delay profile plot for a multipath channel with 3 paths

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.

This article is part of the book
● Wireless Communication Systems in Matlab (second edition), ISBN: 979-8648350779 available in ebook (PDF) format and Paperback (hardcopy) format.

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.

Figure 2: Relation between scattering function, power delay profile, Doppler power spectrum, spaced frequency
correlation function and spaced time correlation function

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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Multipath channel models: scattering function

Understand various characteristics of a wireless channel through multipath channel models. Discuss Wide Sense Stationary channel, uncorrelated scattering channel, wide sense stationary uncorrelated scattering channel models and scattering function.

Introduction

Wireless channel is of time-varying nature in which the parameters randomly change with respect to time. Wireless channel is very harsh when compared to AWGN channel model which is often considered for simulation and modeling. Understanding the various characteristics of a wireless channel and understanding their physical significance is of paramount importance. In these series of articles, I seek to expound various statistical characteristics of a multipath wireless channel by giving more importance to the concept than the mathematical derivation.

This article is part of the book
● Wireless Communication Systems in Matlab (second edition), ISBN: 979-8648350779 available in ebook (PDF) format and Paperback (hardcopy) format.

Complex baseband mutipath channel model:

In a multipath channel, multiple copies of a signal travel different paths with different propagation delays and are received at the receiver at different phase angles and strengths. These rays add constructively or destructively at the receiver front end, thereby giving rise to rapid fluctuations in the channel. The multipath channel can be viewed as a linear time variant system where the parameters change randomly with respect to time. The channel impulse response is a two dimensional random variable – that is a function of two parameters –  instantaneous time and the propagation delay . The  channel is expressed as a set of random complex gains at a given time and the propagation delay . The output of the channel can be expressed as the convolution of the complex channel impulse response and the input

If the complex channel gains are typically drawn from a complex Gaussian distribution, then at any given time , the absolute value of the impulse response is Rayleigh distributed (if  the mean of the distribution or Rician distributed (if  the mean of the distribution . These two scenarios model the presence or absence of a Line of Sight (LOS) path between the transmitter and the receiver.

Here, the values for the channel impulse response are samples of a random process that is defined with respect to time and the multipath delay τ. That is, for each combination of and τ, a randomly drawn value is assigned for the channel impulse response. As with any other random process, we can calculate the general autocorrelation function as

Given the generic autocorrelation function above, following assumptions can be made to restrict the channel model to the following specific set of categories

  • Wide Sense Stationary channel model
  • Uncorrelated Scattering channel model
  • Wide Sense Stationary Uncorrelated Scattering channel model

Wide Sense Stationary (WSS) channel model

In this channel model, the impulse response of the channel is considered Wide Sense Stationary (WSS) , that is the channel impulse response is independent of time . In other words, the autocorrelation function  is independent of time instant \(t\) and it depends on the difference between the time instants where  and . The autocorrelation function for WSS channel model is expressed as

Uncorrelated Scattering  (US) channel model

Here, the individual scattered components arriving at the receiver front end (at different propagation delays) are assumed to be uncorrelated. Thus the autocorrelation function can be expressed as

Wide Sense Stationary Uncorrelated Scattering (WSSUS) channel model

The WSSUS channel model combines the aspects of WSS and US channel model that are discussed above. Here, the channel is considered as Wide Sense Stationary and the scattering components arriving at the receiver are assumed to be uncorrelated. Combining both the worlds, the autocorrelation function is

Scattering function

The autocorrelation function of the WSSUS channel model can be represented in frequency domain by taking Fourier transform with respect one or both variables – difference in time and the propagation delay . Of the two forms, the Fourier transform on the variable gives specific insight to channel properties  in terms of propagation delay and the Doppler Frequency simultaneously. The Fourier transform of the above two-dimensional autocorrelation function on the variable is called scattering function and is given by

Fourier transform of relative time is Doppler Frequency. Thus,  the scattering function is a function of two variables – Dopper Frequency and the propagation delay . It gives the average output power of the channel as a function of Doppler Frequency and the propagation delay .

Figure 1: Relation between scattering function, power delay profile, Doppler power spectrum, spaced frequency
correlation function and spaced time correlation function (click to know more)

Two important relationships can be derived from the scattering function – Power Delay Profile (PDP) and Doppler Power Spectrum. Both of them affect the performance of a given wireless channel. Power Delay Profile is a function of propagation delay and the Doppler Power Spectrum is a function of Doppler Frequency.

Power Delay Profile gives the signal intensity 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

Power Delay Profile can also be obtained from scattering function, by integrating it over the entire frequency range (removing the dependence on Doppler frequency).

Similarly, the Doppler Power Spectrum can be obtained by integrating the scattering function over the entire range of propagation delays.

Fourier Transform of Power Delay Profile and Inverse Fourier Transform of Doppler Power Spectrum:

Power Delay Profile is a function of time which can be transformed to frequency domain by taking Fourier Transform. Fourier Transform of Power Delay Profile is called spaced-frequency correlation function. Spaced-frequency correlation function describes the spreading of a signal in frequency domain. This gives rise to the importance channel parameter – Coherence Bandwidth.

Similarly, the Doppler Power Spectrum describes the output power of the multipath channel in frequency domain. The Doppler Power Spectrum when translated to time-domain by means of inverse Fourier transform is called spaced-time correlation function. Spaced-time correlation function describes the correlation between different scattered signals received at two different times as a function of the difference in the received time. It gives rise to the concept of Coherence Time.

Next Topic : Power Delay Profile

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For further reading

[1] L. Bernadó, T. Zemen, F. Tufvesson, A. F. Molisch and C. F. Mecklenbräuker, “The (in-) validity of the WSSUS assumption in vehicular radio channels,” 2012 IEEE 23rd International Symposium on Personal, Indoor and Mobile Radio Communications – (PIMRC), Sydney, NSW, 2012, pp. 1757-1762, doi: 10.1109/PIMRC.2012.6362634.↗

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QAM modulation: simulate in Matlab & Python

A generic complex baseband simulation technique, to simulate all M-ary QAM modulation techniques is given here. The given simulation code is very generic, and it plots both simulated and theoretical symbol error rates for all M-QAM modulation techniques.

Rectangular QAM from PAM constellation

There exist other constellation shapes (like circular, triangular constellations) that are more efficient (in terms of energy required to achieve same the error probability) than the standard rectangular constellation. Rectangular (symmetric or square) constellations are the preferred choice of implementation due to its simplicity in implementing modulation and demodulation.

In one of the earlier articles, I have discussed the method of constructing constellation for rectangular QAM modulation using Karnaugh-map walks, where the inherent property of Karnaugh-maps is exploited to construct Gray coded QAM symbols.

Any rectangular QAM constellation is equivalent to superimposing two Amplitude Shift Keying (ASK) signals (also called Pulse Amplitude Modulation – PAM) on quadrature carriers. For example, 16-QAM constellation points can be generated from two 4-PAM signals, similarly the 64-QAM constellation points can be generated from two 8-PAM signals.

Figure 1: Signal space constellations for 16-QAM and 64-QAM

The generic equation to generate PAM signals of dimension D is

For generating 16-QAM, the dimension D of PAM is set to . Thus for constructing a M-QAM constellation, the PAM dimension is set as . Matlab code for dynamically generating M-QAM constellation points based on Karnaugh map Gray code walk is given below. The resulting ideal constellations for Gray coded 16-QAM and 64-QAM are shown in Figure 1.

Matlab code

Full Matlab code available in the book Digital Modulations using Matlab – build simulation models from scratch

function [s,ref]=mqam_modulator(M,d)
%Function to MQAM modulate the vector of data symbols - d
%[s,ref]=mqam_modulator(M,d) modulates the symbols defined by the vector d
% using MQAM modulation, where M specifies order of M-QAM modulation and
% vector d contains symbols whose values range 1:M. The output s is modulated
% output and ref represents reference constellation that can be used in demod
if(((M˜=1) && ˜mod(floor(log2(M)),2))==0), %M not a even power of 2
  error('Only Square MQAM supported. M must be even power of 2');
end
  ref=constructQAM(M); %construct reference constellation
  s=ref(d); %map information symbols to modulated symbols
end

Python code

Full Matlab code available in the book Digital Modulations using Python

class QAMModem(Modem):
    # Derived class: QAMModem
    
    def __init__(self,M):
        
        if (M==1) or (np.mod(np.log2(M),2)!=0): # M not a even power of 2
            raise ValueError('Only square MQAM supported. M must be even power of 2')
        
        n = np.arange(0,M) # Sequential address from 0 to M-1 (1xM dimension)
        a = np.asarray([xˆ(x>>1) for x in n]) #convert linear addresses to Gray code
        D = np.sqrt(M).astype(int) #Dimension of K-Map - N x N matrix
        a = np.reshape(a,(D,D)) # NxN gray coded matrix
        oddRows=np.arange(start = 1, stop = D ,step=2) # identify alternate rows
        
        nGray=np.reshape(a,(M)) # reshape to 1xM - Gray code walk on KMap
        #Construction of ideal M-QAM constellation from sqrt(M)-PAM
        (x,y)=np.divmod(nGray,D) #element-wise quotient and remainder
        Ax=2*x+1-D # PAM Amplitudes 2d+1-D - real axis
        Ay=2*y+1-D # PAM Amplitudes 2d+1-D - imag axis
        constellation = Ax+1j*Ay
        Modem.__init__(self, M, constellation, name='QAM') #set the modem attributes

This article is part of the following books
● Digital Modulations using Matlab : Build Simulation Models from Scratch, ISBN: 978-1521493885
● Digital Modulations using Python ISBN: 978-1712321638
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M-QAM demodulation (coherent detection)

Generally the two main categories of detection techniques, commonly applied for detecting the digitally modulated data are coherent detection and non-coherent detection.

In the vector simulation model for the coherent detection, the transmitter and receiver agree on the same
reference constellation for modulating and demodulating the information. The modulators generate the reference constellation for the selected modulation type. The same reference constellation should be used if coherent detection is selected as the method of demodulating the received data vector.

On the other hand, in the non-coherent detection, the receiver is oblivious to the reference constellation used at the transmitter. The receiver uses methods like envelope detection to demodulate the data.

The IQ detection technique is an example of coherent detection. In the IQ detection technique, the first step is to compute the pair-wise Euclidean distance between the given two vectors – reference array and the received symbols corrupted with noise. Each symbol in the received symbol vector (represented on a p-dimensional plane) should be compared with every symbol in the reference array. Next, the symbols, from the reference array, that provide the minimum Euclidean distance are returned.

Let x=(x1,x2,…,xp) and y=(y1,y2,…,yp) be two points in p-dimensional space. The Euclidean distance between them is given by

The pair-wise Euclidean distance between two sets of vectors, say x and y, on a p-dimensional space, can be computed using the vectorized code. The vectorized code returns the ideal signaling points from matrix y that provides the minimum Euclidean distance. Since the vectorized implementation is devoid of nested for-loops, the program executes significantly faster for larger input matrices. The given code is very generic in the sense that it can be easily reused to implement optimum coherent receivers for any N-dimensional digital modulation technique (Please refer the books Digital Modulations using Matlab and Digital Modulations using Python for complete simulation code) .

Matlab code

Full Matlab code available in the book Digital Modulations using Matlab

function [dCap]= mqam_detector(M,r)
%Function to detect MQAM modulated symbols
%[dCap]= mqam_detector(M,r) detects the received MQAM signal points
%points - 'r'. M is the modulation level of MQAM
   if(((M˜=1) && ˜mod(floor(log2(M)),2))==0), %M not a even power of 2
      error('Only Square MQAM supported. M must be even power of 2');
   end
   ref=constructQAM(M); %reference constellation for MQAM
   [˜,dCap]= iqOptDetector(r,ref); %IQ detection
end

Python code

Full Matlab code available in the book Digital Modulations using Python

Performance simulation results

The simulation results for error rate performance of M-QAM modulations over AWGN channel and Rician flat-fading channel is given in the following figures.

Figure 2: Error rate performance of M-QAM modulations in AWGN channel
Figure 3: Error rate performance of 64-QAM modulation in Rician flat-fading channels having different values for Rician K-factors.

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Reference

[1] John G. Proakis, “Digital Communciations”, McGraw-Hill; 5th edition.↗

Related Topics

Digital Modulators and Demodulators - Complex Baseband Equivalent Models
● Introduction
● Complex baseband representation of modulated signal
● Complex baseband representation of channel response
● Modulators for amplitude and phase modulations
 □ Pulse Amplitude Modulation (M-PAM)
 □ Phase Shift Keying Modulation (M-PSK)
 □ Quadrature Amplitude Modulation (M-QAM)
● Demodulators for amplitude and phase modulations
 □ M-PAM detection
 □ M-PSK detection
 □ M-QAM detection
 □ Optimum detector on IQ plane using minimum Euclidean distance
● M-ary FSK modulation and detection
 □ Modulator for M orthogonal signals
 □ M-FSK detection

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Simulation of Rayleigh Fading ( Clarke’s Model – sum of sinusoids method)

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A multipath fading channel  can be modeled as a FIR (Finite Impulse Response) filter with the following impulse response.

$$ h( \tau ; t ) = h_{0}(t) \delta ( \tau – \tau_{0}(t)) + h_{1}(t) \delta ( \tau – \tau_{1}(t)) + . . . + h_{L-1}(t) \delta ( \tau – \tau_{L-1}(t)) $$

where h(τ,t) is the time varying impulse response of the multipath fading channel having L multipaths and hi(t) and τi(t) denote the time varying complex gain and excess delay of the i-th path. The above mentioned impulse response can be implemented as a FIR filter as shown below :

Multipath Fading phenomena – modelled as a Time Varying FIR Filter

The channel under consideration can be modeled as a multipath fading channel in which the impulse response may follow distributions like Rayleigh distribution ( in which there is no Line of Sight (LOS) ray between transmitter and receiver) or as Rician distribution ( dominant LOS path exist between transmitter and receiver), Nagami distribution, Weibull distribution etc.

Different methods of simulation techniques were proposed to simulate/model multipath channels. Some of the models include clarke’s reference model, Jake’s model, Young’s model , filtered gaussian noise model etc.

A Rayleigh fading channel (flat fading channel) is considered in this text.For simplicity we fix the excess delays τi(t) in the above equation and we generate hi(t) that follows Rayleigh distribution. In this simulation Clarke’s Rayleigh fading model is used. This model is also called mathematical reference model and is commonly considered as a computationally inefficient model compared to Jake’s Rayleigh Fading simulator.

Theory of Rayleigh Fading:

Lets denote the complex impulse response h(t) of the flat fading channel as follows :

$$ h(t) = h_{I}(t) + jh_{Q}(t) $$

where hI(t) and hQ(t) are zero mean gaussian distributed. Therefore the fading envelope is Rayleigh distributed and is given by

$$ \left |h(t) \right | = \sqrt{\left |h_{I}(t) \right |^2 + \left |h_{Q}(t) \right |^2} $$

The probability density function (Rayleigh distribution) of the above mentioned amplitude response is given by

$$ f(z)=\frac{2z}{\sigma ^{2}}e^{-\frac{z^{2}}{\sigma ^{2}}} \\ where \; \sigma ^{2} = E\left ( \left | h(t) \right |^{2} \right ) $$

We will use the Clarke’s Rayleigh Fading model (given below) and check the statistical properties of the random process generated by the model against the statistical properties of Rayleigh distribution (given above).

Clarke’s Rayleigh Fading model:

The random process of flat Rayleigh fading with M multipaths can be simulated with the sum-of-sinusoid method described as

Simulation:

1) The rayleigh fading model is implemented as a function in matlab with following parameters:
M=number of multipaths in the fading channel, N = number of samples to generate, fd=maximum Doppler spread in Hz, Ts = sampling period.

function [h]=rayleighFading(M,N,fd,Ts)

% function to generate rayleigh Fading samples based on Clarke's model
% M = number of multipaths in the channel
% N = number of samples to generate
% fd = maximum Doppler frequency
% Ts = sampling period
% Author : Mathuranathan for https://www.gaussianwaves.com
%Code available in the ebook - Simulation of Digital Communication Systems using Matlab

Check this book for full Matlab code.
Simulation of Digital Communication Systems Using Matlab – by Mathuranathan Viswanathan

2)The above mentioned function is used to generate Rayleigh Fading samples with the following values for the function arguments. M=15; N=10^5; fd=100 Hz;Ts=0.0001 second;

Investigation of Statistical Properties of samples generated using Clarke’s model:

3) Mean and Variance of the real and imaginary parts of generated samples are
Mean of real part ~=0
Mean of imag part ~=0
Variance of real part = 0.4989 ~=0.5
Variance of imag part = 0.4989 ~=0.5

The results implies that the mean of the real and imaginary parts are same and are equal to zero.The variance of the real and imaginary parts are approximately equal to 0.5.

4)Next, the pdf of the real part of the simulated samples are plotted and compared against the pdf of Gaussian distribution (with mean=0 and variance =0.5)

Real Part of simulated samples exhibiting Gaussian Distribution characteristics

5)The pdf of the generated Rayleigh fading samples are plotted and compared against pdf of Rayleigh distribution (with variance=1)

PDF of simulated Rayleigh Fading Samples

6) From 4) and 5) we confirm that the samples generated by Clarke’s model follows Rayleigh distribution (with variance = 1) and the real and imaginary part of the samples follow Gaussian distribution (with mean=0 and variance =0.5).

7) The Magnitude and Phase response of the generated Rayleigh Fading samples are plotted here.

The Magnitude and Phase response of the generated Rayleigh Fading samples

See also

[1]Eb/N0 Vs BER for BPSK over Rayleigh Channel and AWGN Channel
[2]Eb/N0 Vs BER for BPSK over Rician Fading Channel
[3]Performance comparison of Digital Modulation techniques
[4]BER Vs Eb/N0 for BPSK modulation over AWGN
[5]Rayleigh Fading Simulation – Young’s model
[6]Introduction to Fading Channels
[7] Chi-Squared distribution

Recommended Books

External Resources

[1]Theoretical expressions for BER under various conditions

Young’s model for Rayleigh fading

Introduction

Young’s fading channel model is a mathematical model used to describe the behavior of a wireless communication channel. It is a type of frequency selective fading channel model that is commonly used to simulate the effects of multipath interference on wireless signals.

The model is based on the assumption that the transmitted signal reaches the receiver through multiple paths, each with a different attenuation and phase shift. The attenuation and phase shift of each path are modeled as independent random variables with specific probability distributions.

The model uses the sum of these attenuated and phase-shifted paths to simulate the received signal. The resulting signal experiences fading due to the constructive and destructive interference of the individual paths.

Young’s fading channel model is useful for simulating the performance of wireless communication systems in a multipath environment. It can help researchers and engineers evaluate the performance of different modulation and coding schemes and develop techniques to mitigate the effects of fading.

Young’s model

In the previous article, the characteristics and types of fading was discussed. Rayleigh Fading channel with Doppler shift is considered in this article.

Consider a channel affected by both Rayleigh Fading phenomena and Doppler Shift. Rayleigh Fading is caused due to multipath reflections of the received signal before it reaches the receiver and the Doppler Shift is caused due to the difference in the relative velocity/motion between the transmitter and the receiver. This scenario is encountered in day to day mobile communications.

A number of simulation algorithms are proposed for generation of correlated Rayleigh random variables. David J.Young and Norman C Beaulieu proposed a method in their paper titled “The Generation of Correlated Rayleigh Random Variates by Inverse Discrete Fourier Transform”[1] based on the inverse discrete Fourier transform (IDFT). It is a modification of the Smith’s algorithm which is normally used for Rayleigh fading simulation. This method requires exactly one-half the number of IDFT operations and roughly two-thirds the computer memory of the original method – as the authors of the paper claims.

Rayleigh Fading can be simulated by adding two Gaussian Random variables as mentioned in my previous post. The effect of Doppler shift is incorporated by modeling the Doppler effect as a frequency domain filter.

The model proposed by Young et.al is shown below.

Rayleigh Fading – Young’s model

The Fading effect + Doppler Shift is simulated by multiplying the Gaussian Random variables and the Doppler Shift’s Frequency domain representation. Then IDFT is performed to bring them into time domain representation. The Doppler Filter used to represent the Doppler Shift effect is derived in Young’s paper.

The equation for the Doppler Filter is :

Matlab Code

Check this book for full Matlab code.
Simulation of Digital Communication Systems Using Matlab – by Mathuranathan Viswanathan

Matlab code Output:

Rayleigh Fading with Doppler Effect

Reference:

[1] D.J. Young and N.C. Beaulieu, “The generation of correlated Rayleigh random variates by inverse discrete fourier transform,” IEEE transactions on Communications, vol. 48, pp. 1114-1127, July 2000.

See also

[1]Eb/N0 Vs BER for BPSK over Rayleigh Channel and AWGN Channel
[2]Simulation of Rayleigh Fading ( Clarke’s Model – sum of sinusoids method)
[3]Performance comparison of Digital Modulation techniques
[4]BER Vs Eb/N0 for BPSK modulation over AWGN
[5]Introduction to Fading Channels

External Resources

[1]Theoretical expressions for BER under various conditions
[2]Capacity of MRC on correlated Rician Fading Channels

Fading channel – complex baseband equivalent models

Keyfocus: Fading channel models for simulation. Learn how fading channels can be modeled as FIR filters for simplified modulation & detection. Rayleigh/Rician fading.

Introduction

A fading channel is a wireless communication channel in which the quality of the signal fluctuates over time due to changes in the transmission environment. These changes can be caused by different factors such as distance, obstacles, and interference, resulting in attenuation and phase shifting. The signal fluctuations can cause errors or loss of information during transmission.

Fading channels are categorized into slow fading and fast fading depending on the rate of channel variation. Slow fading occurs over long periods, while fast fading happens rapidly over short periods, typically due to multipath interference.

To overcome the negative effects of fading, various techniques are used, including diversity techniques, equalization, and channel coding.

Fading channel in frequency domain

With respect to the frequency domain characteristics, the fading channels can be classified into frequency selective and frequency-flat fading.

A frequency flat fading channel is a wireless communication channel where the attenuation and phase shift of the signal are constant across the entire frequency band. This means that the signal experiences the same amount of fading at all frequencies, and there is no frequency-dependent distortion of the signal.

In contrast, a frequency selective fading channel is a wireless communication channel where the attenuation and phase shift of the signal vary with frequency. This means that the signal experiences different levels of fading at different frequencies, resulting in a frequency-dependent distortion of the signal.

Frequency selective fading can occur due to various factors such as multipath interference and the presence of objects that scatter or absorb certain frequencies more than others. To mitigate the effects of frequency selective fading, various techniques can be used, such as equalization and frequency hopping.

The channel fading can be modeled with different statistics like Rayleigh, Rician, Nakagami fading. The fading channel models, in this section, are utilized for obtaining the simulated performance of various modulations over Rayleigh flat fading and Rician flat fading channels. Modeling of frequency selective fading channel is discussed in this article.

Linear time invariant channel model and FIR filters

The most significant feature of a real world channel is that the channel does not immediately respond to the input. Physically, this indicates some sort of inertia built into the channel/medium, that takes some time to respond. As a consequence, it may introduce distortion effects like inter-symbol interference (ISI) at the channel output. Such effects are best studied with the linear time invariant (LTI) channel model, given in Figure 1.

Figure 1: Complex baseband equivalent LTI channel model

In this model, the channel response to any input depends only on the channel impulse response(CIR) function of the channel. The CIR is usually defined for finite length \(L\) as \(\mathbf{h}=[h_0,h_1,h_2, \cdots,h_{L-1}]\) where \(h_0\) is the CIR at symbol sampling instant \(0T_{sym}\) and \(h_{L-1}\) is the CIR at symbol sampling instant \((L-1)T_{sym}\). Such a channel can be modeled as a tapped delay line (TDL) filter, otherwise called finite impulse response (FIR) filter. Here, we only consider the CIR at symbol sampling instances. It is well known that the output of such a channel (\(\mathbf{r}\)) is given as the linear convolution of the input symbols (\(\mathbf{s}\)) and the CIR (\(\mathbf{h}\)) at symbol sampling instances. In addition, channel noise in the form of AWGN can also be included the model. Therefore, the resulting vector of from the entire channel model is given as

\[\mathbf{r} = \mathbf{h} \ast \mathbf{s} +\mathbf{n} \quad\quad (1) \]

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Simulation model for detection in flat fading channel

A flat-fading (also called as frequency-non-selective) channel is modeled with a single tap (\(L=1\)) FIR filter with the tap weights drawn from distributions like Rayleigh, Rician or Nakagami distributions. We will assume block fading, which implies that the fading process is approximately constant for a given transmission interval. For block fading, the random tap coefficient \(h=h[0]\) is a complex random variable (not random processes) and for each channel realization, a new set of complex random values are drawn from Rayleigh or Rician or Nakagami fading according to the type of fading desired.

Figure 2: LTI channel viewed as tapped delay line filter

Simulation models for modulation and detection over a fading channel is shown in Figure 2. For a flat fading channel, the output of the channel can be expressed simply as the product of time varying channel response and the input signal. Thus, equation (1) can be simplified (refer this article for derivation) as follows for the flat fading channel.

\[\mathbf{r} = h\mathbf{s} + \mathbf{n} \quad\quad (2) \]

Since the channel and noise are modeled as a complex vectors, the detection of \(\mathbf{s}\) from the received signal is an estimation problem in the complex vector space.

Figure 3: Simulation model for modulation and detection over flat fading channel

Assuming perfect channel knowledge at the receiver and coherent detection, the receiver shown in Figure 3(a) performs matched filtering. The impulse response of the matched filter is matched to the impulse response of the flat-fading channel as \( h^{\ast}\). The output of the matched filter is scaled down by a factor of \(||h||^2\) which is the total-energy contained in the impulse response of the flat-fading channel. The resulting decision vector \(\mathbf{y}\) serves as the sufficient statistic for the estimation of \(\mathbf{s}\) from the received signal \(\mathbf{r}\) (refer equation A.77 in reference [1])

\[\tilde{\mathbf{y}} = \frac{h^{\ast}}{||h||^2} \mathbf{r} = \frac{h^{\ast}}{||h||^2} h\mathbf{s} + \frac{h^{\ast}}{||h||^2} \mathbf{n} = \mathbf{s} + \tilde{\mathbf{w}} \quad\quad (3) \]

Since the absolute value \(|h|\) and the Eucliden norm \(||h||\) are related as \(|h|^2= \left\lVert h\right\rVert = hh^{\ast}\), the model can be simplified further as given in Figure 3(b).

To simulate flat fading, the values for the fading variable \(h\) are drawn from complex normal distribution

\[h= X + jY \quad\quad (4) \]

where, \(X,Y\) are statistically independent real valued normal random variables.

● If \(E[h]=0\), then \(|h|\) is Rayleigh distributed, resulting in a Rayleigh flat fading channel
● If \(E[h] \neq 0\), then \(|h|\) is Rician distributed, resulting in a Rician flat fading channel with the factor \(K=[E[h]]^2/\sigma^2_h\)

References

[1] D. Tse and P. Viswanath, Fundamentals of Wireless Communication, Cambridge University Press, 2005.↗

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