Tag: Rayleigh

Rician flat-fading channel – simulation

In wireless environments, transmitted signal may be subjected to multiple scatterings before arriving at the receiver. This gives rise to random fluctuations in the received signal and this phenomenon is called fading. The scattered version of the signal is designated as non line of sight (NLOS) component. If the number of NLOS components are sufficiently large, the fading process is approximated as the sum of large number of complex Gaussian process whose probability-density-function follows Rayleigh distribution.

Rayleigh distribution is well suited for the absence of a dominant line of sight (LOS) path between the transmitter and the receiver. If a line of sight path do exist, the envelope distribution is no longer Rayleigh, but Rician (or Ricean). If there exists a dominant LOS component, the fading process can be represented as the sum of complex exponential and a narrowband complex Gaussian process g(t). If the LOS component arrive at the receiver at an angle of arrival (AoA) θ, phase ɸ and with the maximum Doppler frequency fD, the fading process in baseband can be represented as (refer [1])

\[h(t)= \underbrace{\sqrt{\frac{K \Omega}{K +1}}}_\text{A:=} e^{\left( j2 \pi f_D cos(\theta)t+\phi \right)} + \underbrace{\sqrt{\frac{\Omega}{K+1}}}_\text{S:=}g(t)\]

where, K represents the Rician K factor given as the ratio of power of the LOS component A2 to the power of the scattered components (S2) marked in the equation above.

\[K=\frac{A^2}{S^2}\]

The received signal power Ω is the sum of power in LOS component and the power in scattered components, given as Ω=A2+S2. The above mentioned fading process is called Rician fading process. The best and worst-case Rician fading channels are associated with K=∞ and K=0 respectively. A Ricean fading channel with K=∞ is a Gaussian channel with a strong LOS path. Ricean channel with K=0 represents a Rayleigh channel with no LOS path.

The statistical model for generating flat-fading Rician samples is discussed in detail in chapter 11 section 11.3.1 in the book Wireless communication systems in Matlab (see the related article here). With respect to the simulation model shown in Figure 1(b), given a K factor, the samples for the Rician flat-fading samples are drawn from the following random variable

\[h= | X + jY |\]

where X,Y ~ N(μ,σ2) are Gaussian random variables with non-zero mean μ and standard deviation σ as given in references [2] and [3].

\[\mu = g_1 =\sqrt{\frac{K}{2\left(K+1\right)}} \quad \quad \sigma = g_2 = \sqrt{\frac{1}{2\left(K+1\right)}}\]

Kindly refer the book Wireless communication systems in Matlab for the script on generating channel samples for Ricean flat-fading.

Simulation model for modulation and detection over flat fading channel
Figure 1: Simulation model for modulation and detection over flat fading channel

Simulation and performance results

In chapter 5 of the book Wireless communication systems in Matlab, the code implementation for complex baseband models for various digital modulators and demodulator are given. The computation and generation of AWGN noise is also given in the book. Using these models, we can create a unified simulation for code for simulating the performance of various modulation techniques over Rician flat-fading channel the simulation model shown in Figure 1(b).

An unified approach is employed to simulate the performance of any of the given modulation technique – MPSK, MQAM or MPAM. The simulation code (given in the book) will automatically choose the selected modulation type, performs Monte Carlo simulation, computes symbol error rates and plots them against the theoretical symbol error rate curves. The simulated performance results obtained for various modulations are shown in the Figure 2.

Figure 2: Performance of various modulations over Ricean flat fading channel

Rate this article: Note: There is a rating embedded within this post, please visit this post to rate it.

References

[1] C. Tepedelenlioglu, A. Abdi, and G. B. Giannakis, The Ricean K factor: Estimation and performance analysis, IEEE Trans. Wireless Communication ,vol. 2, no. 4, pp. 799–810, Jul. 2003.↗
[2] R. F. Lopes, I. Glover, M. P. Sousa, W. T. A. Lopes, and M. S. de Alencar, A simulation framework for spectrum sensing, 13th International Symposium on Wireless Personal Multimedia Communications (WPMC 2010), Out. 2010.
[3] M. C. Jeruchim, P. Balaban, and K. S. Shanmugan, Simulation of Communication Systems, Methodology, Modeling, and Techniques, second edition Kluwer Academic Publishers, 2000.↗

Books by the author


Wireless Communication Systems in Matlab
Second Edition(PDF)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Python
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Matlab
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart
Hand-picked Best books on Communication Engineering
Best books on Signal Processing

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.

This article is part of the book Wireless Communication Systems in Matlab, ISBN: 978-1720114352 available in ebook (PDF) format (click here) and Paperback (hardcopy) format (click here).

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

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].

Rate this article: Note: There is a rating embedded within this post, please visit this post to rate it.

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

Books by the author


Wireless Communication Systems in Matlab
Second Edition(PDF)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Python
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Matlab
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart
Hand-picked Best books on Communication Engineering
Best books on Signal Processing

Eb/N0 Vs BER for BPSK over Rayleigh Channel and AWGN Channel

The phenomenon of Rayleigh Flat fading and its simulation using Clarke’s model and Young’s model were discussed in the previous posts. The performance (Eb/N0 Vs BER) of BPSK modulation (with coherent detection) over Rayleigh Fading channel and its comparison over AWGN channel is discussed in this post.

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
● Wireless communication systems in Matlab ISBN: 979-8648350779
All books available in ebook (PDF) and Paperback formats

We first investigate the non-coherent detection of BPSK over Rayleigh Fading channel and then we move on to the coherent detection. For both the cases, we consider a simple flat fading Rayleigh channel (modeled as a – single tap filter – with complex impulse response – h). The channel also adds AWGN noise to the signal samples after it suffers from Rayleigh Fading.

The received signal y can be represented as

$$ y=hx+n $$

where n is the noise contributed by AWGN which is Gaussian distributed with zero mean and unit variance and h is the Rayleigh Fading response with zero mean and unit variance. (For a simple AWGN channel without Rayleigh Fading the received signal is represented as y=x+n).

Non-Coherent Detection:

In non-coherent detection, prior knowledge of the channel impulse response (“h” in this case) is not known at the receiver. Consider the BPSK signaling scheme with ‘x=+/- a’ being transmitted over such a channel as described above. This signaling scheme fails completely (in non coherent detection scheme), even in the absence of noise, since the phase of the received signal y is uniformly distributed between 0 and 2pi regardless of whether x[m]=+a or x[m]=-a is transmitted. So the non coherent detection of the BPSK signaling is not a suitable method of detection especially in a Fading environment.

Coherent Detection:

In coherent detection, the receiver has sufficient knowledge about the channel impulse response.Techniques like pilot transmissions are used to estimate the channel impulse response at the receiver, before the actual data transmission could begin. Lets consider that the channel impulse response estimate at receiver is known and is perfect & accurate.The transmitted symbols (‘x’) can be obtained from the received signal (‘y’) by the process of equalization as given below.

$$ \hat{y}=\frac{y}{h}=\frac{hx+n}{h}=x+z $$

here z is still an AWGN noise except for the scaling factor 1/h. Now the detection of x can be performed in a manner similar to the detection in AWGN channels.

The input binary bits to the BPSK modulation system are detected as

$$ \begin{matrix} r=real(\hat{y})=real(x+z) \\ \; \; \hat{d} =1, \; \; if \;r> 0 \\ \; \; \hat{d}=0 , \; \; if \; r< 0 \end{matrix} $$

Theoretical BER:

The theoretical BER for BPSK modulation scheme over Rayleigh fading channel (with AWGN noise) is given by

$$ P_{b} =\frac{1}{2} \left ( 1-\sqrt{\frac{E_{b}/N_{0}}{1+E_{b}/N_{0}}}\right) $$

The theoretical BER for BPSK modulation scheme over an AWGN channel is given here for comparison

$$ P_{b}=\frac{1}{2}erfc(\sqrt{E_{b}/N_{0}}) $$

Simulation Model:

The following model is used for the simulation of BPSK over Rayleigh Fading channel and its comparison with AWGN channel

BPSK Modulation over Rayleigh and AWGN channel

Matlab Code:

Check these books for matlab code

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
● Wireless communication systems in Matlab ISBN: 979-8648350779
All books available in ebook (PDF) and Paperback formats

Simulation Results:

The Simulated and theoretical performance curves (Eb/N0 Vs BER) for BPSK modulation over Rayleigh Fading channel and the AWGN is given below.

Eb/N0 Vs BER for BPSK over Rayleigh and AWGN Channel

See also

[1]Eb/N0 Vs BER for BPSK over Rician Fading 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]Rayleigh Fading Simulation – Young’s model
[6]Introduction to Fading Channels

Books by the author


Wireless Communication Systems in Matlab
Second Edition(PDF)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Python
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Matlab
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart
Hand-picked Best books on Communication Engineering
Best books on Signal Processing

External Resources

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

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

Note: There is a rating embedded within this post, please visit this post to rate it.

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

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

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
● Wireless communication systems in Matlab ISBN: 979-8648350779
All books available in ebook (PDF) and Paperback formats

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.↗

Books by the author


Wireless Communication Systems in Matlab
Second Edition(PDF)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Python
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Matlab
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart
Hand-picked Best books on Communication Engineering
Best books on Signal Processing