Chapter 3 Rayleigh Fading Channel Simulators for SISO
3.3 Karhunen-Loeve Expansion Model
3.4.4 Degree of Chebyshev Approximation
Chebyshev approximation is useful for reducing the computational complexity because of its iterative characteristics. The degree choice for fitting the eigenfunctions is the major factor in effecting precision. In this section, we will discuss the tradeoff between the computational complexity and the precision. Choosing the most complicated eigenfunction of real part and imaginary part respectively, the degree of polynomial curve fitting can be decided under permitting mean square error.
For N = , Figure 3-12 and Figure 3-13 show the polynomial curve fitting for 2 the real component of the 6-th eigenfunction and the imaginary component of the 5-th eigenfunction, respectively. It is shown the relationship of mean-square-error and the degree of curve fitting. As shown in Table 3-3 and Table 3-4, more degree of polynomial will have higher precision for curve fitting. If we would like to have
MSE<10−3, the least degree of real part of the 6-th eigenfunction should select (2N +6), as well as the least degree of imaginary part of the 5th eigenfunction should select (2N +5).
-4 -3 -2 -1 0 1 2 3 4
-0.04 -0.02 0 0.02 0.04 0.06 0.08
Time
Amplitude
Polyfit for Real Part of 6th Eigenfunction
Real Part of 6th Eigenfunction Polyfit using (2N+2) order
Figure 3-12: Polynomial curve fitting for real part of the 6th eigenfunction (N = ) 2
-4 -3 -2 -1 0 1 2 3 4
Polyfit for Imaginary Part of 5th Eigenfuncion
Imaginary Part of 5th Eigenfunction Polyfit using (2N+2) order
Figure 3-13: Polynomial curve fitting for imaginary part of the 5th eigenfunction (N = ) 2
Table 3-3: Polynomial curve fitting of real part of the 6th eigenfunction and its corresponding mean square error (N = ) 2
(2 2)
L = N + L=(2N +4) L=(2N +6)
MSE of Real Part
0.0257 0.0014 0.0001
Degree MSE
Table 3-4: Polynomial curve fitting of imaginary part of the 5th eigenfunction and its corresponding mean square error (N = ) 2
(2 1)
In Chapter3, we propose the modified models for simulating Rayleigh fading channels. In the respect of sum-of-sinusoids method, Wu model is selected because of its less computational complexity. Furthermore, the proposed modified Jakes model improves the accuracy of Wu model but requires almost computational complexity.
The MSE of the proposed method will be lower because of the independent initial random phase.
However, we are not satisfied with the performance of proposed modified Jakes model. K-L expansion method is considered to modify it. We consider the K-L expansion model to be more complete, and compare it with the proposed model using the concept of sum-of-sinusoids. From the simulation results, the proposed modified method using the concept of K-L expansion is better. In terms of statistic properties, autocorrelation and cross-correlation functions of proposed modified K-L expansion model are relatively close to the desired ones. From the viewpoint of computational complexity, our proposed modified K-L expansion model costs less run-time and needs more flexibility.
Degree MSE
In order to reduce more computational complexity, we replace the eigenfunctions of proposed modified K-L expansion model with Chebyshev polynomials and determined weighting matrix. Under the acceptable MSE, the degree of Chebyshev polynomial can be determined. Because of the recursive polynomials and determined weighting matrix, the requirement of computational complexity is less.
The above analysis is under the flat fading environment. In Chapter 4, the proposed Rayleigh fading simulator will extend to the wideband system and MIMO channel.
Chapter 4
Rayleigh Fading Channel Simulators for MIMO System
The channel capacity of a communications link can be greatly increased by deploying multiple transmit and multiple receive antennas. Multiple-input multiple-output (MIMO) capacity were developed based on complex Gaussian independent identically distributed (i.i.d.) channel matrices with additive white Gaussian noise. As well as the performance of realistic receivers, the theoretical capacity depends on the statistical properties of the channel. For realistic thinking of MIMO capacity and algorithms, channel effects such as frequency selective, time variation, spatial and temporal correlations need to be considered.
MIMO channel models can be divided into three classes: ray-tracing, scattering, and correlation models [27]. In the ray-tracing approach, free-space propagation, reflection, diffraction, and scattering are modeled to follow each propagation path through the channel. In the correlation model, MIMO channels are created by multiplying a complex Gaussian i.i.d. matrix by the square roots of the correlation matrices at the receiver and the transmitter [28]-[31]. The scattering model assumes a particular distribution of the scatters and generates channel realizations based on the interaction of scatters and plane wavefronts [32]-[35].
The MIMO communication system is now quite popular, the request for models that simulate the channel effects in MIMO systems becomes more and more important.
For fast development and real-time prototyping of MIMO communication systems, a real-time MIMO channel simulator can be very attractive [36]. The greatest challenge in implementing a real-time MIMO channel system is the large number of independent required Rayleigh faders by using Jakes model. To solve this problem, the proposed Rayleigh fading model using K-L expansion method can have less hardware cost. We will discuss how to extend our proposed Rayleigh fader to MIMO system in the following sections.
4.1 Spatial Channel Model for Mobile Wireless Applications
Spatial channel model (SCM) is adopted by third Generation Partnership Project (3GPP), which is applied for fixed and mobile MIMO wireless communications [37].
The categories of the channel environments are three scenarios: suburban macro-cell (approximated 3km distance BS to BS), urban macro-cell (approximated 3km distance BS to BS), and urban micro-cell (less than 1km distance BS to BS). For urban and suburban macro-cell environments, the cellular radius is 1-6 km and the BS antennas are above rooftop height, which is about 10-80 m and the mean height is 32m. The speed of mobility is 0-250 km/hr. For urban micro-cell environment, the cellular radius is 0.3-0.5 km and the BS antennas are at rooftop height, which has the mean height is about 12.5m. The speed of mobility is 0-120 km/hr. There is a notice that the urban micro-cell environment is relatively sensible for antenna height and scattering environment.
For system level simulation purposes, the fast fading per-path will be evolved in time, although bulk parameters including angle spread (AS), delay spread (DS), lognormal shadow fading (SF), and MS location will remain fixed during its evaluation.
The received signal at MS consists of N time-delayed multipath copies of the transmitted signal. Each scenario is characterized by a set of channel parameters. Each path consists of M subpaths. The parameters are shown in Figure 4-1. The following definitions are used :
v : MS velocity vector.
θ : Angle of the velocity with respect to the MS broadside: v θ =v arg( )v .
ΩBS : BS antenna array orientation, defined as the difference between the
broadside of the BS array and the absolute North (N) reference direction.
ΩMS: MS antenna array orientation, defined as the difference between the broadside of the MS array and the absolute North reference direction.
θBS: LOS angle of departure (AoD) direction between the BS and MS, with respect to the broadside of the BS array.
θMS: Angle between the BS-MS LOS and the MS broadside.
,
at the BS with respect to the BS broadside.
, , n m AoA
θ : Absolute AoA for the m (th m = …1, ,M) subpath of the thn path at the MS with respect to the MS broadside
Figure 4-1: BS and MS angle parameters
4.2 Wideband Fading Channels for SISO System
The proposed Rayleigh fader simulator based on K-L expansion method can be extended to single-input single-output (SISO) system. It only needs to add the effects of multipath delay and fading gain as shown in Figure 4-2. Assuming there are M faders (taps) in SISO system, the mathematical formulation of Rayleigh fading channel simulator r t for SISO system can be simply derived as: ( )
The autocorrelation function for the wideband fading channel needs be derived for verifying the accuracy. The autocorrelation of ( )r t is given by
( ) ( ) where ε is the time delay. The autocorrelation function is also the Bessel function of
delay, which is the same as that of a single fader.
Figure 4-2: Wideband Rayleigh fading channel simulator
4.3 Extension of Proposed Rayleigh Fader Simulator to MIMO System
According to TGn channel model, we suggest a realistic channel environment [38]. The channel environment combines the features of physical and non-physical MIMO channel models. There are L main links for MIMO channels, and each link corresponds to a wideband fading channel which is composed of M independent faders as shown in Figure 4-3. On the other hand, a circular disc (with radius R) of uniformly distributed scatters S is placed around the mobile unit. The channel parameter hnm connecting transmit antenna m and receive antenna n is geometrically constrainted. The base station (BS) is assumed to be elevated and therefore not obstructed by local scattering, while the mobile station (MS) is surrounded by scatters. Figure 4-3 illustrates this scenario where T is the antenna x elements at the BS, R is the antenna elements at the MS. x
In the MIMO channel matrix H for each tap, at one instance of time, the models can be separated into a fixed (constant, LOS) matrix and a variable Rayleigh matrix. For the case of one-ring model there is only the Rayleigh matrix part since the LOS component is not included.
For a 2 X 2 MIMO system, the channel matrix H [39] is receiving and m-th transmitting antenna) which is correlated zero-mean complex Gaussian random variable with unit variance, exp(jφnm) is the element of fixed
matrix H , K is the Rician K-factor, and P is the power of each delay tap. F
In order to correlate the Xnm elements of the matrix X , the Kronecker product of the transmit and receive correlation matrices is performed:
{
1/2}
[ ],[ ]X = ⎡⎢⎣RTX⎤⎥⎦ ⊗⎡⎢⎣RRX⎤⎥⎦ Hiid (4-4) where R and Tx RRx are the receive and transmit correlation matrices respectively,
and Hiid is a vector of independent zero-mean complex Gaussian random variables with unit variance (i.e. the wideband fading signal).
The transmit and receive correlation matrices are expressed as
12 12
where ρTxnm is the complex correlation coefficient between -thn and m-th transmitting antennas, and ρRxnm is the complex correlation coefficients between
-th
m and n-th receiving antennas. According to the above mentioned, the relationships can be reformulated [40].
Letting the MIMO system dimension be MT×MR, the general signal model for covariance matrix R are positive semi-definite Hermitian matrices. t Hiid is the
R T
M ×M identically independent distributed complex Gaussian matrix, in which all
elements are mutually independent Rayleigh fading waveforms, thus MR×MT mutually independent Rayleigh faders are needed. The coefficients ρ in R and r
R matrices could be derived as follows [41]: t
where d is the position vector of the Tx/Rx antenna array, k is the directional vector with main signal propagation, D is the normalized distance between adjacent antennas of the Tx/Rx antenna array, and P( )φ is the power azimuth spectrum (PAS) for each antenna in Tx/Rx.
Using the notations of [42], with /d λ standing for the normalized distance between elements, where d and λ are the element spacing and the wavelength, respectively. The normalized distance between adjacent antenna of the Tx/Rx antenna array D =(2πd)/λ like Figure 4-4, one can easily derives the cross-correlation function between the real and imaginary parts of the complex baseband signals received at two omni-directional antennas separated by the distance d .
The evolution of the correlation coefficients as a function of the distance between the antenna elements mostly depends on the PAS and on the radiation pattern of the antenna elements [43]. The PAS distribution can be divided into three types of different distributions: Uniform, Truncated Gaussian [44], and Truncated Laplacian [45]. Furthermore, Truncated Laplacian distribution is proposed as the best fit to measurement results in urban and rural areas. For the channel environment in this thesis, Truncated Laplacian distribution is the most suitable.
A. Uniform PAS
Consider the multi-cluster uniform PAS modeled as
( ) ,
{ (
0,) (
0,) }
where ε φ denotes the step function and ( ) N is the number of clusters. c
The first step is to normalize the PAS such that it can be thought as a probability distribution. The constants QU k, are derived such that PASU ( )φ fulfills the requirements of a probability distribution function:
( ) 0,
where Δ is the half-domain definition of the PAS (domain is assumed symmetric). φ Equation 4-9 leads to ,
The cross-correlation function between the real parts, which is as the same as the one derived for the imaginary parts, which is written as follows:
( ) ( )
XX YY
R D =R D (4-11)
The autocorrelation function of the real-part is
( ) ( ) ( )
The cross-correlation function between real and imaginary part is ( ) coefficients are defined as
2 2
( ) ( ) ( ) ( ) .
e D f D RXX D jRXY D
ρ ρ + (4-14)
B. Truncated Gaussian PAS
Truncated Gaussian PAS model is defined as:
{ }
The normalization constant QU k, are derived such that
, .
Using above definition and normalization condition, the cross-correlation functions are easily derived, which can be given by
( ) 2 2
C. Truncated Laplacian PAS
Truncated Laplacian PAS model is defined as ( )
The normalization condition is given by
, , .
As well as the cross-correlations is given by
( ) ( ) ( )
Figure 4-3: Propagation channel model for 2 X 2 MIMO system
Figure 4-4: Impinging waves from T to X R X
4.4 Computer Simulations
For the discussion of wideband fading channel for the SISO system, the fader follows the parameters setting of Section 3.4.3. We assume the number of faders
6
Figure 4-5 shows the autocorrelation of a wideband system for SISO. We can see it can approach the desired statistical value. However, the accuracy of wideband fading channel is not as good as that of narrowband fading channel, because the random variable for each fader will not be completely independent in the simulation.
Thus, different faders are not absolutely independent.
For the discussion of MIMO system, we set the transmit antenna spacing
T 4
λ = λ and the receive antenna spacing λR =0.5λ, where λ is the wavelength.
The local scatters uniformly surround receive antennas at BS. For 2× MIMO 2
channel, there are four main links (L = ) and each link is composed of six faders 4 (M = ), so there are twenty-four independent faders in the channel. 6
First to discuss the statistical characteristics of each link of 2× MIMO 2 channel like Figure 4-6, the autocorrelation functions of different links shown in Figure 4-7 are similar. We can support different channel links for MIMO system is almost equivalent. The channel state for different links to transmit signal is fair.
Then, the statistical characteristics between all links of 2× MIMO channel 2 are discussed. We bring up 6 different cases listed in Table 4-1. The cross-correlation functions of two channel links of distinctive cases are compared to verify the channel environment of transmitting signal. For Case A, it describes the cross-correlation function of h and 11 h . Figure 4-8 shows the cross-correlation functions of Case A, 12 Case B, Case C, and Case D. We compare the cross-correlation functions of distinctive cases. Case A and Case B have similar channel environments for transmitting signals. From the standpoint of received signal, the transmitted signal via
h and 11 h is identical with one via 12 h and 21 h . Whereas the transmitting 22 antenna is apart from receive antennas, the receive antennas spacing is small enough for thinking receiving terminal as independent element. Therefore, Case A and Case B have similar channel environment. In the same way, the ratiocination of Case C and Case D can be obtained as above. Case C and Case D also have similar channel environment.
In terms of Case E and Case F, Figure 4-9 shows the cross-correlation functions of these two cases, and compares their statistical properties. Case E and Case F have similar statistical properties. Each receive antenna can be considered independent if the separation between transmitter and receiver are far enough. The transmitted signals via h and 11 h is identical with ones via 22 h and 12 h . When the receive 21 antenna spacing is smaller than the coherence distance 0.5λ, the signals received by
different antennas can be thought of experiencing the same fading. If the receiving antenna spacing is larger than the coherence distance, the received signals of different antennas experience different fading.
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-0.5 0 0.5 1
Normalized Time Delay
Normalized Autocorrelation
Wideband SISO Autocorrelation
Proposed Modified KL Expansion Model Theoretical value
Figure 4-5: Autocorrelation of wideband system for SISO (M = ) 6
Figure 4-6: Links of 2× MIMO system 2
Table 4-1: Cross-correlation between different two links of 2× MIMO system 2
CCaassee AA Case B
Case C Case D
Case E Case F
0 5 10 15 -0.5
0 0.5 1
Normalized Time Delay
Crosscorrelation
Link Autocorreltion
h11 h12 h21 h22
Figure 4-7: Autocorrelation for each link of 2× MIMO system 2
-15 -10 -5 0 5 10 15
Figure 4-8: Cross-correlation functions of Case A, Case B, Case C, and Case D.
-15 -10 -5 0 5 10 15
Figure 4-9: Cross-correlation functions of Case E and Case F.
4.5 Summary
In this chapter, we use the proposed Rayleigh fader using the concept of K-L expansion method to expand to MIMO system. Observing the computer simulations, the statistical properties of the channel links conform with realistic channel environment. Unlike traditional method using sum-of-sinusoids, the advantages for using K-L expansion model are the flexible assignment of maximum Doppler frequency and less computational complexity.
Each fader can be assigned a different maximum Doppler frequency to construct the fading channel for a wideband system. However, Jakes model can just assign single maximum Doppler frequency. Therefore, our proposed Rayleigh fading channel simulator based on K-L expansion method can apply in variable velocity environment.
In terms of computational complexity, the analysis of computational loading has been discussed in Chapter 3. Because of less computational complexity of a single Rayleigh fader, the extended systems such as wideband and MIMO require lower computational complexity.
As above mentioned, the advantages of using the proposed fading channel based on K-L expansion method make real-time MIMO channel modeling simple to implement.
Chapter 5
Conclusion
In this thesis, we have propose two Rayleigh fading simulators based on Jakes model and K-L expansion model, respectively. According to the simulation results, the K-L expansion model is relatively more accurate and faster for simulating Rayleigh faders. We can assign adjustable maximum Doppler frequency to each fader for constructing frequency-selective fading. The proposed fading simulators are more flexible than traditional methods, thus facilitating efficient realization of real time MIMO channel modeling will also be more efficient.
In Chapter 2, the propagation channel models for mobile radio are introduced.
Large and small scale fading is described, and small scale fading becomes the focus for later development. The theme of the thesis is then made clear to be on Rayleigh fading channel simulator development under the condition of WSSUS channels. In Chapter 3, we propose two Rayleigh fading simulators. First, the proposed Rayleigh fader based on sum-of-sinusoids has the same complexity with Wu model, but it can improve the statistical properties to be closer to the desired ones. Afterwards, the proposed fading simulator based on K-L expansion method is shown to have lower computational complexity. The proposed fading simulator based on K-L expansion method is more accurate and faster than the traditional method using sum-of-sinusoids.
In Chapter 4, we extend the proposed Rayleigh fader based on the K-L expansion method to extend to MIMO channels.
To be more specific, the first proposed fading channel simulator is based on the modified Jakes model, which can improve the accuracy of Wu model with essentially the same computational complexity. The second simulator is based on the modified K-L expansion model, which has better accuracy and requires lower computational complexity than conventional ones. Further reduction in computational loading is achieved by substituting the basis functions with Chebyshev polynomials. In addition, the K-L expansion method is more flexible than traditional faders using Jakes model method, because a different maximum Doppler frequency can be assigned to each fader individually. Owing to these advantages, we extend proposed K-L expansion based simulator to MIMO channels. According to simulation results, the developed MIMO simulator meets well the requirement of realistic channel environments.
There are some future works worthy of further research. First, when realizing the fading simulator, fractional symbol rate is usually required for the tap delay. However, fractional tap delay is hard to implement in practice. Second, how to implement a great deal of uncorrelated random sequences efficiently is an interesting issue. Finally,
There are some future works worthy of further research. First, when realizing the fading simulator, fractional symbol rate is usually required for the tap delay. However, fractional tap delay is hard to implement in practice. Second, how to implement a great deal of uncorrelated random sequences efficiently is an interesting issue. Finally,