Vehicular Technologies Increasing Connectivity Part 11 pptx

Moreover,for high-order quadrature amplitude modulation QAM signal constellations especially forcross constellations such as 128-QAM, in which the corner points containing significant pha

Correlation Coefficients of Received Signal I and Q Components 291

(Type 1: exponential distribution , K = -∞ dB)

correlation for K= -∞ dB is slightly lower than the theoretical one We suppose that the difference is due to different conditions between the theoretical and simulation models for

the delay profile, such as the number of waves N and the counted effective amplitude in h i

Therefore, we repeated the simulation, changing from N=10 and effective amplitude of -20

dB to N>100 and effective amplitude of less than -50 dB As a result, the correlation became

close to the theoretical one The above results show that the autocorrelation is independent

of the K factor, while the frequency correlation depends on the K factor The correlation

coefficient in a domain with time and frequency axes for NLOS is shown in Fig 7 Figure 7(a) is a bird’s-eye view of ρ(Δx) obtained from (13), which makes it easy to comprehend the

overall ρ(Δx) We can see that ρ(Δx) has a peak ρ(0)=1 at the origin Δx = 0 and that ρ(Δx)

becomes smaller with increasing Δx or as f’ m T s and Δfσ become larger Furthermore, ρ(Δx)

decreases in a fluctuating manner on the time axis, but decreases monotonically on the frequency axis The high area of ρ(Δx) needed to allocate the pilot signal in coherent

detection, such as ρ(Δx) >0.8, is very small in the domain, whereas the low area needed to

design the diversity antenna, such as ρ(Δx) <0.5, is spread out widely Figure 7(b) also shows

a bird’s-eye view of the simulated correlation; Figs 7(a) and (b) both exhibit almost the same features Figure 7(c) shows the loci of the theoretical and simulated correlations in a small area up to 0.3 on both axes, with contour lines of ρ(Δx) It is easy to compare them The

theoretical value on the time axis agrees well with the simulated one, but the theoretical value on the frequency axis is slightly higher than the simulated one The reason for this is a different model for the delay profile, as described earlier We can see from Fig 7(c) that the theoretical value agrees roughly with the simulated one Furthermore, we note that in Fig 7(c), the locus of the correlation coefficient ρ(Δx) seems to be an ellipse with its major axis on

the time axis and its origin Δx=0 in the domain

ii) Type 2: Delay profile with random distribution

Figure 8 shows the correlation coefficient for a delay profile with random distributions of

K=-∞ and 5 dB The values plotted by symbols of × and ▲ in Figs 8(a) and (b) were

simulated in a similar way to Fig 6, and the fine and broken lines show the theoretical value of ρ(Δx) obtained from (16) by putting Δf =0 for autocorrelation or ρ(Δx)=J0(2πf’ m T s)

and by putting f’ m T s=0 for frequency correlation or

in Fig 8(a), the simulated autocorrelations for both K=-∞ and 5 [dB] have almost the same

features and agree well with the theoretical values, so the correlation seems to be

independent of the K factor On the other hand, the simulated frequency correlations for K=-∞ and 5 dB in Fig 8(b) have different features that also depend on the K factor The

simulated and theoretical values agree well

The correlation coefficient in a domain with time and frequency axes for NLOS is shown

in Fig 9 by a similar method to that for Fig 7 Figure 9(a) is a bird’s-eye view of the theoretical value of ρ(Δx) obtained from (16) We can also see that ρ(Δx) has a peak ρ(0)=1

at the origin Δx = 0, and ρ(Δx) becomes smaller with increasing Δx However, in this case,

ρ(Δx) decreases in a fluctuating manner not only on the time axis, but also on the

frequency axis The high area of ρ(Δx) in the domain, such as ρ(Δx) >0.8, is larger than that

for an exponential distribution; the low area, such as ρ(Δx) <0.5, is spread out widely

Correlation Coefficients of Received Signal I and Q Components 293 Figure 9(b) is the simulated correlation, which exhibits similar features to the theoretical one in Fig 9(a) Figure 9(c) shows the loci of the theoretical and simulated correlations The theoretical value in the domain agrees well with the simulated ones For a delay profile with random distribution, the locus of correlation coefficient ρ(Δx) is also an ellipse

(Type 2: random distribution , K = -∞ dB)

Correlation Coefficients of Received Signal I and Q Components 295

5 Conclusion

An analysis model having a domain with time, frequency, and space axes was prepared to study the correlation coefficients of the I and Q components in a mobile channel, which are needed in order to allocate a pilot signal with M-ary QAM detection, such as in an OFDM system, and to compose antennas in the MIMO technique For a multipath environment, the general correlation coefficient formula was derived on the basis of a delay profile with and without a directive wave, and as examples, the formulas for delay profiles with exponential and random distributions were then derived The formulas exhibit some interesting

features: the autocorrelation on the time axis is independent of the K factor and is expressed

by J0(2πf’ m T s ), but the frequency correlation depends on the K factor and delay profile type

The locus of a fixed value correlation is an ellipse in the domain with time and frequency axes The correlations were also shown in the domain using bird’s-eye views for easy comprehension Furthermore, computer simulation was performed to verify the derived formulas and the theoretical and simulated values agree well Therefore, it is possible to estimate logically the pilot signal allocation of M-ary QAM in OFDM and antenna construction for MIMO in the domain

6 Appendix

6.1 Appendix A derivation of (8)

Under the conditions of the propagation model in 2-A and assuming that N is a large number, <I(0)I(Δx)> is analyzed as follows It is separated into terms of the same ith and other ith arriving waves, as shown in (17)

Furthermore, the second term in (17) vanishes because the values of τi , ξi, and φi for the ith

wave and τj , ξj, and φj for the jth wave are independent of each other and are also random

values, and the sum of the products of cosθi and cosθj then becomes zero With this in mind and considering the small values of Δfτi and f m ’T s , the first term of (17) is modified as (18) to (20) by using, for example, a transforming trigonometric function, τi , ξi, and φi with random values, and an odd function of sine In this procedure, the second terms in (18) and (19) also

vanish, so we finally get (20) as the result for <I(0)I(Δx)>

In the first term A1 in (11), we change A1 to (21) because the directive wave’s amplitude h0 is

usually much larger than that of the nondirective one and because τ0=0 Moreover, the

amplitude h i of the ith arriving wave depends on excess delay time τi according to (10) So

by substituting (10) for h i, we can rewrite (21) as (22)

We try to calculate by replacing the ensemble average of the second term in (22) by

integration with respect to τi assuming a large N As a result, we get (23) assuming τmin is

close to 0 and τmax is large

h h

1

Correlation Coefficients of Received Signal I and Q Components 297

6.3 Appendix C derivation of (15)

We calculate the ensemble average of A2 in (14) in a similar manner to that for (23) by

integrating with the provability density function 1/τmax of τ Considering the independence

of h i and τi and the power of the directive and nondirective waves, or k and 1, we get (26)

[1] ITU Circular Letter 5/LCCE/2, Radiocommunication Bureau, 7 March 2008

[2] Richard van Nee and Ramjee Prasad, OFDM FOR WIRELESS MULTIMEDIA

COMMUNICATIONS, Artech House, 1999

[3] T Hwang, C Yang, G Wu, S Li, and G Y Li, OFDM and Its Wireless Applications: A

Survey, IEEE Trans Veh Technol., Vol 58, No 4, pp 1673-1694, May 2009

[4] G J Foschini and M J Gans, On limits of wireless communications in fading

environments when using multiple antennas, Wireless Personal Commun Vol 6, pp

311-335, 1998

[5] D Shiu, G Foschini, M J Gans, and J Kahn, Fading Correlation and Its Effect on the

Capacity of Multielement Antenna Systems, IEEE Trans Commun., Vol 48, No 3,

pp 502-513, March 2000

[6] Andreas F Molisch, Martin Steinbauer, Martin Toeltsch, Ernst Bonek, and Reiner S

Thoma, Capacity of MIMO Systems Based on Measured Wireless Channel, IEEE

JSAC, Vol 20, No 3, pp 561-569, April 2002

[7] H Nishimoto, Y Ogawa, T Nishimura, and T Ohgana, Measurement-Based

Performance Evaluation of MIMO Spatial Multiplexing in Multipath-Rich Indoor

Environment, IEEE Trans Antennas and Propag., Vol 55, No 12, pp 3677-3689, Dec

2007

[8] Seiichi Sampei and Terumi Sunaga, Rayleigh Fading Compensation for QAM in Land

Mobile Radio Communications, IEEE Trans Veh Technol., Vol 42, No 2, pp

[11] S Kozono, T Tsuruhara, and M Sakamoto, Base Station Polarization Diversity

Reception for Mobile Radio, IEEE Trans Veh Technol., Vol VT33, No 4, pp

301-306, Nov 1984

[12] H Nakabayashi and S Kozono, Theoretical Analysis of Frequency-Correlation

Coefficient for Received Signal Level in Mobile Communications, IEEE Trans Veh Technol., Vol 51, No 4, pp 729-737, July 2002

[13] S Kozono, K Ookubo, and K Yoshida, Study of Correlation Coefficients of Complex

Envelope and Phase in a Domain with Time and Frequency Axes in Narrowband

Multipath Channel, in 69 th IEEE Veh Technol Conf., Barcelona, Spain, April 2009

Jenq-Tay Yuan1and Tzu-Chao Lin2

Graduate Institute of Applied Science and Engineering, Fu Jen Catholic University

Taipei 24205, Taiwan, R.O.C.

1 Introduction

Adaptive channel equalization without a training sequence is known as blind equalization

[1]-[11] Consider a complex baseband model with a channel impulse response of c n The

channel input, additive white Gaussian noise, and equalizer input are denoted by s n , w n

and u n , respectively, as shown in Fig 1 The transmitted data symbols, s n, are assumed toconsist of stationary independently and identically distributed (i.i.d.) complex non-Gaussianrandom variables The channel is possibly a non-minimum phase linear time-invariant filter

The equalizer input, u n = s n ∗ c n+w n is then sent to a tap-delay-line blind equalizer,

f n, intended to equalize the distortion caused by inter-symbol interference (ISI) without atraining signal, where∗denotes the convolution operation The output of the blind equalizer,

y n = f n  ∗ u n = s n ∗ h n+f n  ∗ w n, can be used to recover the transmitted data symbols,

s n, where denotes complex conjugation and h n = f n  ∗ c n denotes the impulse response

of the combined channel-equalizer system whose parameter vector can be written as the

time-varying vector hn= [hn(1), h n(2), ]T with M arbitrarily located non-zero components

at a particular instant, n, during the blind equalization process, where M = 1, 2, 3,

For example, if M = 3 and I M = {1, 2, 5} is any M-element subset of the integers, then

hn= [hn(1), h n(2), 0, 0, h n(5), 0 0]Tis a representative value of hn

Fig 1 A complex basedband-equivalent model

The constant modulus algorithm (CMA) is one of the most widely used blind equalizationalgorithms [1]-[3] CMA is known to be phase-independent and one way to deal with its phaseambiguity is through the use of a carrier phase rotator to produce the correct constellationorientation, which increases the complexity of the implementation of the receiver Moreover,for high-order quadrature amplitude modulation (QAM) signal constellations (especially forcross constellations such as 128-QAM, in which the corner points containing significant phaseinformation are not available), both the large adaptation noise and the increased sensitivity

Multimodulus Blind Equalization Algorithm

Using Oblong QAM Constellations for

Fast Carrier Phase Recovery

17

to phase jitter may make the phase rotator spin due to the crowded signal constellations[10]-[13] Wesolowski [4], [5] Oh and Chin [6], and Yang, Werner and Dumont [7], proposedthe multimodulus algorithm (MMA), whose cost function is given by

R,n − R 2,R]2} + E{[y2

where y R,n and y I,nare the real and imaginary parts of the equalizer output, respectively,

while R 2,R and R 2,I are given by R 2,R =E[s4

R,n]/E[s2

R,n]and R 2,I = E[s4

I,n]/E[s2

I,n], in which

s R,n and s I,n denote the real and imaginary parts of s n, respectively Decomposing the cost

function of the MMA into real and imaginary parts thus allows both the modulus and the

phase of the equalizer output to be considered; therefore, joint blind equalization and carrier phase recovery may be simultaneously accomplished, eliminating the need for an adaptive

phase rotator to perform separate constellation phase recovery in steady-state operation The

tap-weight vector of the MMA, fn, is updated according to the stochastic gradient descent

(SGD) to obtain the blind equalizer output y n=fn Hun

I,n − R 2,I]and L=2l+1 is the tap length of the equalizer

The analysis in [9], which concerns only the square constellations, indicates that the MMA canremove inter-symbol interference (ISI) and simultaneously correct the phase error However,when the transmitted symbols are drawn from a QAM constellation having an odd number

of bits per symbol (N =22i+1 , i= 2, 3, ), the N-points constellations can be arranged into

an oblong constellation [14], [15] so long as E[s2

to be 1.759 instead of 2 as in the conventional cross 128-QAM, so that the average energiesrequired by both cross and oblong constellations are almost identical In this chapter, theoblong constellation illustrated in Fig 2 is used as an example to demonstrate that the MMAusing asymmetric oblong QAM constellations with an odd number of bits per symbol maysignificantly outperform its cross counterpart in the recovery of the carrier phase introduced

by channels, without requiring additional average transmitted power We use the term

asymmetric because the oblong QAM is not quadrantally symmetric, i.e., E[s2R,n ] = E[s2I,n], and

as a consequence E[s2n ] =0 Although reducing the distance between adjacent message points

in the proposed oblong constellation in Fig 2 may increase the steady-state symbol-errorrate (SER) or mean-squared error (MSE) of the adaptive equalizer, this chapter is concernedwith the unique feature of fast carrier phase recovery associated with the MMA using oblongconstellations during blind equalization process owing to its non-identical nature of the realand imaginary parts of the source statistics

2 Analysis of MMA using oblong constellations

This section presents an analysis of the MMA using oblong QAM constellations from theperspective of its stationary points Our analytical results demonstrate that the four saddle

Fig 2 Oblong constellations for 128-QAM sources.

points existing in the square and cross constellations alongθ(k) = π/4, 3π/4, 5π/4, 7π/4

are absent when using oblong constellations Consequently, the frequency of beingattracted towards the vicinity of the saddle points, around which the MMA exhibits slowconvergence, before converging to the desired minimum, is significantly reduced when usingoblong constellations The use of oblong constellations may thus accelerate the magnitudeequalization process during the transient operation

2.1 MMA Cost function of oblong constellations

After some algebraic manipulation, the expansion of the MMA cost function in (1) for

a complex i.i.d zero-mean QAM source (for square, cross, and oblong constellations) with

each member of the symbol alphabet being equiprobable, and a complex baseband channel

excluding additive channel noise, can be written in the combined channel-equalizer space hn

I,n] and k s = E[|s n |4]/σ4

s is the source kurtosis Thefirst term of (3), (1/4) · {E[s4] ·∑i h4(i)}, which is related to the fourth-order statistics (or

301

Multimodulus Blind Equalization Algorithm

Using Oblong QAM Constellations for Fast Carrier Phase Recovery

Fig 3 cross constellations for 128-QAM sources.

fourth-power phase estimator) in [12], [13], [16]-[18] containing the phase information and is

absent from the CMA cost function, allows the MMA to recover a possible phase rotation

of the equalizer output Note that when there is no possibility of confusion, the notation is

simplified by suppressing the time index n, so for example, s ns=s R+js I and h n(k)h(k)

Terms s R and s Iare assumed to be uncorrelated, and both are zero-mean, sub-Gaussian (such

that E [|s|4] − 2E2[|s|2] − |E[s2]|2 < 0) The values of the source statistics of, for example,the oblong (8×16)-QAM can be computed to be E[s2

R] = 65.08 and E[s2

I] = 16.08 Forconvenience of mathematical analysis, (3) can also be expressed in the following polar space

I ] =0 This asymmetric nature makes the major difference between

an oblong constellation and a square (or cross) constellation, since the shape of their resultingMMA cost surfaces would be significantly different, as is revealed later in this chapter

2.2 Stationary points of MMA using oblong constellations

The equalizer is assumed to be either doubly-infinite in length as in [8] or of finite-lengthfractionally spaced as in [3] under the equalizability conditions The set of stationary points of

the MMA for oblong constellations considering M ≥1 can be obtained by setting the gradient

of J MMAin (4) to zero, such that∇J MMA =r∂JMMA

∂r(k) +rΘ(k) ∂JMMA ∂θ(k) =0 The components of thegradients are

is in contrast to the square (or cross) constellations, for which R 2,R =R 2,I and E[s2R] = E[s2I]

such that the last two terms of (6) are both zero Therefore, only sin 4θ(k) =0 is required and,consequently,

4 ,(n=0, , 7) (8)The four stationary points given in (7) are now located Substitutingθ(k) ∈ {0,π}andθ(l) ∈ {0,π}(such that cos 2θ(k) =cos 2θ(l) =1) into (5) yields

Clearly, (9) and (10) give r2+(1) =r2+(2) = =r2+(M)and r2−(1) =r2−(2) = =r2−(M),

respectively Consequently, (9) and (10) given k=M suffice to determine r2+(M)and r2−(M):

Multimodulus Blind Equalization Algorithm

Using Oblong QAM Constellations for Fast Carrier Phase Recovery

The following form for the four stationary points alongθ(k) =0,π/2, π, 3π/2 for each of the

be viewed as the stationary points in terms of the overall vector hnin the MMA cost in (3)

under the common h R(k)and h I(k)space denoting the real and imaginary parts of h(k)

2.3 Two special cases: Square and cross constellations

For the special cases of both square and cross constellations, i.e., R 2,R = R 2,I , E[s2

It has been shown in [9] that the following form for all possible stationary points of the MMA

(for square and cross constellations), except for r(k) =0, can be expressed as

where r2±(M) = E[s4R]/(E[s4R] + [3(M−1)] · E2[s2R])and r2×(M) = 2E[s4R]/(E[s4R] + [3(2M −

1)] · E2[s2R]) Figure 4 depicts the MMA cost surface for a cross 128− QAM input for M=1

Notably, both r2+(M) in (11) and r2−(M) in (12) reduce to r2±(M) when E[s2R] = E[s2I]and

E[s4

R] =E[s4

I]

If the distribution of s n is sub-Gaussian, then all the pre-specified hn (with the associated

I M ) with the stationary points shown in (13), for M ≥ 2, can be shown to be unstableequilibria (saddle points) by applying the concept proposed by Foschini [8] Consequently,

all the vectors hn , M=2, 3, , are saddle points The locations of the four stationary points

at[±r+(M), 0]and[0,±r −(M)]for hnwith the(8×16) − QAM source for different M can be

computed by using (11) and (12) For example, r+(10) =0.249, r −(10) =0.17; r+(5) =0.36,

r −(5) =0.245; r+(2) =0.611, r −(2) =0.416; r+(1) =1, r −(1) =0.68 Clearly, the four saddle

points are located nearer the origin as the number of non-zero components of hn rises As M

decreases during the blind equalization process, the locations of the four saddle points for hn

move dynamically away from the origin in four mutually perpendicular directions The four

saddle points eventually converge as M →1 to[±1, 0]and[0,±r −(1)], where the former twostationary points become the only two global minima, and the latter two are still two saddlepoints

When compared with (8) and (15), the result in (7) and (13) is significant, since it implies

that, for M ≥ 2 (during the transient (or startup) mode operation), the number of saddlepoints is only half those of the square and cross constellations when the oblong constellations

Fig 4 MMA cost surface for 128-cross input as a function of h R(k)and h I(k)for M=1are adopted (i.e., the four saddle points existing in square and cross constellations along

the vicinity of the saddle points, around which it exhibits slow convergence, before converging

to the desired minimum, is significantly diminished when using oblong constellations

Accordingly, using oblong constellations may accelerate the magnitude equalization (or residual

ISI removing) process during the transient mode operation

Computer simulations performed in this study demonstrate that once blind equalization

started, a non-zero component of hn with maximum magnitude square, r2(k), rose, and the

sum of the magnitude squared of the remaining M −1 nonzero components, ∑l=k r2(l),

fell rapidly, eventually diminishing to zero The non-zero component of hn with maximum

magnitude square is called " h(k)with maximum modulus" in the remainder of this chapter This

chapter focuses on the MMA cost in terms of h(k)with maximum modulus, which indicatesthe performance of the MMA during the transient mode operation (M ≥ 2) To discover

how h(k) with maximum modulus evolves during the transient operation when the MMA

adopts the oblong QAM based on the SGD, the MMA cost can be considered in terms of h(k)

with maximum modulus alone by substituting the approximations∑k∑l=k r2(k)r2(l) ∼= 0,

305

Multimodulus Blind Equalization Algorithm

Using Oblong QAM Constellations for Fast Carrier Phase Recovery