International Journal of Electronics, Communication & Instrumentation Engineering Research and Development (IJECIERD) ISSN 2249-684X Vol.2, Issue 3 Sep 2012 1-11 Š TJPRC Pvt. Ltd.,
A NOVEL EQUALIZER STRUCTURE FOR TWO-DIMENSIONAL MODULATIONS WITH COHERENT DEMODULATION CHING-HAUR CHANG Department of Photonics and Communication Engineering, Asia University, Taiwan
ABSTRACT A novel equalizer structure capable of correcting channel distortion for two-dimensional modulations with coherent demodulation was presented.
Based on the fact that the phase of the
in-phase and quadrature sinusoids in the receiver must be synchronized to the phase shift introduced by the communication channel, a structure using a pair of real-valued equalizers, one for the in-phase and the other for the quadrature channels, was developed in this paper.
Conventional approaches in this
regard use structures with complex-valued arithmetic and thus call for a much higher computational complexity.
Due to the fact that only real-valued arithmetic is needed for the new structure, a saving of
50% multiplications, compared to its complex-valued counterpart, is thus achieved even with a better symbol error rate performance.
A more-with-less approach is thus obtained by using the new approach.
Computer simulation results justified the assertions made in this paper.
KEYWORDS: Coherent Demodulation; Non-Coherent Demodulation; Equalizer; Symbol Error Rate. INTRODUCTION Although researches on adaptive equalization of inter-symbol interference channels have been lasting for decades, there is still room in making the structure and computational complexity as simple as possible [1, 2].
This paper proposes an equalizer structure with minimum computational complexity
possible for two-dimensional modulations with coherent demodulation (2DM-CD). Being equalizers for digital modulations with two-dimensional signals, structures with complex-valued arithmetic are almost deemed as a must in the past [3-12].
In fact, a simple approach
can be obtained by taking advantage of the specific phase property in 2DM-CD.
For 2DM-CD, the
phase of the in-phase and quadrature carriers must be synchronized to the carrier phase of the signal received.
Based on this fact, it is proved in this case that the channel consists of only real-valued
parameters and, hence, a pair of real-valued equalizers (RVEs), one for the in-phase and the other for the quadrature channels, is adequate for correcting the distortion encountered.
Basic principle on this
subject is derived and computer simulations are conducted to justify the assertions made in this paper. Computer simulation results show that the use of RVEs has even better symbol error rate (SER)
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Ching-Haur Chang
performance than that using a complex-valued one.
It is advantageous due to a saving of 50%
multiplications is achieved by using the new structure.
An equalizer with benefit of more-with-less is
thus obtained.
METHOD Channel Models The reason why the use of a complex-valued equalizer (CVE) can be avoided for 2DM-CD can be seen from the results of this section. transmitter.
Figure 1 shows the block diagram of a two-dimensional signal
The transmitted signal can be written as
s (t ) = a I ( t )
2 2 cos(2πf c t + ϕ ) − aQ (t ) sin(2πf c t + ϕ ) T T
(1)
where a I (t ) and − aQ (t ) are respectively the in-phase and quadrature components to be transmitted, representing by pulses of duration T , f c is the carrier frequency, and ϕ is the initial phase of the carrier.
Denote the frequency response of the communication channel as H ( f ) = H ( f ) e jθ ( f ) .
Only linear distortion is assumed in the following derivations without loss of generality.
Traditionally,
CVEs are used to overcome the inter-symbol interference (ISI) introduced by the communication channel [13].
Figure 2 shows the structure of a two-dimensional signal receiver with a CVE inserted for
reducing the ISI.
The two sinusoidal signals used to demodulate the band-pass signal are assumed to
have the same phase as that of in the transmitter; that is,
channel and
2 / T cos(2πf c t + ϕ ) for the in-phase
2 / T sin(2πf c t + ϕ ) for the quadrature channel. A pair of low-pass filters (LPFs) is
used to produce the base-band components called x I (t ) and xQ (t ) as shown in Figure 2.
In
response to (1) the two base-band components can be found as
x I (t ) = and
a (t ) a I (t ) H ( f c ) cos[θ ( f c )] − Q H ( f c ) sin[θ ( f c )] T T
(2)
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A Novel Equalizer Structure for Two-Dimensional Modulations with Coherent Demodulation
xQ ( t ) = −
a (t ) a I (t ) H ( f c ) sin[θ ( f c )] − Q H ( f c ) cos[θ ( f c )] T T
where H ( f c ) is the magnitude and
(3)
θ ( f c ) is the phase, both at carrier frequency. It can be seen
from (2) and (3) that x I (t ) and xQ (t ) become interrelated if
θ ≠ 0 . This is indeed the case for
two-dimensional modulations with non-coherent demodulation (2DM-NCD). cause the signal constellation to rotate. to 20o .
Figure 3 depicts the phase rotation for
Non-zero values of
θ
θ changing from 0o
Here (a I (t ) H ( f c ) / T , aQ (t ) H ( f c ) / T ) = ( ±1, ± 1) is assumed for simplicity.
For 2DM-CD, the phase of the in-phase and quadrature sinusoids is synchronized to the phase shift caused by the communication channel through, for example, some means of phase-locked loops. is tantamount to having
This
θ = 0 in (2) and (3). In this case, x I (t ) and xQ (t ) become
x I (t ) =
a I (t ) H ( fc ) T
(4)
and
xQ (t ) = −
aQ (t ) T
H ( fc )
(5)
It is apparent that the two components become independent in this case.
In contrast to the case for
θ ≠ 0 , only amplitude distortion needs to be solved for 2DM-CD. This concludes that real-valued parameters are sufficient to model the communication channels for 2DM-CD, while complex-valued ones are required for those in 2DM-NCD. New Equalizer Structure Define x(n ) = x I (n ) + jxQ (n ) as the input to the equalizer and y (n ) = y I (n ) + jyQ (n) as the output from the equalizer, where x I ( n ) and xQ (n ) denote, respectively, the sampled versions of
x I (t ) and xQ (t ) .
Let a I (n ) and aQ (n ) be the sampled versions of a I (t ) and aQ (t ) ,
respectively, and w( n ) be the impulse response of the equalizer. is assumed for simplicity.
In the following derivations, T = 1
In general, by using the results in (2) and (3), the real and imaginary parts of
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Ching-Haur Chang
the equalizer output can be found as
y I (n) = [a I (n) H ( f c ) ∗ w(n)]cos[θ ( f c )] − [aQ (n) H ( f c ) ∗ w(n)]sin[θ ( f c )]
(6)
and
yQ (n) = −[a I (n) H ( f c ) ∗ w(n)] sin[θ ( f c )] − [aQ (n) H ( f c ) ∗ w(n)] cos[θ ( f c )] where ∗ denotes a convolution sum.
It is apparent from (6) and (7) that CVEs are generally required
to correct simultaneously the phase and amplitude distortions. i.e., for
θ ≠0.
This is really the case for 2DM-NCD;
Let the signal to be transmitted as a ( n ) = a I ( n ) + jaQ (n ) .
Also, denote
aˆ I (n − ∆) and aˆQ ( n − ∆ ) , respectively, as the estimates of a I (n ) and aQ ( n ) with delay ∆ . To equalize the phase and amplitude distortions simultaneously, a CVE with input signal of
x ( n ) = x I ( n ) + jxQ ( n ) and target signal of t (n ) = a I (n − ∆ ) + jaQ (n − ∆ ) is required. For 2DM-CD, y I (n) and yQ (n) can be obtained by substituting
directly by using the results in (4) and (5).
θ = 0 in (6) and (7) or
This reduces to
y I (n) = aI (n) H ( f c ) ∗ w(n)
(8)
yQ (n) = −aQ (n) H ( f c ) ∗ w(n)
(9)
and
Note that y I (n ) and yQ (n) rely, respectively, only on a I (n) H ( f c ) and aQ (n) H ( f c ) . phase distortion has no effect on the two components.
The
This turns out that by using a pair of RVEs, one
for the in-phase channel and the other for the quadrature channel, is thus adequate for correcting the channel distortion in this regard.
Having this, a great saving in computational complexity can be
achieved by using the new structure.
Figure 4 shows the resulting two-dimensional signal receiver
inserted with a pair of RVEs, one with input x I ( n ) and the other with input xQ ( n ) , targeting
A Novel Equalizer Structure for Two-Dimensional Modulations with Coherent Demodulation
5
respectively toward a I ( n − ∆) and aQ ( n − ∆ ) . Any linear or nonlinear structures can be used as the equalizers required in Figures 2 and 4.
For
the purpose of comparison, only linear transversal equalizer with LMS-type algorithms [14] is adopted for justifying the assertion made above.
Generally, for the CVE, the jth tap-weight, w j , is updated
according to the rule given by
wi ( n + 1) = wi ( n ) + µe( n ) x ∗ ( n ), i = 0 , 1, ..., N − 1
(10)
∗
where x (n ) is the complex conjugate of x ( n ) , µ is the learning rate, N is the number of
tap-weights, and
e( n ) = [ a I ( n − ∆ ) − y I ( n )] + j[a Q ( n − ∆ ) − y Q ( n )]
(11)
For the RVEs, the jth tap-weights for the in-phase channel, wI , j , and quadrature channel, wQ , j , are
updated respectively as
w I ,i ( n + 1) = w I ,i ( n ) + µe I ( n ) x I ( n ), i = 0 , 1, ..., N − 1
(12)
wQ ,i ( n + 1) = wQ ,i ( n ) + µeQ ( n ) xQ ( n ), i = 0 , 1, ..., N − 1
(13)
and
where
eI ( n ) = a I ( n − ∆ ) − y I ( n )
(14)
and
eQ ( n) = aQ (n − ∆ ) − yQ ( n)
(15)
Table 1 gives a comparison for the computations required for a pair of RVEs and a CVE. Although the number of additions required for the two approaches is the same, 50% reduction in
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Ching-Haur Chang
multiplications is achieved by using a pair of RVEs, compared to that using a CVE.
SIMULATION RESULTS In this section, approaches using two RVEs and a CVE are compared through a 2DM-CD channel. Due to the fact that only amplitude distortion is confronted for the 2DM-CD channel as derived in Section 2, only channels with real-valued parameters are needed in the simulations.
First, a channel of
the form
H1 ( z ) = 0.3482 + 0.8704z −1 + 0.3482z −2 is assumed.
(16)
Related settings used are: input signal alphabet a (n ) ∈ {± 1 ± j}, symbol duration T = 1 ,
the length of the equalizer N = 5 , and delay ∆ = 1 .
Figure 5 compared the SER performance relative
to the signal-to-noise ratio (SNR) using the two approaches. performs even better than that using CVE.
It can be seen that the case using RVEs
Note that for the case with smaller µ ( µ = 0.005 ) the use
of RVEs performs much better than the other, especially in a higher SNR condition.
This is because the
misadjustment [14] occurred in the adaptation process tends to become smaller with smaller values of
µ . Second, a channel taken from [13] is employed for once again justifying the assertion we made. H 2 ( z ) = 0.07 + 0.03z −1 + 0.21z −2 + 0.36 z −4 + 0.72 z −5
(17)
− 0.5 z −6 − 0.21z −7 + 0.07 z −8 − 0.05z −9 + 0.04 z −10
In this case, except N = 15 and ∆ = 7 , the same conditions as above are used in the simulations. can be seen from Figure 6 that the same results are achieved again.
It
Because a saving of 50% reduction
in multiplications is achieved with a pair of RVEs, the results obtained above support the usefulness of the approach.
CONCLUSIONS A novel equalizer structure for 2DM-CD was proposed in this paper.
Based on the fact that the
two sinusoids used to demodulate the signal received are synchronized in phase to the carrier of the signal received, a simple structure using a pair of RVEs, one for the in-phase and the other for the quadrature channels, was developed.
Since only real-valued arithmetic is required for the new structure,
a saving of 50% multiplications with even better SER performance is achieved compared to its complex-valued counterpart.
Simulation results confirmed the assertions made in this paper.
A Novel Equalizer Structure for Two-Dimensional Modulations with Coherent Demodulation
7
ACKNOWLEDGEMENTS This work was supported by the Grant 100-asia-29 from Asia University, Taiwan.
REFERENCES [1] Allpress, S., & Li, Q. (2010). High performance equalizer having reduced complexity. U.S. Patent 7,656,943 [2] Spinnler, B. (2010). Equalizer design and complexity for digital coherent receivers. IEEE J. Selected Topics in Quantum Electronics, 16, 1180-1192 [3] Qureshi, S. U. H., & Forney Jr., G. D. (1997). QAM receiver having automatic adaptive equalizer. U.S. Patent 4,004,226 [4] Cha, I., & Kassam, S. A. (1995). Channel equalization using adaptive complex radial basis function networks. IEEE Journal on Selected Areas in Communications, 13, 122-131 [5] Oh, K. N., & Chin, Y. O. (1995). Modified constant modulus algorithm: blind equalization and carrier phase recovery algorithm. Proc. IEEE Int. Conf. on Communications, 1, 498-502 [6] You, C., & Hong, D. (1998). Nonlinear blind equalization schemes using complex-valued multilayer feedforward neural networks. IEEE Trans. on Neural Networks, 9, 1442-1455 [7] Chang, C. H., Siu, S., & Wei, C. H. (2000). Complex backpropagation decision feedback equalizer with decision using neural nets. Journal of the Chinese Institute of Electrical Engineering, 7, 63-69 [8] Jianping, D., Sundararajan, N., & Saratchandran, P. (2002). Communication channel equalization using complex-valued minimal radial basis function neural networks. IEEE Trans. on Neural Networks, 13, 687-696 [9] Uncini, A., & Piazza, F. (2003). Blind signal processing by complex domain adaptive spline neural networks. IEEE Trans. on Neural Networks, 14, 399-412 [10] Li, M. B., Huang, G. B., Saratchandran, P., & Sundararajan, N. (2006). Complex-valued growing and pruning RBF neural networks for communication channel equalization. IEE Proc., Vision, Image, and Signal Processing, 153, 411–418 [11] Botoca, C., & Budura, G. (2006). Symbol decision equalizer using a radial basis functions neural network. Proc.7th WSEAS Int. Conf. on Neural Networks, 79-84 [12] Miclau, N. (2005). Complex rival penalized learning for RBF neural network used in communication channel equalization. Proc. Int. Symposium on Signals, Circuits and Systems, 2, 797-800 [13] Proakis, J. (2001). Digital Communications, 3rd Ed. Singapore: McGraw-Hill
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Ching-Haur Chang
[14] Widrow, B., & Stearns, S. D. (1985). Adaptive signal processing. New Jersey : Prentice Hall, Englewood Cliffs
APPENDICES
Figure 1.
A two-dimensional signal transmitter. x I (t )
xI (n)
yI (n)
aˆI (n −∆)
2 cos( 2 π f c t + ϕ ) T
bˆ(n − ∆)
2 sin( 2π f c t + ϕ ) T
xQ (t )
xQ (n)
yQ (n)
aˆQ(n−∆) Figure 2.
A two-dimensional signal receiver with CVE inserted.
2 Ο : θ = 0
o
x : θ = 20
1.5
o
1
20 o Quadrature
0.5 0 -0.5 -1 -1.5 -2 -2
Figure 3.
-1.5
-1
-0.5
0 Inphase
0.5
1
1.5
2
Rotation of signal constellation caused by non-zero value of
θ.
9
A Novel Equalizer Structure for Two-Dimensional Modulations with Coherent Demodulation
xI (t)
xI (n)
yI (n)
aˆI (n−∆)
2 cos(2πf c t + ϕ ) T
bˆ(n − ∆)
2 sin( 2πf c t + ϕ ) T x Q (t )
xQ(n)
yQ (n)
aˆQ(n−∆)
Figure 4.
Two-dimensional signal receiver with novel equalizer structure inserted. -0.5 CVE RVE x 2
-1 -1.5
log(SER)
-2 -2.5 -3 -3.5 -4 -4.5 -5
4
6
8
10
12
14
16 18 SNR (dB)
20
22
24
26
28
(a) 0 CVE RVE x 2 -1
log(SER)
-2
-3
-4
-5
-6
4
6
8
10
12
14
16 18 SNR (dB)
20
22
24
26
28
(b) Figure 5.
Comparison of SER versus SNR using (a) µ = 0.005 and (b) µ = 0.01 using the channel given in (16).
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0 CVE RVE x 2 -1
log(SER)
-2
-3
-4
-5
-6
-7 4
6
8
10
12 SNR (dB)
14
16
18
20
(a) 0 CVE RVE x 2 -1
log(SER)
-2
-3
-4
-5
-6
-7 4
6
8
10
12 SNR (dB)
14
16
18
20
(b)
Figure 6.
Comparison of SER versus SNR using (a) µ = 0.005 and (b) µ = 0.01 using the channel given in (17).
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A Novel Equalizer Structure for Two-Dimensional Modulations with Coherent Demodulation
Table 1
Comparison of Computations Required for RVEs and CVE.
CVE
2 RVEs
Additions
Output/symbol Update of
Multiplications
Additions
Multiplications
Complex
Real
Complex
Real
Real
Real
N-1
2(N-1)
N
4N
2(N-1)
2N
N+1
2(N+1)
~N
~4N
2(N+1)
~8N
4N
~2N
tap-weights/ symbol Total/symbol
2N
4N
~2N
~4N