Comparison between different Adaptive Pre-Equalization ... - CiteSeerX
COMPARISON BETWEEN DIFFERENT ADAPTIVE PRE-EQUALIZATION APPROACHES FOR WIRELESS LAN Edmund Coersmeier1, Ernst Zielinski2 1
Meesmannstrasse 103, 44807, Bochum, Germany, [email protected] Meesmannstrasse 103, 44807, Bochum, Germany, [email protected]
2
Abstract - The IEEE802.11a Wireless LAN standard defines a high system performance and therefor requires a certain signal accuracy for the OFDM transmitter output. Taking the potential analog low, medium IF and RF filter imperfections into account it is necessary to pre-equalize the signal stream digitally before transmitting. The paper will concentrate on the description and comparison between different mathematical approaches for pre-equalization in an OFDM transmitter front-end architecture. For the favorite algorithm the corresponding simulation results will be presented. Keywords – Wireless LAN, Pre-Equalization, analog frontend, LMS, LS, zero-forcing.
transmitter antenna input port which will be compared to the digitally generated envelope at IF = 20 MHz to calculate the error value e[k]. It is assumed that both envelope signals are ideally synchronized. The pre-equalization process can be implemented in different ways whereas in this paper four different proposals (options 1-4) will be explained. Table 1 Four different approaches for pre-equalization process. The favorite approach is defined in option 4. Option
System Identification
Inverse Modelling
1
Least-Squares
Zero-Forcing
I. INTRODUCTION
2
Least-Squares
Least-Mean-Square
The performance of a transmitter output signal is strongly depending on the analog filter accuracy. To recover the potential analog filter imperfections a digital pre-equalizer can be installed. Figure 1 presents a possible heterodyne OFDM transmitter architecture for an IEEE802.11a application. A direct conversion architecture is also possible but to employ real instead of complex mathematical equations in this paper a heterodyne architecture has been chosen.
3
Least-Mean-Square
Least-Mean-Square
4
Maximum Peak Approximation
Approximated Least-Mean-Square
BPSK, QPSK, QAM
IFFT
4x Sample Rate Up Conversion
Envelope Measurement
Analog Baseband
1.5 GHz
Digital IF fIF = 20 MHz
Low Pass Filter
Analog IF
ADC
Adaptive Pre-Equalizer
Envelope Error Detection
Analog RF
3.5-4.5 GHz adjustable
Fig. 1. Example of an IEEE802.11a wireless LAN analog front-end. Using a heterodyne transmitter architecture a digital IF = 20 MHz is installed. The convolution of the digitally preequalized signal stream with the non-ideal analog filters results again in an accurate signal stream. The envelope measurement block provides the envelope signal from the
0-7803-7589-0/02/$17.00 ©2002 IEEE
Table 1 shows that the pre-equalization process can be divided into two classes, the system identification class and the inverse modeling class. For options 1-3 this paper provides for each class two different approaches. Option 4 obtains an exceptional position, because one algorithm can fulfill the whole pre-equalization process. For options 1-3 the system identification class uses the leastsquares and the least-mean-square principles. They are responsible to estimate the parameters of the unknown, time-variant system. The inverse modeling class for options 1-3 contains the zero-forcing and the least-mean-square algorithms. Their goal is to provide an inverted function based on the identified system to compensate certain analog filter imperfections. Option 4 is different, because in fact, it uses only one single algorithm, a least-mean-square algorithm with an approximated gradient for inverse modeling, whereas the system identification process is reduced to an analog filter peak approximation. The paper will first describe the different approaches for system identification and inverse modeling with regards to options 1-3. Afterwards their possible combinations are
PIMRC 2002
described and finally the favorite, option 4, is explained in detail. II. SYSTEM IDENTIFICATION Usually an unknown linear, time-variant system is given by time-continuous response h(t) which can be approximated for the purpose of digital equalization by a set of coefficients h(t) ≈ {h1 h2 … hM} where M denotes the total number of coefficients and can be infinity in an ideal case. Depending on the desired accuracy of the analog filter approximation the number of coefficients will be limited. The system identification will be done via installing an identification model parallel to the analog filter chain. Analog Filter h[n]
Analog Filter h[n]
y~[n]
y~[n] +
d[n] d[n]
Adaptive Filter h^[n]
y[n] -
h [n]
LMS
Fig. 2. System identification process for least-squares (left) and least-mean-square (right). After the system identification process both, the analog filter chain and the identification model provide the same behavior, such that e[n] = y[n]–y~[n]= 0. According to table 1 the identification process can be done via the least-squares (options 1,2) and least-mean-square approach (option 3). Option 4 will be considered in section V. A. System Identification with Least-Squares The least-squares algorithm estimates a set of suitable coefficients to provide via the digital filter model an appropriate analog filter approximation. The algorithm is supplied by the reference signal y~[n] and the input data d[n]. To compute the optimum digital filter coefficients the following steps have to be performed: 1.
Over-determined system creation for L+1 equations
e[n − L] d[n − L] = d[n] e[n]
2.
y[n − L] ~ d[n − (L + M) +1] hˆ0 [n] − ⋅ ˆ d[n − M +1] hM −1[n] ~ y[n]
hˆ[n ]
2 2
⇔ A[n]⋅ hˆ[n ] = b[n ]
The filter vector update is done via equation (5).
[[ ]
1 ˆ ˆ hˆ[n + 1] = hˆ[n ] − µ ∇ J h[n] = hˆ[n ] + µ ⋅ e[n ]⋅ d [n ] 2
III. INVERSE MODELING To realize the complete pre-equalization approach the inverse modeling process has to be introduced. The inverse modeling refers to the reciprocal of the system identification vector h^[n]. Figure 3 provides the principle setup of the inverse modeling process. d[n-τw] Delay
d[n]
x[n] w[n]
Adaptation
(2)
(5)
It is shown that the system identification process can be reduced to an addition of the current system estimate vector h^[n] with the product of a constant µ, the error value e[n] and the actual data vector d[n]. Equation (5) will be calculated iteratively on a sample-by-sample base.
(1)
h^[n]
d[n-τw-τh]
Delay
y[n] Analog Filter h[n]
Analog Filter h[n]
w[n]
Adaptation
x[n] e[n]
-
+
y[n]
h^[n]
Fig. 3. Inverse modeling process for zero-forcing (left) and least-mean-square (right).
Solution of the normal equations
hˆ[n ] = A −1 [n ]⋅ b[n ]
(4)
d[n]
Normal equations creation min D[n ]⋅ hˆ[n] − ~ y
3.
The least-mean-square algorithm approximates the analog filter function h(t) step by step, hence the calculation of vector h^[n] is done iteratively. Each identified filter vector h^[n] corresponds to exactly one single error value e[n]. The least-mean-square algorithm starts from an arbitrary initial value h^[0] and computes the next filter vector h^[1] in the direction of the estimated gradient of the cost function J[h^]. ˆ J hˆ = ∂ e 2 [n] = 2 ⋅ e[n]⋅ d [n ] ∇ ∂ hˆ
e[n] ^
B. System Identification with Least-Mean-Square
[[]
h^[n]
LS
where A[n] = DT[n]⋅D[n] is a matrix of dimension MxM and b[n] = DT[n]⋅y~[n]. Equation (1) shows that first n-error values and (L+M)-data samples have to be collected to solve the equation system respectively to evaluate the optimum coefficient set. The minimization process in equation (2) provides the step from an over-determined system to a system involving exactly as many equations as unknown variables are available. Hence the least-squares approach calculates the coefficients on a block-by-block base.
(3)
In this paper two different ways of inverse modeling are presented for options 1-3 (table 1). On the one hand zero-
forcing on the other hand least-mean-square. The inverse modeling for option 4 is given in section V.
The least-mean-square approach is based on iterative updates because one single error value e[n] corresponds to one set of coefficients.
A. Inverse Modelling with Zero-Forcing The matrix-vector product of the analog filter matrix H and the digital adaptive filter coefficients vector w is given by equation (6). c = H [n ]⋅ w[n ]
(6)
The matrix H is the convolution matrix with the given structure of equation (7). 0 h0 h 0 hM −1 hM −1 0 H = 0 0 0
(7)
Options 1-3 (table 1) will be analyzed in this section. The interaction of the three systems differ in terms of computational amount and the resulting performance. Based on this comparison the next section will provide a new preequalization approach (option 4) which provides significant advantages in terms of computational amount. A. Least-Squares and Zero-Forcing, Option 1
0 0 h0 hM −1
IV. COMPARISON OF COMBINATIONS FOR SYSTEM IDENTIFICATION AND INVERSE MODELING
M + N −1× N
N denotes the total number of pre-equalizer coefficients and M the total number of identified analog filter coefficients. The overall expected system behaviour should be 1 : i = τ w+ h ci = 0 : i ≠ τ w+ h
(8)
To solve equation (6) the following steps have to be done. A = HT ⋅H
(9)
b = HT ⋅c
(10)
w = A −1 ⋅ b
(11)
The combination least-squares + zero-forcing provides a block-based equalizer coefficient calculation. The leastsquares algorithm requires a data matrix containing (L+M) different data samples and provides the best estimation vector h^. This vector is given to the zero-forcing block which inverts the matrix A. The desired pre-equalizer vector w has to be extracted afterwards. This approach delivers optimum equalizer coefficients but requires a huge computational amount. B. Least-Squares and Least-Mean-Square, Option 2 In contrast the combination in option 2 (table 1), leastsquares + least-mean-square, provides the pre-equalizer coefficients iteratively on a sample-by-sample base. Although the identification is accomplished block-by-block the least-mean-square algorithm computes the coefficients at each sample instance n. Because each gradient respectively each coefficient vector is based on only one single error value e[n] the precision of the least-mean-square algorithm is lower than the zero-forcing approach. Finally the computational amount of option 2 has been decreased significantly compared to option 1, because no matrix inversion is necessary. C. Least-Mean-Square and Least-Mean-Square, Option 3
The zero-forcing inverse modeling operates block based and can be calculated after M identified analog filter coefficients are available. Because of the zero-forcing approach the error value e[n] can be assumed to be 0 always.
(12)
Option 3 can perform the system identification and the coefficient calculation on a sample-by-sample base. But from a practical point of view this approach might lead to instabilities due to the fact that the identification process might change its output at the beginning too fast. Hence it can be recommended to reduce the identification process to a very low speed or a block-based approach to guarantee the feedback loop stability. The main advantage of this approach is its simplicity
(13) 1 ˆ ∇[J [w]] = w[n ] + µe[n ]⋅ D[n ]⋅ h[n ] 2
Options 1-3 uses always two algorithms to fulfill the preequalization task. One algorithm identifies the analog filters (system identification) and the second algorithm obtains the inverse modeling. Option 4 (table 1) can handle the preequalization process with one single algorithm and hence
B. Inverse Modeling with Least-Mean-Square To obtain the inverse modeling with the least-mean-square algorithm the gradient vector has to be calculated.
ˆ [J [w]] = 2 ⋅ e[n]⋅ D[n]⋅ h[n] ∇ After that the coefficient update follows. w[n + 1] = w[n ] − µ
has a much lower computational complexity. Option 4 will be explained in the following section V. V. PRE-EQUALIZATION VIA PEAK APPROXIMATION FOR THE SYSTEM IDENTIFICATION PROCESS This pre-equalization approach (option 4, table 1) reduces the system identification process to an analog filter peak approximation. If one assumes an ideal system identification process there could be established a theoretical preequalization similar to figure 4. d[n-τw] Delay
d[n]
d[n-τw-τh] Delay
Adaptive Filter w[n]
x[n]
Analog Filter h[n]
y[n] -
{ } {
ˆ E e 2 [n ] ∇ Jˆ [w] ≈ ∇
x^[n] Adaptation
(14)
The identification process to find the analog filter peak position can be done offline during the filter design already. Alternatively a filter peak position measurement with a single filter prototype is sufficient. Figure 5 provides a possible implementation.
d[n-τw] d[n]
Adaptive Filter w[n]
d[n-τw-τh] Delay
x[n]
Analog + Filter h[n]
y[n] -
+
w[n+1] x^[n] h^[n]
Adaptation
} (17)
= −2 ⋅ e[n ]⋅ d [n − τ h ]
]
Delay
(16)
= −2 ⋅ e[n ]⋅ D[n]⋅ hˆ[n ]
e[n]
A copy of the analog filter is placed in-front of the inverse modeling process. To reduce the amount of system identification the copy of the analog filter can be approximated. Only the analog filter maximum peak position has to be identified, but neither the absolute maximum peak amplitude nor other filter coefficients have to be considered. Equation (14) provides the identified and approximated filter transfer vector h^.
[
(15)
After the system approximation has been described the inverse modeling approach will be explained. To be successfull in updating the pre-equalizer filter coefficients the gradient has to be calculated based on the approximated system identification. Equation (16) provides the gradient of the approximated cost function, equation (17) the approximation of the least-mean-square gradient vector.
{ }
Fig. 4. Theoretical pre-equalization setup after ideal system identification. The identified system can be interpreted as a copy of the analog filters.
T hˆ = 0 0 0...1...0 0
xˆ[n] = D[n]⋅ hˆ[n] = hˆτ h ⋅ d [n − τ h ] = d [n − τ h ]
∇ Jˆ [w] = −2 ⋅ E e[n ]⋅ D[n]⋅ hˆ[n ]
+
w[n+1] Copy of Analog Filter h[n]
Considering such an approximated transfer vector from equation (14) the parameter τh defines the analog filter delay. Signal vector x^[n] is given by equation (15).
e[n]
Fig. 5. Analog maximum filter peak approximation.
The approximated gradient is updated on a sample-bysample base and depends only on the measured error value e[n] and the delayed input signal. The mentioned delay corresponds to the approximated analog filter peak position. Equation (18) – (20) describe the error calculation e[n]. x[n] = d T [n]⋅ w[n]
(18)
y[n] = xT [n]⋅ h[n]
(19)
e[n] = d n − τ
w + h
− y[n] = d n − τ
w + h
− xT [n]⋅ h[n] (20)
Equation (21) provides the pre-equalizer filter coefficient update based on the approximated gradient vector. Here the update is again done via a sample-by-sample base. w[n + 1] = w[n]+ µ ⋅ e[n]⋅ d n − τ h
(21)
The here-described option 4 (table 1) reduces the complex system identification processes to the implementation of a simple delay block, corresponding to the pre-defined analog filter peak position. Expected small variations of this peak position will be handled by the feedback loop automatically. The inverse modeling is done based on an approximated gradient for the least-mean-square algorithm and hence the overall system requires only one adaptive algorithm. This is less computational amount compared to options 1-3 from table 1. Finally the signal performance results are similar to options 2-3 and fulfill practical requirements based on standards like IEEE802.11a.
VI. SIMULATION RESULTS The simulation results refer to option 4 (table 1) and employ a 64-QAM in an IEEE802.11a wireless LAN environment. The receiver is assumed to be ideally synchronized and imperfections are added only by the transmitter analog filters. Figure 6 shows imperfections caused by the transmitter analog filters due to 7 dB amplitude ripple in the pass-band and an additional non-linear filter phase.
Figure 8 provides the 64-QAM IQ-constellation diagram after pre-equalization. The required IEEE802.11a signal accuracy is fulfilled.
Fig. 8. Pre-equalized IQ constellation diagram (option 4). VII. CONCLUSION Fig. 6. Imperfections caused by an elliptic analog filter with amplitude ripple up to 7 dB and additional phase nonlinearities in the pass-band. Figure 7 provides the error measurement of value e[n]. There are plotted about 400000 error samples e[n] whereas the final error values have been reduced to 3% of the starting values. Additional simulation iterations would decrease the remaining error further more.
This paper provides the comparison between 4 different preequalization approaches. Each of options 1-3 (table 1) requires two algorithms to fulfill the system identification and the inverse modeling. Option 1 provides the best performance but has more computational complexity compared to options 2-3. Option 4 provides a similar performance like options 2-3 but requires much less computational complexity, because the system identification process is replaced through a simple delay block. The delay corresponds to the pre-defined analog filter delay. Option 4 fulfills the IEEE802.11a accuracy requirements and hence has been chosen as the pre-equalization favorite. REFERENCES IEEE 802.11a-1999, The Institute of Electrical and Electronics Engineering, Inc. 3 Park Avenue, New York, NY 10016-5997, USA. [2] Adaptive Pre-Equalization for Nonlinear System, John J. Shynk, Final Report 1998-99 for MICRO Project 98138, Department of Electrical & Computer Engineering, University of California, Santa Babara, CA 93106, USA [3] Adaptive Filter Theory, Simon Haykin, Prentice Hall, Third Edition, 1996 [4] High Precision Analog Front-End Transceiver Architecture for Wireless Local Area Network, Edmund Coersmeier, Yuhuan Xu, Ludwig Schwoerer, Ken Astrof, 6th International OFDM-Workshop 2001, Hamburg [1]
Fig. 7. Error curve e[n]. 400000 samples plotted.
Meesmannstrasse 103, 44807, Bochum, Germany, [email protected] Meesmannstrasse 103, 44807, Bochum, Germany, [email protected]
2
Abstract - The IEEE802.11a Wireless LAN standard defines a high system performance and therefor requires a certain signal accuracy for the OFDM transmitter output. Taking the potential analog low, medium IF and RF filter imperfections into account it is necessary to pre-equalize the signal stream digitally before transmitting. The paper will concentrate on the description and comparison between different mathematical approaches for pre-equalization in an OFDM transmitter front-end architecture. For the favorite algorithm the corresponding simulation results will be presented. Keywords – Wireless LAN, Pre-Equalization, analog frontend, LMS, LS, zero-forcing.
transmitter antenna input port which will be compared to the digitally generated envelope at IF = 20 MHz to calculate the error value e[k]. It is assumed that both envelope signals are ideally synchronized. The pre-equalization process can be implemented in different ways whereas in this paper four different proposals (options 1-4) will be explained. Table 1 Four different approaches for pre-equalization process. The favorite approach is defined in option 4. Option
System Identification
Inverse Modelling
1
Least-Squares
Zero-Forcing
I. INTRODUCTION
2
Least-Squares
Least-Mean-Square
The performance of a transmitter output signal is strongly depending on the analog filter accuracy. To recover the potential analog filter imperfections a digital pre-equalizer can be installed. Figure 1 presents a possible heterodyne OFDM transmitter architecture for an IEEE802.11a application. A direct conversion architecture is also possible but to employ real instead of complex mathematical equations in this paper a heterodyne architecture has been chosen.
3
Least-Mean-Square
Least-Mean-Square
4
Maximum Peak Approximation
Approximated Least-Mean-Square
BPSK, QPSK, QAM
IFFT
4x Sample Rate Up Conversion
Envelope Measurement
Analog Baseband
1.5 GHz
Digital IF fIF = 20 MHz
Low Pass Filter
Analog IF
ADC
Adaptive Pre-Equalizer
Envelope Error Detection
Analog RF
3.5-4.5 GHz adjustable
Fig. 1. Example of an IEEE802.11a wireless LAN analog front-end. Using a heterodyne transmitter architecture a digital IF = 20 MHz is installed. The convolution of the digitally preequalized signal stream with the non-ideal analog filters results again in an accurate signal stream. The envelope measurement block provides the envelope signal from the
0-7803-7589-0/02/$17.00 ©2002 IEEE
Table 1 shows that the pre-equalization process can be divided into two classes, the system identification class and the inverse modeling class. For options 1-3 this paper provides for each class two different approaches. Option 4 obtains an exceptional position, because one algorithm can fulfill the whole pre-equalization process. For options 1-3 the system identification class uses the leastsquares and the least-mean-square principles. They are responsible to estimate the parameters of the unknown, time-variant system. The inverse modeling class for options 1-3 contains the zero-forcing and the least-mean-square algorithms. Their goal is to provide an inverted function based on the identified system to compensate certain analog filter imperfections. Option 4 is different, because in fact, it uses only one single algorithm, a least-mean-square algorithm with an approximated gradient for inverse modeling, whereas the system identification process is reduced to an analog filter peak approximation. The paper will first describe the different approaches for system identification and inverse modeling with regards to options 1-3. Afterwards their possible combinations are
PIMRC 2002
described and finally the favorite, option 4, is explained in detail. II. SYSTEM IDENTIFICATION Usually an unknown linear, time-variant system is given by time-continuous response h(t) which can be approximated for the purpose of digital equalization by a set of coefficients h(t) ≈ {h1 h2 … hM} where M denotes the total number of coefficients and can be infinity in an ideal case. Depending on the desired accuracy of the analog filter approximation the number of coefficients will be limited. The system identification will be done via installing an identification model parallel to the analog filter chain. Analog Filter h[n]
Analog Filter h[n]
y~[n]
y~[n] +
d[n] d[n]
Adaptive Filter h^[n]
y[n] -
h [n]
LMS
Fig. 2. System identification process for least-squares (left) and least-mean-square (right). After the system identification process both, the analog filter chain and the identification model provide the same behavior, such that e[n] = y[n]–y~[n]= 0. According to table 1 the identification process can be done via the least-squares (options 1,2) and least-mean-square approach (option 3). Option 4 will be considered in section V. A. System Identification with Least-Squares The least-squares algorithm estimates a set of suitable coefficients to provide via the digital filter model an appropriate analog filter approximation. The algorithm is supplied by the reference signal y~[n] and the input data d[n]. To compute the optimum digital filter coefficients the following steps have to be performed: 1.
Over-determined system creation for L+1 equations
e[n − L] d[n − L] = d[n] e[n]
2.
y[n − L] ~ d[n − (L + M) +1] hˆ0 [n] − ⋅ ˆ d[n − M +1] hM −1[n] ~ y[n]
hˆ[n ]
2 2
⇔ A[n]⋅ hˆ[n ] = b[n ]
The filter vector update is done via equation (5).
[[ ]
1 ˆ ˆ hˆ[n + 1] = hˆ[n ] − µ ∇ J h[n] = hˆ[n ] + µ ⋅ e[n ]⋅ d [n ] 2
III. INVERSE MODELING To realize the complete pre-equalization approach the inverse modeling process has to be introduced. The inverse modeling refers to the reciprocal of the system identification vector h^[n]. Figure 3 provides the principle setup of the inverse modeling process. d[n-τw] Delay
d[n]
x[n] w[n]
Adaptation
(2)
(5)
It is shown that the system identification process can be reduced to an addition of the current system estimate vector h^[n] with the product of a constant µ, the error value e[n] and the actual data vector d[n]. Equation (5) will be calculated iteratively on a sample-by-sample base.
(1)
h^[n]
d[n-τw-τh]
Delay
y[n] Analog Filter h[n]
Analog Filter h[n]
w[n]
Adaptation
x[n] e[n]
-
+
y[n]
h^[n]
Fig. 3. Inverse modeling process for zero-forcing (left) and least-mean-square (right).
Solution of the normal equations
hˆ[n ] = A −1 [n ]⋅ b[n ]
(4)
d[n]
Normal equations creation min D[n ]⋅ hˆ[n] − ~ y
3.
The least-mean-square algorithm approximates the analog filter function h(t) step by step, hence the calculation of vector h^[n] is done iteratively. Each identified filter vector h^[n] corresponds to exactly one single error value e[n]. The least-mean-square algorithm starts from an arbitrary initial value h^[0] and computes the next filter vector h^[1] in the direction of the estimated gradient of the cost function J[h^]. ˆ J hˆ = ∂ e 2 [n] = 2 ⋅ e[n]⋅ d [n ] ∇ ∂ hˆ
e[n] ^
B. System Identification with Least-Mean-Square
[[]
h^[n]
LS
where A[n] = DT[n]⋅D[n] is a matrix of dimension MxM and b[n] = DT[n]⋅y~[n]. Equation (1) shows that first n-error values and (L+M)-data samples have to be collected to solve the equation system respectively to evaluate the optimum coefficient set. The minimization process in equation (2) provides the step from an over-determined system to a system involving exactly as many equations as unknown variables are available. Hence the least-squares approach calculates the coefficients on a block-by-block base.
(3)
In this paper two different ways of inverse modeling are presented for options 1-3 (table 1). On the one hand zero-
forcing on the other hand least-mean-square. The inverse modeling for option 4 is given in section V.
The least-mean-square approach is based on iterative updates because one single error value e[n] corresponds to one set of coefficients.
A. Inverse Modelling with Zero-Forcing The matrix-vector product of the analog filter matrix H and the digital adaptive filter coefficients vector w is given by equation (6). c = H [n ]⋅ w[n ]
(6)
The matrix H is the convolution matrix with the given structure of equation (7). 0 h0 h 0 hM −1 hM −1 0 H = 0 0 0
(7)
Options 1-3 (table 1) will be analyzed in this section. The interaction of the three systems differ in terms of computational amount and the resulting performance. Based on this comparison the next section will provide a new preequalization approach (option 4) which provides significant advantages in terms of computational amount. A. Least-Squares and Zero-Forcing, Option 1
0 0 h0 hM −1
IV. COMPARISON OF COMBINATIONS FOR SYSTEM IDENTIFICATION AND INVERSE MODELING
M + N −1× N
N denotes the total number of pre-equalizer coefficients and M the total number of identified analog filter coefficients. The overall expected system behaviour should be 1 : i = τ w+ h ci = 0 : i ≠ τ w+ h
(8)
To solve equation (6) the following steps have to be done. A = HT ⋅H
(9)
b = HT ⋅c
(10)
w = A −1 ⋅ b
(11)
The combination least-squares + zero-forcing provides a block-based equalizer coefficient calculation. The leastsquares algorithm requires a data matrix containing (L+M) different data samples and provides the best estimation vector h^. This vector is given to the zero-forcing block which inverts the matrix A. The desired pre-equalizer vector w has to be extracted afterwards. This approach delivers optimum equalizer coefficients but requires a huge computational amount. B. Least-Squares and Least-Mean-Square, Option 2 In contrast the combination in option 2 (table 1), leastsquares + least-mean-square, provides the pre-equalizer coefficients iteratively on a sample-by-sample base. Although the identification is accomplished block-by-block the least-mean-square algorithm computes the coefficients at each sample instance n. Because each gradient respectively each coefficient vector is based on only one single error value e[n] the precision of the least-mean-square algorithm is lower than the zero-forcing approach. Finally the computational amount of option 2 has been decreased significantly compared to option 1, because no matrix inversion is necessary. C. Least-Mean-Square and Least-Mean-Square, Option 3
The zero-forcing inverse modeling operates block based and can be calculated after M identified analog filter coefficients are available. Because of the zero-forcing approach the error value e[n] can be assumed to be 0 always.
(12)
Option 3 can perform the system identification and the coefficient calculation on a sample-by-sample base. But from a practical point of view this approach might lead to instabilities due to the fact that the identification process might change its output at the beginning too fast. Hence it can be recommended to reduce the identification process to a very low speed or a block-based approach to guarantee the feedback loop stability. The main advantage of this approach is its simplicity
(13) 1 ˆ ∇[J [w]] = w[n ] + µe[n ]⋅ D[n ]⋅ h[n ] 2
Options 1-3 uses always two algorithms to fulfill the preequalization task. One algorithm identifies the analog filters (system identification) and the second algorithm obtains the inverse modeling. Option 4 (table 1) can handle the preequalization process with one single algorithm and hence
B. Inverse Modeling with Least-Mean-Square To obtain the inverse modeling with the least-mean-square algorithm the gradient vector has to be calculated.
ˆ [J [w]] = 2 ⋅ e[n]⋅ D[n]⋅ h[n] ∇ After that the coefficient update follows. w[n + 1] = w[n ] − µ
has a much lower computational complexity. Option 4 will be explained in the following section V. V. PRE-EQUALIZATION VIA PEAK APPROXIMATION FOR THE SYSTEM IDENTIFICATION PROCESS This pre-equalization approach (option 4, table 1) reduces the system identification process to an analog filter peak approximation. If one assumes an ideal system identification process there could be established a theoretical preequalization similar to figure 4. d[n-τw] Delay
d[n]
d[n-τw-τh] Delay
Adaptive Filter w[n]
x[n]
Analog Filter h[n]
y[n] -
{ } {
ˆ E e 2 [n ] ∇ Jˆ [w] ≈ ∇
x^[n] Adaptation
(14)
The identification process to find the analog filter peak position can be done offline during the filter design already. Alternatively a filter peak position measurement with a single filter prototype is sufficient. Figure 5 provides a possible implementation.
d[n-τw] d[n]
Adaptive Filter w[n]
d[n-τw-τh] Delay
x[n]
Analog + Filter h[n]
y[n] -
+
w[n+1] x^[n] h^[n]
Adaptation
} (17)
= −2 ⋅ e[n ]⋅ d [n − τ h ]
]
Delay
(16)
= −2 ⋅ e[n ]⋅ D[n]⋅ hˆ[n ]
e[n]
A copy of the analog filter is placed in-front of the inverse modeling process. To reduce the amount of system identification the copy of the analog filter can be approximated. Only the analog filter maximum peak position has to be identified, but neither the absolute maximum peak amplitude nor other filter coefficients have to be considered. Equation (14) provides the identified and approximated filter transfer vector h^.
[
(15)
After the system approximation has been described the inverse modeling approach will be explained. To be successfull in updating the pre-equalizer filter coefficients the gradient has to be calculated based on the approximated system identification. Equation (16) provides the gradient of the approximated cost function, equation (17) the approximation of the least-mean-square gradient vector.
{ }
Fig. 4. Theoretical pre-equalization setup after ideal system identification. The identified system can be interpreted as a copy of the analog filters.
T hˆ = 0 0 0...1...0 0
xˆ[n] = D[n]⋅ hˆ[n] = hˆτ h ⋅ d [n − τ h ] = d [n − τ h ]
∇ Jˆ [w] = −2 ⋅ E e[n ]⋅ D[n]⋅ hˆ[n ]
+
w[n+1] Copy of Analog Filter h[n]
Considering such an approximated transfer vector from equation (14) the parameter τh defines the analog filter delay. Signal vector x^[n] is given by equation (15).
e[n]
Fig. 5. Analog maximum filter peak approximation.
The approximated gradient is updated on a sample-bysample base and depends only on the measured error value e[n] and the delayed input signal. The mentioned delay corresponds to the approximated analog filter peak position. Equation (18) – (20) describe the error calculation e[n]. x[n] = d T [n]⋅ w[n]
(18)
y[n] = xT [n]⋅ h[n]
(19)
e[n] = d n − τ
w + h
− y[n] = d n − τ
w + h
− xT [n]⋅ h[n] (20)
Equation (21) provides the pre-equalizer filter coefficient update based on the approximated gradient vector. Here the update is again done via a sample-by-sample base. w[n + 1] = w[n]+ µ ⋅ e[n]⋅ d n − τ h
(21)
The here-described option 4 (table 1) reduces the complex system identification processes to the implementation of a simple delay block, corresponding to the pre-defined analog filter peak position. Expected small variations of this peak position will be handled by the feedback loop automatically. The inverse modeling is done based on an approximated gradient for the least-mean-square algorithm and hence the overall system requires only one adaptive algorithm. This is less computational amount compared to options 1-3 from table 1. Finally the signal performance results are similar to options 2-3 and fulfill practical requirements based on standards like IEEE802.11a.
VI. SIMULATION RESULTS The simulation results refer to option 4 (table 1) and employ a 64-QAM in an IEEE802.11a wireless LAN environment. The receiver is assumed to be ideally synchronized and imperfections are added only by the transmitter analog filters. Figure 6 shows imperfections caused by the transmitter analog filters due to 7 dB amplitude ripple in the pass-band and an additional non-linear filter phase.
Figure 8 provides the 64-QAM IQ-constellation diagram after pre-equalization. The required IEEE802.11a signal accuracy is fulfilled.
Fig. 8. Pre-equalized IQ constellation diagram (option 4). VII. CONCLUSION Fig. 6. Imperfections caused by an elliptic analog filter with amplitude ripple up to 7 dB and additional phase nonlinearities in the pass-band. Figure 7 provides the error measurement of value e[n]. There are plotted about 400000 error samples e[n] whereas the final error values have been reduced to 3% of the starting values. Additional simulation iterations would decrease the remaining error further more.
This paper provides the comparison between 4 different preequalization approaches. Each of options 1-3 (table 1) requires two algorithms to fulfill the system identification and the inverse modeling. Option 1 provides the best performance but has more computational complexity compared to options 2-3. Option 4 provides a similar performance like options 2-3 but requires much less computational complexity, because the system identification process is replaced through a simple delay block. The delay corresponds to the pre-defined analog filter delay. Option 4 fulfills the IEEE802.11a accuracy requirements and hence has been chosen as the pre-equalization favorite. REFERENCES IEEE 802.11a-1999, The Institute of Electrical and Electronics Engineering, Inc. 3 Park Avenue, New York, NY 10016-5997, USA. [2] Adaptive Pre-Equalization for Nonlinear System, John J. Shynk, Final Report 1998-99 for MICRO Project 98138, Department of Electrical & Computer Engineering, University of California, Santa Babara, CA 93106, USA [3] Adaptive Filter Theory, Simon Haykin, Prentice Hall, Third Edition, 1996 [4] High Precision Analog Front-End Transceiver Architecture for Wireless Local Area Network, Edmund Coersmeier, Yuhuan Xu, Ludwig Schwoerer, Ken Astrof, 6th International OFDM-Workshop 2001, Hamburg [1]
Fig. 7. Error curve e[n]. 400000 samples plotted.
