Zero-Forcing Equalizability of FIR and IIR Multi-Channel ... - CiteSeerX
Zero-Forcing Equalizability of FIR and IIR Multi-Channel Systems with and without Perfect Measurements Erwei Bai and Zhi Ding Department of Electrical and Computer Engineering University of Iowa Iowa City, IA 52242 Email: ferwei, [email protected] Phone: 319-335-5949, 319-335-6028(fax)
Abstract
In this paper, we study the linear zero-forcing equalizability of a communication channel. Necessary and sucient conditions are given in terms of zeros of the transfer function. Detailed procedures of equalizer design for minimum sensitivity with respect to modeling errors and white noise are also presented. It is found that the worst-case e ect of channel mismatch on the performance may be modeled as equivalent losses in SNR.
This work was supported in part by grants from the National Science Foundation.
1 Introduction Zero-forcing equalization is a technique often used in high SNR communication receivers for the removal of inter-symbol interferences. Zero-forcing equalizers are essentially channel inverses used to recover the unknown input signal of a linear system from its measurable output signal. Since the goal of the equalizer is to recover the input signal from the output measurements. zero-forcing requires that the input signal be recovered EXACTLY with a nite delay in the absence of noise. One problem is that zero-forcing (ZF) equalizers may not exist. The goal of this correspondence is to de ne the conditions for the existence of ZF equalizer in various multi-channel systems. The approach is from a system theoretic point of view, based on the Smith and McMillian forms, which facilitates the conceptual generalization of ZF equalizers to the IIR cases. Equalizability conditions are characterized in terms of system poles/zeros. The design of zero-forcing equalizers under model uncertainty and additive noise is also addressed.
2 Preliminary and notation We consider communication channels modeled by
Y~ (z) = H(z)U~ (z)
(2.1)
where U~ (z ) is the m-dimensional message signal, Y~ (z ) the n-dimensional received signal and H(z ) the n by m dimensional transfer function. This model represents a large class of communication channels with multiple users and multiple sensor outputs. Two examples are considered here.
2.1 Fractionally over-sampled channels Consider a multi-user quadrature amplitude modulation data communication system. Let N be the number of user channels. Then, the received signal is given by
y(t) =
N X X j =1 k
hj (t ? kT )uj (k)
where T is the symbol baud period, hj (t) the composite impulse response of the j th channel and uj (k) input signal of j th user. Let y(t) be sampled at a rate of such that
pT = q; 1
for some positive integers p and q . It has been shown in [2, 4] that the over-sampled output can be viewed as a pN -input and q -output linear system exactly in the form of equation (2.1).
2.2 Channels with input modulation-induced cyclostationarity For channels with input modulation-induced cyclostationarity [7], the received signal is given by
y(t) =
X l
h(t ? lT )u(l) (l)
where the input signal u(k) is modulated by a deterministic and periodic sequence (k) with period p. For 0 i p ? 1, we de ne
yi (z) = =
X
y(kpT + iT )z ?k =
k pX ?1 X X j =1 l
k
XX l
k
h(kpT + iT ? lT )u(l) (l)z ?k
h(kpT + iT ? (lp + j )T )u(lp + j ) (lp + j )z ?k :
Since (lp + j ) = (j ), we have (letting r = k ? l) pX ?1
yi (z) =
j =1 pX ?1
=
j =1
where
hl(z) = uj (z) =
(j )
r
h(rpT + (i ? j )T )z ?r
X l
u(lp + j )z ?l
(j )hi?j (z)uj (z)
X k X k
X
h(kpT + lT )z ?k ;
?(p ? 1) l p ? 1;
u(kp + j )z ?k ; j = 0; 1; : : :p ? 1:
Thus, for a causal impulse response h(t) and ?p + 1 l ?1, we obtain
hl(z) =
X k
h(kpT + lT )z ?k = z ?1 hp+l (z):
Therefore, we obtain, exactly in the form of (2.1), 0 y0(z) 1 0 h0(z) z?1 hp?1(z) z?1hp?2(z) h1(z) h0 (z) z ?1hp?1 (z) B@ ... CA = B B B .. .. @ ... . . y ( z ) | p?{z1 } h (z) h (z) h (z) Y~ (z)
|
p?1
p?2
{zp?3
H(z)
2
::: ::: ...
:::
z ?1 h1 (z) 1 0 z ?1 h2 (z) C CB
(2.2)
1 CA : .. @ .. C . A . h0(z) } | (p ? 1){zup?1(z) } (0)u0(z) U~ (z)
2.3 Useful results In this paper, we use C [z ] and C (z ) respectively to represent the sets of all polynomial and rational functions in z ?1 with complex coecients. A matrix Q(z ) in C [z ] or C (z ) is said to be unimodular if detQ(z ) = non-zero constant independent of z ?1 : We need the following facts for later use.
Smith Form [3]: For any H(z) 2 C nm [z], there exist some unimodular matrices Q(z) 2 C nn [z] and P (z) 2 C mm [z] such that (1)
0 1(z) 0 BB 0 2(z) BB ... .. . B Q(z)H(z)P (z) = B 0 BB 0 0 BB 0. .. @ .
::: :::
0 0 . . . ... : : : q (z) ::: 0 .. .. . . . . 0 0 ::: 0 for some monic polynomials i (z ); i = 1; 2; :::q minfn; mg.
0 0 .. . 0 0 .. . 0
01 0C C .. C .C C 0C C 0C C . . . ... C A ::: 0
::: ::: ::: ::: :::
(2) 1 (z ) is the greatest common divisor of all entries of H(z ).
McMillan Form [3]: For any H(z) 2 C nm (z), there exist some unimodular matrices Q(z) 2 C nn [z] and P (z) 2 C mm [z] such that (1)
0 1(z)= 1(z) 0 BB 0 2 (z)= 2(z) BB .. .. . . BB Q(z)H(z)P (z) = B 0 0 BB 0 0 B@ .. ..
::: :::
0 0 .. ... . : : : q (z)= q(z) ::: 0 .. .. . . . . 0 0 ::: 0 for some monic polynomials i (z ) and i (z ), i = 1; 2; :::q minfn; mg. (2) 1(z ) is the least common denominator of all entries of H(z ).
De nition 2.1 (see [3].) The roots of i (z ) are called the transmission zeros of H(z ).
3
0 0 .. . 0 0 .. . 0
01 0C C .. C .C C 0C C 0C C . . . ... C A ::: 0
::: ::: ::: ::: :::
3 Single Input Multiple Output Systems 3.1 Useful De nitions When there is only a single user, oversampling by an integer or the use of multiple antenna allows the MIMO system of (2.1) reduces to a single input multiple output (SIMO) system, 0 h (z) 1 1 B (3.3) ~y(z) = @ ... C A u(z) = ~h(z)u(z): hn (z) Thus in SIMO zero-forcing, our goal is to design a vector lter
w~ (z) = (w1(z); w2(z); :::; wn(z))0 such that the equalizer output ub(z ) = w~ (z )0~y(z ) recovers the input signal u(z ) with a possible delay
w~ (z)0~y(z) = w~ (z)0~h(z)u(z) = z ?d u(z): Here, ()0 indicates complex conjugate and transpose. Note that w~ (z ) is either a stable rational transfer function in C n (z ) or a FIR system in C n [z ]. To make the distinction, we use the following de nitions:
De nition 3.1 (1) A transfer function ~h(z) 2 C nm [z] or C nm (z) is said to be stably ZFequalizable if there exists a nite order, stable and causal rational function w~ (z ) 2 C nm (z ) such that
w~ (z)0~h(z) = z ?d I
for some nite integer delay d 0. (2) ~h(z ) 2 C nm [z ] or C nm (z ) is said to be FIR ZF-equalizable if there exists a nite order FIR lter w~ (z ) 2 C nm [z ] such that
w~ (z)0~h(z) = z ?d I for some nite integer delay d 0.
Thus, if ~h(z ) is stably ZF-equalizable, then u(z ) can be recovered exactly by a stable equalizer w~ (z) in the absence of noise. In addition, if ~h(z) is FIR ZF-equalizable, then the zero-forcing equalizer w~ (z ) is an FIR lter. 4
Note that for SIMO systems, the transfer function ~h(z ) 2 C n [z ] or C n (z ). This implies that in the Smith form or in the McMillan form, P (z ) is a constant which can be absorbed by Q(z ), resulting in 0 (z)= (z) 1 0 (z) 1 C B B 0 CC Q(z)~h(z) = B B@ ... CA ; or Q(z)~h(z) = BB@ 0... CCA : 0 0 Clearly, for any SIMO ~h(z ) 2 C n [z ] or C n (z ), transmission zeros are also blocking zeros.
De nition 3.2 A transfer function ~h(z) is said to be minimum phase if all of its transmission zeros are stable.
The de nition of minimum phase SIMO system must be handled with care. In fact, when n > 1, ~h(z) is minimum phase does not imply that every zero hi (z) is stable. In fact, every zero of hi (z) may be unstable for a minimum phase ~h(z ) as long as hi (z )'s do not have common unstable zeros.
3.2 Zero-forcing Equalization of SIMO systems We now present some basic results on stable ZF-equalizability of FIR models for a SIMO system.
Theorem 3.1 If a SIMO system has n outputs with transfer function ~h(z) 2 C n[z], then (1) The transfer function ~h(z ) is stably ZF-equalizable i ~h(z ) is minimum phase; (2) ~h(z ) is FIR ZF-equalizable i (z ) = z ?d for some d 0.
Proof: For (1), if (z) is nonminimum phase, then (z0) = 0 for some jz0j 1. If w~ (z) is stable, then jjw~ (z0)jj < 1 and 0 (z ) 1 0 C B 0 w~ (z0)0~h(z0) = w~ (z0)0Q?1 (z0 ) B B@ ... CCA = 0 6= z0?d :
0 This means that ~h(z ) is not ZF-equalizable by any stable w~ (z ). Thus, ~h(z ) is not stably ZFequalizable, which completes the necessity part of (1). For suciency, suppose (z ) is stable. Let ?d w~ (z) = z(z) Q(z)0(1; 0; :::; 0)0
for some d 0 so that w(z ) is causal. Then it is apparent that ~h(z ) is stably ZF-equalizable as ?d w~ (z)0~h(z) = z(z) (1; 0; :::; 0)Q(z)Q?1(z)( (z); 0; :::; 0)0 = z ?d (z)= (z) = z ?d :
5
For the necessity part of (2), suppose (z ) 6= z ?d . Then (z0 ) = 0 for some jz0j < 1. Consequently, w~ (z0 )0~h(z0 ) = 0 6= z0?d
for any FIR w~ (z ) and ~h(z ) is not FIR ZF-equalizable. The suciency part of (2) is identical to the proof of (1) with (z ) = z ?d . Based on the properties of the Smith and McMillan Forms, we have
Corollary 3.1 ~h(z) is stably ZF-equalizable i fhi(z)g do not have any unstable common zeros,
i.e., their greatest common divisor (z ) is minimum phase. Furthermore, ~h(z ) is FIR ZF-equalizable i fhi (z )g do not have any common zero except at in nity, i.e., (z ) = z ?d ; d 0.
3.3 Relationship to Blind Identi ability There are two important questions in blind channel identi cation and equalization: identi ability and equalizability. Blind identi ability depends on what output information is used while equalizability only depends on the channel characteristics. They are di erent but are closely related. Using only second order statistics with more outputs than inputs, the blind identi ability of single input multiple output (SIMO) and multiple input multiple output (MIMO) channels has been addressed in [1][2][4][8][10][11]. >From Theorem 3.1, we know that for a SIMO system, FIR equalizability is equivalent to that all sub-channels share no common factor except some delay z ?d . With no delay or delay is not an integer multiple of the symbol interval, this condition reduces to that all sub-channels share no common factor which is exactly the condition for blind channel identi ability [10] based on second order statistics. It should be noted that for SIMO channels that are not FIR equalizable, higher order order statistics of the output signals can still be used to identify the channel.
3.4 Determining A Zero-forcing Equalizer If ~h(z ) is stably or FIR ZF-equalizable, a solution nding a stable/FIR equalizer w~ (z ) is direct with the use of the Smith Form. We need to consider two separate cases: (z ) = z ?d and minimum phase (z ).
6
If (z ) = z ?d , it is the greatest common divisor of fhi (z )g. Let 0 h (z) 1 1 ~h(z) = z ?d h(z) = z ?d B @ ... CA hn (z ) P in which hi (z ) = lk=0 hik z ?k ; i = 1; 2; :::n with l = L ? d. De ne the i?th lter in the vector P equalizer w~ (z ) as wi (z ) = lk?=01 wki z ?k ; i = 1; 2; ::n: Then, w~ (z )0h (z ) = 1 and w~ (z )0~h(z ) = z ?d if and only if
0 h1 BB ...0 BB 1 BB hl B@ |
h20
.. . . . ... . . . . h 1 h 2 . . . h 2 0 0 l . . . ... . . . ... h1l h2l
{z
T (h)
hn0
::: ::: ::: ::: :::
.. . . . . hnl . . . ...
0 w1 1 1 BB ..0 CC CC BB w1. CC 0 1 1 CC BB l.?1 CC BB 0 CC n BB .. CC = B .. C : h 0 C C B wn CC @ . A .. C . AB B .0 CC 0 hnl B . } @ wn. A | {zl?1 }
(3.4)
W
The band Toeplitz matrix T (h) has full row rank i fhi (z )g do not have any common zeros, i.e., i ~h(z ) = z ?d h(z ) does not have any blocking zeros. When T (h) has full row rank, zero-forcing solution W always exists but may not be unique. If a stable (z ) 6= z ?d , we can let h(z ) = (z )h(z ) and solve for w~ (z ) to achieve w~ (z )0h (z ) = 1. Then the zero-forcing equalizer is simply z ?d ?1 (z )w~ (z ) for some d 0 such that z ?d ?1 (z ) is causal.
3.5 Modeling Uncertainty and Additive Noise When the channel response is fully known, the zero-forcing equalizer design is simple. However, when modeling uncertainty and channel noise exist, their e ect should be minimized by appropriately utilizing the exibility of non-uniqueness in the solution of (3.4). Suppose that ~h(z ) is the nominal model which may be di erent from the true system ~h (z ). Under additive channel noise vector ~v(k), the true channel output vector is
~y(k) = ~h (z)u(k) + ~v(k): Now consider the case that ~h(z ) is FIR ZF-equalizable which means w~ (z )0~h(z ) = z ?d . We de ne the error signal as the di erence between the true equalizer output and the desired output
e(k) = w~ (z)0~y(k) ? z ?d u(k) 7
and the modeling error as
~h(z ) = ~h(z ) ? ~h (z ):
The total error can then be written as
e(k) = w~ (z)0~y(k) ? w~ (z)0~h(z)u(k) = w~ (z)0~h(z)u(k) + w~ (z)0~v(k): Suppose u(k) and vi (k)'s are i.i.d. zero mean with variance u2 , and v2 respectively, we have
E fe(k)g = 0 E fje(k)j2g = u2 W 0 H 0HW + v2 kW k2; where we have de ned
H = T (~h) and H = T (~h):
Clearly, s2W 0 H 0HW + v2 kW k2 is the variance of the error due to model uncertainty and noise. Note that W satis es
HW = ~ed+1 ; where ~ed+1 = (0; :::; 0; |{z} 1 ; 0:::; 0)0:
(3.5)
(d+1)th
Under the zero-forcing requirement HW = ~ed+1 , we would like to design a W to minimize the e ect of noise and mismodeling. Note that the e ect of uncertainty is re ected in terms of H 0H which is unknown and arbitrary. However, we can minimize the worst case error for all possible max(H 0H ) c, where max is the maximum eigenvalue. The constant c > 0 represents the upper bound on the channel mismatch and can be absorbed into u2 . Without loss of generality, we only need to consider the case max(H 0H ) 1. Mathematically, this is to nd a W that 2 0 H 0HW + 2kW k2 ) = v
min max ( W HW =~ed+1 max (H 0 H )1 s
2
2
min ( + v )kW k HW =~ed+1 s
2:
Theorem 3.2 Assume the nominal model ~h(z) be FIR ZF-equalizable. Let H = U (; 0)V 0 be the singular value decomposition of H in (3.5) and z1 be a solution of
U z1 = ~ed+1 Then,
W = V (z1 ; 0; :::0)0
assumes the minimum norm kW k2 among all W satisfying (3.5).
8
(3.6)
Proof: Let z = zz12 = V 0W . Then, z
HW = U (; 0)V 0 W = U (; 0) zz1 = ~ed+1 : 2
Solution of z = z1 always exists by the assumption that ~h(z ) is FIR ZF-equalizable and moreover 2 z 1 z = z is a solution if and only if U z1 = (0; :::; 0; 1; 0; :::; 0)0 with arbitrary z2 . Since kW k2 = 2 kzk2, the minimum norm kzk2 is obviously achieved by setting z2 = 0 and thus the result follows.
3.6 Equalization of IIR SIMO systems While FIR systems are easy to deal with, IIR SIMO systems require some straightforward modi cations. Most results and derivations are parallel to FIR cases, thus requiring either a short proof or no proof.
Theorem 3.3 Consider an SIMO system with transfer function ~h(z) 2 C n(z) with McMillan Form Q(z)~h(z) = ( (z)= (z); 0; :::; 0): Then, (1) ~h(z ) is stably ZF-equalizable i ~h(z ) is minimum phase. (2) ~h(z ) is FIR ZF-equalizable i it has no blocking zeros.
Proof: For (1), the necessary condition is obvious. For suciency, note from the McMillan Form that
De ne
Q(z) (z)~h(z) = ( (z); 0; :::; 0)0: w(z) = Q(z)0(1; 0; :::; 0)0 and w~ (z) = ((zz)) w(z)0z ?d
for d 0 so that w~ (z ) is causal. Then, w~ (z)0h(z) = ((zz)) z?d (1; 0; :::; 0)Q(z)Q?1(z)( (z); 0:::; 0)0 = z ?d : This completes the rst part. The proof of (2) is similar by de ning
w(z) = Q(z)0(1; 0:::; 0)0 and w~ (z) = (z)w(z): 9
This results in
w~ (z)0h(z) = (z)(1; 0; :::; 0)Q(z)= (z)Q?1(z)(z ?d ; 0:::; 0)0 = z ?d : The calculation of the lter w~ (z ) in the IIR case is almost identical to that of FIR cases. The key is to nd a w~ (z ) satisfying 0 1(z) 1 w~ (z)0 B @ ... CA = (z)z?d n (z) which can be solved in a similar way as in (3.4) replacing i (z ) by hi (z ) and the vector (1; 0; :::; 0)0 by the coecients of (z )z ?d .
4 MIMO systems Theorem 4.1 Consider an n m MIMO system with transfer function H(z). Let n m and H(z) have the Smith Form 0 1(z) 0 : : : 0 1 BB 0 2(z) : : : 0 CC BB ... .. .. C ... . . C C: B B Q(z)H(z)P (z) = B 0 0 : : : p (z ) C CC BB 0 0 : : : 0 C B@ .. .. .. .. C A Then,
. 0
. 0
.
:::
. 0
(1) H(z ) is stably ZF-equalizable i H(z ) is minimum phase (all i (z ) are minimum phase). (2) H(z ) is FIR ZF-equalizable i i (z ) = z ?di for some di 0.
Proof: Note that for MIMO systems, stable(FIR) equalizability implies that there is a stable(FIR) w~ (z) such that w~ (z)h(z) = z ?d I where I is the p p identity matrix. We now show the necessity of (1). Suppose there is a jz0 j 1 and (z0 ) = 0 for some i, rank h(z0 ) < p =) and w~ (z0)h(z0 ) < p. But rankz0?d I = p. This completes the necessity part. For suciency, let 0 1 0 ::: 0 0 ::: 01 1 (z) BB 0 1(z) : : : 0 0 : : : 0 CC 2 ? d CC Q(z) w~ (z) = z P (z) B B@ ... .. . . . .. .. . . ::: 0A . 1 0 0 : : : n (z) 0 : : : 0 10
for some d 0 so that w~ (z ) is causal. Then it is easily veri ed that w~ (z )h(z ) = z ?d I . This completes the suciency part of (1). The proof of (2) is similar.
5 Concluding remarks In this work, the zero-forcing equalizability of MIMO systems is studied. In communication systems where the channel is not FIR equalizable at the baud rate, oversampling may result in an SIMO or an MIMO system which is FIR-equalizable. Both the stable equalizability condition and the FIR equalizability condition of multi-channel systems are provided. The ZF equalizer design for SIMO systems under additive noise and modeling error was also derived.
References [1] K. Abed-Meriam, P. Duhamel, D. Gesbert, L. Loubaton, S. Mayrargue, E. Moulines and D. Slock, \Prediction Error Methods for Time-Domain Blind Identi cation of Multichannel FIR Filters," Proc. 1995 IEEE ICASSP, pp. 1968-1971, Detroit, MI, May 1995. [2] E.W. Bai and M.Y. Fu, \Blind system identi cation and channel equalization of IIR systems without statistical information", IEEE Trans. on Signal Processing, Vol. 47, No.7, pp.19101921, 1999 [3] F.M. Callier and C. A. Desoer, MULTIVARIABLE FEEDBACK SYSTEMS, Springer-Verlag, New York, 1982. [4] Zhi Ding, \Matrix Outer-product Decomposition Method for Blind Multiple Channel Identi cation," IEEE Trans. on Signal Processing, 45(12), pp. 3053-3061, 1997. [5] G.D. Forney,Jr., \Maximum-likelihood sequence estimation of digital sequences in the presence of inter-symbol interference," IEEE Trans. Information Theory, Vol. IT-18, pp.363-378, May 1972. [6] E. Moulines, P. Duhamel, J.-F. Cardoso, and S. Mayrargue, \Subspace Methods for the Blind Identi cation of Multichannel FIR Filters", IEEE Transactions on Signal Processing, SP43:516-525, Feb. 1995.
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[7] E. Serpedin and G. Giannakis, \Blind channel identi cation and equalization with modulation induced cyclostationarity", IEEE Transactions on Signal Processing, SP-46:1930-1944, July, 1998 [8] D. Slock, \Blind Fractionally-Spaced Equalization, Perfect-Reconstruction Filter Banks and Multichannel Linear Prediction", Proc. 1994 IEEE ICASSP, pp. IV:585-588, Adelaide, May 1994. [9] D. Slock, \Blind Joint Equalization of Multiple Synchronous Mobile Users Using Oversampling and/or Multiple Antennas", Proc. 1994 Asilomar Conf. on Signals, Systems, and Computers, pp.:1154-1158, Nov. 1994. [10] L. Tong, G. Xu and T. Kailath, \A new approach to blind identi cation and equalization of multi-path channels," IEEE Trans. on Info. Theory, IT-40:340-349, March 1994. [11] G. Xu, H. Liu, L. Tong and T. Kailath, \A least squares approach to blind channel identi cation", IEEE Trans. on Signal Processing, Vol. 43, No.12, pp.2982-2993, 1995
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Abstract
In this paper, we study the linear zero-forcing equalizability of a communication channel. Necessary and sucient conditions are given in terms of zeros of the transfer function. Detailed procedures of equalizer design for minimum sensitivity with respect to modeling errors and white noise are also presented. It is found that the worst-case e ect of channel mismatch on the performance may be modeled as equivalent losses in SNR.
This work was supported in part by grants from the National Science Foundation.
1 Introduction Zero-forcing equalization is a technique often used in high SNR communication receivers for the removal of inter-symbol interferences. Zero-forcing equalizers are essentially channel inverses used to recover the unknown input signal of a linear system from its measurable output signal. Since the goal of the equalizer is to recover the input signal from the output measurements. zero-forcing requires that the input signal be recovered EXACTLY with a nite delay in the absence of noise. One problem is that zero-forcing (ZF) equalizers may not exist. The goal of this correspondence is to de ne the conditions for the existence of ZF equalizer in various multi-channel systems. The approach is from a system theoretic point of view, based on the Smith and McMillian forms, which facilitates the conceptual generalization of ZF equalizers to the IIR cases. Equalizability conditions are characterized in terms of system poles/zeros. The design of zero-forcing equalizers under model uncertainty and additive noise is also addressed.
2 Preliminary and notation We consider communication channels modeled by
Y~ (z) = H(z)U~ (z)
(2.1)
where U~ (z ) is the m-dimensional message signal, Y~ (z ) the n-dimensional received signal and H(z ) the n by m dimensional transfer function. This model represents a large class of communication channels with multiple users and multiple sensor outputs. Two examples are considered here.
2.1 Fractionally over-sampled channels Consider a multi-user quadrature amplitude modulation data communication system. Let N be the number of user channels. Then, the received signal is given by
y(t) =
N X X j =1 k
hj (t ? kT )uj (k)
where T is the symbol baud period, hj (t) the composite impulse response of the j th channel and uj (k) input signal of j th user. Let y(t) be sampled at a rate of such that
pT = q; 1
for some positive integers p and q . It has been shown in [2, 4] that the over-sampled output can be viewed as a pN -input and q -output linear system exactly in the form of equation (2.1).
2.2 Channels with input modulation-induced cyclostationarity For channels with input modulation-induced cyclostationarity [7], the received signal is given by
y(t) =
X l
h(t ? lT )u(l) (l)
where the input signal u(k) is modulated by a deterministic and periodic sequence (k) with period p. For 0 i p ? 1, we de ne
yi (z) = =
X
y(kpT + iT )z ?k =
k pX ?1 X X j =1 l
k
XX l
k
h(kpT + iT ? lT )u(l) (l)z ?k
h(kpT + iT ? (lp + j )T )u(lp + j ) (lp + j )z ?k :
Since (lp + j ) = (j ), we have (letting r = k ? l) pX ?1
yi (z) =
j =1 pX ?1
=
j =1
where
hl(z) = uj (z) =
(j )
r
h(rpT + (i ? j )T )z ?r
X l
u(lp + j )z ?l
(j )hi?j (z)uj (z)
X k X k
X
h(kpT + lT )z ?k ;
?(p ? 1) l p ? 1;
u(kp + j )z ?k ; j = 0; 1; : : :p ? 1:
Thus, for a causal impulse response h(t) and ?p + 1 l ?1, we obtain
hl(z) =
X k
h(kpT + lT )z ?k = z ?1 hp+l (z):
Therefore, we obtain, exactly in the form of (2.1), 0 y0(z) 1 0 h0(z) z?1 hp?1(z) z?1hp?2(z) h1(z) h0 (z) z ?1hp?1 (z) B@ ... CA = B B B .. .. @ ... . . y ( z ) | p?{z1 } h (z) h (z) h (z) Y~ (z)
|
p?1
p?2
{zp?3
H(z)
2
::: ::: ...
:::
z ?1 h1 (z) 1 0 z ?1 h2 (z) C CB
(2.2)
1 CA : .. @ .. C . A . h0(z) } | (p ? 1){zup?1(z) } (0)u0(z) U~ (z)
2.3 Useful results In this paper, we use C [z ] and C (z ) respectively to represent the sets of all polynomial and rational functions in z ?1 with complex coecients. A matrix Q(z ) in C [z ] or C (z ) is said to be unimodular if detQ(z ) = non-zero constant independent of z ?1 : We need the following facts for later use.
Smith Form [3]: For any H(z) 2 C nm [z], there exist some unimodular matrices Q(z) 2 C nn [z] and P (z) 2 C mm [z] such that (1)
0 1(z) 0 BB 0 2(z) BB ... .. . B Q(z)H(z)P (z) = B 0 BB 0 0 BB 0. .. @ .
::: :::
0 0 . . . ... : : : q (z) ::: 0 .. .. . . . . 0 0 ::: 0 for some monic polynomials i (z ); i = 1; 2; :::q minfn; mg.
0 0 .. . 0 0 .. . 0
01 0C C .. C .C C 0C C 0C C . . . ... C A ::: 0
::: ::: ::: ::: :::
(2) 1 (z ) is the greatest common divisor of all entries of H(z ).
McMillan Form [3]: For any H(z) 2 C nm (z), there exist some unimodular matrices Q(z) 2 C nn [z] and P (z) 2 C mm [z] such that (1)
0 1(z)= 1(z) 0 BB 0 2 (z)= 2(z) BB .. .. . . BB Q(z)H(z)P (z) = B 0 0 BB 0 0 B@ .. ..
::: :::
0 0 .. ... . : : : q (z)= q(z) ::: 0 .. .. . . . . 0 0 ::: 0 for some monic polynomials i (z ) and i (z ), i = 1; 2; :::q minfn; mg. (2) 1(z ) is the least common denominator of all entries of H(z ).
De nition 2.1 (see [3].) The roots of i (z ) are called the transmission zeros of H(z ).
3
0 0 .. . 0 0 .. . 0
01 0C C .. C .C C 0C C 0C C . . . ... C A ::: 0
::: ::: ::: ::: :::
3 Single Input Multiple Output Systems 3.1 Useful De nitions When there is only a single user, oversampling by an integer or the use of multiple antenna allows the MIMO system of (2.1) reduces to a single input multiple output (SIMO) system, 0 h (z) 1 1 B (3.3) ~y(z) = @ ... C A u(z) = ~h(z)u(z): hn (z) Thus in SIMO zero-forcing, our goal is to design a vector lter
w~ (z) = (w1(z); w2(z); :::; wn(z))0 such that the equalizer output ub(z ) = w~ (z )0~y(z ) recovers the input signal u(z ) with a possible delay
w~ (z)0~y(z) = w~ (z)0~h(z)u(z) = z ?d u(z): Here, ()0 indicates complex conjugate and transpose. Note that w~ (z ) is either a stable rational transfer function in C n (z ) or a FIR system in C n [z ]. To make the distinction, we use the following de nitions:
De nition 3.1 (1) A transfer function ~h(z) 2 C nm [z] or C nm (z) is said to be stably ZFequalizable if there exists a nite order, stable and causal rational function w~ (z ) 2 C nm (z ) such that
w~ (z)0~h(z) = z ?d I
for some nite integer delay d 0. (2) ~h(z ) 2 C nm [z ] or C nm (z ) is said to be FIR ZF-equalizable if there exists a nite order FIR lter w~ (z ) 2 C nm [z ] such that
w~ (z)0~h(z) = z ?d I for some nite integer delay d 0.
Thus, if ~h(z ) is stably ZF-equalizable, then u(z ) can be recovered exactly by a stable equalizer w~ (z) in the absence of noise. In addition, if ~h(z) is FIR ZF-equalizable, then the zero-forcing equalizer w~ (z ) is an FIR lter. 4
Note that for SIMO systems, the transfer function ~h(z ) 2 C n [z ] or C n (z ). This implies that in the Smith form or in the McMillan form, P (z ) is a constant which can be absorbed by Q(z ), resulting in 0 (z)= (z) 1 0 (z) 1 C B B 0 CC Q(z)~h(z) = B B@ ... CA ; or Q(z)~h(z) = BB@ 0... CCA : 0 0 Clearly, for any SIMO ~h(z ) 2 C n [z ] or C n (z ), transmission zeros are also blocking zeros.
De nition 3.2 A transfer function ~h(z) is said to be minimum phase if all of its transmission zeros are stable.
The de nition of minimum phase SIMO system must be handled with care. In fact, when n > 1, ~h(z) is minimum phase does not imply that every zero hi (z) is stable. In fact, every zero of hi (z) may be unstable for a minimum phase ~h(z ) as long as hi (z )'s do not have common unstable zeros.
3.2 Zero-forcing Equalization of SIMO systems We now present some basic results on stable ZF-equalizability of FIR models for a SIMO system.
Theorem 3.1 If a SIMO system has n outputs with transfer function ~h(z) 2 C n[z], then (1) The transfer function ~h(z ) is stably ZF-equalizable i ~h(z ) is minimum phase; (2) ~h(z ) is FIR ZF-equalizable i (z ) = z ?d for some d 0.
Proof: For (1), if (z) is nonminimum phase, then (z0) = 0 for some jz0j 1. If w~ (z) is stable, then jjw~ (z0)jj < 1 and 0 (z ) 1 0 C B 0 w~ (z0)0~h(z0) = w~ (z0)0Q?1 (z0 ) B B@ ... CCA = 0 6= z0?d :
0 This means that ~h(z ) is not ZF-equalizable by any stable w~ (z ). Thus, ~h(z ) is not stably ZFequalizable, which completes the necessity part of (1). For suciency, suppose (z ) is stable. Let ?d w~ (z) = z(z) Q(z)0(1; 0; :::; 0)0
for some d 0 so that w(z ) is causal. Then it is apparent that ~h(z ) is stably ZF-equalizable as ?d w~ (z)0~h(z) = z(z) (1; 0; :::; 0)Q(z)Q?1(z)( (z); 0; :::; 0)0 = z ?d (z)= (z) = z ?d :
5
For the necessity part of (2), suppose (z ) 6= z ?d . Then (z0 ) = 0 for some jz0j < 1. Consequently, w~ (z0 )0~h(z0 ) = 0 6= z0?d
for any FIR w~ (z ) and ~h(z ) is not FIR ZF-equalizable. The suciency part of (2) is identical to the proof of (1) with (z ) = z ?d . Based on the properties of the Smith and McMillan Forms, we have
Corollary 3.1 ~h(z) is stably ZF-equalizable i fhi(z)g do not have any unstable common zeros,
i.e., their greatest common divisor (z ) is minimum phase. Furthermore, ~h(z ) is FIR ZF-equalizable i fhi (z )g do not have any common zero except at in nity, i.e., (z ) = z ?d ; d 0.
3.3 Relationship to Blind Identi ability There are two important questions in blind channel identi cation and equalization: identi ability and equalizability. Blind identi ability depends on what output information is used while equalizability only depends on the channel characteristics. They are di erent but are closely related. Using only second order statistics with more outputs than inputs, the blind identi ability of single input multiple output (SIMO) and multiple input multiple output (MIMO) channels has been addressed in [1][2][4][8][10][11]. >From Theorem 3.1, we know that for a SIMO system, FIR equalizability is equivalent to that all sub-channels share no common factor except some delay z ?d . With no delay or delay is not an integer multiple of the symbol interval, this condition reduces to that all sub-channels share no common factor which is exactly the condition for blind channel identi ability [10] based on second order statistics. It should be noted that for SIMO channels that are not FIR equalizable, higher order order statistics of the output signals can still be used to identify the channel.
3.4 Determining A Zero-forcing Equalizer If ~h(z ) is stably or FIR ZF-equalizable, a solution nding a stable/FIR equalizer w~ (z ) is direct with the use of the Smith Form. We need to consider two separate cases: (z ) = z ?d and minimum phase (z ).
6
If (z ) = z ?d , it is the greatest common divisor of fhi (z )g. Let 0 h (z) 1 1 ~h(z) = z ?d h(z) = z ?d B @ ... CA hn (z ) P in which hi (z ) = lk=0 hik z ?k ; i = 1; 2; :::n with l = L ? d. De ne the i?th lter in the vector P equalizer w~ (z ) as wi (z ) = lk?=01 wki z ?k ; i = 1; 2; ::n: Then, w~ (z )0h (z ) = 1 and w~ (z )0~h(z ) = z ?d if and only if
0 h1 BB ...0 BB 1 BB hl B@ |
h20
.. . . . ... . . . . h 1 h 2 . . . h 2 0 0 l . . . ... . . . ... h1l h2l
{z
T (h)
hn0
::: ::: ::: ::: :::
.. . . . . hnl . . . ...
0 w1 1 1 BB ..0 CC CC BB w1. CC 0 1 1 CC BB l.?1 CC BB 0 CC n BB .. CC = B .. C : h 0 C C B wn CC @ . A .. C . AB B .0 CC 0 hnl B . } @ wn. A | {zl?1 }
(3.4)
W
The band Toeplitz matrix T (h) has full row rank i fhi (z )g do not have any common zeros, i.e., i ~h(z ) = z ?d h(z ) does not have any blocking zeros. When T (h) has full row rank, zero-forcing solution W always exists but may not be unique. If a stable (z ) 6= z ?d , we can let h(z ) = (z )h(z ) and solve for w~ (z ) to achieve w~ (z )0h (z ) = 1. Then the zero-forcing equalizer is simply z ?d ?1 (z )w~ (z ) for some d 0 such that z ?d ?1 (z ) is causal.
3.5 Modeling Uncertainty and Additive Noise When the channel response is fully known, the zero-forcing equalizer design is simple. However, when modeling uncertainty and channel noise exist, their e ect should be minimized by appropriately utilizing the exibility of non-uniqueness in the solution of (3.4). Suppose that ~h(z ) is the nominal model which may be di erent from the true system ~h (z ). Under additive channel noise vector ~v(k), the true channel output vector is
~y(k) = ~h (z)u(k) + ~v(k): Now consider the case that ~h(z ) is FIR ZF-equalizable which means w~ (z )0~h(z ) = z ?d . We de ne the error signal as the di erence between the true equalizer output and the desired output
e(k) = w~ (z)0~y(k) ? z ?d u(k) 7
and the modeling error as
~h(z ) = ~h(z ) ? ~h (z ):
The total error can then be written as
e(k) = w~ (z)0~y(k) ? w~ (z)0~h(z)u(k) = w~ (z)0~h(z)u(k) + w~ (z)0~v(k): Suppose u(k) and vi (k)'s are i.i.d. zero mean with variance u2 , and v2 respectively, we have
E fe(k)g = 0 E fje(k)j2g = u2 W 0 H 0HW + v2 kW k2; where we have de ned
H = T (~h) and H = T (~h):
Clearly, s2W 0 H 0HW + v2 kW k2 is the variance of the error due to model uncertainty and noise. Note that W satis es
HW = ~ed+1 ; where ~ed+1 = (0; :::; 0; |{z} 1 ; 0:::; 0)0:
(3.5)
(d+1)th
Under the zero-forcing requirement HW = ~ed+1 , we would like to design a W to minimize the e ect of noise and mismodeling. Note that the e ect of uncertainty is re ected in terms of H 0H which is unknown and arbitrary. However, we can minimize the worst case error for all possible max(H 0H ) c, where max is the maximum eigenvalue. The constant c > 0 represents the upper bound on the channel mismatch and can be absorbed into u2 . Without loss of generality, we only need to consider the case max(H 0H ) 1. Mathematically, this is to nd a W that 2 0 H 0HW + 2kW k2 ) = v
min max ( W HW =~ed+1 max (H 0 H )1 s
2
2
min ( + v )kW k HW =~ed+1 s
2:
Theorem 3.2 Assume the nominal model ~h(z) be FIR ZF-equalizable. Let H = U (; 0)V 0 be the singular value decomposition of H in (3.5) and z1 be a solution of
U z1 = ~ed+1 Then,
W = V (z1 ; 0; :::0)0
assumes the minimum norm kW k2 among all W satisfying (3.5).
8
(3.6)
Proof: Let z = zz12 = V 0W . Then, z
HW = U (; 0)V 0 W = U (; 0) zz1 = ~ed+1 : 2
Solution of z = z1 always exists by the assumption that ~h(z ) is FIR ZF-equalizable and moreover 2 z 1 z = z is a solution if and only if U z1 = (0; :::; 0; 1; 0; :::; 0)0 with arbitrary z2 . Since kW k2 = 2 kzk2, the minimum norm kzk2 is obviously achieved by setting z2 = 0 and thus the result follows.
3.6 Equalization of IIR SIMO systems While FIR systems are easy to deal with, IIR SIMO systems require some straightforward modi cations. Most results and derivations are parallel to FIR cases, thus requiring either a short proof or no proof.
Theorem 3.3 Consider an SIMO system with transfer function ~h(z) 2 C n(z) with McMillan Form Q(z)~h(z) = ( (z)= (z); 0; :::; 0): Then, (1) ~h(z ) is stably ZF-equalizable i ~h(z ) is minimum phase. (2) ~h(z ) is FIR ZF-equalizable i it has no blocking zeros.
Proof: For (1), the necessary condition is obvious. For suciency, note from the McMillan Form that
De ne
Q(z) (z)~h(z) = ( (z); 0; :::; 0)0: w(z) = Q(z)0(1; 0; :::; 0)0 and w~ (z) = ((zz)) w(z)0z ?d
for d 0 so that w~ (z ) is causal. Then, w~ (z)0h(z) = ((zz)) z?d (1; 0; :::; 0)Q(z)Q?1(z)( (z); 0:::; 0)0 = z ?d : This completes the rst part. The proof of (2) is similar by de ning
w(z) = Q(z)0(1; 0:::; 0)0 and w~ (z) = (z)w(z): 9
This results in
w~ (z)0h(z) = (z)(1; 0; :::; 0)Q(z)= (z)Q?1(z)(z ?d ; 0:::; 0)0 = z ?d : The calculation of the lter w~ (z ) in the IIR case is almost identical to that of FIR cases. The key is to nd a w~ (z ) satisfying 0 1(z) 1 w~ (z)0 B @ ... CA = (z)z?d n (z) which can be solved in a similar way as in (3.4) replacing i (z ) by hi (z ) and the vector (1; 0; :::; 0)0 by the coecients of (z )z ?d .
4 MIMO systems Theorem 4.1 Consider an n m MIMO system with transfer function H(z). Let n m and H(z) have the Smith Form 0 1(z) 0 : : : 0 1 BB 0 2(z) : : : 0 CC BB ... .. .. C ... . . C C: B B Q(z)H(z)P (z) = B 0 0 : : : p (z ) C CC BB 0 0 : : : 0 C B@ .. .. .. .. C A Then,
. 0
. 0
.
:::
. 0
(1) H(z ) is stably ZF-equalizable i H(z ) is minimum phase (all i (z ) are minimum phase). (2) H(z ) is FIR ZF-equalizable i i (z ) = z ?di for some di 0.
Proof: Note that for MIMO systems, stable(FIR) equalizability implies that there is a stable(FIR) w~ (z) such that w~ (z)h(z) = z ?d I where I is the p p identity matrix. We now show the necessity of (1). Suppose there is a jz0 j 1 and (z0 ) = 0 for some i, rank h(z0 ) < p =) and w~ (z0)h(z0 ) < p. But rankz0?d I = p. This completes the necessity part. For suciency, let 0 1 0 ::: 0 0 ::: 01 1 (z) BB 0 1(z) : : : 0 0 : : : 0 CC 2 ? d CC Q(z) w~ (z) = z P (z) B B@ ... .. . . . .. .. . . ::: 0A . 1 0 0 : : : n (z) 0 : : : 0 10
for some d 0 so that w~ (z ) is causal. Then it is easily veri ed that w~ (z )h(z ) = z ?d I . This completes the suciency part of (1). The proof of (2) is similar.
5 Concluding remarks In this work, the zero-forcing equalizability of MIMO systems is studied. In communication systems where the channel is not FIR equalizable at the baud rate, oversampling may result in an SIMO or an MIMO system which is FIR-equalizable. Both the stable equalizability condition and the FIR equalizability condition of multi-channel systems are provided. The ZF equalizer design for SIMO systems under additive noise and modeling error was also derived.
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[7] E. Serpedin and G. Giannakis, \Blind channel identi cation and equalization with modulation induced cyclostationarity", IEEE Transactions on Signal Processing, SP-46:1930-1944, July, 1998 [8] D. Slock, \Blind Fractionally-Spaced Equalization, Perfect-Reconstruction Filter Banks and Multichannel Linear Prediction", Proc. 1994 IEEE ICASSP, pp. IV:585-588, Adelaide, May 1994. [9] D. Slock, \Blind Joint Equalization of Multiple Synchronous Mobile Users Using Oversampling and/or Multiple Antennas", Proc. 1994 Asilomar Conf. on Signals, Systems, and Computers, pp.:1154-1158, Nov. 1994. [10] L. Tong, G. Xu and T. Kailath, \A new approach to blind identi cation and equalization of multi-path channels," IEEE Trans. on Info. Theory, IT-40:340-349, March 1994. [11] G. Xu, H. Liu, L. Tong and T. Kailath, \A least squares approach to blind channel identi cation", IEEE Trans. on Signal Processing, Vol. 43, No.12, pp.2982-2993, 1995
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