IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. 2, FEBRUARY 1996 143
Quadrilevel Periodic Sequences
for
Fast
Start-up Equalization and Channel Estimation
Ging-Shing Liu and Che-Ho WeiAbstract-A class of four-level periodic sequences derived from
the binary m-sequences are presented. These sequences can be used as the training sequences in adaptive channel estimation and equalization with fast start-up for multilevel PAM or QAM transmission. The sequence generator can be implemented by simple hardware.
I. INTRODUCTION
DAPTIVE signal processing has been widely applied
A
to channel equalizatiodestimation of band-limited chan- nels. A training process with a predetermined reference pattern known also by the receiver during the start-up period is often required. Mueller and Spaulding [2] have proposed a fast cyclic training algorithm by using periodic sequences for fast start-up adaptive equalization. The problem can be considered from the viewpoint of optimization of the training sequences [1]-[6], where several candidates have been de- signed to achieve rapid convergence. In this letter, our attention is concentrated on the construction of the four-level {f
1, f3) periodic training sequences with desirable correlation property.11. FOUR-LEVEL SEQUENCES FOR FAST START-UP
To speed up the convergence rate, the eigenvalue spread of the periodic training pattern a = [a0 a 1 a2
.
.
~ ~ - 1 1 ofN symbols must be small. That is, the pattern with a flat spectrum (or equivalently, an impulse-like autocorre- lation function) is a good choice. Based on this criterion, self-orthogonal sequences, constant-amplitude and zero- autocorrelated (CAZAC) sequences and PN sequences are usually considered as candidates. If a sequence which is orthogonal to any cyclically shifted version a(,) of itself, i.e., satisfying a
.
a(,)* = AS, (A: sequence energy), then it is called a self-orthogonal sequence [4]. A CAZAC [51, [7] sequence provides constant spectrum amplitude and its correlation satisfies a.
a ( J ) = 6,. The CAZAC sequence can be used to reduce the variance of the estimate such that the convergence is accelerated. However, the CAZAC sequences with N>
4 cannot be constructed by the symbols in { f l , f3,.. .
,
f 2 M - 1) [7]. Another solution is to use a short pseudo-random binary sequence of length N = 2m-1Paper approved by N. A. Zervos, the Editor for Fadinfiualization of the IEEE Communications Society. Manuscript received September 10, 1992; revised June 15, 1994 and March 15, 1995. This work was supported by the
National Science Council of the Republic of China under Grant NSC81-0404-
EOO9-009
G.4. Liu is with the National Space Program Office, Hsinchu, Taiwan, R.O.C.
C.-H. Wei is with the Institute of Electronics and Center for Telecommuni- cations Research, National Chiao-Tung University, Hsinchu, Taiwan, R.O.C.
Publisher Item Identifier S 0090-6778(96)01612-1.
4
4
4
binary nr-sequence generator
(ck = -&L(&-z)
Fig. 1. Proposed sequence generator.
for cyclic training [ l ] , [2], [6]. They are usually generated by an m-bit ( m = llog2(N
+
1)) maximum-length sequence (or referred to as m-sequence) generator. Although they are not self-orthogonal and zero-autocorrelation, they are often used in practice as the approximated ones. For high speed transmission, the multilevel signaling method has been widely adopted to achieve bandwidth efficiency. Thus, multilevel cyclic sequences wiith better correlation properties become useful in channel equalization and/or channel estimation in these cases.For a four-level m-sequence, it can be generated directly from Galois field GF(4), but they are not widely accepted due to the complexity problem [8]. A four-level sequence generator based on a binary m-sequence generator with generating poly- nomial g ( x ) is presented and shown in Fig. 1. The incoming
binary m-sequence is stored in a buffer with depth of L. The four-level sequence { c k } is related to the binary symbols
{+l, -1). The symbol C, is represented equivalently by a
pair of binary digits ( a k - L , a k ) . This operation of mapping and buffering can be further simplified by Boolean algebra and realized by combinational logic gates. By using the computer- aided search method., several generators with sequences having an impulse-like autocorrelation function are tabulated in Table I. These sequence generators are uniquely characterized by
g(z), denoted by an octal representation, and the memory
depth L. There are two choices of memory depth L for each case. Due to the hardware complexity consideration, the time span (N) of the equalizer or channel estimator usually considered is less than 128. Thus, those four-level sequence generators with N
<
128 are listed. Two other longer a k - L and a k by c k = - a k - L ( a k - 2), where a k - L , a k E144 EEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. 2, FEBRUARY 1996 TABLE I
FOUR-LEVEL SEQUENCE GENERATORS “
Generator Polynomial Buffer Length
g(x)(in octal) L Degree L“;gth m 3 7 13* 4 s 4 15 23* 11 12 5 31 45* 13 14 75 11 12 67 12 13 6 63 103* 57 58 147 38 39 155 55 56 7 127 211* 217 235 367 277 325 203* 313 345 96 97 40 41 9 10 54 55 108 109 20 21 120 121 88 _ _ x9 113 i i 4 8 255 435 230 231 9 511 1021* 381 382
*: only two feedback connections
sequences ( N
>
128) are also supplemented. In general, the autocorrelation function {y3} for a periodic sequence can be defined by y3 $ a.
~ ( 3 1 , j = 0 , 1 , 2 , .. .
,
N - 1. Based onthis definition, the autocorrelation of the four-level sequences obtained from Table I satisfies
j = 0 (mod N )
y3 =
{
2 - p ,:;+
j = l ( m o d N ) . (1) otherwiseAs the sequence length N increases, its correlation approaches that of a four-level CAZAC sequence but with a correlation
time confined among two symbol periods. Although there is no CAZAC sequence on { + 3 , f l , -1, - 3 ) with length N
>
4, we actually find the approximated ones. Since the sequence generator can be realized simply by several storage elements and logic gates, it is very efficient in the hardware implementation.111. CONCLUSION
A class of four-level periodic sequences with special cor- relation properties are presented. They can be used for fast start-up channel equalization or estimation. Several sequence generators with useful length are obtained by means of a computer search.
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