On the autocorrelation properties of truncated maximum-length sequences and their effect on the power spectrum Marco Baldi, Member, IEEE, Franco Chiaraluce, Member, IEEE, Noureddine Boujnah and Roberto Garello, Senior Member, IEEE Abstract Truncated maximum-length binary sequences are studied in this paper. The impact of truncation on their autocorrelation properties and power spectral density is investigated. Several new analytical results are given and validated through simulation. The first and second order statistics of the periodic autocorrelation function and the spectral peak amplitudes over the ensemble of all possible starting seeds are analyzed. Explicit bounds are found for the mean square of the periodic autocorrelation function. An analytical technique for evaluating the maximum spectral peak values is derived. As a case study, high data rate space links using LFSR randomizers are considered. Truncation may induce high peaks in the spectrum, requiring suitable margins to comply with power flux density constraints. The new results allow to analytically estimate the margin, providing useful information for the link design.
Index Terms Maximum-length sequences, linear feedback shift registers, periodic autocorrelation function, truncation noise.
I. I NTRODUCTION Linear feedback shift registers (LFSRs) are widely used for generating maximum-length binary sequences (in short, L-sequences) with very good correlation properties. It is known that, through a suitable Copyright (c) 2010 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to
[email protected]. M. Baldi and F. Chiaraluce are with Dipartimento di Ingegneria Biomedica, Elettronica e Telecomunicazioni, Università Politecnica delle Marche, via Brecce Bianche 12, 60131 Ancona, Italy (e-mail:
[email protected],
[email protected]). N. Boujnah and R. Garello are with Dipartimento di Elettronica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy (e-mail:
[email protected],
[email protected]).
1
2
choice of the feedback coefficients [1], an LFSR with L cells produces a sequence of length N = 2L − 1 bits, with nearly-ideal properties, i.e., similar to those of a sequence of independent and identically distributed binary random variables. Just because of these properties, L-sequences can be found in several signal processing and telecom applications, including: rapid acquisition in radar ranging systems, spreading sequences in Code Division Multiple Access systems, randomization of binary data, and many others [2]. Their performance can be compared with that of other solutions, like Gold codes, Kasami codes, or chaotic sequences [3]. However, the nearly-optimal correlation properties of L-sequences are strictly dependent on their exact length. In fact, just cutting out one bit in the designed length N can significantly change the correlation properties. Truncated L-sequences (TL-sequences) are segments of L-sequences where the last (or, equivalently, the first) C bits have been cut. Pruning causes the appearance of a truncation noise in their autocorrelation function, that may considerably decrease the system performance. On the other hand, the need to shorten the L-sequence often appears in practical applications. As an example, a segment of a long L-sequence can be used as the correlator reference signal in synchronization systems, in order to decrease the acquisition time. In [4], it was shown that the crosscorrelation properties of TL-sequences make them not useful for multi-user ranging. A study on global navigation satellite systems, focusing on Galileo E1 signals [5], demonstrated that random codes generated through genetic algorithms can outperform TL-sequences. In data randomization, the L-sequence can be shortened to match the data field size [6], [7]; as showed in [8], this truncation may generate high peaks in the power spectral density (PSD) of transmitted signals. Previous work on TL-sequences started in the 70’s (see [9], [10] and the references therein). In [9], expressions were derived for the moments of the distribution of weights of TL-sequences, and an algorithm was presented for the computation of the third moment. In [10] the procedure was extended to the fourth moment and numerical results were given for subsequence length ≤ 100 bits. Later, an interest was devoted to triple correlation [11], that allows to determine the period of the mother L-sequence and the characteristic polynomial of the LFSR which generates it (with potential applications in cryptography). Triple correlation can also be used in receivers for complete or truncated direct-sequence spread-spectrum signals in the presence of noise and multiple-access interference; in [12], it was verified that the quality of the data estimates depends on the length of the TL-sequences. In this paper, we will focus on the properties of: •
The periodic autocorrelation function. The correlation properties of different truncated sequences can exhibit significant differences, depending on both the LFSR characteristic polynomial and the
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starting seed. New results will be provided, characterizing the first and second order means of the autocorrelation function, averaged over all possible starting seeds. •
The power spectral density. The mean and the variance of the peak amplitudes will be analyzed, and a technique for estimating their maximum values will be provided.
Apart from their theoretical interest, these results can find useful application in practice. As a case study, we will focus on the adoption of LFSR for data randomization in space links. On board, L-sequences generated by an LFSR randomizer are used to increase the randomness of transmitted data and truncation is adopted to match the frame size [13]. Unfortunately, when all-zero sequences are transmitted, this may induce high peaks in the power spectral density. To overcome these deviations from the ideal behavior, and to satisfy the limits on the receiver power flux density, high margins are usually introduced in a heuristic way, and the transmitted power is strongly reduced with possible impact on the error rate performance. Up to now, the evaluation of this link budget margin has been faced only through measurements or qualitative considerations. The results presented in this paper allow to analytically obtain a good estimation of the required margin. The organization of the paper is as follows. In Section II we briefly recall the basic properties of Lsequences. In Section III we investigate the first and second order means of the autocorrelation function for TL-sequences. In Section IV we study the properties of their power spectral density. In Section V we present the application of the new results to space links. Finally, Section VI concludes the paper. II. M AXIMUM - LENGTH
SEQUENCES AND THEIR TRUNCATED VERSIONS
−1 Let us consider an N -bit binary sequence u = {uk }N k=0 , uk ∈ {0, 1}, and its bipolar representation −1 b = {bk }N k=0 , bk ∈ {−1, +1}, obtained by uk = 0 → bk = +1 and uk = 1 → bk = −1. The
normalized periodic autocorrelation function of the sequence (interpreted as the principal period of an infinite sequence) is: Ru (τ ) =
N −1 1 X bk bk+τ , N
(1)
k=0
where τ is an integer: 0 ≤ τ < N , and k + τ ≡ (k + τ ) mod N . Let us denote by u(d) the cyclically shifted version of u by d bits, from left to right (according with this notation, u(0) represents the sequence u without any shift). The autocorrelation function is invariant to shifting operations, i.e.: Ru(d) (τ ) = Ru (τ ).
(2)
An “ideal” long random sequence, with even length, should have Ru (τ ) = 0, ∀τ 6= 0, i.e., it should
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c(1) x
Fig. 1.
s(1)
c(2)
s(2)
c(L-1)
c(3) s(L-1)
s(3)
s(L)
c(L) u
The LFSR structure.
be orthogonal to all its shifted versions. (It is a common guess that ideal random sequences do not exist for non-trivial lengths N > 4 bits.) A. L-sequences An LFSR is shown in Fig. 1 and consists of: •
L binary cells, whose contents will be denoted by s(1), s(2), ...s(L) ∈ {0, 1}. At time t = 0 the
LFSR contains the initial non-zero seed: s0 = [s0 (1), s0 (2), ..., s0 (L)]. •
L potential feedback connections c(1), c(2), ...c(L) ∈ {0, 1}. (The i-th connection exists if c(i) = 1
and does not exist if c(i) = 0.) They define the coefficients of the characteristic polynomial g(D) = P L−i , that can be used to describe the LFSR structure in a compact way. DL + L i=1 c(i)D
The resulting binary sequence u is periodic and, nominally, of infinite length. If (and only if) the
characteristic polynomial g(D) is primitive, the sequence has period N = 2L − 1. (The number of φ(N ) L ,
primitive polynomials of degree L is equal to
where φ is the Euler’s totient function [1].) In the
following, we will call L-sequence the length-N principal period of an LFSR sequence with a primitive polynomial. It is well known that, when u is an L-sequence, the autocorrelation function is nearly-ideal. In fact, for any starting seed we have: 1 Ru (τ ) = −1
N
B. Truncated L-sequences
if τ = 0,
(3)
if 1 ≤ τ ≤ N − 1.
Given an L-sequence u with length N , let us suppose to cut out its last C bits: the new sequence has length M = N − C , and we call it a TL-sequence (u, M ). Its autocorrelation function is given by: M −1 1 X bk bk+τ Ru,M (τ ) = M
(4)
k=0
where 0 ≤ τ < M and k + τ ≡ (k + τ ) mod M . Unfortunately, a TL-sequence can lose its nearly-ideal properties in terms of autocorrelation function.
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TABLE I T RUNCATED SEQUENCES FOR L = 3 AND M = 4 BITS AND THEIR AUTOCORRELATION PROPERTIES (u, 4)
deleted bits
Ru,4 (1)
Ru,4 (2)
Ru,4 (3)
+1 + 1 − 1 − 1
−1 + 1 − 1
0
−1
0
−1 + 1 + 1 − 1
−1 − 1 + 1
0
−1
0
+1 − 1 + 1 + 1
−1 − 1 − 1
0
0
0
−1 + 1 − 1 + 1
+1 − 1 − 1
−1
1
−1
−1 − 1 + 1 − 1
+1 + 1 − 1
0
0
0
−1 − 1 − 1 + 1
−1 + 1 + 1
0
0
0
+1 − 1 − 1 − 1
+1 − 1 + 1
0
0
0
Let us denote by Rs0 ,M (τ ) the autocorrelation function (4) for a specific initial seed. Because of the cyclic property of the L-sequence, changing the initial seed is equivalent to consider a cyclically shifted version of the original L-sequence u. Since each seed s0 is univocally related to a shift value d, in the following we will also use the following equality: Ru(d) ,M (τ ) = Rs0 ,M (τ ).
(5)
Example 1 Let us consider an LFSR with L = 3 cells and characteristic polynomial g(D) = D3 +D2 +1. It produces the L-sequence u = 0011101 with N = 7 or a cyclically shifted version of it, depending on the initial seed. Table I considers all these possible L-sequences. By cutting out the last C = 3 bits, we obtain the seven TL-sequences (u, 4) shown in the left column of the table. The last three columns report the values of the autocorrelation function Ru,4 (τ ) of the TL-sequences for τ = 1, 2, 3.
⋄
Although very simple, Example 1 shows that (i) the autocorrelation function of a TL-sequence is significantly different from that of the original L-sequence and (ii) it depends on the starting seed. III. T HE AUTOCORRELATION
FUNCTION OF
TL- SEQUENCES
For an L-sequence we have R(τ ) = −1/N , when τ 6= 0, for any starting seed. Instead, the autocorrelation function of the truncated sequence depends on the initial seed value. A number of recurrence relations, that simplify computation of Rs0 ,M (τ ) of a TL-sequence, have been reported in [14]. Even more relevant, however, it is interesting to investigate the statistical properties of the autocorrelation function. Since the correlation values, for a given τ , depend on the starting seed s0 , we will compute their average behavior with respect to s0 . The mean value of Rs0 ,M (τ ) can be easily characterized by the following Lemma, obtained by adapting a result from [9]:
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Lemma III.1 Given an L-sequence whose length is reduced from N to M bits by truncation, the mean value of the periodic autocorrelation function over the ensemble of all possible seeds for the L-sequence, is equal to
1 Rs0 ,M (τ ) = − N
∀τ 6= 0.
(6)
Proof: Let us denote by b(d) the bipolar sequence that corresponds to u(d) . By using (4) and (5), and by applying the shift-and-multiply group property of L-sequences [15], according to which the product (d)
(d)
of an L-sequence (bk ) with a shifted version of the same sequence (bk+τ , 1 ≤ τ ≤ M − 1) gives again (d)
the sequence but differently shifted (bm ), together with the observation that the bipolar L-sequence has one more (−1) than (+1), we have:
Rs0 ,M (τ ) =
N −1 M −1 M −1 N −1 X X X X 1 X 1 1 (d) (d) (d) (d) Rs0 ,M (τ ) = bk bk+τ = bk bk+τ N s N ·M N ·M d=0 k=0
0
=
1 N ·M
−1 M −1 N X X k=0 d=0
b(d) m =
1 N ·M
M −1 X
k=0 d=0
(−1) = −
k=0
1 . N
(7)
So, the average value of Rs0 ,M (τ ) is independent of τ and is equal to the ideal value of the original L-sequence. (Note that the TL-sequence length is changed to M ≤ N .) The difference is that, for TL-sequences, (6) represents the mean value of R(τ ) over all possible seeds. It should also be noted that (6) is independent of the particular characteristic polynomial adopted. Changing the characteristic polynomial, the mother L-sequence changes but its “structural” properties (for example the run lengths) remain the same. The average over all seeds only depends on these global structural properties and not on the specific polynomial. A. Exact computation of the mean square autocorrelation function
The computation of the second order statistical average of Rs0 ,M (τ ), i.e., the mean square [Rs0 ,M (τ )]2 ,
is more involved. Obviously, it can be obtained through direct calculation, though this can require long
elaboration time in the case of large L. Anyway, through an extension of the approach presented in [9] and [10], it is possible to develop a semi-analytical algorithm that highly simplifies its computation. Lemma III.2 Given an L-sequence whose length is reduced from N to M bits by truncation, the mean square value of the periodic autocorrelation function over the ensemble of all possible seeds for the L-sequence is given by −1 −1 N N −1 M −2 M X X X
2 1 X 2 1 (d) (d) (d) (d) bj bk bj+τ bk+τ , Ru(d) ,M (τ ) = + [Rs0 ,M (τ )]2 = N M N · M2 d=0
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(8)
j=0 k=j+1 d=0
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−3
5
x 10
4.5
® [Rs0 ,M (τ)]2
4 3.5 3 2.5 2 1.5 1 0
Fig. 2.
50
100
150 τ
200
250
300
Example of mean square autocorrelation function over all possible seeds; LFSR with L = 9 and M = 300 bits.
where the sum over d at the right side can assume only two values (−1 and N ) and can be computed through the semi-analytical algorithm reported in Appendix A. Example 2 An example of exact computation of the mean square autocorrelation function by (8) is shown in Fig. 2 for L = 9, that is N = 511, g(D) = D9 + D8 + D4 + D + 1, and M = 300 bits: the curve exhibits a characteristic behavior, with a triangular envelope.
⋄
Differently from (6), expression (8) cannot be simplified in such a way as to become independent of
the sequence bits. So, [Rs0 ,M (τ )]2 depends on the characteristic polynomial g(D). However, as it will
be shown in the following section, lower and upper bounds on [Rs0 ,M (τ )]2 can be obtained, that are independent of the characteristic polynomial.
B. Properties and bounds of the mean square autocorrelation function Though the actual behavior of the mean square autocorrelation function cannot be predicted explicitly in analytical terms, a number of properties can be derived (see Appendix B). Some of these properties are discussed next. Lemma III.3 Given an L-sequence whose length is reduced from N to M bits by truncation, the mean square of the periodic autocorrelation function over the ensemble of all possible seeds for the L-sequence, given by (8), is symmetric with respect to τ = M/2. Proof: See Appendix B. August 17, 2010
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More noticeably, the following Theorem III.1 gives lower and upper bounds on the mean square autocorrelation function of a TL-sequence. Theorem III.1 Given an L-sequence whose length is reduced from N to M bits by truncation, the mean square of the periodic autocorrelation function over the ensemble of all possible seeds for the L-sequence, when τ 6= 0, can be bounded as:
M −1 1 1− ≤ [Rs0 ,M (τ )]2 ≤ M N
1 M 1 M
Proof: See Appendix B.
M −1 N M −1 N
1− 1−
+ 2 Mτ 2 NN+1
+ 2 MM−τ 2
N +1 N
for 0 < τ ≤ for
M 2
M 2 ,
(9)
< τ < M.
Obviously, [Rs0 ,M (0)]2 = 1. Note that for M = N (L-sequences) the lower bound gives the correct
value 1/N 2 , that is the only one to be assumed for this deterministic case. Eq. (9) explains the triangular behavior shown in Fig. 2 and permits us to extend its validity for arbitrary M and N . IV. T HE
SPECTRUM OF A TRUNCATED
L- SEQUENCE
From the knowledge of the autocorrelation function properties, it is possible to study the power spectrum of the TL-sequences. Let us suppose that the binary sequence u is characterized by a bit rate Rb [bit/s]. Each bit uk lasts for a bit time Tb = 1/Rb [s] and is mapped into a pulse bk g(t) with positive or negative amplitude depending on the bit value. Then, the sequence u is mapped into a waveform s(t) = P k bk g(t − kTb ). The simplest case is the so-called non-return-to-zero (NRZ) representation, where g(t)
is a rectangular pulse of duration Tb : each bit is mapped into a constant signal of duration Tb with positive or negative amplitude.
We are interested into studying the signal spectrum S(f ) of the waveform s(t), which tells us how the power is distributed on the frequency axis. It depends on both the autocorrelation function of the binary sequence and the pulse g(t), and is given by [16]: S(f ) =
where Su (f ) =
+∞ X
|G(f )|2 Su (f ) Tb
Ru (τ ) exp(−j2πf τ Tb )
(10)
(11)
τ =−∞
is the sequence spectrum and G(f ) is the Fourier transform of the pulse. (As an example, for the NRZ representation |G(f )|2 = [Tb sinc(f Tb )]2 .)
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A. The power spectrum of ideal sequences and TL-sequences For an ideal random sequence of infinite length, Ru (0) = 1 and zero elsewhere. Then Su (f ) = 1 and the ideal signal spectrum is given by: SI (f ) =
|G(f )|2 . Tb
(12)
Given a binary sequence of infinite length obtained by repeating a TL-sequence of length M ≤ N , the sequence spectrum is equal to: ∞ X
Rb mRb Su (f ) = Qm δ f − , M2 M m=−∞
(13)
M −1 X
(14)
with
Qm =
rk ω km .
k=0
In this expression, ω = exp(−j2π/M ) is the M -th root of unity in the complex plane, and rk are the values of the non-normalized autocorrelation function. By (10), the signal spectrum S(f ) is a comb of Dirac deltas, separated by: ∆=
Rb M
[Hz]
(15)
and shaped by the |G(f )|2 function. B. The mean of TL-sequences spectral peak amplitudes
As proved in Section III, for a TL-sequence of length M ≤ N we have Ru (τ ) = 1 for τ = 0
and Ru (τ ) = −1/N elsewhere, regardless of the value of M . As a consequence, by considering the
average value of (14) we can state the following Lemma:
Lemma IV.1 The average sequence spectrum of a TL-sequence is given by:
Rb (N + 1) X Rb Rb X Su (f ) = δ f −k − δ(f − jRb ). N ·M M N k
(16)
j
When M = N (non-truncated L-sequences) Eq. (16) holds for each starting seed. For M < N (TLsequences) it represents an average behavior over all possible starting seeds. Putting (16) into (10) provides the exact signal spectrum of L-sequences and the average signal spectrum of TL-sequences. C. The power contained in a bin Let us consider a frequency interval (a bin) of generic amplitude w [Hz]. In practical applications, we are usually interested into evaluating the power contained in bins with w 100 Mbit/s.) Given the ideal August 17, 2010
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spectrum, the power PI contained in a bin can be obtained by integrating the signal spectrum over it. Noting by f ∗ the initial frequency, if Rb >> w, we can write: Z f ∗ +w |G(f ∗ )|2 PI = 2 SI (f )df ≈ 2SI (f ∗ )w = 2 w. Tb f∗
(17)
For a TL-sequence, the power P contained in a bin depends on the number of Dirac deltas included in the interval, that is: Z = ⌈w/∆⌉,
(18)
where ⌈y⌉ denotes the smallest integer larger than or equal to y . By putting (16) into (10) we obtain the average signal spectrum of a TL-sequence; then, we can compute the average power contained in a bin. Assuming Rb >> w we have: Z f ∗ +w |G(f ∗ )|2 Rb (N + 1) |G(f ∗ )|2 Rb hS(f )i df ≈ 2 hP i = 2 Z ≈2 Z . Tb N ·M Tb M f∗
(19)
(Here we have considered a bin that does not include a multiple of Rb , where the situation is more favorable because, according with (16), the delta coefficients are smaller.) Let us denote by ρ = P/PI the extra-power factor contained in a bin induced by the adoption of a periodic TL-sequence instead of the ideal random sequence. Taking into account (17), we can compute the average extra-power factor: hρi =
⌈w⌉ hP i ∆ Rb =Z = ∆ ≈Z w . PI wM w ∆
(20)
As expected, the extra-power factor does not depend on the shaping filter g(t) but only on the binary sequence properties. It is important to distinguish two cases. When the frequency separation is not greater than the bin amplitude (∆ =
Rb M
≤ w) each bin contains at least a delta. By (20), if ∆ w some bins contain no delta, but some other bins contain a delta with a large extra-power factor. This is the most critical case. Correspondingly, in fact, Z = 1 and by (20) we have hρi =
∆ w
> 1: this value can be very large when ∆ >> w. Moreover, when the L-sequence is
truncated (M < N ), the separation ∆ becomes larger and, according with (20), the extra-power factor increases. Example 3 Let us consider an L-sequence with M = N = 215 − 1 bits (i.e., an LFSR with L = 15 cells) and w = 4 kHz. By (15) we have w/∆ ≈ 1.31 · 108 /Rb , that is smaller than 1 for Rb > 131 Mbit/s. As an August 17, 2010
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example, for Rb = 260 Mbit/s, we have w/∆ ≈ 0.5 and in each bin the average extra-power factor is hρi ≈ 2 ≡ 3 dB. By truncating the sequence at M = 16352 bits, for w = 4 kHz and Rb = 260 Mbit/s,
the average extra-power factor increases to hρi ≈ 6 dB, just as a consequence of the shorter length. On the other hand, by assuming M = N = 232 − 1 bits (i.e., an LFSR with L = 32 cells) we have w/∆ ≈ 1.72 · 1013 /Rb , that is very large even for very high bit rate. For Rb = 260 Mbit/s we have hρi ≈ 1 ≡ 0 dB. This means that, in this case, the power contained in a bin is almost ideal.
⋄
D. The power margin In many wireless systems, regulation issues establish that the maximum received power contained in a bin of w Hz cannot exceed a prefixed value (see [17] and the next section for examples of actual values). On the other hand, design is usually made by considering the ideal spectrum. As seen in Section IV-C, when an actual sequence u is transmitted, a bin can contain some extra-power factor, due to the non-ideality of the sequence spectrum. As a consequence, to comply with the regulations, it is necessary to reduce the transmitted power by a margin ξ . Clearly, the value of this link budget margin should be as small as possible: a high value of ξ strongly reduces the transmitted power, and then the received power, with an impact on the error rate performance. The margin ξ should be chosen equal to the largest extra-power factor ρ in the considered bins: ξ = max ρ. bins
(21)
Given an L-sequence, for each starting seed and each bin we have the same extra-power factor ρ = hρi. Then, for an L-sequence, we can choose a margin ξ = hρi. For TL-sequences, (20) represents the extra-power factor in a generic bin averaged over all starting seeds. It provides some indication on the required margin, but rather coarse, since (20) does not refer to a given sequence and a given bin. Fixed a starting seed, according to (21) the margin must be computed on the largest extra-power factor over all bins. Then the average of the margins over all seeds will be larger than (20). However, in spite of its coarseness, (20) makes evidence of a first impact of the truncation on the margin ξ , due to the reduction of the sequence length. Since the length diminishes from N to M , the separation ∆ becomes larger than the designed value. Then, according with (20), the average extra-power factor increases, and the margin must be increased as well. The second impact is the truncation noise, which causes the dispersion of the extra-power factor around its mean value and will be studied next. According to the analysis of the previous section, we will focus on the most critical case with ∆ > w and Z = 1 (the bin contains one delta). August 17, 2010
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1 0.9
Cumulative Probability
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 3.5
Fig. 3.
4
4.5
5
5.5 ξ [dB]
6
6.5
7
7.5
Cumulative distribution function of the margin values (ξ) over all possible g(D) and seeds; LFSR with L = 9 and
M = 300 bits; Rb = 1.2 Mbit/s.
The sequence spectrum of a TL-sequence is given by (13), and the signal spectrum by (10). By integrating over a bin between f ∗ and f ∗ + w containing the m-th delta we have: P =2
|G(f )|2 Rb Qm . Tb M 2
(22)
Then the extra-power factor in that bin is equal to ρ=
P Qm = . PI wM 2 Tb
(23)
Since, according to (21), the margin must be equal to (or greater than) the largest extra-power factor, for studying the margin it is necessary to investigate the properties of Qm . By (14), Qm can be interpreted as the weighted sum, through the coefficients ω km , of the values rk of the autocorrelation function. The exact computation of (14), for any value of m : 0 ≤ m ≤ M − 1 and any possible initial seed and characteristic polynomial, permits us to find all possible spectra, compare them and select the sequences for which the margin is minimum. Example 4 In order to give an example of exhaustive margin analysis, we have considered the same parameters as in Example 2, that is, L = 9 and M = 300 bits. For a bit rate Rb = 1.2 Mbit/s, we have computed the value of ξ for all possible g(D) and initial seeds. As shown in Fig. 3, that reports the cumulative distribution function of the ξ values for the considered case, the margin varies in a range of about 3 dB (more precisely, between 3.91 dB and 7.24 dB). The minimum value has been found with g(D) =
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D9 + D6 + D5 + D3 + D2 + D + 1 and s0 = (001100110); the maximum value has been found with g(D) = D9 + D8 + D7 + D2 + 1 and s0 = (111111010).
⋄
It is quite evident, however, that an exhaustive search of this kind is possible only for rather small values of N and M : the time (and the memory) required for an exhaustive analysis becomes rapidly huge for increasing L. Example 5 For our simulation tool (implemented by using a Matlab code over a PC equipped with a 1.86 GHz Intel Core2 CPU and 1 GB RAM), assuming M = 16352 and N = 32767 (these two values will be of interest in the following section), we have estimated a CPU time of about 100 days for an exhaustive analysis of all sequences. Moreover, the memory occupancy is also large (in the order of 225 MB for this case) and can become intolerably high for longer LFSRs.
⋄
E. The variance of TL-sequences spectral peak amplitudes In this section we will address deviations of TL-sequence spectral peak amplitudes from their mean value computed in Section IV-B. By exploiting the symmetry properties of the sequence {rk }, we can write:
⌊ M2−1 ⌋
Qm = 2
X
rk cos
k=1
2πkm + r0 + rM/2 cos πm, M
(24)
where the last term exists, and must be considered, only in the case of even M , and ⌊x⌋ denotes the integer part of x. The terms at the right side cannot be considered uncorrelated, at least in general. So, 2 , requires the evaluation of the covariance terms. The computation of the variance of Qm , noted by σQ m 2 is discussed in Appendix C. Unfortunately, the procedure becomes very complex exact calculus of σQ m
for large values of M . 2 is provided. At the expense of overestimating the true value, In Appendix D, an upper bound for σQ m
it can be expressed by a closed-form formula and, therefore, is very simple to compute. We limit here to report the upper bound for the case of even M , that will be useful in the following: M N +1 2 2 σQm ≤ σ QM/2 = M 3M − 4 + (3 − 2M ) . N N
(25)
The computation details are in Appendices C and D, where the case of odd M is also discussed. From (25) we observe that, for a given M , the value of the upper bound σ 2QM/2 increases for larger N . This suggests that it is not convenient to truncate an L-sequence longer than the one strictly necessary, as it yields a larger margin (see also Example 7). August 17, 2010
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F. A technique for estimating the largest spectral peaks and the actual margin Until now we have analyzed the mean value and the variance of the spectral peaks for TL-sequences. By (23), these results provide the first and the second order statistics of the extra-power factor ρ contained in a bin. In this section we present a method for estimating the maximum value of Qm and ρ and then, by (21), of the margin ξ . The estimation provided is probabilistic and is aimed at finding a threshold that is overcome by ξ with very low probability, over all m values. To find such threshold, the probability Pχ = Pr{Qm ≥ hQm i + χσQm } can be evaluated. In the rest of this paragraph we report some values
of such probability, calculated through exhaustive analysis. Denoting by χ∗ a value of χ for which Pχ is sufficiently low, because of (21) and (23)-(25), we can say that, with high probability, the margin ξ satisfies the following relationship:
QM/2 + χ∗ σ QM/2 M NN+1 + χ∗ σ QM/2 ξ≤ = . M 2 wTb M 2 wTb
(26)
Example 6 Table II shows the Pχ values for 7 ≤ L ≤ 10 and M/N ≈ 0.5. This ratio will be of interest in the following. The probabilities in the table have been obtained by considering all possible seeds and characteristic polynomials, and all significant values of m : 0 ≤ m ≤ M − 1. It is also interesting to investigate the corresponding probability density functions (p.d.f.). These are plotted in Fig. 4, for the considered values of L. In the figure, Qm has been replaced with Q, since the p.d.f. has been estimated over all possible m values. It is interesting to observe that the p.d.f.’s have similar shapes but, for increasing L (and, then, increasing M ) they involve larger and larger Q’s. On the other hand, when increasing M ,
the average value increases as well and this justifies the fact that Pχ does not change significantly with L. In practice, we can say that the normalized random variables, obtainable by subtracting the average
value and dividing by the standard deviation, have nearly the same p.d.f.. TABLE II VALUES OF Pχ
FOR DIFFERENT
L AND M/N ≈ 0.5 ( EXHAUSTIVE ANALYSIS )
L
P1
P2
P3
P4
P5
P6
7
1.65 · 10−1
4.82 · 10−2
9.08 · 10−3
9.84 · 10−4
5.47 · 10−5
0
8
1.66 ·
10−1
4.70 ·
10−2
8.66 ·
10−3
1.02 ·
10−3
3.83 ·
10−5
3.80 · 10−6
9
1.67 ·
10−1
4.71 ·
10−2
8.44 ·
10−3
9.45 ·
10−4
4.91 ·
10−5
2.90 · 10−6
10
1.66 · 10−1
4.48 · 10−5
9.00 · 10−7
4.72 · 10−2
8.34 · 10−3
8.29 · 10−4
Carrying out an exhaustive analysis for L > 10 is very hard because of the large processing time required. However, for longer sequences (like L = 15, for example, that will be of interest in the August 17, 2010
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0.012 L=7, M=64 L=8, M=128 L=9, M=256 L=10, M=512
0.01
p.d.f.
0.008
0.006
0.004
0.002
0 0
200
400
600
800
1000
Q
Fig. 4.
Estimated p.d.f. of Q for different L and M/N ≈ 0.5 (exhaustive analysis).
subsequent sections) Montecarlo evaluations can be performed to confirm such results. As we see from Table II that P6 < 10−5 , we assume χ∗ = 6. Inserting this value in (26) and assuming Rb = 260 Mbit/s, N = 32767 bits and M = 16352 bits we find: ξ ≤ 15.77 dB.
(27)
As a first confirmation of this result, we have considered the truncated sequence generated by the LFSR with g(D) = D15 + D14 + 1 and s0 = (111111111111111) (this characteristic polynomial and seed will also be considered in Section V). The simulated spectrum exhibits a margin of 13.16 dB, that agrees with condition (27). Note that there are 1800 possible characteristic polynomials, each with 32767 possible starting seed. We have performed a Montecarlo experiment with 60000 cases. The results are reported in Fig. 5, where the estimated p.d.f. for the margin is plotted. In the considered simulations the limit (27) has never been exceeded, and it reveals to be a tight upper bound on the margin value.
⋄
The probabilities Pχ slightly change with the ratio M/N and decrease with its value. This means that the preliminary analysis on their evaluation should be tailored on the M/N value of interest. This, combined with the observation on the value of σ 2QM/2 (see Section IV-E), confirms that it is better to truncate an L-sequence with the shortest admissible length, as shown by the following example. Example 7 Table III shows the Pχ values for 7 ≤ L ≤ 11 and M/N ≈ 0.25. As we see from the table that P8 < 10−5 , we assume χ∗ = 8. Inserting this value in (26) and assuming Rb = 260 Mbit/s, N = 65535 bits and M = 16352 bits we find: ξ ≤ 17.35 dB. This margin is larger than that obtained in Example 6 for the August 17, 2010
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1.6 1.4 1.2
p.d.f.
1 0.8 0.6 0.4 0.2 0 12
12.5
13
13.5 ξ [dB]
14
14.5
15
Simulated p.d.f. for the margin ξ in the case of Rb = 260 Mbit/s, N = 32767 bits and M = 16352 bits.
Fig. 5.
TABLE III VALUES OF Pχ
FOR DIFFERENT
L AND M/N ≈ 0.25 ( EXHAUSTIVE ANALYSIS )
L
P1
P2
P3
P4
P5
P6
P7
P8
7
1.55 · 10−1
5.46 · 10−2
1.51 · 10−2
3.53 · 10−3
6.29 · 10−4
1.64 · 10−4
0
0
8
1.51 ·
10−1
10−2
10−2
10−3
10−4
10−5
9
1.48 · 10−1
5.15 · 10−2
1.60 · 10−2
3.77 · 10−3
7.24 · 10−4
1.08 · 10−4
8.90 · 10−6
0
10
1.49 · 10−1
5.17 · 10−2
1.51 · 10−2
3.68 · 10−3
7.15 · 10−4
1.08 · 10−4
1.13 · 10−5
8.00 · 10−7
11
1.49 · 10−1
5.19 · 10−2
1.55 · 10−2
3.76 · 10−3
7.23 · 10−4
1.07 · 10−4
1.08 · 10−5
5.00 · 10−7
5.26 ·
1.56 ·
3.54 ·
4.90 ·
6.89 ·
7.70 ·
10−6
0
same M , thus confirming the convenience to truncate an L-sequence with the shortest admissible length. As the Pχ values are smaller and smaller for increasing M/N , (27) can be used for any 0.5 ≤ M/N < 1, with higher and higher confidence.
⋄
As mentioned in the Introduction, other solutions can be employed as an alternative to TL-sequences. The following example compares the required margin of TL-, chaotic and Kasami sequences. Example 8 Let us consider Rb = 260 Mbit/s and M = 16352 bits. It is interesting to compare the performance of TL-sequences with that of chaotic sequences obtained by using the generator proposed in [18]. Chaotic generators do not suffer the presence of a periodicity in the output sequence; so they seem particularly suitable for obtaining sequences of arbitrary length. Although not being based on L-sequences, the generator in [18] exploits an LFSR of length L = 32. So, for a fair assessment, we compare its performance with that of TL-sequences with L = 32. As a further comparison, we have considered a set of Kasami sequences based on a 32-bit LFSR. For all generators, a Montecarlo simulation has been August 17, 2010
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TABLE IV S IMULATED EXTREMES AND STATISTICAL MOMENTS OF THE MARGIN ξ WHEN USING
FOR
Rb = 260 M BIT / S AND M = 16352 BITS
32- BIT TL-, CHAOTIC AND K ASAMI SEQUENCES .
Minimum
Maximum
Mean
Std. Dev.
TL-sequences
14.18 dB
19.14 dB
15.80 dB
0.55 dB
Chaotic sequences
14.37 dB
18.82 dB
15.80 dB
0.55 dB
Kasami sequences
14.23 dB
18.93 dB
15.80 dB
0.55 dB
carried out on 30000 different sequences. Table IV reports the values of the extremes and the statistical moments found for the margin ξ in our Montecarlo simulations. From their comparison we observe that, at least for this specific choice of the parameters, the performance of the three types of sequences is quite similar. It should be noted that in the example under study the value of M/N is small, that is most of the bits in the mother sequences are truncated. Through several other simulations, performed for different values of the parameters, we have verified that the performances of the considered sequences remain very similar one each other when truncation is strong. On the contrary, when only a relatively small number of bits are truncated and the ratio M/N is rather high (for example M/N ≥ 0.25), the conclusions may be different because the unequal correlation properties of the mother sequences become more important. For example, a comparison under these conditions between TL-sequences and truncated Kasami sequences reveals that ⋄
the former ones are generally better. V. A PPLICATION
TO SPACE LINKS
The Power Flux Density (PFD) at the Earth’s surface produced by emissions from a spacecraft must not exceed prefixed values, for all conditions and transmission schemes. In particular, the regulations established by the International Telecommunication Union (ITU) [17] fix maximum levels for the PFD measured over a 4 kHz bin (so, expressed in dBW/m2 /4 kHz) for a given elevation angle. Recently confirmed values for these limits can be found in [19]. Some regional regulations (for example in the USA) are even more stringent than the ITU mask. When designing a space mission, compliance with these limits is usually conjectured on the basis of the ideal PSD, under the hypothesis of transmitting equiprobable and uncorrelated symbols. As shown in the previous section, if the transmitted sequences are not ideal, extra-peaks may appear in the PSD, increasing the power contained in some 4 kHz bins. To cope with sub-optimality, a margin is introduced and the transmitted power is reduced for not exceeding August 17, 2010
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s(1)
Fig. 6.
s(2)
s(3)
s(4)
s(5)
s(6)
s(7)
s(8)
The CCSDS randomizer.
the mask. This may have a negative impact on the error rate performance of the system; so, a proper dimensioning of the margin is an important issue. In order to improve the randomness of the transmitted binary sequences, thus approaching the ideal behavior, suitable randomizers are used. All international space agencies (NASA, ESA, CNES, etc.) design their space links according to the Recommendations issued by the Consultative Committee for Space Data Systems (CCSDS). The CCSDS randomizer is the LFSR with L = 8 cells depicted in Fig. 6 [20]. The connections (1, 3, 5, 8) (corresponding to the primitive polynomial D8 + D7 + D5 + D3 + 1) guarantee the generation of a maximum-length sequence with period N = 28 − 1 = 255 bits. All the LFSR cells are preset to 1 at the beginning of each frame. Hence, the LFSR sequence summed to each frame is always the same. (This is a very popular choice for practical applications because it allows to acquire simultaneously both frame sync and LFSR sync.) The CCSDS information sequences are segmented into transfer frames (TFs). A detailed description of the CCSDS Telemetry transfer frames can be found in [13]: the structure details will not be reported here as they are unessential for the current analysis. A key role is played by the TF length, that will be denoted by F . In line of principle, any length F ≤ 16384 bits is admissible, but a set of typical lengths are usually fixed by the coding technique adopted. The transfer frames can be encoded by using one (or more) of four coding options described in [20]. In our analysis, we have focused on the TF lengths obtained by using the Reed-Solomon (RS) code, but the procedure can be extended to the other coding schemes. The RS code has codeword length n = 255 bytes, and it is used together with a convolutional interleaver with depth I = 1, 2, 3, 4, 5 or 8 blocks. This corresponds to a TF length ranging from F = 2040 bits to 16320 bits.
The content of a TF frame is EX-ORed with the sequence generated by the LFSR. Finally, an attached sync mark (ASM), with length depending on the code option, is added to acquire block synchronization at the receiver side. In the case of RS coding, the ASM is 32 bits long. As for the modulator within the Physical Layer, 2-PSK, 4-PSK and Offset 4-PSK are currently the most used modulation formats in CCSDS recommendations. The use of 8-PSK is also allowed in conjunction with trellis coded modulation. In this paper, we focus on the 2-PSK modulation format, which is still the most used one in space missions. A rectangular filter of time duration Tb is adopted. The PSD for this scheme can be studied with reference August 17, 2010
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to its equivalent baseband representation, that coincides with the spectrum of the NRZ representation. In this case, with reference to the notation used in Section IV, |G(f )|2 = [Tb sinc(f Tb )]2 . A. The Only Idle Data Frames When payload TFs are transmitted, the CCSDS LFSR is usually sufficient to provide good randomness, and the PSD is very similar to the ideal one given by (12). However, only idle data (OID) frames can be inserted within transmission to preserve frame continuity, when no payload transfer frames are ready to be transmitted. These idle frames are typically filled with zero bits: this choice is very useful for having a complete separation between the Data Link layer and the Code and Sync layer. Since the OID frames are always the same and the LFSR is restarted at the beginning of each frame, all the frames entering the Physical Layer are equal and consist of an ASM followed by F bits generated by the LFSR. As a consequence, the theory developed in the previous section can be applied: some extra-peaks appear in the PSD. These peaks may exceed the ITU requirements in some bins: for this reason a margin ξ must be included. In absence of a suitable theory (like the one proposed in this paper), this margin is usually designed by measurements or qualitative considerations: as an example, a typical value adopted by many missions is 15 dB. B. Numerical examples In this subsection we give some examples of simulated power spectra, showing that the theoretical analysis, developed in Section IV, and the related results give useful information for designing the margin. As mentioned above, a key role in the analysis is played by the TF length F , in comparison with the length N of the L-sequence. In fact, in case of OID frames: •
If F ≥ N , the sequence consists of some periods of the L-sequence (plus a truncated part).
•
If F < N , the sequence is a TL-sequence of length F .
Example 9 Let us fix Rb = 260 Mbit/s and F = 16352 bits (corresponding to the application of an RS code with I = 8), that are values of interest in some typical high data rate space missions. Let us suppose to use
the CCSDS randomizer of Fig. 6. Its period, N = 255 bits, is much smaller than F , and each transmitted frame includes more than 64 repetitions of the L-sequence. So, when OID frames are transmitted, the frame sequence is (almost) periodic with period N = 255 bits, i.e., N Tb s. The truncation noise, that involves only the last period, is negligible. The analysis based on the spectral peaks mean should apply, and ξ = hρi (see Section IV-D).
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30 20
Normalized PSD [dB]
10 0 -10 -20 -30 -40 -50 -60 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 Fequency [GHz]
Fig. 7.
0.4
0.5
PSD with OID frames, for Rb = 260 Mbit/s and F = 16352 bits, using the CCSDS LFSR.
s(1)
Fig. 8.
0.3
s(2)
s(3)
s(4)
s(5)
s(6)
s(7)
s(8)
s(9)
s(10)
s(11)
s(12)
s(13)
s(14)
s(15)
The CNES randomizer.
The PSD for Rb = 260 Mbit/s is shown in Fig. 7. The actual PSD is plotted by assuming the ideal curve (bold line) as a reference. For simplicity, the spectra are drawn after normalization, i.e. the maximum value of the ideal spectrum is 0 dB. This way, the required margin can be directly read as the overextension of the actual PSD against the ideal one. For the considered numerical values, (20) gives hρi = 24 dB, and such margin is confirmed by the simulation, that gives ξ ≈ 24.33 dB. The slight difference can be ascribed to the actual structure of the CCSDS transfer frame, that has been considered in the simulations. ⋄
A margin greater than 20 dB is very high and often unacceptable in practice (the Sentinel mission, for example, adopts a margin of 12.5 dB at 5 degrees elevation). It should be noted that this margin is not due to the truncation effects, but it is determined by the short length of the adopted L-sequence. To overcome the problem of high spurious at high data rates, CNES proposed to use, for high data rate space missions, the longer LFSR (L = 15) shown in Fig. 8 [21]. The connections (1, 15) (corresponding to the primitive polynomial D15 + D14 + 1) guarantee to generate a binary sequence with period N = 32767 bits. Since N is much greater than before, the spurious separation should be significantly reduced and, according
with the analysis in Section IV, the margin ξ should be lowered as well. More precisely, continuing to assume Rb = 260 Mbit/s, (20) gives hρi = 3 dB, that is a very small extra-power factor.
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21
30 20
Normalized PSD [dB]
10 0 -10 -20 -30 -40 -50 -60 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 Frequency [GHz]
Fig. 9.
0.3
0.4
0.5
PSD with OID frames for Rb = 260 Mbit/s and F = 16352 bits, using the CNES LFSR.
Unfortunately, in this case, the TF length is shorter than the LFSR period N (i.e., it contains less than one LFSR period) and the spurious separation is imposed by the value of F [21]. To evaluate the margin, it is possible to apply the theory developed for TL-sequences. The following condition holds: ∆=
Rb , F
(28)
that replaces (15) and gives a first contribution to the increase of the required margin. But a major effect is expected because of the modification of the autocorrelation properties of the randomizer sequence, due to truncation. So, in order to find a tighter margin, the theory in Section IV-F must be applied. Example 10 The PSD for Rb = 260 Mbit/s, F = 16352 bits and the CNES LFSR is shown in Fig. 9; coherently with the CCSDS LFSR, we have adopted an all-one initial condition. From the simulation we obtain a margin of 13.5 dB, that satisfies condition (27), derived in Section IV-F with the same parameters. We note this value is only slightly greater than that obtained by neglecting the presence of the ASM (see Example 6). ⋄
C. Practical Considerations Even by using a long LFSR, the margin that TL-sequences require is quite large. To eliminate the truncation noise is impossible, since the frame lengths are typically fixed and not coincident with the L-sequence lengths. On the contrary, some action can be conceived to vary the spurious separation, this way providing a first contribution to the reduction of ξ . In (28), the value of ∆ is determined by the channel symbol frame length. This is the consequence of two facts: A) the OID frames are filled by all-zero bit patterns; August 17, 2010
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B) the LFSR is re-initialized for each frame. The only way to make less stringent the channel symbol frame constraint is to remove at least one of these facts. Unfortunately, both of them are useful for simplifying the system design. In [8], however, we proposed some solutions that, modifying A) or B), permit us to reduce significantly the margin value. VI. C ONCLUSION To determine analytically the properties of truncated maximum-length sequences is considered a very difficult task. For this reason, a common procedure is resorting to numerical evaluation. However, the computational complexity rapidly grows with increasing sequence length. In this paper, we have shown that an analytical approach is feasible for studying the statistical behavior of the autocorrelation function. By averaging over all possible seeds, a number of properties of the first and second order statistics of the periodic autocorrelation function have been demonstrated. In particular, we have found lower and upper bounds on its mean square, that hold regardless of the characteristic polynomial adopted. Starting from these results on the autocorrelation function, new results on the spectrum of truncated L-sequences have been obtained and discussed. The average and the variance of the spectral peak values have been analyzed, and a method for estimating the largest amplitudes of these peaks has been presented. As an example of application, we have discussed the design of a space link margin. This margin is required to comply with the limitations imposed on the power levels emitted by a spacecraft or a satellite. The problem is particularly critical for high data rate missions, where the adoption of truncated sequences may lead to very large margin values. The results obtained on the spectral peaks allow to estimate the required margin by using simple formulas and procedures. A number of examples and simulations have been presented throughout the paper for validating the analytical results. Future research could consist in further developing the approach here proposed, including for example: the extension of the statistical analysis to higher order moments, the development of faster algorithms for exactly computing the variance of the autocorrelation function and the power peaks, and the extension of the analysis of the truncation impact to other classes of sequences. ACKNOWLEDGMENT The authors wish to thank Gian Paolo Calzolari and Enrico Vassallo of ESA for having suggested the topic and for helpful technical discussion. Part of this work has been sponsored by the European Space Agency, under the ESOC Contract No. 20959/07/D/MRP. They are also indebted with the reviewers for their valuable comments and suggestions. August 17, 2010
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A PPENDIX A S EMI - ANALYTICAL ALGORITHM TO COMPUTE THE MEAN SQUARE AUTOCORRELATION FUNCTION P P The j k at the right side of (8) involves M (M − 1)/2 contributions, and each of them equals −1
or N . Let us denote by ζ the number of contributions equal to N ; by elaborating (8), we can write:
1 M −1 2 N +1 2 [Rs0 ,M (τ )] = . (29) 1− +ζ 2 M N M N Computation of ζ , that depends on τ , can be done in the following terms.
We observe that a first contribution to ζ comes from the satisfaction of the following conditions: k = j + τ,
j = k + τ. PN −1 h (d) i2 h (d) i2 bk+τ , that is always equal to N . In this case, in fact, the sum over d in (8) becomes d=0 bj+τ
However, taking into account that both j + τ and k + τ must be considered mod M , it is easy to verify that this can occur only when: τ=
M M M , 0≤j≤ − 1, k = j + . 2 2 2
Explicitly, this means that such circumstance takes place only for even M , providing M/2 contributions equal to N in the middle of the shift range. The sum over d can also take the value N when the product of three involved sequences equals the (d)
fourth sequence. In practice, noting by bi
the sequence b(d) shifted by i positions, the problem consists
in finding the integers A, B, C, E , with 0 ≤ A < B < C < E ≤ M − 1 that satisfy the condition: (d)
(d)
(d)
(d)
bA · bB · bC = bE .
(30)
In [9], it was demonstrated that the combinations of four shifts satisfying (30) are such that the quadrino mial f (D) = Dd 1 + DE−C + DE−B + DE−A is divisible by the characteristic polynomial g(D). So,
the problem is reduced to the evaluation of the number of quadrinomials of this type. To this purpose, an efficient algorithm was presented in [9], [10]; details are here omitted for saving space. A PPENDIX B P ROPERTIES
OF THE MEAN SQUARE AUTOCORRELATION FUNCTION
In this appendix a number of properties for the mean square autocorrelation function of a truncated sequence (Eq. (8)) are highlighted and demonstrated. Property 1 - The mean square autocorrelation function of a truncated sequence is symmetric with respect to τ = M/2. August 17, 2010
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Proof: The property holds since the autocorrelation function is also symmetric. In fact, every 1 PM −1 (d) (d) term in Rs0 ,M (τ ) = M j=0 bj bj+τ , for τ < M/2, has an identical term in Rs0 ,M (M − τ ) = P M −1 (d) (d) 1 j ′ =0 bj ′ bj ′ +M −τ . For the proof, two distinct cases need to be considered. If 0 ≤ j < M − τ , we M
set j ′ = j + τ ; this way, in fact, we have: (d) (d)
(d)
(d)
(d)
(d)
bj ′ bj ′ +M −τ = bj+τ bj+M = bj+τ bj .
If M − τ ≤ j < M − 1, we set j ′ = j + τ − M ; this way, in fact, we have: (d) (d)
(d)
(d)
bj ′ bj ′ +M −τ = bj+τ −M bj
(d)
(d)
= bj+τ bj .
Property 2 - The mean square autocorrelation function of a truncated sequence can be lower bounded as follows:
1 [Rs0 ,M (τ )]2 ≥ M
M −1 1− . N
(31)
Proof: The minimum value can be found by setting ζ = 0 in (29). Property 3 - Verification of (30) with (j ,k , j +τ , k+τ ) in place of (A, B , C , E ) implies that j +τ < M and k + τ ≥ M . Proof: With the considered positions, (30) can also be rewritten as follows: (d)
(d)
(d)
(d)
bj · bk = bj+τ · bk+τ .
(32)
Let us set, w.l.o.g., d = 0 (and omit the apex); the shift-and-multiply group property of L-sequences implies that a shift L exists for which b0 · bk−j = bm . On the other hand, obviously, bj · bk = bm+j . If k + τ < M (that implies j + τ < M , too), we have bj+τ · bk+τ = bm+j+τ and, being 1 ≤ τ ≤ M − 1,
condition (32) can never be satisfied. Verification is similar for j + τ ≥ M (that implies k + τ > M , too). On the contrary, for k + τ ≥ M and j + τ < M , (32) can be rewritten as bj · bk = bj+τ · bk+τ −M and, given τ , this condition can be satisfied for some pairs (j , k ). It must be noticed that the above property fixes a necessary, though not sufficient, condition for satisfying (32). According to it, the values of j and k that are potentially able to satisfy (32) are: j ≤ min[(M − τ − 1, M − 2)] = M − τ − 1,
M − τ ≤ k ≤ M − 1.
(33)
Property 4 - Eq. (32) cannot be satisfied for even M and τ = M/2. Proof: By assuming k + τ ≥ M , j + τ < M (according with Property 3), and setting τ = M/2 (d)
(d)
(d)
(d)
(d)
(d)
(acceptable only for even M ), we have bj+τ · bk+τ −M = bj+M/2 · bk−M/2 = bj+M/2 · bk+M/2 . On the (d)
(d)
(d)
(d)
(d)
(d)
other hand, we can write bj+M/2 · bk+M/2 = bm+j+M/2 6= bm+j = bj · bk . August 17, 2010
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Based on this property and on (29) with ζ = M/2, we can conclude that, in the case of even M :
M 1 2 2 [Rs0 ,M (M/2)] = − 1+ . (34) M N 2N Property 5 - For a given τ , the pairs (j , k ) satisfying (32) cannot have elements in common.
Proof: Let us denote by (j , k ) a pair satisfying (32); according with the approach described in Appendix A, this means that a polynomial f (D) = Dd 1 + DE−C + DE−B + DE−A exists, such that g(D)|f (D), where the combination (A, B , C , E ) coincides with the four elements (j , k , j + τ , k + τ ),
properly ordered. Two cases are possible: i) j + τ > k → E = j + τ , ii) k > j + τ → E = k . The order of the other elements is not important. In case i), the polynomial f (D) can be written as: f1 (D) = 1 + Dj−k+τ + Dτ + Dj−k+M = Dj+τ t(D),
with: t(D) = DN −j + DN −k + DN −j−τ + DN −k−τ +M .
(35)
In case ii), instead, the polynomial f (D) can be written as: f2 (D) = Dk t(D).
Let us consider two pairs (j , k ) and (j ′ , k ). The first pair corresponds to (35) while, for the second pair, we must consider: t′ (D) = DN −j +DN −k +DN −j −τ +DN −k−τ +M = t(D)+DN −j +DN −j −τ +DN −j +DN −j−τ . (36) ′
′
′
′
Let us suppose that one pair satisfies (32), which means that g(D) divides t(D) or t′ (D). The same would be for the second pair iff g(D) divides: ′ ′ ′ r(D) = DN −j + DN −j −τ + DN −j + DN −j−τ = DN −j−τ (1 + Dτ ) 1 + Dj−j .
(37)
′ So, for such purpose, g(D) should be a factor of (1 + Dτ ) or 1 + Dj−j ; however, this is impossible, as
g(D) is primitive and the smallest integer p such that g(D) divides (1 + Dp ) results in p = N = 2L − 1.
Property 6 - For a given τ , the maximum number of pairs (j , k ) satisfying (32) is equal to τ for i < M/2 and to M − τ for i > M/2.
Proof: By exploiting Property 5, the pairs (j , k ) satisfying (32) are all distinct. The maximum number of distinct pairs can be obtained from (33) by considering that: •
when τ < M/2, the range of variability of k is less extended than the range of variability of j and it includes τ elements;
August 17, 2010
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26
•
when τ > M/2, the range of variability of j is less extended than the range of variability of k and it includes M − τ elements.
Property 7 - An upper bound for the mean square autocorrelation function of the truncated sequence is as follows:
[Rs0 ,M (τ )]2 ≤
1 M 1 M
M −1 N M −1 N
1− 1−
+ 2 Mτ 2 NN+1
+ 2 MM−τ 2
N +1 N
for 0 < τ ≤ for
M 2
M 2 ,
(38)
< τ < M.
Proof: According with Property 6, in order to find the upper bound, we must consider ζ = τ for τ < M/2 and ζ = M − τ for τ > M/2; these values, replaced in (29), give the upper bound for τ 6= M/2. For τ = M/2 the result (34) holds, that can be obtained from (38) as a particular case.
A PPENDIX C VARIANCE
OF
Qm
Let us consider (24), and define: x0 = r0 = M,
xk = 2rk cos 2πkm for k = 1, 2, ..., ⌊ M2−1 ⌋, M , x M/2 = rM/2 cos πm, for even M.
(39)
Moreover, let us set P = ⌊M/2⌋; so, (24) can be rewritten as: Qm =
P X
xk .
P X
xk .
(40)
k=0
Since x0 is a constant, for a given truncation, and we are interested in computing the variance of Qm , we can equivalently consider: Q′m
=
(41)
k=1
The significant (in this sense) xk ’s can be collected in a vector of random variables: X = [x1 x2 ... xP ]T .
The covariance matrix of X is defined as follows: cx 1 x 2 . . . σ2 x1 cx2 x1 σx22 ... CX = .. .. .. . . . cx P x 1 cx P x 2 . . .
(42)
cx1 xP cx2 xP .. . σx2P
(43)
where σx2k is the variance of xk , and cxk xh = hxk xh i − hxk i hxh i is the covariance of (Xk , Xh ), with
2πkm for k = 1, 2, ..., ⌊ M2−1 ⌋ and xM/2 = − M hxk i = −2 M N cos M N cos πm.
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The variance of Qm then results in: 2 σQ m
=
2 σQ ′ m
=
P X
σx2k
+2
P −1 X
P X
cx k x h .
(44)
k=1 h=k+1
k=1
The first term in (44) can be approximated by using the lower bound and the upper bound in (9). Through simple but tedious calculations, by using the lower bound (lb) in (9), one finds: •
for even M , P X
σx2k
k=1
•
!
lb
for odd M , P X
σx2k
k=1
i h (2M − 3) M 1 − M −1 − M22 , for m = 0 or M , N N 2 h i = 2 (M − 3) M 1 − M −1 − M , for m = 6 0 or M N N2 2 ;
!
lb
(45)
i h (2M − 2) M 1 − M −1 − M22 , for m = 0, N N i h = (M − 2) M 1 − M −1 − M 2 , for m = 6 0. N N2
(46)
Looking at (45)-(46), we see that:
i) the difference between the cases of even M and odd M is negligible (for non-trivial M ); ii) the value for m = 0 (or m = M/2 in case of even M ) is almost twice the value for m 6= 0. Similarly, by using the upper bound (ub) in (9), one finds: •
for even M , P X k=1
•
σx2k
!
ub
for odd M , P X k=1
σx2k
!
ub
P P 2 σ + M (M − 1) NN+1 for m = 0 or M k=1 xk 2 , lb = P P 2 + M (M2 −2) NN+1 for m 6= 0 or M k=1 σxk 2 ;
(47)
lb
P P 2 σ + (M 2 − 1) NN+1 , for m = 0, x k=1 k lb P = 1+cos( 2πm (M 2 −1) P N +1 2 M ) + N − 2 3+4 cos 2πm +cos 4πm , for m 6= 0. k=1 σxk 2 ( M ) ( M ) lb
(48)
In the case of even M , we observe that the value of the additional term for m = 0, M/2 is almost twice the value of the additional term for m 6= 0, M/2. As remark ii) holds for the lower bound, we can conclude that the upper bound for m = 0, M/2 is always greater than the upper bound for m 6= 0, M/2. This conclusion has been used in the analytical framework (see (25), in particular). Computing (or even bounding) the second contribution at the right side of (44) is much more difficult. Taking into account the covariance definition, this must be written as: 2
P −1 X
P X
k=1 h=k+1
August 17, 2010
cx k x h = 2
P −1 X
P X
k=1 h=k+1
hxk xh i −
P −1 X
P X
k=1 h=k+1
!
hxk i hxh i .
(49)
DRAFT
28
800 600 400 200 0 -200 -400 -600 -800 100
Fig. 10.
80
60
40
20
0
0
20
40
60
80
100
Covariance matrix for N = 255 bits, M = 200 bits and m = M/2 = 100 bits.
The second term at the right side can be evaluated by hand. For the first term, we have: PM PM −2 −1 2 2 hrk rh i cos 2πkm cos 2πhm 8 k=1 h=k+1 M M P P −1 X X P M2 −1
hxk xh i = 2 rk rM/2 cos 2πkm , for even M, +4 cos πm k=1 M M M −3 P M −1 P k=1 h=k+1 8 2πkm 2 2 cos 2πhm , for odd M. k=1 h=k+1 hrk rh i cos M M
(50)
So, the problem is reduced to the calculus of hrk rh i, for which the semi-analytical algorithm outlined in Appendix A can be applied. This, however, could require very long elaboration times. Thus, in Appendix D, we present an approximation that drastically reduces the computation effort. A PPENDIX D A PPROXIMATE
COMPUTATION OF
2 σQ m
A pictorial sketch of the covariance matrix (43) is shown, for an example, in Fig. 10. As qualitatively expected, the behavior of the covariance elements, out of the diagonal, is almost unpredictable. However, it is interesting to observe that the triangular behavior shown by the upper bound on the mean square autocorrelation function (see (9) and Fig. 2 for an example), along the line k = h, now translates into a quarter-pyramidal surface. The development of an analytical approach to foresee, or even to limit, the values of cxk xh is very hard; so, we can resort to an approximation. Let us consider the case of even M , that is more frequent in practice. In Appendix C we have obtained P (45) and (47), that provide simple analytical expressions for the lower and upper bounds on Pk=1 σx2k . The latter has the meaning of the trace of the covariance matrix and an appealing idea could be to approximate the variance (44) with such trace. The feasibility of this idea can be validated by plots
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5
10
4
10
3
10
2
10
Variance Trace Lower Bound Upper Bound
1
10
0
10
50
100
150
200
250
M
Fig. 11.
2 Exact and approximate evaluation of σQ , as a function of M , for N = 255 bits and m = M/2. m
like that in Fig. 11, comparing, for a given N , the true variance (inclusive of all the elements of the covariance matrix) with the trace, as functions of M . Fig. 11 considers the case of an LFSR with L = 8 and g(D) = D8 + D4 + D3 + D2 + 1, and the even values of M in the range [2; N − 1]. The difference between the variance and the trace depends on the value of M , but it is always limited. The figure also reports the upper and lower bounds on the trace computed in Appendix C for m = M/2; the upper bound, in particular, overestimates the trace and is always greater than the variance. Therefore, 2 and to use it in the computation of it is acceptable to adopt this upper bound to estimate the value of σQ m
the margin ξ , according with the procedure described in Section IV-F. Then, following this approximation, (25) has been obtained by elaborating the first expression in (47). For the sake of clarity, we observe that (25) implies neither the true variance of Qm is independent of m nor it has its maximum for m = M/2, but it provides an upper bound holding for any m. Moreover, the possibility to approximate the variance with the trace does not mean that the variables (Xk , Xh ) are uncorrelated. On the contrary, correlation can also be rather strong, but its sign alternates
between positive and negative values, in such a way that its global effect on the sum results in a rather small contribution. The approximation cannot be used for m = 0; in this case, in fact, the variance is always much larger than the trace. This can be explained by considering that S0 is related with the spectrum contents at f = 0 and, therefore, with the mean value of each truncated sequence. The latter, however, is almost null also in the truncated sequence, since the number of +1’s remains close to the number of −1’s; so, S0 never corresponds to a large spectral peak.
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