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Time measurement, synchronization and time-resolution										185

Table 6.6 Parameters of ternary sequences with perfect periodic ACF

N	p	n	q			N	p	n	q

13	3	3	3	1.444	292	¼ 4 73	2	3	8	1.141
21	2	3	4	1.312	307		17	3	17	1.062
31	5	3	5	1.240	341	¼ 4 91	2	5	4	1.332
52 ¼ 4 13	3	3	3	1.444	364	¼ 4 91	3	3	9	1.123
57	7	3	7	1.163	381	¼ 4 133	19	3	19	1.055
73	2	3	8	1.141	532	¼ 4 133	11	3	11	1.099
84 ¼ 4 21	2	3	4	1.312	553		23	3	23	1.045
91	3	3	9	1.123	651	¼ 4 183	5	3	25	1.042
121	3	5	3	1.494	732	¼ 4 183	13	3	13	1.083
124 ¼ 4 31	5	3	5	1.240	757		3	3	27	1.038
133	11	3	11	1.099	781		5	5	5	1.250
183	13	3	13	1.083	871		29	3	29	1.036
228 ¼ 4 57	7	3	7	1.163	993		31	3	31	1.033
273	2	3	16	1.066	1057		2	3	32	1.032

In contrast to what has been presented, construction of ternary sequences for q ¼ 2w, and proving the perfection of their periodic ACF, involves much more sophisticated mathematical concepts, such as quadrics in finite fields [43].

If any of the considered ternary sequences is multiplied symbol-wise with a unique binary sequence 1, 1, 1, 1 having perfect periodic ACF, the resulting ternary sequence will have quadrupled length with no effect on the peak factor or ACF perfection. In the same way, a symbol-wise product of two perfect ACF ternary sequences of co-prime lengths N1, N2 has again perfect ACF, length N ¼ N1N2 and peak-factor ¼ 1 2, where i stands for the peak-factor of the ith sequence, i ¼ 1, 2.

Table 6.6 summarizes the lengths and values of the peak-factor of sequences generated as described along with parameters q, p, n for the range N 1057. The rows where lengths are presented as products correspond to symbol-wise products of initial ternary sequences with the binary sequence 1, 1, 1, 1. In this case parameters p, n, q characterize the initial ternary sequence. As is seen, very small to negligible values of are characteristic of many of the listed codes, giving the designer rather solid alternatives to the best binary sequences, wherever perfect periodic ACF is desirable.

6.12 Suppression of sidelobes along the delay axis

Suppose that the designer is not inclined to abandon binary f1g sequences and at the same time is dissatisfied with the achievable level ( p, max 1/N) of their periodic ACF sidelobes. In such a situation an effective way to settle these contradictory trends is to ‘imitate’ the perfect periodic ACF by way of rejecting matched filtering in favour of a special mismatched processing, which suppresses sidelobes all over the signal period. Very close ideas underlie the reduction or suppression of aperiodic sidelobes [39,44,45] and combating intersymbol interference with the aid of zero-forcing equalizers [2,5,7], but in the most transparent form they are visible when applied to periodic signals [39,46,47].

186 Spread Spectrum and CDMA

6.12.1 Sidelobe suppression filter

Consider some sequence . . . , ai 1, ai, aiþ1, . . . of period N, which manipulates chips of duration D and a finite impulse response (FIR) filter running the summation of N signal replicas delayed by iD and weighted by coefficients bi, i ¼ 0, 1, . . . , N 1 as shown in Figure 6.20. In principle, what is presented below may be accomplished for sequences of arbitrary alphabet; however, it seems reasonable to restrict ourselves to only the binary f 1g alphabet, since beyond this constraint there exist many sequences with perfect periodic ACF, depriving the sidelobe suppression task of a solid motivation. Accordingly, filter coefficients bi, i ¼ 0, 1, . . . , N 1 are all assumed real.

When fed by a sequence ai, i ¼ . . . , 1, 0, 1, . . . the filter responds by the sequence ci, i ¼ . . . , 1, 0, 1, . . . whose elements are found as a convolution:

N 1

ci ¼	ai lbl; i ¼ . . . ; 1; 0; 1; . . .	ð6:36Þ
	l¼0

With periodic input sequence ai ¼ aiþN , i ¼ . . . , 1, 0, 1, . . . the output will also be periodic with the same period N: ci ¼ ciþN , i ¼ . . . 1, 0, 1, . . . . Then N elements c0, c1, . . . cN 1 specify the output sequence exhaustively, and (6.36) becomes a cyclic convolution where subtraction in the index is fulfilled modulo N.

Let us impose on the filter a requirement:

c0 6¼0; ci ¼ 0; i ¼ 1; 2; . . . ; N 1

ð6:37Þ

meaning physically that the filter output signal has non-zero mainlobes repeated with the period ND, while all the sidelobes between them are zero. This filter, the sidelobe suppression filter (SLSF), imitates by its response the perfect periodic ACF. Since for a binary code perfect ACF is not achievable (with a single trivial exception), the SLSF is a mismatched filter, and therefore yields in SNR to the matched filter.

The fastest way to come to an explicit expression for the filter coefficients is to use discrete Fourier transform (DFT). A sequence ai, i ¼ 0, 1, . . . , N 1 and its DFT spectrum a~k, k ¼ 0, 1, . . . , N 1 are related to each other by the direct and inverse DFT:

a~k

¼ i 0

ai exp j

; k ¼ 0; 1; . . . ; N 1

N 1

2 ik

a~k exp j N

; i ¼ 0; 1; . . . ; N 1

ai ¼ N k¼0

1 N 1

2 ik

...., ai – 1, ai, ai + 1,....

∆

bN – 1

...., ci – 1, ci, ci + 1,....

∑

Figure 6.20 FIR filter for a sequence of length N

Time measurement, synchronization and time-resolution	187

Our goal is to get to the discrete delta function (6.37) at the filter output having only one non-zero element per period. Its spectrum is uniform: c~k ¼ c0, k ¼ 0, 1, . . . , N 1. Then on the strength of the convolution theorem [1] the spectrum of the sequence (6.36) at the

~	, k	¼ 0, 1, . . . , N 1, where the spectrum of the sequence of
filter output, c~k ¼ a~kbk ¼ c0	, k	¼ 0, 1, . . . , N 1, where the spectrum of the sequence of
~
the filter coefficients bk is nothing but the SLSF transfer function:
	~		c0
	bk ¼			; k ¼ 0; 1; . . . ; N 1	ð6:38Þ
	bk ¼		a~k	; k ¼ 0; 1; . . . ; N 1	ð6:38Þ

As is seen, the SLSF transfer function is inverse to the signal spectrum, which is why filters of this sort are often called inverse filters. An inverse filter just transforms an input spectrum to make it uniform. As the last equation shows, SLSF is physically realizable for any periodic sequence whose DFT spectrum has no zero components. Applying the inverse DFT to (6.38) gives an explicit form of SLSF coefficients:

bi ¼ N k¼0		a~k exp j	N	; i ¼ 0; 1; . . . ; N 1	ð6:39Þ
c0	N 1	1	2 ik
	X

6.12.2 SNR loss calculation

The matched filter coefficients (see Figure 6.20) would be (ignoring immaterial common

factor) mirror-like to

the

input

sequence:6

i ¼

N i

, i

0, 1,

. . .

, N

1 and

output

2 NP

sequence peak Amf ¼

sequence is binary. For the input

i¼0

i ¼ N, since the input

noise having correlation spread within D and variance

, the output matched filter

variance mf ¼

N 1

aN i ¼ N . Thus, the power SNR qmf

i¼0

at the

matched filter

output:

qmf2

Amf2

ð6:40Þ

In a similar manner for the SLSF output sequence peak Asl ¼ c0 and noise variance:

N 1

2sl ¼ 2 b2i i¼0

Due to the Parseval theorem and the time shift property of DFT, the periodic ACF of an

arbitrary sequence u , u

, . . . , u

N 1

of period N is linked to the sequence energy spectrum

ju~0j

, ju~1j

, . . . , ju~N 1j

by the inverse DFT:

; m ¼ 0; 1; . . . ; N 1

ð6:41Þ

ðmÞ ¼ i¼0

uiui m

¼ N k¼0

ju~kj2exp

N 1

1 N 1

j2 mk

6 A cyclic shift of coefficients against the case of the aperiodic signal (bi ¼ aN 1 i) serves to make (6.38) more compact by excluding the linear phase exponent. For a periodic signal this shift means an adequate cyclic shift of the periodic output signal, and thereby has no effect on its shape or output SNR.

188 Spread Spectrum and CDMA

In particular:

N 1	1		N 1
X			X
Rpð0Þ ¼	juij2¼	N	ju~kj2
i¼0			k¼0

Using this along with (6.38) in the result for the filter output noise variance gives:

	2		N 1		2		c2 2		N 1	1
2	¼		X	~		¼	0		X j j
sl	¼	N		bk		¼		N		a~k 2
			k¼0						k¼0

and output SLSF power SNR (signal peak at the SLSF output Asl ¼ c0):

qsl	¼ sl2		¼ 2		N 1	a~k		2	!	1
		A2		N	N 1	1	j			1
2		sl			X j		j

k¼0

Now we may find energy loss of the SLSF versus the matched filter as:

	qmf2	N 1		1
		X
¼ qsl2 ¼		X	j	j	2	ð6:42Þ
¼ qsl2 ¼		k¼0		a~k	2	ð6:42Þ
		k¼0

To better comprehend the last result, note that:

	N 1		1			N 1
	X					X
N ¼		ai2 ¼				ja~kj2 ¼ ja~j2
N ¼	i¼0	ai2 ¼		N		ja~kj2 ¼ ja~j2
	i¼0					k¼0
and		ja~kj2	!			¼ ja~j 2
N k¼0		ja~kj2	!			¼ ja~j 2
1	N 1	1			1		1
	X

represent, respectively, the arithmetic and harmonic averages of a sequence energy spectrum ja~kj2, k ¼ 0, 1, . . . , N 1. The harmonic average of any set of non-negative numbers never exceeds the arithmetic one, and they coincide only if the averaged numbers are all equal. Thereby the ratio of these entities may serve as some measure of the range of scattering of the averaged numbers. But in our case this ratio:

ja~j2

N 1 1

j j

¼ k

0 a~k 2 ¼

is precisely the energy loss of the SLSF. Therefore, SNR loss is determined by the non-uniformity of the sequence energy spectrum ja~kj2, k ¼ 0, 1, . . . , N 1 estimated in terms of the distinction between its harmonic and arithmetic averages.

The possibility of suppressing all periodic sidelobes offers a new criterion for designing binary sequences, which is alternative to the one of minimizing the maximal sidelobe

Time measurement, synchronization and time-resolution	189

p, max. Indeed, what is the point of worrying about the sidelobe level, if all the sidelobes may be successfully nullified? It is much more natural to minimize the penalty paid for their elimination—and this penalty is, of course, SNR loss . Consider first a binary sequence (hypothetical if N 6¼4) whose energy spectrum is uniform: ja~kj2¼ N, k ¼ 0, 1, . . . , N 1. In the light of (6.41) this means that the sequence has perfect periodic ACF. It is absolutely predictable that, since there is nothing to suppress, SLSF in this case should coincide with the matched filter, having no SNR loss. Equation (6.42) confirms this, giving ¼ 1 or—measured in decibels— dB ¼ 10 lg ¼ 0 dB. Because binary sequences with perfect periodic ACF do not exist, their sidelobe suppression is done in exchange for SNR loss ( dB > 0 dB), justifying the introduction of the design criterion ¼ min.

As in many problems concerning binary sequences (see Section 6.4), a binary globally optimum sequence of fixed length N with minimum loss may be found only by an exhaustive search. This work has been run up to length N ¼ 30 [48]. Certainly, due to the exponential growth of a computational resource, this search cannot continue far beyond the above mentioned range. However, many regular rules of generating binary sequences of lengths as great as one likes and having very small loss (although their global optimality is not guaranteed) are now known.

Let us take a special class of binary sequences having two-level periodic ACF, i.e. constant level R of sidelobes:

RpðmÞ ¼	( R;;m	¼	0 modN	ð6:43Þ
	N m		0 modN
		6¼

All minimax binary sequences, among others, are of this type. The energy spectrum of such a sequence, as (6.41) shows, is the direct DFT of ACF:

ja~kj2¼ m 0	RpðmÞ exp	N	¼ N R þ R m 0 exp	N	; k ¼ 0; 1; . . . ; N 1
N 1		j2 mk	N 1	j2 mk
X			X
¼			¼

The last sum has already appeared in Section 6.11.2 and, as was proved, equals N if k ¼ 0 and zero otherwise. Thus:

j~kj

( N

þ R; k

Þ0

a 2

ðN 1

R; k ¼ 0

6¼

Substituting this into (6.42) gives:

N 1

1 þ ðN 2Þ

6:45

N þ ðN 1ÞR þ N R ¼ ð1 Þ½1 þ ðN 1Þ &

where ¼ R/N is the normalized ACF sidelobe. To come to a SLSF structure, rewrite (6.39) as:

bi ¼ N k¼0				a~kk	2 exp	N	; i ¼ 0; 1; . . . ; N 1
	c0	N 1		a~		j2 ik
		X	j	j

190 Spread Spectrum and CDMA

and substitute (6.44) into this equation:
	c0	"	a~0	a~0	1		N 1	j2 ik	#; i ¼ 0; 1; . . . ; N 1
bi ¼					þ		X a~k exp
bi ¼	N		N þ ðN 1ÞR	N R	þ	N R	X a~k exp	N
							k¼0

The sum in k here produces coefficients of the matched filter, since it may be presented as:

1 N 1		a~k exp j2 ð			N	i	k
N k¼0		a~k exp j2 ð			N Þ
	X									# ¼ aN i; i ¼ 0; 1; . . . ; N 1
		1 N 1		a~k exp j2 ð		N		i	k
	¼ "N k 0			a~k exp j2 ð		N Þ
			X

where use is made of the fact that binary sequence elements are real. Since c0 is an arbitrary scaling factor, let us choose it equal to N R. Then:

bi ¼ aN i	a~0	; i ¼ 0; 1; . . . ; N 1	ð6:46Þ
	1 þ ðN 1Þ

The first term here corresponds to the sequence faig read from right to left, i.e. coefficients of the matched filter. Therefore, for the sequences possessing two-valued ACF (6.43), SLSF is obtained by a slight modification of the matched filter: subtracting a constant from all coefficients. Moreover, for binary sequences of this sort coefficients of SLSF take on only two possible values:

1 a~0

1 þ ðN 1Þ

where a~0 is a sequence constant component, i.e. the difference between the numbers of plus and minus ones over the period a~0 ¼ Nþ N .

As is known from the previous material, lots of binary sequences exist with ACF (6.43) where R ¼ 1 (m-sequences, Legendre sequences and other minimax ones with ACF (6.12)). Evaluating their SLSF loss by (6.45) results in ¼ 2N/(N þ 1), i.e. 2 (3 dB) for lengths that are of practical interest. It is seen then that the most popular minimax binary sequences are of no great value in the light of the criterion of SLSF loss: half of their energy is lost under SLSF processing.

On the other hand, when R is positive and low enough as compared to N< 1/(1 ), i.e. may appear rather small. So-called Singer codes [34,48] are a good example of such binary sequences. A Singer code exists for any length of the form N ¼ (qn 1)/(q 1) and has two-valued ACF (6.43) with R ¼ N 4qn 2, where q ¼ pw is a natural power of prime p and n is natural. The most interesting modification of Singer codes in our context corresponds to q ¼ 3, in which case ¼ (3n 2 1)/(3n 1) and < (3n 1)/(8 3n 2) < 9/8 ¼ 1:125, i.e. dB < 0:51 dB. As is seen, these codes, being processed by SLSF, are rather attractive, since SNR loss accompanying a total sidelobe suppression for them is small.

Example 6.12.1. Let us take a periodic version of the binary Barker code of length N ¼ 5 from Table 6.1: þ1, þ1, þ1, 1, þ1, for which Nþ ¼ 4, N ¼ 1 and constant component a~0 ¼ 3. Its periodic ACF, as may be checked by direct evaluation, obeys (6.43) with R ¼ 1 ( ¼ 1/5). Actually, this sequence is a Singer one with q ¼ 4, n ¼ 2. Figure 6.21b shows the ACF (i.e. the

Time measurement, synchronization and time-resolution	191

matched filter response) of the periodic signal modulated by this sequence (Figure 6.21a). As is seen from (6.46), a matched filter for this sequence is easily transformed into an SLSF by changing all coefficients þ1 to þ2/3 and 1 to 4/3, which after appropriate scaling means changing 1 to 2 with no change of þ1 (see Figure 6.22). The SLSF response to the same signal is constructed in Figure 6.23, where the waveform labels correspond to the points in Figure 6.22. The filter output has the desired form, i.e. with zero level of sidelobes. Using (6.45), it is easy to find the energy loss of the SLSF as ¼ 10/9 ¼ 1:1111 . . . (0.46 dB).

(a)

(b)

Figure 6.21 Binary signal of length N ¼ 5 and its periodic ACF

s(t)		∆			∆			∆			∆


			2
			2
	+			–			+			+	+			r(t)
	+			–			+			+	+
1			2			3			4		5		Chip MF
1			2			3			4		5
						∑						6		7
												6		7

Figure 6.22 SLSF for a sequence of length N ¼ 5

2 t

Figure 6.23 SLSF response for the binary sequence of length N ¼ 5

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