2.3 M-ary data transmission: deterministic signals

growth of SNR. As a matter of fact, the complementary error function Q(x) drops approximately as exp ( x2/2) when x is sufficiently large, and even a small increment of large x may reduce Q(x) to a negligible level in comparison to its initial value. Due to this, when SNR is large enough only the closest signal pairs may contribute perceptibly to the sum in (2.22), and if dmin is the minimum distance over all signal pairs occurring nmin times among them, estimation (2.22) transforms asymptotically into the following:

Approximation (2.23) points, above all, to an asymptotic convergence of the union bound to the genuine value of the error probability, SNR increasing. To explain it physically, return to Figure 2.3 and note that when the noise level is very low only the signal vectors which are nearest to the true one are at risk of being erroneously mistaken for the latter. This means that asymptotically only signal pairs with distance dmin determine the true error probability Pe itself (not only its upper bound!), which entails closeness between Pe and its union bound.

Result (2.23) underlies one of the possible and most important formulations of the signal design problem: maximization of minimum distance between M signals. As was already mentioned in Section 2.1, such a task is geometrically equivalent to packing M vectors in such a manner that the closest pairs of them have maximal achievable distance: dmin ¼ max. Various limitations can be imposed on a signal (vector) constellation. First of all, some energy constraint should be prescribed, allowing

selection procedure may be called volume packing. Very frequently, however, it is required that no energy be involved in mapping messages onto signals, i.e. that all energies be the same Ek ¼ E ¼ const, k ¼ 1, 2, . . . , M. In this case all signal vectors have equal lengths, i.e. lie on the sphere surface, hence the name spherical packing.

The other typical limitation in signal design is the dimension ns of signal space, inside which signal vectors are packed. The physical content of this constraint is again associated with a very practical limit on the bandwidth resource. To explain the interconnection between them, consider first the case of baseband signals and suppose that the total (two-sided) bandwidth and time interval which can be allocated to all M signals together are limited to Wt and Tt, respectively. The first of these restrictions allows for bandwidth saving, while the second reflects the desire to transmit necessary data during an acceptable time period, i.e. with acceptable transmission rate R ¼ log M/Tt. Then, according to the sampling theorem, only about WtTt independent samples are at our disposal to synthesize M signals, each signal being thereby treated as a vector in the space of dimension ns ¼ WtTt. Some caution in estimation of the number of independent samples is caused by the impossibility of the energy of any signal being concentrated within finite intervals of both the time and frequency domains simultaneously. But in the first-approximation estimates this theoretical fact can be ignored.

20 Spread Spectrum and CDMA

When ns ¼ 2, finding the optimal constellation with volume packing becomes more

difficult and may even lead to asymmetrical patterns, while spherical packing is trivial p

and is performed by a uniform placing of M points on the circle of radius E. Widely practised in modern digital communications, M-ary quadrature amplitude modulation (QAM) gives an example of symmetrical volume-packed two-dimensional constellations which are not necessarily theoretically optimal but convenient from a hardware implementation point of view (Figure 2.6b, M ¼ 16). On the other hand conventional M-ary phase shift keying (PSK) constellations have uniformly spaced points on the circle and are optimal in terms of spherical packing (Figure 2.6c, M ¼ 8).

The problem of optimal packing in spaces of higher dimension ns > 2 is very complex and has no general mathematical solution so far. Many useful particular results are scattered over the range of books and papers (see, e.g. the bibliography in [11] and web site [12]).

Let us now try to find an upper limit on the minimum distance in the loosest statement imposing no preliminary binding on the signal space dimension ns, and estimate the minimal value of ns, which allows this limit to be achieved. Restricting our attention to the spherical packing (Ek ¼ E, k ¼ 1, 2, . . . , M), calculate the sum of all

M2 squared distances, including trivial ones (from any signal to itself). Cosine theorem (2.16) gives:

with kl being the correlation coefficient of the kth and lth signals. To estimate the sum of all correlation coefficients use definition (2.14) for kl, change the order of integration and summation, and note that the double sum in the integrand has separable summation indexes k and l, which transforms it into a product of two identical sums:

Since the integral of a square is never negative, it follows from (2.26):

M

X

d2ðsk; slÞ 2M2E

k;l¼1

At the same time, the sum above is no smaller than M(M 1)dmin2 . Combining this inequality with the preceding one results in the upper border on the minimum distance:

If M signals achieving this upper bound existed they would be quite fairly called optimal in terms of the minimum distance criterion. To show that they do exist, take M vectors uk, k ¼ 1, 2, . . . , M having zero pairwise inner products and unit lengths: (uk, ul) ¼ kl, k, l ¼ 1, 2, . . . , M, where kl ¼ 0, k 6¼ l; kk ¼ 1 is the Kronecker delta

function. Such vectors, called orthonormal, exist in any vector space whose dimension is no smaller than M. Now form M new vectors vk, k ¼ 1, 2, . . . , M, subtracting from

each of uk the sum u ¼ PM uk weighted by a coefficient 1/M: vk ¼ uk u/M. Calcu-

k¼1

late the inner product of vk and vl. Due to its linearity:

where use is made of the orthonormality of vectors uk. Let us change the lengths of p

vectors vk, multiplying them by ME/(M 1), and take the resulting vectors as signal ones:

Then the squared distance between two signals according to cosine theorem (2.16) and equation (2.29) is:

which coincides with the right-hand side of (2.27). Hence, signals lying on the bound (2.27) really exist. More than this, the distances between any two of them are the same, i.e. these signals fall into the category of equidistant ones. They are widely known under the special name of simplex signals. Directly from their definition it follows that:

meaning that simplex signals are linearly dependent, unlike initial orthonormal vectors uk. It is easily verified that the dimension ns ¼ M 1, i.e. smaller by one than the number of signals, is necessary and sufficient for constructing M simplex signals.

The property of equidistance of simplex signals also entails equality of correlation coefficients kl for any pair. Evaluation of kl with the help of (2.14), (2.29) and (2.28) results in:

showing that the angles between any two simplex signals are the same and greater than/2. For the simplest sets of M ¼ 2, 3, 4 simplex signals (see Figure 2.7), the values of the correlation coefficient equal 1 (antipodal signals), 1/2 and 1/3, respectively, which in turn correspond to angles 180 , 120 and approximately 110 . When M ¼ 4, simplex vectors form the simplest regular polyhedron (tetrahedron), which explains the name of the signals: simplex is Latin for ‘simple’.

Figure 2.7 Examples of simplex signals

For any equidistant signal set d(sk, sl) ¼ dmin for all pairs of distinct vectors so that in (2.23) nmin ¼ M(M 1), and this is also the number of summands in (2.22). Substitution of this along with (2.30) in equation (2.23) gives an approximation of the asymptotic error probability achievable with simplex signals, which according to (2.22) is at the same time the upper bound of the error probability:

s !

ME

Pe;min ðM 1ÞQ ð2:31Þ ðM 1ÞN0

Since simplex signals are optimal as for minimum distance, the right-hand side of the latter expression presents simultaneously the minimum possible asymptotic error probability for M signals of fixed and equal energies E.

The orthogonal signals, which are another example of equidistant signals, are practically as effective as the simplex ones when the number of signals M is sufficiently large. Indeed,

the correlation coefficient of orthogonal signals is zero and the distance between any p

two of them d(sk, sl) ¼ dmin ¼ 2E. This, used in (2.23), produces an asymptotic error probability for M orthogonal signals, which again borders the exact error probability from above:

r

E

Pe;ort ðM 1ÞQ ð2:32Þ

N0

Comparing (2.32) and (2.31) shows that to equalize the error probabilities in both cases, orthogonal signals should be of M/(M 1) times higher energy than simplex signals, i.e. energy loss of the first to the second ones is defined as ¼ M/(M 1). When M 1 this loss is negligible and orthogonal signals can be considered optimal; e.g. for M ¼ 64 ¼ 64/63, which corresponds to an increase of energy of orthogonal signals against simplex ones by less than 0.07 dB (or 2%). This discrepancy is certainly of no practical significance, and whenever M is large enough orthogonal and simplex signals can be used interchangeably depending on implementation or other reasons.

Talking about M-ary orthogonal signalling (in the literature the terms orthogonal modulation and orthogonal coding are also used), let us remember that the maximal number of orthogonal signals is exactly equal to the signal space dimension: M ¼ ns. Therefore, within the fixed total bandwidth Wt and duration Tt, up to WtTt baseband or 2WtTt bandpass orthogonal signals can be accommodated. Additional physical