force MAI to zero will inevitably make a useful effect (the first term) in (10.10) vanish too. At the same time, linear independence means that K N, in which case the most adequate choice of signatures is an orthogonal set (see previous subsection) entailing optimality of a single-user receiver and automatically rejecting MAI with no SNR loss and no special decorrelation processing. In the case of oversaturation (K > N) linear independence of signatures is impossible and the decorrelating algorithm cannot be used.

10.1.3 Minimum mean-square error detection

Let us again exploit the idea of mismatched processing in a correlator tuned to the first user’s signal, but this time, instead of forcing MAI to zero, we will try to minimize the overall corrupting effect of MAI and noise. Coming back to (10.7), we may note that only the term A1b1 in it is a useful component, the other two presenting an overall interference (MAI plus noise). In this light it is natural to look for a linear operation (10.9), imitating a useful contribution with minimum mean-square error (MMSE). To formalize the problem we first rewrite (10.9) in a vector form, substituting u(t) from (10.11):

N 1

X

the appropriate moment (see (2.68)), allowing to look at ~y as a vector of observations after the chip matched filtering. Our task now is to minimize mean square deviation "2 of &1 from A1b1 by an appropriate choice of a reference code vector u:

"2 ¼ jA1b1 &1j2 ¼ A1b1 u~yT 2 ¼ min u

Note that no a priori normalization of the reference u is necessary. After squaring and term-wise averaging, the mean-square error takes the form:

where we use elementary matrix algebra (commutativity of multiplication by a scalar and associativity of vector–matrix multiplication, commutativity of a scalar product u~yT ¼ ~yuT ) and non-randomness of u. The ith component of ~y after substitution of (10.6) and then (10.11) becomes:

of repetition periods D are orthogonal (e.g. if chip duration is no longer than D, those chips do not overlap). Now we see that yib1 ¼ A1a1, i, because bits of different users are

In a similar manner we calculate the matrix ~yT ~y, whose elements are simply correlation moments yiyj of samples yi. Then, according to (10.16) and allowing for non-correlatedness of noise samples after the chip matched filter:

2 being the variance of a noise component of yi. Thus, the N N correlation matrix R of the observation vector ~y is:

where IN is the Nth order identity matrix. Substituting (10.17) in (10.15) after discarding the first term independent of u gives the following scalar function to minimize by adjusting u:

At the point u of extremum of f (u) the gradient of f (u), i.e. vector, whose components are derivatives of f (u) with respect to every component of the vector u, should be a zero vector. The gradient of f (u) is readily found (see Problem 10.3) as 2(uR A21a1). Thus, with invertible matrix R the vector u delivering an extremum to f (u) is defined by the equation:

where R is given by (10.18). The reader is challenged to check that the extremum just found is really a minimum of (10.19) (Problem 10.3).

Clearly, this algorithm does not rest on the invertibility of the signature correlation matrix C ¼ AT A; just the observation correlation matrix (10.18) should be invertible, which is practically always true. Hence, the solution (10.20), in contrast to (10.13), is universal regardless of the relation between K and N. At the same time, at least in one important particular case the solution (10.20) degenerates to the single-user algorithm. Let the signature set be a Welch-bound one, meaning that the rows of the signature matrix A are orthogonal (see Section 7.2.2), i.e. AAT ¼ IN . If all signals have the same intensity A, G2 ¼ A2IK , and the observation correlation matrix (10.18) becomes the simplest, R ¼ (A2 þ 2)IN , resulting in u ¼ [A2/(A2 þ 2)]a1, which reproduces a scaled first signature, i.e. the reference of a conventional receiver. Thus, no special MMSE processing exists for the Welch-bound signatures of equal power. This fact is rather trivial if K N, since then such signatures are orthogonal and a conventional receiver

eliminates MAI fully with best noise filtering, but for the oversaturation scenario (K > N) the statement is not that predictable.

In the literature the result (10.20) is often given in another form including explicitly the signature correlation matrix C ¼ AT A [99–101]. Deriving it is possible, for instance, through the matrix inversion lemma given here in the form fitted to the context:

Proof of this result consists in a direct check (Problem 10.4). Note that it works whenever the matrix G is invertible, which is observed automatically if all users’ amplitudes are nonzero. Making use of (10.21) and equation A21a1 ¼ e1G2AT in (10.20) results in:

u ¼ e1G2AT R 1 ¼ 12 e1G2hAT AT AðAT A þ 2G 2Þ 1AT i

¼ 12 e1G2hAT ðAT A þ 2G 2ÞðAT A þ 2G 2Þ 1AT i þ e1ðAT A þ 2G 2Þ 1AT

and eventually:

Returning back to (10.14), we write the final form of the decision rule on the first user’s bit as:

Spreading this rule to the receiver of the kth user’s data is again immediate: ek has to replace e1.

Emphasizing again that the rule under study is universal independently of signature correlation matrix invertibility, it is nevertheless noteworthy that if C is non-singular (K N is a necessary condition of it) and thermal noise diminishes, the MMSE detector converges asymptotically to the decorrelating one:

u ¼ e1ðC þ 2G 2Þ 1AT ! e1C 1AT :

2!0

To demonstrate the efficiency of MMSE it is appropriate to compare the signal- to-interference-plus-noise ratio (SINR) for the receiver effect &1 in cases of reference

where q2kb ¼ A2k/ 2 is power SNR per data bit for the kth user.

314 Spread Spectrum and CDMA

Example 10.1.1. Consider an oversaturated synchronous CDMA with Welch-bound signatures. Binary Welch-bound ensembles exist for any K > N allowing the existence of a K K Hadamard matrix. The signatures then are just K columns of this matrix after discarding any K N rows. In the light of the aforesaid, the case of equal powers will not

Rayleigh variable to imitate the Rayleigh channel. Figure 10.1 presents the dependence of SINR (10.24) for MMSE and conventional detectors on bit SNR under some ‘benign’ (wellscattered) amplitude pattern. The curves show that the gain of MMSE sometimes appears significant (in Figure 10.1 up to about 10 dB). Still, it should be remembered that such a profit is just a matter of chance: for some amplitude patterns it may appear even bigger, but the more uniform the amplitude pattern, the smaller is the difference in SINR versus the conventional receiver. One more remark is that the MMSE detector has much better resistance to the scatter of users’ intensities (if signatures are non-orthogonal, of course) in comparison with the conventional receiver, making it especially attractive wherever the power control is not perfect.

1st user bit SNR (dB)

Figure 10.1 Example SINR curves for MMSE and single-user receivers

10.1.4 Blind MMSE detector

Although the computational complexity of the MMSE algorithm (as well as the decorrelating one) is not at all practically prohibitive there is still one implementation issue motivating further research. As (10.20) shows, the key operation of the MMSE algorithm is inversion of the observation correlation matrix R defined by (10.18). To perform it the receiver of the kth user should know, along with its own signature, also