Diss / (Springer Series in Information Sciences 25) S. Haykin, J. Litva, T. J. Shepherd (auth.), Professor Simon Haykin, Dr. John Litva, Dr. Terence J. Shepherd (eds.)-Radar Array Processing-Springer-Verlag

The rotations, as applied to the general (off-diagonal) elements, may then be expressed in the form

(5.48)

(5.49)

It is clear that (5.48) contains no square-root operations, while (5.49) permits the freedom of choosing the updated scale factor b' to be

The systolic array implementation of this square-root-free Givens rotation algorithm is almost the same as that described in Sect. 5.4.3 for the more conventional form. The cells, of course, store and process different quantitiesthe internal cells update elements of the unit upper-triangular matrix R(n), while the squares of the diagonal elements of R(n) are updated in the boundary nodes. The corresponding cell instructions are given explicitly in Fig. 5.5a. Note that the cosine and sine parameters, broadcast along each row, have been replaced by the modified expressions

(5.51)

and

(5.52)

which again contain no square-roots. From (5.50) and (5.51), it can be seen that the update for b may be performed in the boundary node and takes the particularly simple form

The updated scale factors b are passed diagonally from one boundary cell to the next as demonstrated in Fig. 5.5a. To compensate for the input data skew, the value of b must be delayed for one clock cycle between each pair of boundary nodes, the corresponding storage elements or latches being represented by the black dots.

Golub (private communication in [5.18]) suggested a more efficient form for the square-root-free algorithm derived by Gentleman. This version, which requires only two multiplications and two additions to be performed in each

according to (5.53). For the purposes of normal least-squares processing, this factor is initialized to unity so that, on input to the array, i(n) = x(n) and on subsequent rows, b1/ 2 Xk = Xk' Gentleman [5.18] has noted that more general values of the initial scale factor b serve to weight the input data and this leads to a simple implementation of the weighted least-squares algorithm. The consequences of this observation are explored further in Appendix 5.C.

As with the conventional Givens rotation array described in Sect. 5.4.3, cells in the right-hand column of the square-root-free least-squares processor perform the same function as those internal to the main triangular array. Factorization applies for these also and so they update elements of the vector u(n), where

Thus the whole array updates the matrices R(n} and u(n). From (5.27), (5.42), and (5.56), it is clear that the least-squares weight vector w(n) is simply given by

and hence the method of back-substitution, as described in Sect. 5.4.3, could also be used in conjunction with the square-root-free algorithms.

5.4.5 Sensitivity to Arithmetic Precision

One of the main reasons for choosing the method of QR decomposition rather than the conventional Sample Matrix Inversion technique for adaptive beamforming is its superior numerical property. This should lead to improved performance under conditions of limited arithmetic precision. We conclude this section, therefore, by presenting some typical results from a detailed computer simulation which was carried out to compare the effect of reduced arithmetic wordlength on the performance of two adaptive cancellation processors - one based on Sample Matrix Inversion and the other based on recursive QR decomposition. The results indicate quite distinctly the improved performance offered by the QR decomposition method under conditions oflimited arithmetic precision. In both cases the sequence of data samples was generated and applied to a constraint pre-processor, (to be described in Sect. 5.7). The pre-processor applied a look direction constraint towards the desired signal and was implemented at full computer precision. The transformed data were then truncated to the chosen arithmetic wordlength which was retained throughout the subsequent QR decomposition and Sample Matrix Inversion computation. To ensure a fair comparison between the two basic approaches, the Sample Matrix Inversion solution as defined by (5.14) was actually computed by performing a QR decomposition on the estimated covariance matrix. The only distinction, therefore, is that the recursive QR decomposition algorithm applies its orthogonal transformation to the original data matrix rather than the associated covariance matrix. In both cases the back-substitution was performed using full computer precision.

174 T.J. Shepherd and J.G. McWhirter

Fig. 5.6. Comparison of signal to noise ratios computed using QR decomposition and Sample Matrix Inversion

Figure 5.6 shows a typical comparative result generated using a 24-bit floating point wordlength (16-bit mantissa and 8-bit exponent). The expected signal to noise ratio at the output of an 8-element array is plotted as a function of the number of data samples used to compute the least-squares weight vector. In this particular example we have modelled the effect of three equal power jamming signals received individually at levels of0 dB relative to a thermal noise floor of - 50 dB at the antenna array elements. The complex envelope of each jammer was described by an independent, narrowband Gaussian process. The model also incorporated a desired signal received by the array at a level of 15 dB above the thermal noise floor, but approximately 40 dB below the total received jamming.

From Fig. 5.6 it can be seen that the initial rate of adaptation is extremely rapid for both the Sample Matrix Inversion and the data domain QR decomposition algorithms. In each case, a substantial degree of jamming cancellation is obtained after about 10 to 20 data samples. However, with Sample Matrix Inversion there is clear evidence of an unstable weight vector as reflected by extreme fluctuations in the adaptive response curve. In contrast, the data domain QR decomposition method shows no sign of numerical instability and it is found that, over the timescale shown on these plots, the signal to noise ratio performance gets progressively better as the statistical accuracy (in the form of the updated R matrix) increases with time.

For this particular example, it was found that the Sample Matrix Inversion technique required a floating point wordlength of 32 bits (24-bit mantissa and 8-

176 T.J. Shepherd and J.G. McWhirter

where A(n) is a (p - 1) x (p - 1) matrix, l,._p denotes the (n - p) x (n - p) l;!:nit matrix, ,(n) and ",(n) are (p - 1) x 1 vectors and yen) is a scalar. Now since Q(n) represents a sequence of Givens rotations, it may also be expressed as a product of the form

in which all off-diagonal elements are zero except those in the (i, n) and (n, i) locations. It follows directly that

;=1

i.e., yen) is the product of the cosine terms in the p - 1 rotations represented by Q(n). It can also be shown that

,(n) = [s!(n), s!(n)c1(n), s~(n)c1(n)c2(n),

(5.64)

and that A(n) is a (p - 1) x (p - 1) lower triangular matrix, with diagonal elements [c1(n), c2(n), ... Cp-1 (n)]. Note that the determinant ofA(n) is equivalent to the scalar yen) defined above.

Relationships between the constituent submatrices of Q(n) may be derived by exploiting the unitarity of Q(n). From (5.60) and the property

Equations (5.68) and (5.72) show that the vectors qI(n) and ",(n) share the same norm.

Some insight into the significance of these submatrices of Q(n) may be gleaned by substituting the expression (5.60) for Q(n) into the update equation (5.40). This furnishes the relationships

Multiplying both sides of (5.40) by QH(n) and using (5.60) and (5.69) leads directly to the relationships

Hermitian conjugate) may be regarded as an operator which updates the triangular matrix pR(n - 1) to R(n). Note, however, that elements of A -H(n) are functions of the elements of pR(n - 1) and xT(tn ).

178T.J. Shepherd and J.G. McWhirter

5.5.3A Posteriori Residual Extraction

For the purposes ofdirect a posteriori residual extraction, we must derive for the residual e(n, n) defined in (5.58) an alternative expression which does not depend explicitly on the least-squares weight vector wen). Substituting for x T(t.) from (5.78) leads to the expression

Now recall that (5.78) was derived by multiplying both sides of (5.40) by QH(n). We perform the same operation on (5.41) to obtain the relationship

This expression, which does not involve the weight vector wen), permits the residual to be computed as a simple by-product ofthe recursive QR decomposition process using the systolic array in Fig. 5.4. As noted in Sect. 5.4.3, when the conventional Givens rotation algorithm is employed, the parameter a(n) is generated quite naturally within the triangularization process and simply emerges from the bottom cell of the right-hand column in Fig. 5.4 eight (in general 2p - 2) clock cycles after the first element of x(t.) enters the array. The scalar yen), as given by (5.63), may also be computed very simply. The product of cosine terms is generated recursively by the parameter y as it passes from cell to cell along the chain of boundary processors. As with the b parameter in Fig. 5.5, correct timing of the systolic array requires this parameter to be delayed by one clock cycle at the output of each boundary celL The simple product required to form the a posteriori residual in (5.83) may then be computed by the final processing cell F in Fig. SA.

The a posteriori residual may be computed in a similar manner when the square-root-free algorithm is employed. The square-root-free algorithm delivers from the bottom cell in the right-hand column a scalar a(n) given by

and it has been assumed that ben) is initialized to unity on input to the array. From (5.83) and (5.84), it follows that

where y(n) is the product of cosine terms which arise in the conventional Givens rotation algorithm. However, Ci(n), as computed by the boundary processors for the square-root-free algorithm, is simply the product of all the cterms and it can easily be shown that this is equivalent to the product of the squares of the conventional cosine parameters. Thus,

(5.86)

and

Hence the a posteriori residual e(n, n) may also be obtained directly from the square-root-free processor array, using a final multiplier cell F as illustrated in Fig.5.5a.

Avoiding the need to derive an explicit solution for the least-squares weight vector w(n), and hence the problems associated with back-substitution, greatly simplifies the overall processor architecture as is evident from Figs. 5.4 and 5.5. It also enhances the numerical stability of the adaptive combiner since the solution of(5.27) could, in general, be ill-conditioned. Note, for example, that the processor array in Fig. 5.4 or 5.5 will produce the correct (zero) residual even if n < p and the matrix X(n) is not of full rank. This sort of unconditional stability is most important in the design of real time signal processing systems. In many respects, the systolic array in Fig. 5.4 or 5.5 constitutes an extremely sophisticated digital ruter which acts like a funnel, taking in several streams of sampled data and outputting a single stream of processed data. The fact that an advanced numerical algorithm can be mapped onto such a simple and regular structure highlights the benefit of coupling the algorithm and architecture designs as closely as possible. This hybrid discipline is well described by the term "Algorithmic Engineering", further examples of which will be found later in this chapter.

5.5.4 A Priori Residual Extraction

The a priori residual e(n, n - 1) as defined in (5.59) may also be obtained directly from the triangular processor array as we shall now show. By substituting for xT(tn ) from (5.74) into (5.59), we obtain the expression

Now recall that (5.74) was derived by substituting the expression (5.60) for Q(n) into the basic update relationship (5.40). In a similar manner, (5.41) may be used to derive the corresponding equation

180 T.J. Shepherd and J.G. McWhirter

and substituting this expression for yen) y(tn ) into (5.88) allows the a priori residual to be written in the form

In the context ofleast-squares lattice and fast Kalman algorithms, a scalar ofthe form of a(n) has been termed a "rationalized" residual [5.23], and the quantity [1 - y2(n)] a "log-likelihood" variable [5.20], owing to its occurrence in the description of multivariate Gaussian processes.

As regards the square-root-free algorithms, it follows from (5.84), (5.86), and (5.91) that

and so the scalar which emerges naturally from the bottom cell in the right-hand column of Fig. 5.5 is the corresponding a priori residual. Clearly, no final multiplier cell is required if this residual constitutes the desired output of the processor array.

5.6 Weight Freezing and Flushing

5.6.1 Basic Concept

In Sect. 5.5 it was shown that if a data vector [xT(tn), y(tn )] is input to the triangular array in Fig. 5.4, the corresponding a posteriori least-squares residual e(n, n) emerges from the final cell F after the appropriate number of clock cycles. In order to achieve this result, the array must effectively perform two distinct functions: (1) generate the updated triangular matrix R(n) and the corresponding vector u(n) [or D(n), R(n) and u(n) for the square-root-free algorithm] and hence, implicitly, the updated weight vector wen); (2) act as a simple linear filter which applies the updated weight vector to the input data according to (5.58).

If the array is subsequently "frozen" by suppressing any update of the stored values, but allowed to function normally in all other respects, it will continue to