Many beamforming schemes have been devised to cancel an interference source that is correlated with the signal. In principle, these work by restoring the rank of R. In this section, some of these are briefly reviewed [God97].
In some earlier work [Wid82, Gab80], a mechanical movement of the array perpendicular to the look direction was suggested to reduce the signal cancelation effect by the correlated interference. The scheme generally known as the spatial dither algorithm works on the principle that as the movement is perpendicular to the look direction, the signal induced in the array is not affected, whereas the interference that arrives from a direction different from that of the signal gets modulated with this motion. This causes a reduction in interference, as noted in [Cho87] where the dither algorithm is further developed such that a mechanical movement is not required.
The spatial smoothing scheme [Eva81] uses a notion of spatial averaging by subdividing the array into smaller subarrays, and estimates the array correlation matrix by averaging the correlation matrices estimated from each such subarray. The use of spatial smoothing for beamforming is discussed in [Sha85, Red87] showing that the use of this method reduces effective correlation between the interference and desired signal resulting in reduced signal cancelation caused by optimal beamforming. Details on spatial smoothing
FIGURE 5.1
Output SNR vs. the uncorrelated noise power for a four element linear array with one-half wavelength spacing for various values of 冨δ冨, pI = 1, θI = 85°, pS = 1, θ0 = 90°, δp = 45°. (From Godara, L.C., IEEE Trans. Acoust. Speech Signal Process., 38, 1–15, 1990. ©IEEE. With permission.)
The spatial smoothing method uses uniform averaging of all matrices obtained from various subarrays, that is, each matrix is weighted equally. This results in an estimate of the matrix that is not as good as the one that could have been obtained from given subarray matrices. Ideally in the absence of correlation, the array correlation matrix for a uniformly spaced linear array has a Toeplitz structure, that is, elements of the matrix along each diagonal are equal, and the estimated matrix by the spatial smoothing scheme is not the closest to the Toeplitz matrix. An estimated matrix that is closest to a Toeplitz matrix is obtained by a spatial averaging technique [Tak87, Lim90]. This technique weighs each subarray matrix differently and then optimize the weights such that it minimizes the mean square error between the weighted matrix and a Toeplitz matrix. When this matrix is used to estimate the weights of the beamformer, the resulting system reduces more interference than that given by the uniform weighted matrix estimate.
It should be noted that the number of rows and columns in the estimated matrix is equal to the number of elements in the subarray and not equal to the number of elements in the full array. Thus, the weights estimated by this matrix could only be applied to one of the subarrays. Consequently, not all array elements are used for beamforming. This reduces the array aperture and its degrees of freedom. For an environment consisting of M − 1 direction interferences and the desired signal, the subarray size should be at least M + 1 and the number of subarrays should be at least M(M − 1) + 1 [Tak87].
A scheme that does not reduce the degrees of freedom of the array is described in [God90]. It decorrelates the sources by structuring the correlation matrix as the Toeplitz type by averaging along each diagonal, and uses the resulting matrix to estimate the weights of the full array. An adaptive algorithm to estimate the weights of an array based
FIGURE 5.2
Output SNR vs. the magnitude of the correlation coefficient for a four-element linear array with one-half wavelength spacing for various values of σ2n, pI = 100, θI = 85°, pS = 1, θ0 = 90°, δp = 45°. (From Godara, L.C., IEEE Trans. Acoust. Speech Signal Process., 38, 1–15, 1990. ©IEEE. With permission.)
on this principle is presented in [God91], and the concept is extended to broadband beamforming in [God92]. Details are provided in Section 5.7 and Section 5.8.
A beamforming scheme [Wid82] based on master and slave concepts cancels the corre- lated arrival by the use of two channels. In one channel, the look direction signal is blocked, and then weights are estimated by solving the constrained beamforming problem. These weights are then used on the second channel. As the signal is not present at the time of weight estimation, the beamformer does not cancel the signal. However, the process only works for one correlated interference. It is extended for the multiple correlated interference case in [Lut86] where an array of 2M − 1 elements is required to cancel M − 1 interferences.
Other schemes that require some knowledge of the interference, such as direction or the correlation matrix due to interference only, are discussed in [Han86, Han88, Qia95, Wil88, Han92]. Many of the schemes discussed above improve the array performance in the presence of correlated arrivals by treating the correlated components as interferences and canceling them by forming nulls in their directions using beamforming techniques. These methods do not utilize the correlated components as is done in the diversity-combining techniques discussed in Chapter 7. In diversity combining, various components are added in a way to improve the signal level.
The RAKE receiver [Vau88, Tur80, Pri58, Faw64] achieves this increase in signal level for a CDMA system by using a number of demodulators operating in parallel to track each component employing the user code for that signal. The signal delay is identified by sliding the code sequence to obtain the maximum correlation with the received component.
The signals are added at the baseband after appropriate delay and amplitude scaling. The receiver, however, does not cancel unwanted interference by shaping the beam pattern.
FIGURE 5.3
Output SNR vs. the magnitude of the correlation coefficient for a ten-element linear array with one-half wave- length spacing for various values of σn2, pI = 100, θI = 85°, pS = 1, θ0 = 90°, δp = 45°. (From Godara, L.C., IEEE Trans. Acoust. Speech Signal Process., 38, 1–15, 1990. ©IEEE. With permission.)