Discrete-time beamforming

In order to implement beamforming in digital systems, we must sample the received signals in time, at t = nD. Now in our digital delay-and-sum, our time delays t_i must be quantized to the nearest sample. (Note that in these diagrams, alphas are the shading coefficients rather than the slowness vector.)

These quantization errors in the time delay perterb the array pattern. In general, the sensors must be sampled at a rate much greater than the Nyquist rate to approximate the time delays required for beam steering. To handle the high bandwidth and large number of elements needed for high resolution sonar, the necessary sampling rate is quite large. These high sampling rates impose requirements on the A/D converters and on the bandwidth of the cables which connect the A/D converters to the beamformer. Large amounts of memory are also required to handle the long delays associated with large arrays and high sample rates. Because of the high implementation cost, digital interpolation beamformers are generally used instead.

Digital Interpolation Beamforming

Now we sample at just above the Nyquist rate, and achieve the desired time-delay resolution by digital interpolation of the sampled data.

In the above digital system, the sensors are sampled at the interval D, needed to satisfy the Nyquist criterion. In the beamformer, digital interpolation is performed to the interval d, where D = L d, and L is an integer larger than one. Now time delays are quantized to integer multiples of d, i.e., t_i = N_id.

In a wideband sonar, the digital interpolation beamformer technique relaxes specifications on the A/D conversion rate and data transmission bandwidth at the expense of additional computation for digital interpolation. Beam degradation introduced by interpolation is controllable and quite small for an interpolation filter of modest design.

Digital Interpolation

Digital interpolation is a two-step process: zero-insertion and then lowpass filtering. Since the filter and the summation are both linear, the filter can be placed either before or after the beamformer summation.

The first step of interpolation, zero-insertion, involves transforming the digital stream x_m(nD) by inserting L-1 zeros after each sample. The resulting stream x_m(nd) has L times more samples, and has had its sampling period reduced by a factor of L.

To complete the interpolation, an ideal digital lowpass filter with a cutoff frequency at w = ¹/L is required. Filtering the stream x_m(nd) yields the interpolated approximation to the input sampled at the interval d. Since FIR filters are not ideal, error is introduced at the beamformer output. Increasing the number of filter coefficients reduces this error, so there is a tradeoff between accuracy and computational complexity.

Frequency-Domain Beamforming for Discrete-Time Signals

Again, frequency-domain beamforming is inherently narrowband. The discrete-time frequency-domain beamformer output is given by

Note that the factor R_i(nD,w)exp(jwnD) can be efficiently calculated for many values of w with a 1-D FFT.

Since the beam has been formed in the frequency domain, the steering delays t_i do not have to be quantized.

For the special case of equally spaced linear arrays, the discrete frequency-domain beamformer output can be computed by applying a 2-D DFT directly to the receiver signals r_i(nD).