delayandsum
Delay-and-sum is a basic beamforming technique used in array signal processing to extract signals from a desired direction by compensating for differences in arrival times across sensors and then summing the signals. In an array of N sensors, each sensor i records a signal x_i(t) which contains the target source along with noise and interference. For a specified direction, delays τ_i are computed so that signals from that direction align in time; the delay-and-sum output is y(t) = sum_{i=1}^N x_i(t - τ_i). In the frequency domain, the operation corresponds to applying a phase shift e^{-j ω τ_i} to each sensor’s spectrum before summation.
The delays τ_i depend on the array geometry and the assumed direction of arrival. The method is
Advantages include simplicity, robustness, and low computational cost, enabling real-time processing on modest hardware. Limitations include