Beamforming enables you to find problems in your system before they occur. As Martin explains， you can perform your own investigation with a computer， a dual-channel A/D card， a pair of microphones， and some software.
Tensions are high and a hostile submarine lurks in the waters just off the coast. A patient exhibits troubling symptoms-do layers of tissue mask a tumor？ A driver checks his mirrors and begins to change lanes but never sees the truck hiding in his blind spot.
It‘s better to go looking for trouble than to have it find you. But how do you do that？ Whether its subhunting sonar， medical ultrasound，or collision-avoidance radar， how do you uncloak danger？
One important arrow in your quiver is beamforming. Think of it as a way to focus and measure the energy falling on an array of sensors as you "look" in different directions. You can do your own handson investigation with only a computer，a dual-channel A/D card， a pair of microphones， and the software developed in this article.
Beamforming techniques come in two flavors： adaptive and fixed. Adaptive methods are haute cuisine. Fixed methods are burgers with fries. Each has its place， but blue-collar fare will be served up here.
TIME DOMAIN BEAMFORMINGTo understand how beamforming works， picture a line of identically spaced microphones. Next， imagine a wavefront consisting of noise plus the signal you want to detect， striking this uniform line array. Finally，make two common simplifications，valid for many applications. One，assume that the wavefront originated in the far field so that it can be approximated as a flat surface propagating toward the array and any curvature it possessed near the source can be disregarded. Second， ignore the three-dimensional quality of wave propagation and restrict your attention to the two-dimensional plane. With this in mind， Figure 1 shows what it would look like if you wanted to focus the energy arriving perpendicular (broadside)to the array.
Each phone is excited by the same wavefront at the same time. To get the total energy falling on the array at any instant， sum together the data from each phone. Refer to Figure 2 if you want to focus the energy from some other direction.
To sum the energy across a given wavefront， you need to insert delays because the wavefront has to travel a different distance to strike each phone. The extra distance the wavefront travels from one phone to the next is d sin θ， where d is the uniform spacing between the phones and θ，the look direction， is by convention measured clockwise from broadside to the array. If the wavefront travels through the medium(e.g.， air or water) at speed c， then the difference between the time it strikes one phone and the next is (d sin θ)/c. Generalizing， from Figure 2 the wavefront strikes phone p at (pd sin θ)/c units of time before it strikes the reference phone. So， for phone p and time t， if you call the phone‘s response yp(t)， then the time-shifted summed output of the line array is given by：
where P is the number of phones in the line array， numbered from 0 to P - 1.
Great， you e done. Just sweep around the array and find the direction that gives the maximum response， right？ Actually， in practice，this time domain， delay， and sum beamformer has drawbacks because the data acquired is sampled， not continuous.
For an ADC clocking in data at some constant interval
