Phased Array Beamforming

Phased array beamforming is a technique for measuring waves and determining their DOA. A simple microphone phased array uses several microphones placed along a flat plane. When sounds arrive at the microphones, depending on the DOA, the phase on the signal at each microphone is slightly different. This is because the microphones are in different places and so the sound will reach each microphone at a different time. We can calculate the time delay between the sound reaching each successive microphone using equation 7.1.

delay =L sin θ

c (7.1)

Figure 7.5 shows how this equation relates to the physical microphone array. The distance between the microphones is L and measured in metres, c is the speed of sound, which is 343 m/s, and θ is the DOA of the sound. L sin θ is therefore the distance, in metres, that the sound wave must travel before it reaches the next microphone, and dividing this by the speed of sound gives the time delay.

We can effectively steer the direction in which the microphone array “listens” by applying a delay to the signal from each microphone. These delays are calculated such that the sound will reach each microphone at the same time only if the sound source is in the direction we steered the array. The resulting delayed signals are then summed. If the sound source is in the direction in which the array was steered then the summation will give a peak, and the summation will be noise if the DOA was from a different angle. This process is called “delay and sum beamforming” and is illustrated in figure 7.6.

Benefits of Phased Array Beamforming

The principle of phased arrays can be applied to measuring any kind of waveform that propagates through a medium; including electromagnetic and acoustic waves. Phased array DOA estimation techniques are a useful area of research because of their potential applications: for example RADAR, SONAR or ultrasound scanning [48]. By using delay and sum to “point” the soundboard in any direction we can theoretically achieve

Σ

(a)

z

-2

z

-1

z

0

Σ

(b)

Figure 7.6: As a sound wave moves from left to right, it reaches each microphone in turn. Without delay (a), the sum of the signals is noisy. By delaying the signal from the microphones (b), the signals can reach the summation module at the same time, resulting in a peak in the summation. The amount to delay the signals by varies depending on the DOA of the sound wave.

a high degree of accuracy in DOA estimation, as has been demonstrated in the scientific literature.

In Valin et. al. [90] the authors create an array of 8 microphones on the corners of 50 x 40 x 36 cm cuboid. This is mounted onto a robot and is used for navigating the robot towards a sound source. The robot was tested by playing tones at it with a speaker, and the results are presented as the mean azimuth prediction error as it varies with distance from the source. The system gave an error of up to 3◦over a distance of 3 metres. Dumbacher et. al. [26] compare several DOA estimation techniques, including delay and sum phased array beamforming, by using them to detect the sound source in a car wheel moving on a rolling road dynamometer. The results were not statistically analysed but presented as a image of the car tyre with the measured sound intensities super-imposed on top. The results correctly located the sound source as the lower front edge of the tyre. In [100] a four microphone phased array used to locate noise sources in a car. The equipment is capable of measuring the DOA in a three dimensional space, so gives not just the DOA azimuth, but its elevation too. The authors claim that “in general, detection can achieve more than 90% correct rate” [100].

The main disadvantage of delay and sum beamforming is that it requires a high sample rate on the microphones. The discretisation caused by the sampling means that we can only apply time delays that are some factor of the sample rate. With a higher sample rate, the duration of a single sample is shorter, so delays are more able to closely match the delay needed to steer the phased array. According to [63] “frequently the inputs sample rate is five to ten times that required for waveform reconstruction”. If our maximum expected frequency is 10 kHz then we need to sample at at least 20 kHz to be able to reconstruct the waveform, so for delay and sum beamforming our actual

sample rate must be at least 100 kHz to 200 kHz.

Our intended application of phased array beamforming is smaller than any we have found in the literature. To fit on top of an e-puck the microphones need to be close together, requiring very short beamforming delays and hence a very high sample rate. As we are using a PIC to do all our processing, this presents an upper limit on the capabilities of the hardware.

Delay and sum beamforming is the most basic form of phased array DOA estimation [63]. Considerable work has gone into developing more advanced signal processing techniques. Krim and Viberg [48] give a thorough review of these up until 1996, the most notable of which is the Multiple Signal Classification (MUSIC) algorithm [81]. MUSIC is capable of detecting multiple signals in the environment, and performs significantly better at DOA estimation than other popular algorithms [48, 81].

Despite MUSIC being a superior DOA estimator to delay and sum beamforming, we have chosen to use the simpler algorithm in our work. This is because the focus of our work is not on DOA estimation but the evolution of swarm robotic behaviour. The aim of this chapter is to develop hardware that is adequate for measuring a sound’s frequency and DOA, without investing too much time on its development. Advanced signal processing is outside the scope of this work and investigation into this field is left as future research.