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In this module we discuss the theory behind delay and sum beamforming with a two-dimensional array, the background of our ELEC 301 project.

1. introduction

Beamforming is the discipline that takes a set of microphones, usually in an array, and a set of point source signals (in a space that we assumed to be R3 orvery nearly so) and tries to focus on one signal source to the exclusion of both noise and other signal sources. In this project , we assume a single source and use the old and powerful technique of delay and sum beamforming implemented over a 2-dimensional array arranged as a 3 by 3 square with thecenter missing. Having a 2-dimensional array allows the location of a source to be determined up to an ambiguity of reflection across plane of the array.

2. delay and sum beamforming

Delay and sum beamforming is quite true to its name as it merely takes the set of signals, delays and maybe weights them by varying amounts, and thenadds them all together. The size of the delays is determined by the direction (for farfield) or point (for nearfield) at which the set of microphones is aimed. Despite its simplicity, delay and sum manages to achieve optimal noisesuppression for the case of a point source in a background of white noise. Of course, normal signal processing applies, and one can do better than just delayand sum if information about the signal other than location is known a priori. For example, if it is known that the signal is bandlimited and baseband, thena suitable lowpass filter can be applied to further suppress noise.

2.1 nearfield processing

Though not implemented, nearfield calculations are both more computationally intensive and accurate. If it is assumed that the microphones have somesort of center for distance, then the center can be designated as the origin for the coordinate system. A point source at a point (xs, ys, zs) would thenemit a signal s(t). A microphone at a point (xm, ym, zm) would then receive a signal m(t). Assuming that signal propogates uniformly with speed v andthat signal strength is equal to the original signal strength divided by the square of the distance, we can conclude that the received signal is:

2.2 farfield processing

In this project, it was assumed that the array was always operating in farfield, an approximation in which the source is assumed to be far enough away thatthe spherical waves it emits can be approximated with plane waves. It is accurate in the limit where the distance between the microphones and the sourceis large enough so that the angle between the source and each microphone does not change significantly.

3. complications

3.1 time quantization

Since all of the processing is done in a digital environment, we must work with samples of the signals and not the signals themselves. Because of this,it is not possible to implement an arbitrary time shift as any shift must be done in increments of the sample period. To remedy this, the signals wereinterpolated digitally by upsampling them and then putting them through a lowpass filter with cutoff corresponding to the amount of upsampling. Anequiripple filter was chosen for the lowpass filter as there appears to be no constraints as to the exact shape of the filter and because an equiripplefilter would avoid the Gibb’s phenomena found in a direct approximation of an ideal lowpass filter. Using this interpolation, greater resolution canbe achieved in the time shifts, though the drawback is the large amount of additional data that must now also be processed. In fact, even though theconcept of delay and sum is incredibly simple, the amount of computation that must be done because of the upsampling is often prohibitively high. It isimpossible for the amount of interpolation to be too high, but if it is too low, then it is entirely possible that the direction of the source will be inaccurateor entirely wrong as the algorithm will be unable to shift the signals to where they match enough.

3.2 aliasing, resolution, and sampling frequency

Given a fixed sampling frequency, there is always the ”normal” aliasing associated the Nyquist Theorem, restricting the fully reconstructable signalsto those that are bandlimited to half of the sampling frequency. Something similar occurs with array spacing, and if proper care is not taken, aliasingmay occur in spatial dimensions. Using the spatial analogue of the Nyquist Theorem, the minimum spacing between microphones must be at most halfthe wavelength corresponding to highest frequency present. Thus, to achieve any resolution at all for higher frequency signals, smaller arrays must be used;however, with a smaller array, the precision with which a direction can be determined is diminished. It appears that there is an uncertainty principalat odds with beamforming in its spatial dimensions.

3.3 unknown source location

This is the main focus of the project: to try to locate a source using an array of microphones and then focus the array in the direction ofthe source, obtaining greater suppression of noise than would be possible using only one microphone. Since the direction of the source is unknown,we decided to scan for the source by sweeping all possibilities. This is where the far field approximation significantly reduces computational complexity.Using nearfield, any algorithm would be forced to evaluate all possible combinations of three coordinates. With farfield, there are only two angles todeal with as opposed to three coordinates so there is far less to compute.

Due to lack of computing power, we were forced to make a few, less-than-desirableassumptions in order to make the algorithm run at all without crashing. One of these simplifications was using only three of the microphones toperform the sweep of possible angles. A further simplification was to assume that the three microphones could be broken into two pairs in the calculationsfor determining the pair of angles from which the maximum was coming from. Further, hardware and computer limitations limited sampling to a rate of 8000 Hz from each of eight microphones and made the processing cost of upsampling prohibitive beyond a factor of around 10.

Questions & Answers

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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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Source:  OpenStax, Investigation of delay and sum beamforming using a two-dimensional array. OpenStax CNX. Feb 04, 2006 Download for free at http://cnx.org/content/col10315/1.3
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