Discrete-time beamforming

Beamforming is a signal processing technique used to spatially select propagating waves (most notably acoustic and electromagnetic waves). In order to implement beamforming on digital hardware the received signals need to be discretized. This introduces quantization error, perturbing the array pattern. For this reason, the sample rate must be generally much greater than the Nyquist rate.[1]

Introduction

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Beamforming aims to solve the problem of filtering signals coming from a certain direction as opposed to an omni-directional approach. Discrete-time beamforming is primarily of interest in the fields of seismology, acoustics, sonar and low frequency wireless communications. Antennas regularly make use of beamforming but it is mostly contained within the analog domain.

Beamforming begins with an array of sensors to detect a 4-D signal (3 physical dimensions and time). A 4-D signal   exists in the spatial domain at position   and at time  . The 4-D Fourier transform of the signal yields   which exists in the wavenumber-frequency spectrum. The wavenumber vector   represents the 3-D spatial frequency and   represents the temporal frequency. The 4-D sinusoid  , where   denotes the transpose of the vector  , can be rewritten as   where   , also known as the slowness vector.

Steering the beam in a particular direction requires that all the sensors add in phase to the particular direction of interest. In order for each sensor to add in phase, each sensor will have a respective delay   such that   is the delay of the ith sensor at position   and where the direction of the slowness vector   is the direction of interest.

Discrete-time weighted delay-and-sum beamforming

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Source:[2]

The discrete-time beamformer output   is formed by sampling the receiver signal   and averaging its weighted and delayed versions.

 

where:

  •   is the number of sensors
  •   are the weights
  •   is the sampling period
  •   is the steering delay for the ith sensor

Setting   equal to   would achieve the proper direction but   must be an integer. In most cases   will need to be quantized and errors will be introduced. The quantization errors can be described as  . The array pattern for a desired direction given by the slowness vector   and for a quantization error   becomes:

 

Interpolation

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Source:[3]

 
Flowchart of upsample and linear filter for discrete-time beamforming

The fundamental problem of discrete weighted delay-and-sum beamforming is quantization of the steering delay. The interpolation method aims to solve this problem by upsampling the receiving signal.   must still be an integer but it now has a finer control. Interpolation comes at the cost of more computation. The new sample rate is denoted as  . The beamformer output   is now

 

The sampling period ratio   is set to an integer to minimize the increase in computations. The samples   are interpolated from   such that

 

After   is upsampled and filtered, the beamformer output   becomes:

 

At this point the beamformer's sample rate is greater than the highest frequency it contains.

Frequency-domain beamforming

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Source:[4]

As seen in the discrete-time domain beamforming section, the weighted delay-and-sum method is effective and compact. Unfortunately quantization errors can perturb the array pattern enough to cause complications. The interpolation technique reduces the array pattern perturbations at the cost of a higher sampling rate and more computations on digital hardware. Frequency-domain beamforming does not require a higher sampling rate which makes the method more computationally efficient.[5]

The discrete-time frequency-domain beamformer is given by

 

For linearly spaced sensor arrays  . The discrete short-time Fourier transform of   is denoted by  . In order to be computationally efficient it is desirable to evaluate the sum in as few calculations as possible. For simplicity   moving forward. An effective method exists by considering a 1-D FFT for many values of  . If   for   then   becomes:

 

where  . Substituting the 1-D FFT into the frequency-domain beamformer:

 

The term in brackets is the 2-D DFT with the opposite sign in the exponential

 

if the 2-D sequence   and   is the (M X N)-point DFT of   then

 

For a 1-D linear array along the horizontal direction and a desired direction:

 

where:

  •   and   are dimensions of the DFT
  •   is the sensor separation
  •   is the frequency index between   and  
  •   is the steering index between   and  

  and   can be selected to "steer the beam" towards a certain temporal frequency and spatial position

References

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  1. ^ Sonar Beamforming users.ece.utexas.edu. Retrieved November 12, 2015
  2. ^ Dudgeon, Dan; Mersereau, Russel (1983). Multidimensional Signal Processing. Prentice-Hall. pp. 303–307. ISBN 0-13-604959-1.
  3. ^ D. Dudgeon and R. Mersereau, Multidimensional Digital Signal Processing, Prentice-Hall, First Edition, pp. 307 - 309, 1983.
  4. ^ D. Dudgeon and R. Mersereau, Multidimensional Digital Signal Processing, Prentice-Hall, First Edition, pp. 309 - 311, 1983.
  5. ^ Camargo, Hugo Elias (4 May 2010). "A Frequency Domain Beamforming Method to Locate Moving Sound Sources". hdl:10919/27765.