Array Theory

Loudspeaker arrays are not new. The first designs can be found dating back to the 1930s. Due to the limited frequency range, the frequency-dependent directional behavior and the resulting irregular off-axis response, loudspeaker columns were mainly used for speech reinforcement. With the development of high-quality horns in the 1970s, loudspeaker columns lost even more ground.

The (ongoing) rapid development of micro-electronics in the last decades, in particular the development of digital signal processors (DSP's), offered a whole new range of possibilities for the control of loudspeaker arrays. In the early 1990s, JBL’s Duran Audio group, introduced the first commercially available active, DSP-controlled loudspeaker line array (AXYS® Octavox and Intellivox series).

Probably, the success of these arrays in a wide field of applications (airports, train stations, churches, sport stadiums, musical and opera shows etc.) has contributed to today's increasing interest of manufacturers in loudspeaker arrays for the concert touring market.

Sound is a wave phenomenon. In an acoustic wave, energy (i.e. a sound signal) is transported from one region in space to another. This means that a sound wave has both temporal properties (i.e. the contents of the signal) and spatial properties (i.e. the directional behavior).

The most fundamental acoustic source is the monopole. A monopole is a point source, which may be considered as an infinitely small radially pulsating sphere. The acoustic wave pattern of a monopole is spherically symmetrical, which means that the energy is radiated equally in all directions (i.e. omni-directional). Although a real point source does not exist, it is a basic tool to describe the radiation pattern of more complex sources, e.g. a vibrating loudspeaker diaphragm or an array.

An array can be simply defined as a spatial distribution of loudspeakers along a line (i.e., a line array) or at a surface (i.e., a planar array). By driving the array elements independently, both the temporal and the spatial response of an array can be controlled electronically.

Array Directivity
The response Ptotm of an array at receiver position m can be mathematically described as a superposition of the individual contributions Pnm of the array elements n.

Note that Eq. 1 is a complex addition, since at any point in space the wave field of each loudspeaker has a certain amplitude and a phase. So, the total response of an array is always the result of interference of the individual wave fields, not just a combination of the separate energy contributions.

Far Field
In acoustics the response of loudspeakers and arrays is often approximated by its far field response. Many room acoustic parameters, like the critical distance, Alcons, STI etc. can be predicted using a far field assumption. The far field condition (or Fraunhofer condition) is given by

in which r is the array-to-receiver distance, L the size of the array and λ the wavelength. For example, for a 5 m line array, producing a tone of 100 Hz, the far field starts at approx. 3.7 m, while for a 10 kHz tone at around 370 m. 

In the far field, the complex radiation pattern (i.e. wave field) of an array may be described as the response of one single point source with a directivity function Garray

where φ and θ are the azimuth and elevation angles respectively, f the frequency and c the speed of sound.

If all loudspeakers have the same far field directivity function Gs, the directivity function Garray of the total array in the far field is given by

where F[W(rn,f)] denotes the spatial Fourier transform of the driving function W (i.e. the driving signal for each loudspeaker at position rn).

Near Field
In many situations (e.g. using large arrays and/or at high frequencies), the far field condition is not fulfilled. In order to calculate the near field response, Eq. 1 has to be applied. If it is assumed that the listeners are in the far field of each individual loudspeaker, though in the near field of the total array, Eq. 1 can be approximated by:

where rm,n is distance between loudspeaker n and receiver m. Obviously, Eq. 5 is also valid in the far field of the array.

The main parameters that affect the directional behavior of an array are: Array Size & Loudspeaker Spacing.

Array Size
The effect of array size is illustrated using the following example.

Consider a vertical line array, consisting of an increasing number of monopoles (N= 2 to 16). The distance Δz between the monopoles is fixed (0.17 m, i.e. λ/2 @ 1 kHz), which means that the length of the array (NΔz) is a variable in this example. All monopoles are fed with the same source signal, successively a sine of 125 Hz, 250 Hz, 500 Hz and 1 kHz. The driving signals have been normalized for each situation, which means that the far field on-axis (positive y-axis) response is kept constant. For each situation the SPL is calculated on a grid of 200x200 points in the y-z plane (i.e. in a vertical plane through the array). The results are shown in Fig.1.

A few important observations can be made from the plots in Fig.1:  

For a fixed frequency (i.e. looking along one column), the main beam becomes narrower for increasing array lengths.

For a fixed array length (i.e. looking along one row), the main beam becomes narrower for increasing frequencies.

Note that the far field directivity pattern is constant along the diagonal (lower left to upper right corner). For these situations the array length is constant relatively to the wavelength. From the results it can also be verified that the angular array response is distance-dependent, especially for large arrays and/or high frequencies, as may be expected from Eq. 2. Apart from the main lobe, also some side lobes exist. These side lobe are a result of the finite size of the array, i.e. the discontinuity of the array. Tapering of the amplitude near the edges of the array, i.e. 'softening' the edges, can reduce these side lobe effects.

Loudspeaker Spacing
The effect of loudspeaker spacing is illustrated in Fig.2. Again, consider a vertical line array, consisting of a decreasing number of monopoles (N= 16 to 2). The total length of the array NΔz is kept constant (2.72 m), which means that the loudspeaker spacing Δz is a variable now.

Note that the situations in the upper row of Fig.2 and the lower row of Fig.1 are identical. From Fig.2.the following main observation can be made: For increasing frequencies and/or increasing loudspeaker spacing (i.e. going from the upper left corner to the lower right corner) an increasing number of so-called grating lobes occur.

These grating lobes are a result of spatial under-sampling at high frequencies and/or sparse arrays. Note that for a fixed frequency (i.e. looking along one column), the width of the main beam is unaffected by the loudspeaker spacing. Grating lobes do not occur if the spatial Nyquist-criterion is fulfilled.

Using l=c/f the spatial anti-aliasing criterion of Eq. 6 can be reformulated as:

Below the spatial Nyquist frequency (fnq=c/2Δz) no grating lobes will occur.