Technology
Digital Directivity Synthesis (DDS)


What is DDS?
Digital Directivity Synthesis is an advanced, versatile array control optimization concept. Using DDS, any desired 3D radiation pattern can be synthesized within the physical constraints of a pre-defined array (e.g. transducer distance, array length etc.).


DDS is based on a unique, specially adapted 'constrained weighted least-squares' optimization algorithm. Starting from a desired direct SPL distribution in a hall or space, the optimum output filter for each array channel is calculated. In other words, the desired 'illumination' of the hall or space is 'mapped' back' to the array, instead of mapping the array response to the hall. Note that the physical array configuration itself is not optimized by DDS.

Using the WinControl software the output filters can be uploaded to the on-board DSP hardware, which takes care of the real-time signal processing. Apart from the DSP, each array unit is equipped with a micro controller that takes care of all the surveillance routines, DSP management, storage and logging and RS-485 network functionality.


THE FORWARD ARRAY MODEL

Consider the 3D loudspeaker array configuration as shown in Fig.1. As shown, the array geometry is not restricted to a (curved or bent) line, but can also be (curved) planar.

The array has N independent output channels. Each channel is driving one or more loudspeakers. Channel n is processed by output filter Wn(f). The total array response Pm(f) at a certain receiver position m can be calculated by the summing all channel responses:

The response Hm,n(f) of channel n at receiver m is given by the sum over loudspeaker 1 to Ln in channel n:

in which rm,l is the distance between receiver m and the transducer l in channel n and k representing the wave number, k=2πf/c, with c the speed of sound. It is assumed that all loudspeakers in channel n are identical and can be fully described by their far field complex directivity response Glls, which is a function of azimuth φm,l, elevation θm,l and frequency f.

Eqs.(1-2) describe the forward array model. This means that given a set of output filters Wn(f), the array response Pm(f) at any receiver position m can be calculated. Note that this model is also valid for receivers located in the near field of the array. The only assumption that has been made is that far field conditions for the individual loudspeakers are satisfied.


In general, the output filters Wn(f) are unknown. Let Dm(f) represent the user-defined desired response of the array at frequency f for a set of M receivers (m =1..M). By replacing Pm(f) by the desired response Dm(f), Eq (1) can be re-written as a system of M equations with N unknowns. An exact and unique solution does not exist since in general there are more equations than unknowns (M>N).

However, it is still possible to find a 'best fit' for Wn(f) by applying a 'weighted least-squares' error criterion. This implies minimizing the difference between the desired response Dm(f) and the realized response Pm(f). In addition error weights are used which indicate the 'importance' or 'priority' of different receiver positions. Receiver positions that have a relatively high weighting factor, are favored above others.

DDS utilizes a unique, specially adapted variant of the weighted least squares algorithm. Besides an optimum match between desired and realized array response, the DDS design goals were:

Minimum array sound power. 
Optimum array sensitivity.
Insensitivity to small deviations in position, directivity, and sensitivity of the individual loudspeakers. 
Stable and robust output filters with accurate amplitude and phase response. Therefore, the output filters are implemented as Finite  Impulse Response (FIR) filters.


An overview of the DDS optimization process in practice is shown in Fig. 2.


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