10. Driving functions for LSFS

The reproduction accuracy of WFS is limited due to practical aspects. For the audible frequency range the desired sound field can not be synthesized aliasing-free over an extended listening area, which is surrounded by a discrete ensemble of individually driven loudspeakers. However, it is suitable for certain applications to increase reproduction accuracy inside a smaller (local) listening region while stronger artifacts outside are permitted. This approach is termed LSFS in general.

The implemented Local Wave Field Synthesis method utilizes focused sources as a distribution of virtual loudspeakers which are placed more densely around the local listening area. These virtual loudspeakers can be driven by conventional SFS techniques, like e.g. WFS or NFC-HOA. The results are similar to band-limited NFC-HOA, with the difference that the form and position of the enhanced area can freely be chosen within the listening area.

The set of focused sources is treated as a virtual loudspeaker distribution and their positions \({\x_\text{fs}}\) are subsumed under \(\mathcal{X}_{\mathrm{fs}}\). Therefore, each focused source is driven individually by \(D_\text{l}({\x_\text{fs}}, \w)\), which in principle can be any driving function for real loudspeakers mentioned in previous sections. At the moment however, only WFS and NFC-HOA driving functions are supported. The resulting driving function for a loudspeaker located at \(\x_0\) reads

(10.1)\[D(\x_0,\w) = \sum_{{\x_\text{fs}}\in \mathcal{X}_{\mathrm{fs}}} D_{\mathrm l}({\x_\text{fs}}, \w) D_{\mathrm{fs}}(\x_0,{\x_\text{fs}},\w),\]

which is superposition of the driving function \(D_{\mathrm{fs}}(\x_0,{\x_\text{fs}},\w)\) reproducing a single focused source at \({\x_\text{fs}}\) weighted by \(D_\text{l}({\x_\text{fs}}, \w)\). Former is derived by replacing \(\xs\) with \({\x_\text{fs}}\) in the WFS driving functions and for focused sources. This yields

(10.2)\[D_{\mathrm{fs}}(\x_0,{\x_\text{fs}},\w) = \frac{1}{2\pi} A(\w) w(\x_0) \i\wc \frac{\scalarprod{\x_0-\x_\text{fs}}{\n_{\x_0}}} {|\x_0-{\x_\text{fs}}|^{\frac{3}{2}}} \e{\i\wc |\x_0-{\x_\text{fs}}|}\]


(10.3)\[D_{\mathrm{fs,2.5D}}(\x_0,{\x_\text{fs}},\w) = \frac{g_0}{2\pi} A(\w) w(\x_0) \sqrt{\i\wc } \frac{\scalarprod{\x_0-\xs}{\n_{\x_0}}}{|\x_0-\xs|^{\frac{3}{2}}} \e{\i\wc |\x_0-\xs|}\]

for the 2.5D case. For the temporal domain, inverse Fourier transform yields the driving signals

(10.4)\[d(\x_0,t) = \sum_{{\x_\text{fs}}\in \mathcal{X}_{\mathrm{fs}}} d_{\mathrm l}({\x_\text{fs}}, t) * d_{\mathrm{fs}}(\x_0,{\x_\text{fs}}, t),\]

while \(d_{\mathrm{fs}}(\x_0,{\x_\text{fs}}, t)\) is derived analogously to from or . At the moment \(d_{\mathrm l}({\x_\text{fs}}, t)\) does only support driving functions from WFS.