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This paper presents a direct formula for reproducing a sound field consisting of higher-order circular harmonics with polar phase variation. Sound fields with phase variation can be used for synthesizing various spatial attributes, such as the perceived width or the location of a virtual sound source. To reproduce such a sound field using a linear loudspeaker array, the driving function of the array is derived in the format of an integral formula. The proposed function shows fewer reproduction errors than a conventional formula focused on magnitude variations. In addition, analysis of the sweet spot reveals that its shape can be asymmetric, depending on the order of harmonics.


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