Signal processing algorithms rely on separating the direct path component from

Signal processing algorithms rely on separating the

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waves that are reflected by the surfaces of the room. Signal processing algorithms rely on separating the direct-path component from reverberation and noise since it parameterizes the location of the talker. Direct-path propagation is easily derived from the wave equation [62]. The wave-field at distance r from the sound source s ( t ) can be expressed as follows: ) ( ) , ( c r direct t s r a t r f = This expression shows the wave field as a scaled and time-delayed version of the original source signal. The attenuation factor is inversely proportional to the distance from the source, and the time delay is equal to the ratio of this distance to the speed of sound, c . The constant a depends on the medium and the system of units used. In the three-dimensional Cartesian coordinate system illustrated by Figure 2.1, the distance between the source and the microphone indexed by m is defined as: 10
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Mic m Source x y z y d ) ( s y d x d ) ( s x d ) ( s z d ) ( s d r m r m d r z d Figure 2.1 A source and microphone located in a Cartesian coordinate system. ) ( s m m d d r r r m d r and ) ( s d r are 3-element vectors, which are defined by the Cartesian coordinates of microphone m and the source, respectively: ) ( ) ( ) ( ) ( s z s y s x s z y x m d d d d d d d d r r The propagation time from the source to microphone m can be defined as follows: c r m m τ Hence, the wave-field at location m d r , which is produced by a single source located at ) ( s d r , can also be expressed as follows: ) ( ) , , ( ) ( m m s m direct t s r a t d d f τ = r r (2.1) 11
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2.3 Multi-Path Propagation and the Room Impulse Response In the presence of sound-reflecting surfaces, the sound waves produced by a single source propagate along multiple acoustic paths. This gives rise to the familiar effects of reverberation; sounds reflect off objects and produce echoes. The walls of most rooms are reflective enough to create significant reverberation. While it is not always noticeable to the occupants, even mild reverberation can severely impact the performance of speech-array systems. Hence, multi-path propagation must be incorporated into the signal- processing model. The walls of a room delineate an acoustic enclosure. Acoustic enclosures have been modeled extensively as linear systems [99], and the same techniques have been applied to room acoustics [65][71][47][94]. It has been shown that the wave field at a particular location inside a reverberant room mat also be considered to be linearly related to the source signal, s ( t ). This relationship can be expressed in terms of the convolution of s ( t ) with a room impulse response as follows: ) , , ( * ) ( ) , , ( ) ( ) ( t d d h t s t d d f s m s m r r r r = (2.2) The impulse response, ) , , ( ) ( t d d h s m r r , characterizes all acoustic paths from the source to the location m d r , including the direct path. It is a function of m d r as well as the source location, ) ( s d r , and is highly dependent on these parameters.
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