## The Panning Problem

An interesting illustration of the difference between coherent and
noncoherent signal addition comes up in the problem of *stereo
panning* between two loudspeakers. Let
and
denote
the signals going to the left and right loudspeakers, respectively,
and let
and
denote their respective gain factors (the
``panning'' gains, between 0 and 1). When
, sound
comes only from the left speaker, and when
, sound
comes only from the right speaker. These are the easy cases. The
harder question is what should the gains be for a sound directly in
front? It turns out that the answer depends upon the listening
geometry and the signal frequency content.

If the listener is sitting exactly between the speakers, the ideal
``front image'' channel gains are
,
*provided* that the shortest wavelength in the signal is much
larger than the ear-to-ear separation of the listener. This
restriction is necessary because only those frequencies (below a few
kHz, say), will combine *coherently* from both speakers at each
ear. At higher frequencies, the signals from the two speakers
*decorrelate* at each ear because the propagation path lengths
differs significantly in units of wavelengths. (In addition, ``head
shadowing'' becomes a factor at frequencies this high.) In the
perfectly uncorrelated case (*e.g.*, independent white noise coming from
each speaker), the energy-preserving gains are
. (This value is typically used in practice since the
listener may be anywhere in relation to the speakers.)

To summarize, in ordinary stereo panning, decorrelated high frequencies are attenuated by about 3dB, on average, when using gains dB. At any particular high frequency, the actual gain at each ear can be anywhere between 0 and 1, but on average, they combine on a power basis to provide a 3 dB boost on top of the dB cut, leaving an overall dB change in the level at high frequencies.

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The Short-Time Fourier Transform

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Processing Gain