is the phase angle of H, then it is well known that
the group delay introduced by the transfer function is
given by In particular, suppose that x(t) is the sound pressure level of a source at one location, and y(t) is the resulting response at another location. Then, if the impulse response h(t) of the acoustic channel is known, we can find y(t) from the convolution integral
- It allows us to find the response to multiple sound sources by considering them one at a time and adding the seperate responses.
- It allows us to determine the response to an arbitrary signal from knowing the response to an impulse.
- It allows us to use the Convolution Theorem to interptret behavior in the frequency domain.
The Fourier transform of the impulse response h(t) is called the transfer
function H(f). If X(f) is the Fourier transform of the input x(t),
and if Y(f) is the Fourier transform of the output, then the Convolution
Theorem yields the important result
Thus, the spectrum of the received signal is a
simple product of the spectrum of the source and the spectrum of the channel.*
The channel alters the quality of the sound by increrasing (and time shifting)
some components of the spectrum of the source, and reducing (and time shifting)
others. The magnitude of H(f) reveals the degree of change, while the phase
of H(f) provides information about the time shift. To be more specific,
if is the phase angle of H, then it is well known that
the group delay introduced by the transfer function is
given by
.
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