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Since we are working in a "real-world"
reverberant environment, any wave emitted by the loudspeaker will be reflected
off surfaces in the room, we need a method that will consider only the "direct
path" waves--those emitted only by the loudspeaker. In other words, we
need to *filter* out the extraneous sounds that are contributed by the
room--we need a way to accept only the direct wave from the loudspeaker into
our microphone that is measuring the system response.

We accomplish this by creating a bandpass filter tuned to a given frequency, leaving the system on for the time it takes for the wavefront to reach the microphone from the loudspeaker, and then turning the system off. We then shift the loudspeaker to a new frequency, re-tune the filter to the new frequency, and then repeat.

Alas, we would like to test the loudspeaker system response
for a given *range* of frequencies (say 20 Hz to 20 kHz, the range of human
hearing). Using this system, we would need to repeat the experiment for *every*
frequency in the range. We need a practical method to measure every frequency
in a given range.

One way around this obstacle is to use a *linearly
swept sine wave* (also called an *FM chirp*) as our input signal. A
swept sine wave has the formula ** x**(

Since our input signal sweeps, our filter must
also sweep upward at the same rate. However, our filter must always sweep a
certain amount of time *behind* the input signal. In other words, we must
account for the time it takes for the emitted wavefront to reach the microphone.
This delay is usually constant, given that the microphone and speaker do not
move relative to one another. This delay is a time offset, but it is usually
referred to as the *frequency offset*, since the center frequency of the
bandpass filter is always slightly behind the frequency of the signal emitted
by the speaker.

We can quantitatively determine certain variables
within our system, including the time offset of our tracking filter and the
bandwidth of the tracking filter. Assume that the speaker and the microphone
are a distance ** X** apart, and let

** Fr** -

Here, ** c** is the

If our tracking filter has a bandwidth *B**Hz*, then our sweep tone will travel some ** dX** in space while
within the bandwidth of the filter. Since our filter is not ideal, we must further
define

** dx** =

We can further expand the concept of time delay
spectrometry to measure the *complete* system response of a sound system,
*including the reverberant environment*. We must simply slow down the sweep
signal so that some of the acoustical scattering from the room surfaces leaks
through our tracking filter. Essentially, our goal is to let the room reach
steady-state equilibrium at each frequency before moving to a new frequency.
By taking measurements at progressive time intervals, we obtain a plot of sound
pressure as a function of *both* frequency and time.

Since we are assuming that our system is LTI,
we may think of each reflecting surface as a *loudspeaker image*. Our system
is a series of loudspeaker images, each system having its own unique transfer
function. Each transfer function is the product of a *spectral energy distribution*
from each surface, ** S**(

.

The impulse response of the system is given by
the Fourier transform of * R*(

.

Now that we have the system response, we can obtain a response for any given input signal. This input signal is often called the program material, and is denoted by p(t) or P(w). In the time domain, the output for any given program source p(t) is the convolution integral of the program signal and the room impulse response:

.

Note that this analysis is valid only for a given
combination of speaker and observer. *Our system response is not a general
characteristic for a room *but it is related to the "true" room
response. In most practical situations, the system response given a stationary
observer and loudspeaker system is all that is needed.

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