Handbook for Sound Engineers

Test and Measurement 1613

known and better developed than acoustical measure-
ment methods.

46.3.2.3 The Transfer Function

The effect that a filter has on a waveform is called its
transfer function. A transfer function can be found by
comparing the input signal and output signal of the fil-
ter. It matters little if the filter is an electronic compo-
nent, loudspeaker, room, or listener. The time domain
behavior of a system (impulse response) can be dis-
played in the frequency domain as a spectrum and phase
(transfer function). Either the time or frequency descrip-
tion fully describes the filter. Knowledge of one allows
the determination of the other. The mathematical map
between the two representations is called a transform.
Transforms can be performed at amazingly fast speeds
by computers. Fig. 46-13 shows a domain chart that
provides a map between various representations of a
system’s response. The measurer must remember that
the responses being measured and displayed on the ana-
lyzer are dependent on the test stimulus used to acquire
the response. Appropriate stimuli must have adequate
energy content over the pass band of the system being
measured. In other words, we can’t measure a sub-
woofer using a flute solo as a stimulus. With that crite-
ria met, the response measured and displayed on the
analyzer is independent of the program material that
passes through a linear system. Pink noise and sine
sweeps are common stimuli due to their broadband
spectral content. In other words, the response of the sys-
tem doesn’t change relative to the nature of the program
material. For a linear system, the transfer function is a
summary that says, “If you put energy into this system,
this is what will happen to it.”

The domain chart provides a map between various
methods of displaying the system’s response. The utility
of this is that it allows measurement in either the time or
frequency domain. The alternate view can be deter-
mined mathematically by use of a transform. This
allows frequency information to be determined with a
time domain measurement, and time information to be
determined by a frequency domain measurement. This
important inverse relationship between time and
frequency can be exploited to yield many possible ways
of measuring a system and/or displaying its response.
For instance, a noise immunity characteristic not attain-
able in the time domain may be attainable in the
frequency domain. This information can then be viewed
in the time domain by use of a transform. The Fourier
Transform and its inverse are commonly employed for

this purpose. Measurement programs like Arta can

display the signal in either domain, Fig. 46-14.

46.3.3 Measurement Systems

Any useful measurement system must be able to extract

the system response in the presence of noise. In some

applications, the signal-to-noise requirements might

actually determine the type of analysis that will be used.

Some of the simplest and most convenient tests have

poor signal-to-noise performance, while some of the

most complex and computationally demanding methods

can measure under almost any conditions. The measurer

must choose the type of analysis with these factors in

mind. It is possible to acquire the impulse response of a

filter without using an impulse. This is accomplished by

feeding a known broadband stimulus into the filter and

reacquiring it at the output. A complex comparison of

the two signals (mathematical division) yields the trans-

fer function, which is displayed in the frequency domain

as a magnitude and phase or inverse-transformed for dis-

play in the time domain as an impulse response. The

impulse response of a system answers the question, “If I

feed a perfect impulse into this system, when will the

energy exit the system?” A knowledge of “when” can

characterize a system. After transformation, the spec-

trum or frequency response is displayed on a decibel

scale. A phase plot shows the phase response of the

device-under-test, and any phase shift versus frequency

becomes apparent. If an impulse response is a measure

of when, we might describe a frequency response as a

measure of what. In other words, “If I input a broadband

stimulus (all frequencies) into the system, what frequen-

cies will be present at the output of the system and what

will their phase relationship be?” A transfer function

includes both magnitude and phase information.

46.3.3.1 Alternate Perspectives

The time and frequency views of a system’s response

are mutually exclusive. By definition the time period of

a periodic event is

(46-1)

where,

T is time in seconds,

f is frequency in hertz.

Since time and frequency are reciprocals, a view of

one excludes a view of the other. Frequency information

cannot be observed on an impulse response plot, and

T^1

f

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