Excitation and response 33
we must be able to calculate these functions, analytically, numerically or by using
empirical methods. However, it should be observed that these variables normally are
space dependent. As an example, the sound pressure level, both inside and outside the
enclosure, could be expected to vary with position. One has then to deal with a
representative set of transfer functions between the variables.
On the other hand, the noise problem could be an existing one. A solution then
requires the analogous possibility using measurement techniques to map the noise field.
The complexity of such mapping will be highly dependent on the number of sources, i.e.
if there is just one dominating source or if there is a complex interaction of many
sources.
2.3 Transfer function. Definition and properties
When using the term transfer function we will, as pointed out above, assume that the
actual system is linear and stable. The purpose of such a function is that when known,
one may not only determine the response of a harmonic input with a given frequency but
also find the response following an arbitrary input time function x(t). This could be a
transient or stochastic time function or a combination of such functions. To make it
simple we shall as much as possible use a harmonic signal input for the illustrations in
this chapter.
2.3.1 Definitions
Strictly speaking, the term transfer function applies to the ratio of the Laplace transforms
of the input and output signals, the frequency response function H(f) or H(ω) being a
special case. When writing
Yf Hf Xf()= () ()⋅ , (2.1)
Y(f) and X(f) are the Fourier transform of the output signal y(t) and the input signal x(t),
respectively, as shown in Figure 2.2. Normally we shall use the term transfer function for
the function H(f) but we shall also sometimes apply the term frequency response. The
latter is often used when the input signal amplitude is frequency independent but has a
constant arbitrary value.
X(f) Y(f)Excitation
SYSTEMResponse(Input) (Output)x(t) y(t)H(f)h(τ)Figure 2.2 A system having one input and one output. Mathematical representations in time and frequency
domains.