34 Building acoustics
A description giving the relationship between the input and output in the time
domain is also appropriate. The impulse response gives us the relation between the time
signals x(t) and y(t). The x(t) being an infinite sharp pulse, i.e. a Dirac δ-function, the
signal y(t) will give us the impulse response. The transfer function H(f) and the impulse
response h(τ) is also a Fourier transform pair and they therefore contain the same
information. Modern measurement technique using signal types such as MLS or swept
sine, utilize this property as they may easily determine the impulse response and from
there use the Fourier transform to find the corresponding transfer function. Concerning
the different types of test signals being used, see section 1.5.2.
A transfer function H(f) is normally a complex function, which in other words tells
us that the variables describing the input and output have a phase difference. We may
therefore choose between having the H(f) expressed by its real and imaginary parts or by
its modulus and phase:
(^) j( ) 2 2 j( )
() Re{ ()} jIm{ ()} () j ()
or () () ff() () ,
H fHf HfAfB
Hf Hf eθA f B f e
=+⋅=+⋅
=⋅= +⋅
f
θ (2.2)where θ(f) is given by
()
tan( ( )).
()
Bf
f
Afθ = (2.3)The modulus |H(f)| is often referred to as the gain factor of the system and θ(f) the
corresponding phase factor. It is important to note that H(f) is a function of frequency
only, which is a consequence of assuming the system to be linear and stable. If the
system does not fulfil these conditions the transfer function will, in the former case,
depend on the input amplitude. In the latter case, there will be a time dependency.
2.3.2 Some important relationships
Knowing the transfer function we may calculate the response (output) for any type of
excitation (input) if the conditions, as mentioned above, are fulfilled. A Fourier series or
integral may express the excitation and the response will be a sum of the responses for
each of the components in the excitation. From the response we may then calculate the
mean square value or the RMS-value.
Associated with the general treatment of oscillations or signals in Chapter 1 we
shall state some important relationships concerning “two-signal” or “two-channel”
analysis. We will again assume that the system has one input and one output as shown in
Figure 2.2. For systems having several inputs and/or several outputs the reader should
consult the specialized literature on the subject, e.g. Bendat and Piersol (2000).
2.3.2.1 Cross spectrum and coherence function
Broadband stochastic signals are suitable for investigating an actual system. Assuming
that the excitation (input signal) is such a signal we may show that the following two
relationships are valid, equations linking the transfer function and the spectral density: