Building Acoustics

Excitation and response 35

2

GHfGyy= ()⋅ xx (2.4)

and GHfGxy=⋅( ) xx. (2.5)

The first equation links together the power spectrum, or more correctly the power
spectral density, on the input and output side of the system. Only the gain factor appears
in the equation and all quantities are real. Equation (2.5), however, is complex and here
we have got a new spectral function Gxy. This is the cross spectral density function,
shortened to cross spectrum and defined by

*

2

xy() lim T() (),T

T

Gf XfYf

⇒∞ T

⎡ ⎤

=⋅⎢ ⎥

⎣ ⎦

(2.6)

where the star symbol * signifies that the complex conjugate of the Fourier transform of
the input signal shall be used. A very important application of this function is to
determine the transfer function H(f) by way of Equation (2.5). This is the preferred
instrument technique instead of using the direct definition given in Equation (2.1). The
reason is that it may be shown that with the former the expected value will be more
correct, which is related to the ideal situation expressed by the Equations (2.4) and (2.5).
Apart from the assumptions concerning linearity and time invariance, we have tacitly
assumed that there are no external noise signals either in the input or output to disturb
our measurements. In practice, there certainly will be such disturbances. A method to
control this, i.e. finding out if the signal in the output really is caused by the excitation
signal and not by any disturbing signal, is by measuring the so-called coherence function
γxy, given by

2

()

() 0 () 1.

() ()

xy

xy xy

xx yy

Gf

ff

GfGf

γ=

⋅

≤γ≤ (2.7)

A coherence function identically equal to 1.0 implies that the output signal is caused by
nothing other than the applied input signal. If less than 1.0, this means that there are
systematic and/or random errors in the measured transfer function. This need not be
caused by external noise only but could be linked to the problem of using too few lines in
the discrete Fourier transform, i.e. too few lines when the transfer function varies
strongly within narrow frequency intervals. An example is when the system exhibits
strong resonances (see below). This type of systematic error, which is caused by
inadequate frequency resolution, is called leakage, a subject treated in section 1.4.4.

2.3.2.2 Cross correlation. Determination of the impulse response

In an analogous way as the power spectral density G (power spectrum) has its time
domain equivalent description in the autocorrelation function R(τ), the cross spectrum
Gxy has its equivalent, its Fourier transform, in the cross correlation function Rxy(τ). This
is defined by

0

1

( ) lim ( ) ( ) d.

T

Rxxyτ =⋅+T⇒∞T∫ tytττ (2.8)

One might say that this function experienced its renaissance in measurements of sound
and vibration by the introduction of MLS as a test signal. Using such signals one may