Page 719 - Automotive Engineering Powertrain Chassis System and Vehicle Body

CHAP TER 2 1. 1 Interior noise: Assessment and control

The normalised cross spectral density is called the Re-arranging:
coherence between signals x(t) and y(t). Therefore,
ð ð
N N
2 R yy ðsÞ¼ E hðs 1 Þhðs 2 Þ
S xy ðf Þ
2
g ðf Þ¼ (C21.1.28) 0 0
xy
S xx ðf ÞS yy ðf Þ
xðt s 1 Þxðt þ s s Þds 1 ds 2 (D21.1.6)
2
When one-sided power and cross spectral densities are
used, which is after substitution of

G xy ðf Þ 2 R xx ðsÞ¼ E½xðtÞxðt þ sފ (D21.1.7)
2
g ðf Þ¼ G xx ðf ÞG yy ðf Þ (C21.1.29) ð N ð N
xy
R yy ðsÞ¼ hðs 1 Þhðs ÞR xx ðs þ s 1 s Þds 1 ds 2
2
2
If x(t) and y(t) are linearly related and the only signals 0 0
in a two-port system, then the coherence will be unity. (D21.1.8)
The coherence will be zero if those two signals are The Fourier transform of this equation is:
completely unrelated by linear relationships.
2
S yy ðf Þ¼ jHðf Þj S xx ðf Þ (D21.1.9)
Appendix 21.1D: The frequency where H( f ) is the system frequency response function,
related to the system impulse-response function thus:
response function
ð N
Take a linear, time-invariant system with input x( t ) and Hðf Þ¼ hðsÞe ið2pfsÞ ds (D21.1.10)
output y( t ). 0
The response due to an input starting at time t 0 is S yy ( f ) is the power spectral density function.
given by (Fahy and Walker, 1998) as: The above can be repeated, this time using the cross
correlation function R xy for random signals as T / N,
ð t
yðtÞ ¼ hðt t 1 Þxðt 1 Þdt 1 (D21.1.1) R yy ðsÞ¼ E½xðtÞyðt þ sފ (D21.1.11)
ð
t 0
N
where h(t) is the impulse response function of the R xy ðsÞ¼ E xðtÞ hðs 1 Þxðt þ s s 1 Þds 1
0
system. This equation is the causal version of the con- (D21.1.12)
volution integral. ð
If y(t) is stationary then N
R xy ðsÞ¼ hðs 1 ÞR xx ðs s 1 Þds 1 (D21.1.13)
ð t 0
yðtÞ¼ hðt t 1 Þxðt 1 Þdt 1 (D21.1.2) S xy ðf Þ¼ Hðf ÞS xx (D21.1.14)
N
S xy ( f ) is the cross spectral density function.
or
ð
N
yðtÞ¼ hðsÞxðt sÞds (D21.1.3)
0 Appendix 21.1E: Plane waves
Now, for random signals as T / N, the autocorrela- in a tube with a termination
tion function R yy is: impedance

R yy ðsÞ¼ E yðtÞyðt þ sÞ (D21.1.4) If the frequency of sound in a tube of length L is suffi-
ciently low so that only plane waves propagate, then the
solution to the 1-D linear acoustic wave equation has the
one can substitute for y(t) and for y(t þ T) to give
form (Kinsler et al., 1982):
ð
N
R yy ðsÞ¼ E hðs 1 Þxðt s 1 Þds 1 i½wtþkðL xފ i½wt kðL xފ
0 pðxÞ¼ Ae þ Be (E21.1.1)
ð
N
hðs Þxðt þ s s Þds 2 (D21.1.5) At x ¼ L the impedance of the acoustic wave equals
2
2
0 the mechanical impedance of the termination Z mL .
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