Introduction to Acoustics

14.7.5 Dispersion Relations
The causality requirement on the impulse response,
h(t) = 0 for t < 0, has implications for the transfer
function. Causality means that the real and imaginary
parts of the transfer function are Hilbert transforms
of one another. Specifically, if the real and imaginary
parts of H are defined as H(ω) = HR(ω) + iHI(ω)
then
and
HR(ω) = 1
π P
HI(ω) = −1
π P
�∞
−∞
�∞
−∞
dω ′ HI(ω ′ )
, (14.111)
ω − ω ′
dω ′ HR(ω ′ )
.
ω − ω ′
The symbol P signifies that the principal value of a divergent
integral should be taken. In many cases, this
requires no special steps, and definite integrals from
integral tables give the correct answers.
These equations are known as dispersion relations.
They arise from doing an integral in frequency space
to calculate the impulse response for t < 0. The fact
that this calculation must return zero means that H(ω)
must have no singularities in the complex frequency
plane for frequencies with a negative imaginary part.
Similar dispersion relations apply to the natural log of
the transfer function, relating the filter gain to the phase
14.8 The Cepstrum
The cepstrum (pronounced kepstrum) is the inverse
Fourier transform of the natural logarithm of the spectrum.
Because it is the inverse transform of a function
of frequency, the cepstrum is a function of a time-like
variable. But just as the word cepstrum is an anagram
of the word spectrum, the time-like coordinate is called
the quefrency, an anagram of frequency. The field of
cepstrology is full of word fun like this.
Fig. 14.8 The cepstrum of an original signal to which is
added a delayed version of the same signal, with a delay of
2ms(a = 1). The original signal is the sum of two female
talkers
shift as in (14.100)
and
Γ (ω) = Γ (0) − ω2
0.1151π P
Φ(ω) =
Acoustic Signal Processing 14.8 The Cepstrum 517
0.1151 ω
P
π
�∞
−∞
�∞
−∞
dω ′
dω ′ Γ (ω′ )
ω ′2 − ω 2 .
Φ(ω′ )
ω ′ (ω ′2 − ω 2 )
(14.112)
Because Γ (ω) isevenandΦ(ω) is odd, both integrands
are even in ω ′ , and these integrals can be replaced by
twice the integral from zero to infinity. The second equation
above is particularly powerful. It says that, if we
want to find the phase shift of a system, we only have to
measure the gain of the system in decibels, multiply by
0.1151, and do the integral. Of course, it is in the nature
of the integral that in order to find the phase shift at any
given frequency we need to know the gain over a wide
frequency range.
The dispersion relations for gain and phase shift also
arise from a contour integral over frequencies with a negative
imaginary part, but now the conditions on H(ω)
are more stringent. Not only must H(ω) have no poles
for Im(ω) < 0, but ln H(ω) must also have no poles.
Consequently H(ω) must have no zeros for Im(ω) < 0.
A system that has neither poles nor zeros for Im(ω) < 0
is said to be minimum phase. The dispersion relations in
(14.112) only apply to a system that is minimum phase.
Cepstrum
2
1
0
–1
–2
0
1 2 3 4 5 6 7
Quefrency (ms)
Part D 14.8