(1.1)
1. Mathematical definition of phase distortion [1]
The mathematical foundation of this thesis research will now be developed. The introduction of terms relevant to this research and mathematical definition of them will aid to better understand and identify the nature of phase distortion present in audio signals.
The impulse response h(t) and complex frequency response H(w ), which characterize a causal, linear, and time-invariant system are interrelated by the Fourier transform pair
|
|
(1.1)
|
 |
(1.2)
|
The complex frequency response in Eqs. (1.1) and (1.2) can be expressed in polar form as
 |
(1.3)
|
where |H(w )| is the magnitude response (or gain for sinusoidal, steady-state operation) and f (w ) is the phase response (or steady-state phase shift of output relative to input). The requirement in the time domain for distortionless signal processing (preservation of waveform shape) is for h(t) to have the form
 |
(1.4)
|
where d is
the unit impulse and constants K > 0 and T 
0. Eq. (1.4) and the convolution theorem together imply that a distortionless
system scale the input signal by a constant K and delays the signal by
T seconds. The output is therefore a delayed version of the input. Substituting
Eq. (1.4) into Eq. (1.1) gives the corresponding restrictions on the frequency
response
 . |
(1.5)
|
Comparing Eqs. (1.5) and (1.3) indicates that there is a twofold frequency-domain requirement for distortionless processing; constant magnitude response |H(w )|=K and phase response proportional to frequency f (w ) = -w T. Distortion of the waveform or linear distortion is caused by deviations of |H(w )| from a constant value K. Phase distortion denotes departures of f (w ) from the linearly decreasing characteristic -w T.
Due to causality, there is a minimum amount of phase shift, or minimum phase f m(w ) necessarily associated with a given magnitude response |H(w )|. This is described by the Hilbert transform
 |
(1.6)
|
If f (w ) = f m(w ), as given in Eq. (1.6), then h(t) will be zero for negative values of time. Due to sources described next, additional, or excess phase can exist as well, therefore in general the total phase shift is
 . |
(1.7)
|
A practical definition for excess phase is
 |
(1.8)
|
where q 0 is a constant and q a(0) = 0. In Eq. (1.8), q a(w ) is the frequency-dependent phase shift of a (nontrivial) all-pass network, -w T represents pure time delay as in Eq. (1.5), and q 0 is a frequency-independent phase shift caused by polarity reversal between input and output of a Hilbert transformer, for example. Comparing the substitution of Eqs. (1.8) with (1.7) and Eq. (1.3) to Eq. (1.5), it can be seen that the terms f m(w ), q a(w ), and q 0 contribute to the phase distortion of the system.
Two frequency-domain measures related to the phase response are phase delay t p(w ) and group delay t g(w ) defined as
 |
(1.9)
|
 |
(1.10)
|
For an amplitude-modulated carrier wave, the difference between phase delay and group delay is displayed in Fig. 1.1.
Fig. 1.1. Phase delay and group delay for amplitude-modulated wave. [D. Preis, "Phase Distortion and Phase Equalization in Audio Signal Processing — A Tutorial Review," J. Audio Eng. Soc., vol. 30, pp. 774-794 (1982 Nov.), Fig. 1.]
The lower waveform here has a positive t
p(w ) and t
g(w ) with respect
to the upper waveform at carrier frequency w
. From Eqs. (1.3) and (1.5), the absence of phase distortion requires that the
phase and group delays in Eqs. (1.9) and (1.10) each equal the overall time
delay T
0, or
 |
(1.11)
|
Therefore, deviations of either t p(w ) or t g(w ) from the constant value T indicate the presence of phase distortion. The substitution of Eqs. (1.7) and (1.8) into Eq. (1.10) shows the three terms that contribute to the total group delay
 |
(1.12a)
|
 |
(1.12b)
|
Comparing Eqs. (1.11) and (1.12a), group-delay distortion is defined as D t g(w ) = t g(w ) - T or by using Eq. (1.12b),
 |
(1.13)
|
Eq. (1.13) implies that D
t g(w
) = 0 is a necessary condition for no phase distortion and peak-to-peak excursions
of D t
g(w ) is a useful quantitative
measure of phase distortion. Although the all-pass group delay t
ga(w )
0, the minimum-phase group delay t gm(w
) can be negative or positive. Therefore, D t
g(w ) from Eq. (1.13)
can be of negative or positive value. When t
g(w ) is calculated
from f (w
) using Eq. (1.10), only the phase-slope information is preserved. The phase
intercept information, f (0) = f
m(0) + q 0 is lost
through differentiation. This result implies that when D
t g(w
) = 0 in Eq. (1.13), some phase distortion is possible, if for example, f
(w ) = f
m(w ) = -p
/2 [H(w ) is an ideal integrator]
or f (0) = q
0 = p/2 [H(w
) contains a Hilbert transformer]. Thus D t
g(w ) = 0 and f
(0) 
0 (or a multiple of p) implies no group-delay
distortion but a different form of phase distortion known as phase-intercept
distortion. In general, the total phase distortion produced by a linear
system consists of both group-delay and phase-intercept distortion. As indicated
in Eq. (1.13), the two sources of group-delay distortion D
t g(w
) are the minimum-phase response and the frequency-dependent all-pass portion
of the excess phase response.
Fig. 1.2 shows the frequency and time-domain responses of a minimum-phase low-pass second-order resonant system whose transfer function is
 |
(1.14)
|
where s is the complex frequency variable (s = jw ) and b > a > 0.
Fig. 1.2. Responses of minimum-phase low-pass resonant system. (a) Frequency domain. (b) Time domain. [D. Preis, "Phase Distortion and Phase Equalization in Audio Signal Processing — A Tutorial Review," J. Audio Eng. Soc., vol. 30, pp. 774-794 (1982 Nov.), Fig. 2.]
As the peaking in the magnitude response increases by the reduction of the damping parameter value a , the group delay response peaks near resonance. In the time domain, the impulse response h(t) = e(-a t)sin(b t) oscillates at frequency ƒ = b /2p Hz and is exponentially damped with time constant 1/a . The pronounced ringing is due to the frequencies in the vicinity of b /2p Hz are strongly emphasized. The lack of symmetry in the impulse response is due to group-delay distortion. For this case, from Eq. (1.12b), t g(w ) = t gm(w ).
Fig. 1.3 shows the frequency and time-domain responses of a first-order all-pass network transfer function that has the form
 . |
(1.15)
|
Fig. 1.3. Responses of first-order, all-pass system. (a) Frequency domain. (b) Time domain. [D. Preis, "Phase Distortion and Phase Equalization in Audio Signal Processing — A Tutorial Review," J. Audio Eng. Soc., vol. 30, pp. 774-794 (1982 Nov.), Fig. 3.]
The magnitude M(w ) of the complex frequency response is constant. The group delay t g(w ), which is always positive in value, shows a decrease as frequencies increase. For this case, from Eq. (1.12b), t g(w ) = t ga(w ).
Fig. 1.4 illustrates phase distortion caused by a frequency-independent phase shift or phase-intercept distortion.
Fig. 1.4. Band-limited square wave (dotted), its Hilbert transform (dashed), and sum of dotted and dashed curves (solid). [D. Preis, "Phase Distortion and Phase Equalization in Audio Signal Processing — A Tutorial Review," J. Audio Eng. Soc., vol. 30, pp. 774-794 (1982 Nov.), Fig. 4.]
The dotted curve represents a band-limited square wave (sum of the first four
nonzero harmonics). The dashed curve is the Hilbert transform of the square
wave obtained by the shifting the phase of each harmonic by p/2
radians, or 90
. This constant phase shift of each of the harmonics yields a linearly distorted
waveform having greatly increased peak factors. The solid curve is the sum of
the square wave and its Hilbert transform. Since the corresponding harmonics
in this sum are of equal amplitude and in-phase quadrature, the solid curve
could have been obtained by scaling the magnitude of the amplitude spectrum
of the original square wave by 
and rotating its phase spectrum by 45.
For this case, from Eq. (1.8), f x(w
) = q 0 = p/4
rad. Mathematically, the Hilbert transform of the time function f(t)
is itself a function of time which is defined by the convolution of -1/p
t with f(t):
 |
(1.16)
|
The integral in Eq. (1.16) is understood in the sense of principal value. Assuming
the Fourier transform of f(t) to be F(w
) and noting that the Fourier transform of -1/p
t is e(jp/2) = j
= 
for
positive frequencies w > 0, spectrally,
the Hilbert transform of f(t) corresponds to a perfect p
/2 rad (or 90
) shift in the positive-frequency phase spectrum of F(w
). Two successive Hilbert transformations of f(t) yield -f(t),
which is the simple polarity reversal in the time domain or 180
phase shift of F(w ) in the frequency
domain. The impulse response -1/p t of
a Hilbert transformer is not causal, meaning it is non-zero for negative values
of time. The response is also of infinite duration in time, which implies that,
in practice, an approximation is only possible with a Hilbert transformer.
This chapter introduced the mathematical foundation of the thesis research. Phase distortion and group delay were defined in mathematical terms. Frequency and time-domain characteristics of a minimum-phase second-order low-pass and first-order all-pass systems were investigated. Finally, an example of phase-intercept distortion was presented. These results will aid in the understanding and identification of phase distortion of audio signals in mathematical terms. The next chapter will investigate phase distortion from a human perception standpoint.