LR Rabiner and RW Schafer, June 3

DRAFT: L. R. Rabiner and R. W. Schafer, June 3, 2009
8.2. HOMOMORPHIC SYSTEMS FOR CONVOLUTION 433
From Eq. (8.13) it follows that any integer multiple of 2π can be added to the
phase arg{X(e jω )} without change of the value of the complex exponential.
When computing the phase angle of X(e jω ), the principal value computation
only uses the real and imaginary parts of the DTFT at a single frequency ω.
For example, in going from ω to ω ± ɛ, the complex value of X(e jω ) can jump
from the third quadrant of the z-plane to the second quadrant so that an angle
that could be defined as slightly more negative than −π radians (in order to
maintain continuity) would instead be assigned a positive value that is slightly
less than π radians (in order to maintain the principal value). Thus, the principal
value jumps by 2π whenever the phase angle exceeds π or falls below −π. This
is illustrated in Figure 8.7 where the phase becomes increasingly negative as
ω increases so that the principal value phase displays discontinuities at the
frequencies at which the phase enters and leaves the band between −3π <
arg{X(e jω )} ≤ −π, etc. The phase is said to “wrap around modulo 2π” when
the principal value phase is computed. This is seen by comparing the top graph
in Figure 8.7 to the bottom graph. The principal value phase clearly does not
satisfy the additivity condition; i.e.,
ARG{X(e jω )} = ARG{X1(e jω )} + ARG{X2(e jω )}, (8.14a)
since the discontinuity frequencies of the sum will not, in general, be the same
as those of either of the two individual principal value functions. If however, the
phase is computed as a continuous function arg{X(e jω )} as in the lower plot in
Figure 8.7, then it follows that since angles add in a product,
arg{X(e jω )} = arg{X1(e jω )} + arg{X2(e jω )}. (8.14b)
Given that it is possible to compute the complex logarithm so as to satisfy
Eq. (8.14b), the inverse transform of the complex logarithm of the Fourier
transform of the input is the output of the characteristic system for convolution,
i.e.,
ˆx[n] = 1
π jω jω jωn
log |X(e )| + j arg{X(e )} e dω. (8.15)
2π −π
If the input signal x[n] is real, then it follows that log |X(e jω )| is an even function
of ω and arg{X(e jω )} an odd function of ω. This means that the real and
imaginary parts of the complex logarithm in Eq. (8.15) have the appropriate
symmetry for ˆx[n] to be a real sequence. Furthermore, this means that ˆx[n] can
be represented as
ˆx[n] = c[n] + d[n], (8.16a)
where c[n] is the inverse DTFT of log |X(e jω )| and the even part of ˆx[n]
c[n] =
ˆx[n] + ˆx[−n]
, (8.16b)
2
while d[n] is the inverse DTFT of arg{X(e jω )} and the odd part of ˆx[n]
d[n] =
ˆx[n] − ˆx[−n]
. (8.16c)
2