Digital Signal Processing Reference
In-Depth Information
be delayed by an arbitrary amount, the output signal will be composed of
sinusoids whose relative alignment may be very different from the original.
Phase alignment determines the shape of the signal in the time domain, as
we have seen in Section 4.7.4. A filter with unit magnitude across the spec-
trum, which does not affect the amplitude of the sinusoidal components,
but whose phase response is not linear, can completely change the shape of
a filtered signal.
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Linear Phase.
A very important type of phase response is
linear phase
:
e
j
ω
)=
e
−j
ω
d
H
(
(5.23)
Consider a simple system which just delays its input, i.e.
y
[
n
]=
x
[
n
−
D
]
with
D
∈
; this is obviously an LTI system with impulse response
h
[
n
]=
δ
[
e
−j
ω
D
. This means that, if the
value
d
in (5.23) is an integer, (5.23) defines a pure delay system; since the
magnitude is constant and equal to one, this is an example of an allpass
filter. If
d
is not an integer, (5.23) still defines an allpass delay system for
which the delay is fractional, and we should interpret its effect as explained
in the previous Section. In particular, if we think of the original signal in
terms of its Fourier reconstruction formula, the fractionally delayed output
is obtained by stepping forward the initial phase of
all
oscillators by a non-
integer multiple of the frequency. In the discrete-time domains, we have
a signal which takes values “between” the original samples but, since the
relative phase of any one oscillator, with respect to the others, has remained
the same as in the original signal, the shape of the signal in the time domain
is unchanged.
For a general filter with linear phase we can always write
]
(
e
j
ω
)=
n
−
D
and frequency response
H
H
e
−j
ω
d
e
j
ω
)=
e
j
ω
)
(
(
H
In other words, the net effect of the linear phase filter is that of a cascade of
two systems: a zero-phase filter which affects only the spectral magnitude of
the input and therefore introduces no phase distortion, followed by a (pos-
sibly fractional) delay system (which, again, introduces just a delay but no
phase distortion).
Group Delay.
When a filter does not have linear phase, it is important
to quantify the amount of phase distortion both in amount and in location.
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In all fairness, the phase response of a system is not very important inmost audio appli-
cations, since the human ear is largely insensitive to phase. Phase is however extremely
important in data transmission applications.
