Table 1: Filter specifications
f
p
f
p

0.1 
f
s
f
s

0.12 
δ
p
δ
p

0.5 dB 
δ
s
δ
s

60 dB 
Δ
τ
g
Δ
τ
g

∀
f
,f≤
f
p
:Δ
τ
g
≤9
f
f
f
p
Δ
τ
g
9

Please refer to Pictoralial filter
specification for a pictorial description of how the
various filter constraints correspond to the frequency
response of a lowpass filter. Note that the Pictoralial filter specification
does not imply that the system response must be equiripple.
The constraints on the system response, to be discussed
shortly, are on the maximum allowable peak errors. However,
in general, an equiripple solution to a given filter
specification will require the lowest filter order.
The full filter specification is given in filter specifications. The
parameters
f
p
f
p
and
f
s
f
s
define the end of the passband and the beginning of
the stopband, respectively, on the frequency axis that has
been normalized to 1. The parameters
δ
p
δ
p
and
δ
s
δ
s
define the maximum allowable ripple, in Decibels, in
the passband and the stopband, respectively. The parameter
∀
f
,f≤
f
p
:Δ
τ
g
≤9
f
f
f
p
Δ
τ
g
9
defines the maximum allowable deviation from a group
delay of 9 in the passband. The parameter
τ
g
τ
g
is the group delay of the
system and is defined by the following:
τ
g
ω=−d∠Hωd
x
τ
g
ω
x
H
ω
(1)
The parameter
Δ
τ
g
Δ
τ
g
is defined as:
Δ
τ
g
ω=max
τ
g
ω
ω≤
ω
p
−min
τ
g
ω
ω≤
ω
p
Δ
τ
g
ω
ω
ω
p
τ
g
ω
ω
ω
p
τ
g
ω
(2)
Thus,
∀
f
,f≤
f
p
:Δ
τ
g
≤9
f
f
f
p
Δ
τ
g
9
states that the group delay is not allowed to
deviate more than 9 samples in the passband. A system that
has constant group delay will have a phase response that is
generalized linearphase. Thus, deviation from constant group
delay is a measure of the deviation of the phase response from
linear. The MATLAB command grpdelay
computes the
group delay of a system as a function of frequency given the
filter coefficients
a
a and
b
b.
For which of the three techniques discussed in
filtering techniques must we verfiy
explicitly that the group delay specification is met? All of
them, some of them, or none of them?
Why do we only specify the filter coefficients for only the
positive frequencies? What are we assuming? What does this
imply about the coefficients
a
a and
b
b of the lowpass filter?