Implementing IIR/FIR Filters

delayed by two cycles (4π) and the first harmonic is
delayed by four cycles (8π). The group delay of a
system, τ g , is defined by taking the derivative of φ(θ):
dφ
τ
g
≡ – -----dθ
x 1 (t) = sin ωt
x 2 (t) = sin 2ωt
Composite input
pulse shape
Filter
Figure 7-32 Signal Data through a FIR Filter
Eqn. 7-21
Δφ 1 = 4π-8π = -4π
Δφ 2 = 8π-16π = -8π
Legend:
Represents the filtered wave
Represents the unfiltered wave (i.e., no delay)
Stretched pulse within filter Composite output
pulse shape
Since θ is the normalized frequency, τ g in Eqn. 7-21
is a dimensionless quantity and can be related to
the group delay in seconds by dividing by the sample
frequency, f s . For a linear-phase system, τ g is
independent of θ and is equal to τ. This fact can be
seen by substituting Eqn. 7-20 into Eqn. 7-21.
In this example, τ g is two cycles per Hz. Note that
the pulse shape within the filter has been retained
MOTOROLA 7-5
t
t
t