Unformatted text preview: p Delay 83 Group Delay • When the input is composed of many
sinusoidal components with different
frequencies that are not harmonically
related, each component will go through
different phase delays
• In this case, the signal delay is determined
using the group delay defined by
dθ(ω)
τ g (ω) = −
dω
Copyright © 2005, S. K. Mitra • In defning the group delay, it is assumed
that the phase function is unwrapped so that
its derivatives exist
• Group delay also has a physical meaning
only with respect to the underlying
continuoustime functions associated with
y[n] and x[n]
84
Copyright © 2005, S. K. Mitra 14 Phase and Group Delays Phase and Group Delays • A graphical comparison of the two types of
delays are indicated below • Example  The phase function of the FIR
filter y[n] = α x[n] + β x[n − 1] + α x[n − 2]
is θ(ω) = −ω
• Hence its group delay is given by τ g (ω) = 1
verifying the result obtained earlier by
simulation θ(ω) θ(ω o ) Group delay
_ τ (ω o)
g
_ τ p(ω o) Phase delay ω ωo 85 86
Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Phase and Group Delays Phase and Group Delays • Example  For the Mpoint movingaverage
filter
1/ M , 0 ≤ n ≤ M − 1
h[n] = ⎧
⎨ 0,
otherwise
⎩
the phase function is
M/2
( M − 1)ω
2π k ⎞
θ(ω) = −
+ π ∑ µ⎛ ω −
⎟
⎜
M⎠
2
k =0 ⎝
• Hence its group delay is τ g (ω) = M −1
2 87 88 • Physical significance of the two delays are
better understood by examining the
continuoustime case
• Consider an LTI co...
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 Fall '13
 Digital Signal Processing, Signal Processing, S. K. Mitra

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