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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 continuous-time 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 M-point moving-average 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 continuous-time case • Consider an LTI co...
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