Ch3Handouts_2

# K mitra x a j 1 a j c 2 a j c

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online