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All-pass Systems
System with unit magnitude response Basic building Block:
Pole in a zero in 1/a*Pole in a , zero in 1/aReal co-efficient all-pass:
Basic LP system
ωαω jj eeH −)(
Constant group delay α
(
Impulse response symmetric about 2α if 2α integerNo symmetry if Ζ∉α2
Generalized Linear Phase
Generalized linear phase allows a constant phase shift beside a constant delay.
Summary Analysis Part
Discrete-time representation of signalsLTI tLTI systemDTFT AnalysisZ-transform AnalysisDiscrete-time SamplingDiscrete time Sampling Fourier Transform Analysis of systems with rational system functionwith rational system functionLinear Phase System
Inverse System
1)()( =zHzH i
∏ −M
zd 1 )1(
][][*][ nnhnh i δ=
∏
∏−
=
−
−⋅== N
k
kk
i
zc
zd
ab
zHzH
1
1
0
0
)1(
)1(
)(1)(
=k 1
Inverse system, stable if all poles and zeros, inside the unit circleminim m phase s stemminimum-phase system
)1()90(50)()90()(5.01)((e q) 11− −
−
nununhzH nn )1()9.0(5.0)()9.0()( ,9.01
)( (e.q) 1 −−=−
= − nununhz
zH
)1()50(90)()50()(9.01)( 11− −
−
hzH nn )1()5.0(9.0)()5.0()( ,5.01
)( 11 −−=
−= − nununh
zzH nn
ii