EE123 Spring 2015
Discussion Section 11
Frank Ong
Plan
•
Phase response
•
System analysis
Why do we care?
Linear difference equations :
N
X
ak y[n
k=0
b 0 + b1 z
H(z) =
a0 + a1 z
k] =
M
X
bk x[n
k]
k=0
1
+ . . . + bM z
1 + ... + a z
N
M
N
QM
b0 k=1 (1
=
QN
a0 k=1 (1
ck z
1
dk z
1)
)
Flipping poles and zeros
?
?
?
Flipping poles and zeros
Flipping poles and zeros
Maximum phase
(Maximum group
delay)
Minimum phase
(minimum group delay)
Group delay
To characterize general phase response, look at the group delay:
j!
grd[H(e )] =
For narrowband signals, phase response
looks like a linear phase
d
{arg[H(ej! )]}
d!
Can be negative!
M. Lustig, EECS UC Berkeley
Group delay
j!
grd[H(e )] =
Input
w1
d
j!
{arg[H(e )]}
d!
w2
Output
w2
w1
For narrowband signals, phase response
looks like a linear phase
M. Lustig, EECS UC Berkeley
All pass system
All pass system
Stable/causal
Minimum phase system
•
Poles and zeros inside the unit circle
Minimum phase system
8h such that
All pass/minimum phase
decomposition
H(z) = Hmin (z) · Hap (z)
1. Pick zeros outside the unit circle. Flip them inside
as poles. Put them together as Hap
2. Construct Hmin taking in the zeros and poles inside
the unit circle and compensating the poles
created by Hap
Problem 1
•
Do minimum phase/ all pass decomposition:
1. Pick zeros outside the unit
circle. Flip them inside as
poles. Put them together as
Hap
2. Construct Hmin taking in the
zeros and poles inside the
unit circle and compensating
the poles created by Hap
Problem 2
along with the all pass/minimum phase decomposition
Solution 2
Solution 2
Problem 3
Problem 4
Solution 4
Problem 5