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EE264
Digital Signal Processing
Lecture 6
Polyphase Implementations
and Quantization in A-to-D
October 8, 2008
Ronald W. Schafer
Department of Electrical
Engineering
Stanford University
STANFORD UNIVERSITY, EE264
Administrative
• HW 2
due on Tuesday, Oct. 7 by 5pm in EE264 drawer
on 2
nd
floor Packard. HW 3 posted by Oct. 7.
• READ: Section 4.6 of DTSP and
supplemental notes
posted on website.
• I will drop your lowest homework grade in figuring the final
homework average.
• Review Sessions: Thursdays 4:15 - 5pm Gates B03
(available online)
• Office Hours:
– RWS: Mon./Weds 10-11, and 12:15-12:45
– Raunaq: Mon. 5-7pm, Tues. 1:30-3:30pm
– Rahim: Friday 4-6pm
• Grader:
Pegah Afshar
STANFORD UNIVERSITY, EE264
Overview of Lecture
• READ: Sections 4.6 and 4.8 of DTSP
• Polyphase implementations
• A-to-D conversion
• Probabilistic analysis of quantization
– Model
– Signal-to-noise ratio
• Oversampling (if time permits)
STANFORD UNIVERSITY, EE264
Polyphase Decomposition of
h
[
n
]
]
[
]
[
k
nM
h
n
e
k
+
=
∑
=
∑
=
−
=
−
−
=
−
1
0
1
0
)
(
)
(
)
(
M
k
k
M
k
M
k
k
k
z
z
E
z
z
H
z
H

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Polyphase Implementation of h[n]
]
[
]
[
k
nM
h
n
e
k
+
=
∑
=
−
=
−
1
0
)
(
)
(
M
k
k
M
k
z
z
E
z
H
)
(
z
H
Polyphase implementations
are useful when combined
with sampling rate changes.
STANFORD UNIVERSITY, EE264
Polyphase Decimation - I
M
↓
]
[
]
[
nM
y
n
w
=
)
(
z
H
Downsampling commutes
with addition.
N
multiplies and
N
-1 additions per
output sample (FIR filter).
STANFORD UNIVERSITY, EE264
Polyphase Decimation - II
M(N/M) multiplies and
M(N/M-1) +M-1 adds.
Polyphase filters clocked
at low rate.
STANFORD UNIVERSITY, EE264
Non-Integer (
T’
=
TM/L
) Rate Change - I
Why should the interpolator go first?

STANFORD UNIVERSITY, EE264
Non-Integer (
T’
=
TM/L
) Rate Change - II
x
e
[
n
]
=
x
[
n
/
L
],
n
=
0,
±
L
,
…
0,
otherwise
⎧
⎨
⎩
⇔
X
e
(
e
j
ω
)
=
X
(
e
j
L
)
˜
x
d
[
n
]
=
˜
x
i
[
nM
]
⇔
˜
X
d
(
e
j
)
=
1
M
˜
X
i
(
e
j
(
−
2
π
r
)/
M
)
r
=
0
M
−
1
∑
)
(
)
(
)
(
~
]
[
]
[
]
[
~
j
e
j
j
i
k
d
i
e
X
e
H
e
X
k
n
h
k
x
n
x
=
⇔
∑
−
=
∞
−∞
=
STANFORD UNIVERSITY, EE264
Non-Integer (
T’
=
TM/L
) Rate Change - III
X
(
e
j
)
=
1
T
X
c
j
T
−
2
k
T
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
k
=−∞
∞
∑
X
e
(
e
j
)
=
X
(
e
j
L
)
T
T
L
STANFORD UNIVERSITY, EE264
Non-Integer (
T’
=
TM/L
) Rate Change - IV
˜
X
d
(
e
j
)
=
1
M
˜
X
i
(
e
j
(
−
2
r
M
)
r
=
0
M
−
1
∑
′
T
=
3
T
/ 2
L
T
T
L
T
L
TM
L
STANFORD UNIVERSITY, EE264
Can the sampling rate be different between
sampling and reconstruction? Problem 4.23
1
1
/
1
s
f
T
=
2
2
/
1
s
f
T
=
C/D
D/C
)
(
t
x
c
]
[
n
x
)
(
t
y
c
M
↓
D/C
C/D
1
1
/
1
s
f
T
=
2
2
/
1
s
f
T
=
]
[
n
x
]
[
n
x
d
)
(
t
x
c
)
(
t
y

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