48
1 Multi-Rate Digital Signal Processing Y. C. Jenq, Ph.D. Department of Electrical & Computer Engineering Portland State University Portland, Oregon 97207 [email protected]
| Date post: | 21-Dec-2015 |
| Category: |
Documents |
| View: | 218 times |
| Download: | 1 times |
1
Multi-RateDigital Signal Processing
Y. C. Jenq, Ph.D.Department of Electrical & Computer Engineering
Portland State UniversityPortland, Oregon 97207
2
Decimation System (Down Sampling)
M xd[n]x[n]
M=3
3
xd[n] = x[nM], where M is an integer
Xd(z) =n xd[n]z-n
=(1/M)m=0,(M-1) X(z(1/M)e-jm2/M)
Xd(ej) =(1/M)m=0,(M-1) X(ej(-m2/M)
Decimation System (Down Sampling)
4
Decimation System (Down Sampling)
0
0
2
M=3
0
X(ej)X(ej)
X(ej()
X(ej()
Xd(ej) =(1/M)m=0,(M-1) X(ej(-m2/M)
4
2
5
Sampling Rate Reduction System
M yd[n]x[n]
Low-pass filter
with cutoff at /M
y[n]
6
Interpolation System (Up Sampling)
L xu[n]x[n]
L=3
7
Interpolation System (Up Sampling)
xu[n] = x[n/L], n = 0, ±L, ±2L, … 0, otherwise
Xu(z) =n xu[n]z-n = X(zL)
Xu(ej) = X(ejL)
8
Interpolation System (Up Sampling)
0 2
L=3X(ejL)X(ej)
Xu(ej) = X(ejL)
9
L
Sampling Rate Increase System
y[n]x[n]
Low-pass filter
with cutoff at /L
xu[n]
10
Decimation and Interpolation
x[n] M M y[n]M=3
11
Decimation and Interpolation
x[n] M M
Let WM = e-j2/M
Y(z) = (1/M)m=0,(M-1) X(ze-jm2/M)
= (1/M)m=0,(M-1) X(zWMm)
Y(ej)= (1/M)m=0,(M-1) X(ej(-m2/M))
y[n]
12
Decimation and Interpolation
Y(ej)= (1/M)m=0,(M-1) X(ej(-jm2/M))
0
0
2
M=3
0
X(ej)
X(ej()
X(ej()2
13
Fractional Sampling Rate Change
L y[n]x[n]
Low-pass filter
with cutoff at
min(/L,/M)
M
14
Block Interconnection Identities
M C MC
M M
M
Multiply by a Constant
15
Block Interconnection Identities
M M
M
16
L C LC
L L
L
Block Interconnection Identities
17
Block Interconnection Identities
L L
L
18
Multi-Rate Identities
MH(zM) M H(z)
L H(zL) L H(z)
19
Multi-rate Switch Models
3
3
3
n = 0, 3, 6,…
n = -1, 2, 5,…
n = -2, 1, 4,…
x[n]
20
Multi-rate Switch Models
3
3
3
x[n]
x[3n]
x[3n-1]
x[3n-2]
Ser
ial t
o P
aral
lel
Con
vert
er
x[n]
21
Multi-rate Switch Models
n = 2, 5, 8,…
n = 1, 4, 7,…
n = 0, 3, 6,…
3
3
3
x1[n]
x2[n]
x3[n]
22
Multi-rate Switch Models
3
3
3
x1[n]
x2[n]
x3[n]
x[3n+2]
x[3n+1]
x[3n] Par
alle
l to
Ser
ial
Con
vert
er
x[n]
23
Poly-phase Structure of Decimation Filter
3
x[n]D0(z3)
D2(z3)
D1(z3)
H(z) = D0(z3)+ z-1D1(z3)+z-2D2(z3)
y[n] y[3n]
H(z) 3x[n] y[3n]y[n]
24
3
3
Poly-phase Structure of Decimation Filter
x[n]D0(z3)
D2(z3)
D1(z3) 3 y[3n]
H(z) = D0(z3)+ z-1D1(z3)+z-2D2(z3)
25
3
3
x[n]D0(z)
D2(z)
D1(z) 3 y[3n]
Poly-phase Structure of Decimation Filter
H(z) = D0(z3)+ z-1D1(z3)+z-2D2(z3)
26
x[3n]
x[3n-1]
x[3n-2]Ser
ial t
o P
aral
lel
Con
vert
er
x[n]
Poly-phase Structure of Decimation Filter
D0(z)
D2(z)
D1(z) y[3n]
H(z) = D0(z3)+ z-1D1(z3)+z-2D2(z3)
27
Poly-phase Structure of Interpolation Filter
3
x[n]
H(z) = z-2I0(z3)+ z-1I1(z3)+I2(z3)
y[n]
I0(z3)
I2(z3)
I1(z3)
H(z) 3x[n] y[n]
28
Poly-phase Structure of Interpolation Filter
3x[n]
H(z) = z-2I0(z3)+ z-1I1(z3)+I2(z3)
y[n]
I0(z3)
I2(z3)
I1(z3) 3
3
29
Poly-phase Structure of Interpolation Filter
3x[n]
H(z) = z-2I0(z3)+ z-1I1(z3)+I2(z3)
I0(z)
I2(z)
I1(z) 3
3 y[n]
30
Poly-phase Structure of Interpolation Filter
x[n]
H(z) = z-2I0(z3)+ z-1I1(z3)+I2(z3)
I0(z)
I2(z)
I1(z)
y[3n+2]
y[3n+1]
y[3n] Par
alle
l to
Ser
ial
Con
vert
er
y[n]
31
Poly-phase Structure of Fractional Sampling Rate Filter
3
x[n]F0(z3)
F2(z3)
F1(z3) 4 y[n]
H(z) = z-2F0(z3)+ z-1F1(z3)+F2(z3)
(M=4, L=3)
32
Poly-phase Structure of Fractional Sampling Rate Filter
(M=4, L=3) 3
x[n]F0(z)
F2(z)
F1(z) 4 y[n] 3
3
H(z) = z-2F0(z3)+ z-1F1(z3)+F2(z3)
33
Poly-phase Structure of Fractional Sampling Rate Filter
(M=4, L=3)
3
x[n]F0(z)
F2(z)
F1(z) 4y[n]
3
3
H(z) = z-2F0(z3)+ z-1F1(z3)+F2(z3)
34
Poly-phase Structure of Fractional Sampling Rate Filter
(M=4, L=3)
3
x[n]F0(z)
F2(z)
F1(z)y[n]
3
3
H(z) = z-2F0(z3)+ z-1F1(z3)+F2(z3)
4
4
4
35
Poly-phase Structure of Fractional Sampling Rate Filter
(M=4, L=3)
3
x[n]F0(z)
F2(z)
F1(z)y[n]
3
3
H(z) = z-2F0(z3)+ z-1F1(z3)+F2(z3)
4
4
4
36
Poly-phase Structure of Fractional Sampling Rate Filter
(M=4, L=3)
3
x[n]F0(z)
F2(z)
F1(z)y[n]
3
3
H(z) = z-2F0(z3)+ z-1F1(z3)+F2(z3)
4
4
4
37
Poly-phase Structure of Fractional Sampling Rate Filter
(M=4, L=3)
x[n] F0(z)
F2(z)
F1(z) y[n]
H(z) = z-2F0(z3)+ z-1F1(z3)+F2(z3)
4
4
4 Par
alle
l to
Ser
ial
Con
vert
er
38
Poly-phase Structure of Fractional Sampling Rate Filter
(M=4, L=3)
x[n] F0(z)
F2(z)
F1(z) y[n]
H(z) = z-2F0(z3)+ z-1F1(z3)+F2(z3)Fk(z) = Fk0(z4)+ z-1Fk1(z4)+z-2Fk2(z4)+z-3Fk3(z4)
4
4
4 Par
alle
l to
Ser
ial
Con
vert
er
39
Poly-phase Structure of Fractional Sampling Rate Filter
(M=4, L=3)
x[n]
F2(z)
F1(z)
y[n]
H(z) = z-2F0(z3)+ z-1F1(z3)+F2(z3)Fk(z) = Fk0(z4)+ z-1Fk1(z4)+z-2Fk2(z4)+z-3Fk3(z4)
4
4 Par
alle
l to
Ser
ial
Con
vert
er
Ser
ial t
o P
aral
lel
Con
vert
er
Fk0
Fk1
Fk2
Fk3
40
Poly-phase Structure of Fractional Sampling Rate Filter
(M=4, L=3)
x[n]
F2(z)
F1(z)
y[n]
H(z) = z-2F0(z3)+ z-1F1(z3)+F2(z3)Fk(z) = Fk0(z4)+ z-1Fk1(z4)+z-2Fk2(z4)+z-3Fk3(z4)
4
4 Par
alle
l to
Ser
ial
Con
vert
er
Ser
ial t
o P
aral
lel
Con
vert
er
Fk0
Fk1
Fk2
Fk3
41
Efficient Design for Very Narrow-band Filters
G(zM)
x[n] y[n]H(z)
x[n] y[n]F(z)
42
Efficient Design for Very Narrow-band Filters
G(zM)
H(z)
F(z)
0
G(z)0
0
0
ps
p
s
s
p s
Desiredpassband Images
p
43
Efficient Design for Very Narrow-band Filters
G(zM)x[n] y[n]F(z) M
G(z)x[n] y[n]F(z) M
G(z)
x[n]
y[n]
F0(z) Mz-1
z-1
M F1(z) +
44
Efficient Design for Very Narrow-band Filters
G(z)x[n]
y[n]
F0(z)
F1(z) +
Ser
ial t
o P
aral
lel
Con
vert
er
FM-1(z)
45
Multi-stage Decimation System
H(z)x[n] y[n] M
G(z)x[n] y[n]F(z) M1
G(zM1)x[n] y[n]F(z) M1M2
M2
46
Multi-stage Decimation System
x[n]y[n]
F0(z)
F1(z) +
Ser
ial t
o P
aral
lel
Con
vert
er
FM1 -1(z)
G0(z)
G1(z) +
Ser
ial t
o P
aral
lel
Con
vert
erGM2 -1
(z)
47
Multi-stage Interpolation System
H(z)x[n] y[n] L
G(z)x[n] y[n]F(z) L1
G(zL1)x[n] y[n]F(z) L1L2
L2
48
Multi-stage Interpolation System
x[n]
y[n]
F0(z)
F1(z)
FL1-1(z)
G0(z)
G1(z)P
aral
lel t
o S
eria
l C
onve
rter
GL2-1(z) Par
alle
l to
Ser
ial
Con
vert
er
