Unformatted text preview: Sampling rate change operations:
upsampling and downsampling;
fractional sampling; interpolation Downsampling
Definition: M
x[0]
x[1]
(↓2)
x[2]
x[3]
x[4]
M
As a matrix operation:
L1
L0
L0 0
0
0 M
0
1
0
M 0
0
0 0L
0L
1L M
x[0]
x[2]
x[4]
M
& = M
x[0]
x[1]
x[2]
x[3]
x[4]
M = :
x[0]
x[2]
x[4]
:
:
2 Upsampling
Definition:
(↑2) M
x[0]
x[1]
x[2]
M = M
x[0]
0
x[1]
0
x[2]
0
M 3 As a matrix operation:
L1
L0
L0
L0
L0
L0 M
0
0
1
0
0
0
M 0
0
0
0
1
0 L
L
L
L
L
L M
x[0]
x[1]
x[2]
:
:
: = M
x[0]
0
x[1]
0
x[2]
0
M 4 x[n] Downsampling
Downsampling by 2
x[n]
→ L y[n]
→ ↓2 012 n y[n]
L
n L y[n] = x[2n]
Y(ω) = ∑ L
34n 0 x[2n]eiωn 12 = ∑ x[m]eiωm/2
m even = ½ ∑ {1 + (1)m} x[m]eiωm/2
m = ½ {∑
m x[m]eiωm
2 +∑
m ω
–i( 2 + π )m}
i(
x[m]e ; (1)m = eiπm = ½ {X (ω/2) + X (ω/2 + π )}
5 Downsampling by M
x[n]
y[n]
→
↓M
→
y[n] = x[Mn] Y(ω) = ∑ x[m]eiωm/M = x[m]eiωm/M ;
1
M M1 –i2πm)k
∑ (e M
k=0 123
123 m = nM
nM
M1
1
2π
= M ∑ { ∑ e–i M km }
m k=0 1 if m = nM
=
0 if m ≠ nM M1
1
ω + 2 πk)
M ∑ X(
M
k=0
6 Upsampling
Upsampling by 2
x[n]
→
↑2 123
123 y[n] = L
y[n]
→ x[n/2] ; n even
0 x[n]
L ; n odd 012
1
L
0 12 n
y[n]
L
34n Y(ω) = ∑ x[n/2]eiωn
n even = ∑ x[m]eiω2m
m = X(2ω) 7 Upsampling by L
123
123 y[n] = x[n/L] ; n = mL
0 x[n]
→ ↑L y[n]
→ ; n ≠ mL Y(ω) = ∑ x[n/L] eiωn
n=mL = X(Lω) 8 Downsampling X(ω)
x[n]
L L
2 1 0 1 2 3 L L
n ω0 0 ω0  2π 2π ½ Xs(ω) xs[n]
L
nL L
2 1 0 1 2 3
x[0] x[2]
x[2]
L L L
 2π π π 0 Y(ω) y[n] 1 0 1
y[n] = (↓2) x[n] = x[2n] π π
 2π ↑ 2ω0 ω 2π ½
nL ω 0 Y(ω) = ½ { X( 2π ↑ 2ω0
ω)+
2 L X( ω
2 + π)} ω
9 Upsampling
1 X(ω) x[n]
L L L x[0] x[1]
x[1]  2π n 1 0 1 L L
π ω0 0 ω0 π L 3 2 1 0 1 2 3 L L
n 123
123 x[n/2] ; n even
y[n] =
0 ; n odd ω Y(ω) 1 y[n] 2π  2π π ω0
2 ω0
2 π 2π ω Y(ω) = X(2ω)
10
10 Interpolation
Use lowpass filter after upsampling
x[n] u[n] ↑L H(ω) y[n] X(ω)  2π ω0 2π 0 ω
U(ω)  2π  2π 0 ω0 2π
L 0 ω0
L ω
Y(ω) 2π ω
11
11 Fractional Sampling
Consider
x[n]
u[n]
↑L
Y(ω) =
= y[n] ↓M M1
ω + 2 πk
1
∑ U( M )
M k=0
1 M1
2
∑ X ( ω +M πk L)
M
k=0 What about x[n] ↓M d[n] ↑L y[n] Y(ω) = D(ωL)
= 1 M1
M∑X
k=0 +
( ωL M 2πk )
12
12 Basic filters, upsampling and
downsampling. 14
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 Spring '03
 GilbertStrang

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