Downsampling
Section 6, Nick Antipa, 3/9/2018
Based on slides by Jon Tamir
Some notes from Giulia Fanti, Frank Ong, Michael Lustig, A
Midterm 1
• ! " = ∑()*+
%&' , - .
*/0%1/)
)*+ 5)*+ ()*+
= 3 , - . */0%1/) + 3 , - . */0%1/) + 3 , - . */0%1/)
%&' %&) %&5)
• What is the problem with this?
My favorite example of aliasing
And
• http://i.imgur.com/8X8Fcoy.gifv
Downsampling
x[n] N y[n] = x[nN ]
• Compresses in the time domain
• Expands in the frequency domain
Alternative derivation of
downsampling DTFT
• Recall: A N-periodic sequence has a discrete Fourier
series (DFS):
N
X1
1 DFS Representation
s̃(m) = S̃k ej2⇡mk/N
N
k=0
N
X1
j2⇡km/N
S̃k = s̃(m)e DFS Coefficients
m=0
Alternative derivation of
downsampling DTFT
• Impulse train: X
s̃(m) = (m kN )
k2Z
• DFS Coefficients (check):
S̃k = 1
• DFS Representation: N
1 X1
s̃(m) = ej2⇡mk/N
N
k=0
Alternative derivation of
downsampling DTFT
x[n] N y[n] = x[nN ]
Alternative derivation of
downsampling DTFT
x[n] N y[n] = x[nN ]
Downsampling x[n] y[n]
N
&'
1. Stretch ! " #$ to !(" )
(
2. Create (N-1) copies of the stretched versions
3. Shift each copy by successive multiples of 2+ and add
4. Divide by N
Decimation (filtering and
downsampling)
Chirp
Q: for a chirp of length L seconds, ranging from !" to !# , what is the expression for
$% & , the continuous-time signal?
!# − !"
A: $% & = sin 2. !" + & &
1
Q: How fast do we need to sample this signal?
A: !2 > 2!#
Q: What is 45 6Ω ? (ignore effects of windowing)
Ω
A: 9:;&
2Ω"
Chirp
*- − *+
!" # = sin 2) *+ + # #
/
Q: What is x[n] if we sample exactly at Nyquist?
*- − *+
! 0 = sin 2) *+ + 10 10
/
Ω+ 0 0
= sin 0) 1− +
Ω- 4 4
omega
Q: Plot the frequency vs n
)
Ω+
)
Ω-
n
4
Downsample by 2 (without AA
filter)
() - -
Recall: ! " = sin "' (*
1−. +.
() 1- 1-
A: ! 2" = sin 2"' (*
1− .
+ .
Q: What is the frequency vs n now?
2'
Ω3 2" 2"
2' 1− +
Ω4 5 5
Ω3
2' '
Ω4
Ω3
'
Ω4
n
5 5
2
Chirp demo
Imaging example
image of an important plot
144
146 . . . . . .
148 . . . . . .
150 . . . . . .
152 . . . . . .
154 . . . . . .
156 . . . . . .
158 . . . . . .
160 . . . . . .
162 . . . . . .
164
90 92 94 96 98 100
Decimated by 2
decimated by 2
72 . .
decimated by 2
.
73
74 . . .
75
76 . . .
77
78 . . .
79
80 . . .
81
82
45 46 47 48 49 50
Decimated by 2 again (4 total)
decimated by 4 decimated by 4
36
36.5
37
37.5
38
38.5
39
39.5
40
40.5
41
22.5 23 23.5 24 24.5 25
Lowpass filter first
blurred image of an important plot
144
146 . . .
148
150 . . .
152
154 . . .
156
158 . . .
160
162 . . .
164
90 92 94 96 98 100
Resampling looks fine!
properly resampled image properly resampled image
36
36.5
37
37.5
38
38.5
39
39.5
40
40.5
41
22.5 23 23.5 24 24.5 25
Upsampling
x[n] N y[n]
• Expands in the time domain
• Compresses in the frequency domain
Y (ej! ) = X(ej!N )
Interpolation (Upsampling and
filtering)
Interpolation
1. Smooth discrete-time signal
• Low frequency content
2. Upsample by 3
• No longer smooth! Contains high frequencies
3. LPF
• Removes high frequencies
