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The Hong Kong University of Science and Technology
Department of Electronic and Computer Engineering
Prof. Matthew R. McKay
ELEC 212 Chapter 4:
Sampling of ContinuousTime Signals
(Part II)
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0910 Fall ELEC 212
Sampling of CT Signals (Part II)
Changing Sampling Rate using DiscreteTime Processing
±
Discretetime signal
x
[
n
] is obtained by extracting the information from a
continuoustime signal
x
c
(
t
)
every
T
seconds:
±
Often necessary to
change the sampling rate
of a discretetime signal, such that
±
Approach 1:
–
Reconstruct
x
c
(
t
)
from
x
[
n
]
, then resample
x
c
(
t
)
with period
T'
to obtain
x'
[
n
]
±
Problems in practice due to nonideal analog reconstruction filters, ADC and DAC
±
Approach 2:
–
Changing the sampling rate using
only discretetime operations
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0910 Fall ELEC 212
Sampling of CT Signals (Part II)
Sampling Rate Reduction by an Integer Factor
π
2
∑
∞
−∞
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
−
=
r
c
j
T
k
T
j
X
T
e
X
ω
2
1
)
(
]
[
nM
x
M
copies of the periodic DTFT
, frequencyscaled
by
M
and shifted by integer multiples of
.
(original sequence)
(downsampled sequence)
)
(
j
e
X
∑
∞
−∞
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
−
=
r
c
j
T
k
T
j
X
T
e
X
2
1
)
(
∑
∞
−∞
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
−
=
r
c
j
d
MT
r
MT
j
X
MT
e
X
2
1
)
(
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0910 Fall ELEC 212
Sampling of CT Signals (Part II)
Sampling Rate Reduction by an Integer Factor
(Time Normalization)
Exact reconstruction possible in both cases!
Note: Sampling here is 2x
the Nyquist rate.
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0910 Fall ELEC 212
Sampling of CT Signals (Part II)
Sampling Rate Reduction by an Integer Factor
Exact reconstruction not possible due to aliasing!
Question:
How do we avoid aliasing due to downsampling … ?
6
0910 Fall ELEC 212
Sampling of CT Signals (Part II)
Sampling Rate Reduction by an Integer Factor
±
Aliasing due to downsampling can be avoided with
prefiltering
(ie. reduce the BW
before downsampling)
Note the LPF cutoff frequency
No Aliasing!
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 Spring '11
 Prof.MatthewMcKay
 Digital Signal Processing, Signal Processing, sampling rate, 0910 Fall ELEC

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