Digital Signal Processing Reference
In-Depth Information
x
(
t
)
x
s
(
t
)
x
(
t
),
x
s
(
t
)
Sample & Hold (S&H)
x
(0)
x
(1)
x
(2)
x
(3)
FT
x
s
(
t
)
x
(
t
)
h
(
t
)
↔
H
(
f
)
t
, sec
T
s
2T
s
3T
s
0
H
(
f
)=|
T
s
sinc(
T
s
f
)|
T
s
h
(
t
)
1
t
, sec
f
, Hz
0
T
s
0
Fig. 2.4
Sample-and-Hold: operation, impulse response, and transfer function
2.2.2.2 Stage 2
Note that the DAC converts the digital signal into an analog stair-step waveform,
which contains many high-frequency artefacts. This is because the sample and
hold is effectively a filter with a very slow roll off, and it does not fully eliminate
the energy in all the unwanted spectral images. To reconstruct a smooth replica of
the original signal, the stair-step function must be passed through a low-pass filter,
often called an anti-imaging filter. This filter ideally removes all frequency content
above the upper frequency limit, B.
For perfect reconstruction of the analog signal to occur within this two stage
process, three criteria must be satisfied. The first criteria is that there is no
quantization noise in the digital signal. The second criterion is that the sampling
frequency should be greater than twice the signal bandwidth B, i.e., greater than
twice the highest frequency present in the signal:
B
f
s
2
:
ð
2
:
4
Þ
This requirement is necessary so that the repeating spectra shown in Fig. (
2.3
)do
not ''run into other''. The minimum sampling rate needed to avoid reconstruction
errors (f
s
= 2B) is called the Nyquist rate. The third criteria is that the filtering
provided by the combination of the sample and hold function and the subsequent
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