Page 70 - Mechatronics with Experiments

56 MECHATRONICS
is aliasing, it is impossible to recover the original signal even with an ideal re-construction
filter. Here we will consider the following two rescontruction D/A converters:

1. Ideal reconstruction filter (Shannon’s Reconstruction).
2. ZOH (Zero-Order-Hold).

In order to focus on D/A functionality and its ability to reconstruct the original signal from
its samples, let us assume that the time delay and quantization errors are negligable and
that the sampled signal is sent out to the D/A converter without any modification.

(i) Ideal Reconstruction Filter D/A Converter
In order to recover the original signal from the sampled signal, we need to recover the
original frequency content of the signal from the frequency content of the sampled
signal (Figure 2.7). Therefore, we need an ideal filter, an ideal reconstruction filter,
which has the following frequency response,

{ [ ]
,
⎧ T; w ∈ − s s
⎪|H(j )| = 2 2
0; otherwise (2.37)
⎨
⎪ ∠H(j ) = 0; ∀
⎩
Let us take the inverse Fourier transform of this filter transfer function to determine
what kind of impulse response such a filter would have.

−1
F (H(jw)) = 1 T T.e jwt .dw
2 ∫
−
T

T T jωt
= e .dw (2.38)
2 ∫
−
T
( )
2 w t
s
h(t) = ⋅ sin (2.39)
w t 2
s
This is the impulse response of an ideal reconstruction filter. Notice that the impulse
response of the ideal reconstruction filter is non-causal (Figure 2.8).
In order to practically implement it, one must introduce a time delay into the
system that is large enough compared to sampling period. It cannot be implemented in
closed loop control systems due to the stability problem the time delay would cause.
The original signal could, in theory if not in practical applications, be reconstructed
from its samples as follows,
∞
∑ (t − kT)
y(t) = y(kT) ⋅ sin c (2.40)
T
k=−∞
(ii) D/A: Zero-Order-Hold (ZOH)
The great majority of D/A converters operate as zero-order-hold functions. The signal
is kept at the last value until a new value comes. The change between the two values
is a step change. Let us try to obtain a transfer function for a zero-order-hold (ZOH)
D/A converter. To this end, consider that a single unit pulse is sent to the ZOH D/A
(Figure 2.9). The output of the ZOH D/A would be a single pulse with unit magnitude