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&= rect \left (\frac{t - \tfrac {T}{2} }{T}\right) * {\color{Blue}\sum_k x(kT)\ \delta (t - kT)} \\ | &= rect \left (\frac{t - \tfrac {T}{2} }{T}\right) * {\color{Blue}\sum_k x(kT)\ \delta (t - kT)} \\ | ||
&= rect \left (\frac{t - \tfrac {T}{2} }{T}\right) * {\color{OliveGreen} X_s(t)}\\ | &= rect \left (\frac{t - \tfrac {T}{2} }{T}\right) * {\color{OliveGreen} X_s(t)}\\ | ||
| − | |||
\end{align} | \end{align} | ||
</math> | </math> | ||
</div> | </div> | ||
| + | |||
| + | <div style="font-family: Verdana, sans-serif; font-size: 12px; text-align: center; width: 30%; margin: 0px; border: 1px solid #aaa; padding: 1em;"> <math>\therefore\ x_r(t) = h_{zo} (t) * X_s(t)</math> | ||
| + | </div> | ||
| + | |||
| + | |||
<math>*** Image \quad goes \quad here***\,\!</math> | <math>*** Image \quad goes \quad here***\,\!</math> | ||
| + | |||
In the frequency domain, | In the frequency domain, | ||
| − | <math>hahaha\ | + | <math>hahaha\,\!</math> |
<math>*** Image \quad goes \quad here***\,\!</math> | <math>*** Image \quad goes \quad here***\,\!</math> | ||
So even though <math>x_r(t)\,\!</math> is not band limited, its higher frequency components are attenuated because the <math>|H_{zo}(f)|\ \,\!</math> decreases as <math>|f|\ \,\!</math> increases. | So even though <math>x_r(t)\,\!</math> is not band limited, its higher frequency components are attenuated because the <math>|H_{zo}(f)|\ \,\!</math> decreases as <math>|f|\ \,\!</math> increases. | ||
Revision as of 06:26, 23 September 2009
LECTURE on September 11, 2009
The perfect reconstruction of $ {x(t)}\,\! $ from $ x_s(t)\,\! $ is possible if $ X(f) = 0\,\! $ when $ |f| \ge \frac{1}{|2T|} $
PROOF: Look at the graph of $ X_s(f)\,\! $
$ *** Image \quad goes \quad here***\,\! $
To avoid aliasing,
$ \frac{1}{T}\ - f_M \ge f_M $ $ \quad\iff\quad $ $ \frac{1}{T}\ \ge 2f_M $
To recover the signal, we will require a low pass filter with gain $ T\,\! $ and cutoff, $ \frac{1}{2T} $
Let $ x_r(t)\,\! $ be the reconstructed signal. Then,
$ X_(f) = H_r(f) X_s(f)\,\! $
where,
$ H_r(f) = T rect(f)\,\! $
So,
$ \begin{align} x_r(t) &= h_r(t) * {\color{OliveGreen} X_s(t)} \\ &= sinc \left (\frac{t}{T}\right) * {\color{Blue} \sum_k X(kT) \delta(t-kT)} \\ &= \sum_k X(kT) sinc \left (\frac{t}{T}\right) * \delta(t-kT) \\ &= \sum_k X(kT) sinc \left (\frac{t - kT}{T}\right)\\ \end{align} $
Recall, $ \quad sinc(x) = 0 \quad \iff \quad x = \pm 1, \pm 2, \pm 3 ... \,\! $
$ *** Image \quad goes \quad here***\,\! $
At all integer multiples of T,
$ x_r(nT) = X(nT)\,\! $
If Nyquist is satisfied, $ \quad x_r(nT) = X(nT)\quad \forall \quad 't'\,\! $
Contrast this reconstruction with the zero-order hold,
$ *** Image \quad goes \quad here***\,\! $
$ \qquad \Rightarrow piecewise\ construct\ approximation\ $
$ \begin{align} x_r(t) &= \sum_k x(kT)\ rect \left (\frac{t - \tfrac {T}{2} - kT}{T}\right) \\ &= \sum_k x(kT)\ rect \left (\frac{t - \tfrac {T}{2} }{T}\right) * \delta (t - kT)\\ &= rect \left (\frac{t - \tfrac {T}{2} }{T}\right) * \sum_k x(kT)\ \delta (t - kT) \qquad and\ if\ we\ look\ clearly,\ \\ &= rect \left (\frac{t - \tfrac {T}{2} }{T}\right) * {\color{Blue}\sum_k x(kT)\ \delta (t - kT)} \\ &= rect \left (\frac{t - \tfrac {T}{2} }{T}\right) * {\color{OliveGreen} X_s(t)}\\ \end{align} $
$ *** Image \quad goes \quad here***\,\! $
In the frequency domain,
$ hahaha\,\! $
$ *** Image \quad goes \quad here***\,\! $
So even though $ x_r(t)\,\! $ is not band limited, its higher frequency components are attenuated because the $ |H_{zo}(f)|\ \,\! $ decreases as $ |f|\ \,\! $ increases.
