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| − | == Sampling- A Bridge Between CT and DT == | + | == [[Sampling_Theorem|Sampling]]- A Bridge Between CT and DT == |
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Discrete signals are desirable to work with due to the ability to process them with program such as MATLAB. By use of sampling a continuous signal can be converted to a discrete signal, manipulated via a computer program and then converted back into a continuous time form. | Discrete signals are desirable to work with due to the ability to process them with program such as MATLAB. By use of sampling a continuous signal can be converted to a discrete signal, manipulated via a computer program and then converted back into a continuous time form. | ||
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Xp(t) is an impulse train that has amplitudes of impulses equal to the samples of X(t) at intervals spaced by T. | Xp(t) is an impulse train that has amplitudes of impulses equal to the samples of X(t) at intervals spaced by T. | ||
| − | <math>x_p(t) = x(t)p(t)\!</math> | + | <math class="inline">x_p(t) = x(t)p(t)\!</math> |
| − | where <math>p(t) = \sum^{\infty}_{n = -\infty} \delta(t - nT)\!</math> | + | where <math class="inline">p(t) = \sum^{\infty}_{n = -\infty} \delta(t - nT)\!</math> |
| − | and <math> x(t)\! </math> is the function being sampled. | + | and <math class="inline"> x(t)\! </math> is the function being sampled. |
| − | Since <math>x(t) \delta(t - t_0) = x(t_0) \delta(t-t_0)\!</math>, | + | Since <math class="inline">x(t) \delta(t - t_0) = x(t_0) \delta(t-t_0)\!</math>, |
<math>x_p(t) = \sum^{\infty}_{n = -\infty} x(nT)\delta(t - nT)\!</math> | <math>x_p(t) = \sum^{\infty}_{n = -\infty} x(nT)\delta(t - nT)\!</math> | ||
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<math> X_p(j\omega) = \frac{1}{T} \sum^{\infty}_{k = -\infty}X(j(\omega - K*\omega_s)) \!</math> | <math> X_p(j\omega) = \frac{1}{T} \sum^{\infty}_{k = -\infty}X(j(\omega - K*\omega_s)) \!</math> | ||
| − | which is a scaled and shifted copy of <math>X(j\omega)\!</math> | + | which is a scaled and shifted copy of <math class="inline">X(j\omega)\!</math> |
X(t) can be recovered exactly from Xp(t) by using a low pass filter with gain T and cut off frequency greater than Wm but less than Ws - Wm. | X(t) can be recovered exactly from Xp(t) by using a low pass filter with gain T and cut off frequency greater than Wm but less than Ws - Wm. | ||
| + | ---- | ||
| + | [[Main_Page_ECE301Fall2008mboutin|Back to ECE301 Fall 2008]] | ||
Revision as of 15:53, 30 November 2010
Sampling- A Bridge Between CT and DT
Discrete signals are desirable to work with due to the ability to process them with program such as MATLAB. By use of sampling a continuous signal can be converted to a discrete signal, manipulated via a computer program and then converted back into a continuous time form.
Sampling involves a function known as an impulse train. An impulse train is a series of impulses that are spaced out by a period T, known as the Sampling Period.
Xp(t) is an impulse train that has amplitudes of impulses equal to the samples of X(t) at intervals spaced by T.
$ x_p(t) = x(t)p(t)\! $ where $ p(t) = \sum^{\infty}_{n = -\infty} \delta(t - nT)\! $ and $ x(t)\! $ is the function being sampled.
Since $ x(t) \delta(t - t_0) = x(t_0) \delta(t-t_0)\! $,
$ x_p(t) = \sum^{\infty}_{n = -\infty} x(nT)\delta(t - nT)\! $
Taking the Fourier Transform of this function yields,
$ X_p(j\omega) = \frac{1}{T} \sum^{\infty}_{k = -\infty}X(j(\omega - K*\omega_s)) \! $
which is a scaled and shifted copy of $ X(j\omega)\! $
X(t) can be recovered exactly from Xp(t) by using a low pass filter with gain T and cut off frequency greater than Wm but less than Ws - Wm.
