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<math> =\sum^{\infty}_{n = -\infty}x(nT)\delta(t - nT)\!</math> | <math> =\sum^{\infty}_{n = -\infty}x(nT)\delta(t - nT)\!</math> | ||
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| + | Now, in the time domain, <math>x_p(t)\!</math> looks like a group of shifted deltas with magnitude equal to the value of <math>x(t)\!</math> at that time, <math>nT\!</math>, in the original function. In the frequency domain, <math>X_p(\omega)\!</math> looks like shifted copies of the original <math>X(\omega)\!</math> that repeat every <math>\omega_s\!</math>, except that the magnitude of the copies is <math>\frac{1}{T}\!</math> of the magnitude of the original <math>X(\omega)\!</math>. | ||
Revision as of 12:55, 9 November 2008
Impulse-train Sampling
One type of sampling that satisfies the Sampling Theorem is called impulse-train sampling. This type of sampling is achieved by the use of a periodic impulse train multiplied by a continuous time signal, $ x(t)\! $. The periodic impulse train, $ p(t)\! $ is referred to as the sampling function, the period, $ T\! $, is referred to as the sampling period, and the fundamental frequency of $ p(t)\! $, $ \omega_s = \frac{2\pi}{T}\! $, is the sampling frequency. We define $ x_p(t)\! $ by the equation,
Graphically, this equation looks as follows,
$ x(t)\! $ ----------> x --------> $ x_p(t)\! $ ^ | | $ p(t) = \sum^{\infty}_{n = -\infty} \delta(t - nT)\! $
By using linearity and the sifting property, $ x_p(t)\! $ can be represented as follows,
$ x_p(t) = x(t)p(t)\! $
$ = x(t)\sum^{\infty}_{n = -\infty} \delta(t - nT)\! $
$ =\sum^{\infty}_{n = -\infty}x(t)\delta(t - nT)\! $
$ =\sum^{\infty}_{n = -\infty}x(nT)\delta(t - nT)\! $
Now, in the time domain, $ x_p(t)\! $ looks like a group of shifted deltas with magnitude equal to the value of $ x(t)\! $ at that time, $ nT\! $, in the original function. In the frequency domain, $ X_p(\omega)\! $ looks like shifted copies of the original $ X(\omega)\! $ that repeat every $ \omega_s\! $, except that the magnitude of the copies is $ \frac{1}{T}\! $ of the magnitude of the original $ X(\omega)\! $.
