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The impulse train of a signal, denoted as <math>x_p(t)</math>, is the original signal multiplied by <math>p(t) = \sum_{n=-\infty}^\infty \delta(t-nT)</math>. This creates a new signal, <math>x_p(t)</math>, which consists of a series of equally spaced impulses with spacing T and area <math>x_c(t)</math>. | The impulse train of a signal, denoted as <math>x_p(t)</math>, is the original signal multiplied by <math>p(t) = \sum_{n=-\infty}^\infty \delta(t-nT)</math>. This creates a new signal, <math>x_p(t)</math>, which consists of a series of equally spaced impulses with spacing T and area <math>x_c(t)</math>. | ||
| − | This kind of signal almost looks like a discrete time signal. The reason it is not, however, is because the index of a discrete time signal needs to be an integer. To | + | This kind of signal almost looks like a discrete time signal. The reason it is not, however, is because the index of a discrete time signal needs to be an integer. To changes this, all that needs to be done is a transformation of the independent variable <math>t = nT</math>. This in effect compresses <math>x_p(t)</math> by T. The same effect happens in the frequency domain. |
The formula to convert between the two is <math>x_p[n]=x_c(nT)</math> | The formula to convert between the two is <math>x_p[n]=x_c(nT)</math> | ||
Revision as of 13:37, 29 July 2009
Conversion of Impulse Train to DT
The impulse train of a signal, denoted as $ x_p(t) $, is the original signal multiplied by $ p(t) = \sum_{n=-\infty}^\infty \delta(t-nT) $. This creates a new signal, $ x_p(t) $, which consists of a series of equally spaced impulses with spacing T and area $ x_c(t) $.
This kind of signal almost looks like a discrete time signal. The reason it is not, however, is because the index of a discrete time signal needs to be an integer. To changes this, all that needs to be done is a transformation of the independent variable $ t = nT $. This in effect compresses $ x_p(t) $ by T. The same effect happens in the frequency domain.
The formula to convert between the two is $ x_p[n]=x_c(nT) $
