(New page: Energy and Power Classifications The total energy of a continuous-time signal x(t), where x(t) is defined for −∞ < t < ∞, is 2( ) lim 2( ) T T T E x t dt x t dt ∞ ∞ <math>−∞ ...) |
|||
| Line 1: | Line 1: | ||
| − | + | <math>Insert formula here</math>The total energy of a continuous-time signal x(t), where x(t) is defined for −∞ < t < ∞, | |
| − | The total energy of a continuous-time signal x(t), where x(t) is defined for −∞ < t < ∞, | + | |
is | is | ||
2( ) lim 2( ) | 2( ) lim 2( ) | ||
| Line 8: | Line 7: | ||
∞ | ∞ | ||
∞ | ∞ | ||
| − | + | −∞ →∞ − | |
| − | =� = � | + | =� = � |
| + | In many situations, this quantity is proportional to a physical notion of energy, for | ||
example, if x(t) is the current through, or voltage across, a resistor. If a signal has finite | example, if x(t) is the current through, or voltage across, a resistor. If a signal has finite | ||
energy, then the signal values must approach zero as t approaches positive and negative | energy, then the signal values must approach zero as t approaches positive and negative | ||
infinity. | infinity. | ||
The time-average power of a signal is | The time-average power of a signal is | ||
| − | + | lim 1 2( ) | |
2 | 2 | ||
T | T | ||
| Line 21: | Line 21: | ||
∞ T | ∞ T | ||
→∞ − | →∞ − | ||
| − | = � | + | = � |
| + | For example the constant signal x(t) =1 (for all t) has time-average power of unity. | ||
With these definitions, we can place most, but not all, continuous-time signals into one of | With these definitions, we can place most, but not all, continuous-time signals into one of | ||
two classes: | two classes: | ||
| Line 27: | Line 28: | ||
x(t) = 0, for all t are energy signals. For an energy signal, P∞ = 0 . | x(t) = 0, for all t are energy signals. For an energy signal, P∞ = 0 . | ||
• A power signal is a signal with finite, nonzero P∞ . An example is x(t) =1, for all t, | • A power signal is a signal with finite, nonzero P∞ . An example is x(t) =1, for all t, | ||
| − | though more interesting examples are not obvious and require analysis. For a | + | though more interesting examples are not obvious and require analysis. For a |
| − | + | ||
| − | + | ||
Revision as of 18:04, 5 September 2008
$ Insert formula here $The total energy of a continuous-time signal x(t), where x(t) is defined for −∞ < t < ∞, is 2( ) lim 2( ) T T T E x t dt x t dt ∞ ∞ −∞ →∞ − =� = � In many situations, this quantity is proportional to a physical notion of energy, for example, if x(t) is the current through, or voltage across, a resistor. If a signal has finite energy, then the signal values must approach zero as t approaches positive and negative infinity. The time-average power of a signal is lim 1 2( ) 2 T T T P x t dt ∞ T →∞ − = � For example the constant signal x(t) =1 (for all t) has time-average power of unity. With these definitions, we can place most, but not all, continuous-time signals into one of two classes: • An energy signal is a signal with finite E∞ . For example, x(t) =e−|t| , and, trivially, x(t) = 0, for all t are energy signals. For an energy signal, P∞ = 0 . • A power signal is a signal with finite, nonzero P∞ . An example is x(t) =1, for all t, though more interesting examples are not obvious and require analysis. For a
