Are all energy signals also power signals

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At a Energy signal In signal theory, it is a real or complex-valued signals(t) with finite signal energy.

Table of Contents

Definition for continuous signals


A complex-valued continuous signal s is called an energy signal if and only if:

\ ({\ displaystyle \ int \ limits _ {- \ infty} ^ {\ infty} {s (t) \ cdot s ^ {*} (t) dt} = \ int \ limits _ {- \ infty} ^ { \ infty} {| s (t) | ^ {2} dt} <\ infty.} \)

For purely real values, the equation simplifies to

\ ({\ displaystyle \ int \ limits _ {- \ infty} ^ {\ infty} {s ^ {2} (t) dt} <\ infty.} \)

Definition of discrete signals


A complex-valued discrete signal s is called an energy signal if:

\ ({\ displaystyle \ sum _ {n = - \ infty} ^ {\ infty} {s (n) \ cdot s ^ {*} (n)} <\ infty} \)

In the case of purely real-valued discrete signals s, the following applies accordingly:

\ ({\ displaystyle \ sum _ {n = - \ infty} ^ {\ infty} {s ^ {2} (n)} <\ infty} \)

As usual, \ ({\ displaystyle s ^ {*}} \) denoted the complex number conjugated to s. The value of the respective integral or the respective sum is called the signal energy.

Typical energy signals


Typical energy signals are all signals that represent finite signal values ​​and that are switched on and off at some point. Examples include decay processes or individual, time-limited pulses.

Typical non-energy signals are all power signals. The Dirac impulse, which is also not an energy signal, has a special position in theory. The integral over the signal function results in the normalized value 1, but not the integral over the square of the signal function.

Physical background


The terms of the signal processing are based on physics and electrical engineering. If, for example, a current i flowing through a resistor R is viewed as a signal, the instantaneous power is calculated as well

\ ({\ displaystyle p (t) = u (t) \ cdot i (t) = R \ cdot i ^ {2} (t)} \)

and accordingly the total converted energy

\ ({\ displaystyle E = R \ cdot \ int \ limits _ {t = - \ infty} ^ {\ infty} i ^ {2} (t) dt} \).

Signal theory background


In signal theory, the set of energy signals together with the addition of functions and multiplication with a real or complex number forms a vector space with an infinite number of basis vectors. Sine and cosine functions in particular come into consideration as base vectors. The Fourier transform is the essential element of the vector or signal space equipped with these basis vectors.

literature


  • Hans Dieter Lüke: Signal transmission. 6th edition. Springer Verlag, 1995, ISBN 3-540-54824-6.

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Categories:Signal processing




Status of information: 11/23/2020 6:32:26 AM CET

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