Energy signal

In a power signal is in the signal theory to a real- or complex-valued signal s (t) with finite signal energy .

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:

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

For purely real-valued discrete signals s, the following applies accordingly:

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

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

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

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

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 the 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 .

Web links

• Energy signals and power signals (accessed July 12, 2018)
• Correlation Technique (accessed July 12, 2018)
• Digital signal processing I / II (accessed on July 12, 2018)
• Time-Frequency Transformations (accessed July 12, 2018)