Download ECE 3323 Section 2.1 Signals Power and Energy

ECE 3323
Principles of Communication
Systems
Section 2.1
Signals Power and Energy
1
Definition of a Signal
A Signal, denoted x(t), is assumed to be either a
voltage or a current in an electrical system. Signals
will be assumed to be Real Functions (as opposed to
Complex, with real and imaginary parts) unless noted.
2
Signal Power
A Signal is assumed to be measured across a 1 Ω
resistance (in the case of a voltage) or measured
through a 1 Ω resistance (in the case of a current) such
that the instantaneous power of the signal is
𝑃= 𝑥 𝑡
2
Signals are assumed to be real valued, so the
instantaneous power is
𝑃 = 𝑥2 𝑡
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Finite Energy Signals
The energy contained in a signal is the time integration
of the instantaneous power. If this quantity is bound
such that
∞
𝐸=
∞
𝑃 𝑑𝑡 =
−∞
𝑥 𝑡
2
𝑑𝑡 < ∞
−∞
the signal is known as an Energy Signal. This
requires that the signal have non-negligible magnitude
for only a finite amount of time.
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Finite Power Signals
If a signal exists for all time, the total energy in the
signal may not be bounded. In these cases the average
power in the signal is
𝑇
𝑃𝑎𝑣𝑒
1
= lim
𝑇→∞ 𝑇
2
𝑥 𝑡
2
𝑑𝑡 < ∞
−𝑇 2
𝑃𝑎𝑣𝑒 = 𝑥 𝑡
2
<∞
The signal is said to be a Power Signal.
5
Signal Classification
Based on these two definitions, there are two classes
of signals:
1. x(t) is an Energy Signal if and only if 0 < E < ,
implying that that P = 0.
2. x(t) is a Power Signal if and only if 0 < P < ,
implying that E = .
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Finite Energy Signals
7
Finite Power Signals
8
Periodic Signals
A signal is Periodic if and only if
𝑥 𝑡 + 𝑛𝑇0 = 𝑥 𝑡
for all t and integer n. The smallest positive number
that satisfies this condition is the period T0. All other
signals are Non – Periodic.
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Periodic Signals
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Deterministic Signals
Signals whose values are defined for their entire
existence are called Deterministic Signals. An
explicit mathematical expression can be written for
these signals.
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Random Signals
Signals whose values are not defined, or not
predictable, are called Random Signals. A
mathematical expression cannot be written. Random
signals can be described by their statistical properties.
12
Continuous, Discrete, Analog or Digital?
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