Power and Energy of a Signal

7/25/2019 Power and Energy of a Signal

1/5

(9votes, average: 3.89out of 5)

Calculating the energy and power of a signal was discussed in one of

the previous posts. I have received numerous queries on this topic. I

intend to elaborate on this topic a little more here.

In signal processing, a signal is viewed as a function of time. The termsizeof a signal is used to represent strength of the signal. It is

crucial to know the size of a signal used in a certain application. For

example, we may be interested to know the amount of electricity

needed to power a LCD monitor as opposed to a CRT monitor. Both

of these applications are different and have different tolerances. Thus

the amount of electricity driving these devices will also be different.

A given signals size can be measured in many ways.Given a

mathematical function (or a signal equivalently), it seems that the

area under the curve, described by the mathematical function, is a

Mathuranathan December 20, 2013 Latest Articles, Signal Processing,Tips & Tricks, Tutorials 5 Comments

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2/5

good measure of describing the size of a signal. A signal can have

both positive and negative values. This may render areas that are

negative. Due to this effect, it is possible that the computed values

cancel each other totally or partially, rendering incorrect result. Thus

the metric function of area under the curve is not suitable for

defining the size of a signal. Now, we are left with two options :either 1) computation of the area under the absolute value of the

function or 2) computation of the area under the square of the

function. The second choice is favored due to its mathematical

tractability and its similarity to Euclidean Norm which is used in

signal detection techniques (Note: Euclidean norm otherwise called

L2 norm or 2-norm is often considered in signal detection

techniques on the assumption that it provides a reasonable measure

of distance between two points on signal space. It is computed as

Euclidean distance in detection theory).

Going by the second choice of viewing the size as the computation

of the area under the square of the function, the energy of a

continuous-time complex signal is defined as

If the signal x(t) is real, the modulus operator in the above equation

does not matter.

This is called Energy in signal processing terms. This is also a

measure of signal strength. This definition can be applied to any

signal (or a vector) irrespective of whether it possesses actual energy

(a basic quantitative property as described by physics) or not. If the

signal is associated with some physical energy, then the above

definition gives the energy content in the signal. If the signal is an

electrical signal, then the above definition gives the total energy of the

signal (in Joules) dissipated over a 1 Ohm resistor.

To know the actual energy of the signal , one has to know the value

of load the signal is driving and also the nature the electrical signal

[1]

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3/5

(voltage or current). For a voltage signal, the above equation has to be

scaled by a factor of .

For current signal, it has to be scaled by .

Here, is the impedance driven by the signal , is the signal

energy (signal processing term) and is the Energy of the signal

(physical quantity) driving the load

In the discrete domain, the energy of the signal is given by

The energy is finite only if the above sum converges to a finite value.

This implies that the signal is squarely-summable. Such a signal is

called finite energy signal.

What if the given signal does not decay with respect to time (as in a

continuous sine wave repeating its cycle infinitely) ? The energy will

be infinite and such a signal is not squarely-summable in other

words. We need another measurable quantity to circumvent thisproblem.This leads us to the notion of Power

Power is defined as the amount of energy consumed per unit time.

This quantity is useful if the energy of the signal goes to infinity or the

signal is not-squarely-summable. For non-squarely-summable

signals, the power calculated by taking the snapshot of the signal over

a specific interval of time as follows

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4/5

1) Take a snapshot of the signal over some finite time duration

2) Compute the energy of the signal

3) Divide the energy by number of samples taken for computation

4) Extend the limit of number of samples to infinity . This

gives the total power of the signal.

In discrete domain, the total power of the signal is given by

Following equations are different forms of the same computation

found in many text books. The only difference is the number of

samples taken for computation. The denominator changes according

to the number of samples taken for computation.

A signal can be classified based on its power or energy content.

Signals having finite energy are energy signals. Power signals have

finite and non-zero power.

A finite energy signal will have zero TOTAL power. Lets investigate

this statement in detail. When the energy is finite, the total power will

be zero. Check out the denominator in the equation for calculating

the total power. When the limit , the energy dilutes to zero

over the infinite duration and hence the total power becomes zero.

Signals whose total power is finite and non-zero. The energy of the

power signal will be infinite. Example: Periodic sequences like

sinusoid. A sinusoidal signal has finite, non-zero power but infinite

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Karnaugh Map walks

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over Fading Channels

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system over a Rayleigh Fading

channel Simulation in

Matlab

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a fading channel

Characterizing a MIMO

channel Channel State

Information (CSI) and

Condition number

MIMO Diversity and Spatial

Multiplexing

Introduction to Multiple

Antenna Systems

Chirp Signal Frequency

Sweeping FFT and power

spectral density

Generating Basic Signals

Gaussian Pulse and Power

Spectral Density using FFT

Generating Basic signals

Rectangular Pulse and Power


Generating Basic signals

Square Wave and Power


sakshama ghoslya - Hello Sir,

> Your final expression for

transmi...

Mathuranathan - Thanks for

spotting that. Will correct the
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5/5

energy.

A signal cannot be both an energy signal and a power signal.

Signals can also be a cat on the wall neither an energy signal nor a

power signal. Consider a signal of increasing amplitude defined by

For such a signal, both the energy and power will be infinite. Thus, it

cannot be classified either as an energy signal or as a power signal.

Calculation of power and verifying it through Matlab is discussed

here.

For details on RMS value of a signal (another measure for the

strength of the signal) click here

[1] Sanjay Lall,Norm and Vector spaces,Information Systems

Laboratory,Stanford University

( ) =

mistake...

PTM - I think there is a little

mistake in code comments...


your comment...

Mathuranathan - Thanks foryour comment....

Hao Yi - the two sources of

spectral leakage have confused

...


pointing it out...

Mathuranathan - I was

plotting the NON-normalized

magnitude spectr...

Yamuna - Hi Sir, In the above

example when I gave the ...

JAYAPRAKASH P - Hi, i am

doing my m tech final year
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Power and Energy of a Signal