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Complex signals can be built from elemental signals, including the complex exponential, unit step, pulse, etc. This module presents the elemental signalsin brief.

Elemental signals are the building blocks with which we build complicated signals. By definition, elemental signals have a simple structure. Exactly what wemean by the "structure of a signal" will unfold in this section of the course. Signals are nothing more thanfunctions defined with respect to some independent variable, which we take to be time for the most part. Very interestingsignals are not functions solely of time; one great example of which is an image. For it, the independent variables are x and y (two-dimensional space). Video signals are functions of three variables: two spatialdimensions and time. Fortunately, most of the ideas underlying modern signal theory can be exemplified with one-dimensional signals.

Sinusoids

Perhaps the most common real-valued signal is the sinusoid.

s t A 2 f 0 t φ
For this signal, A is its amplitude, f 0 its frequency, and φ its phase.

Complex exponentials

The most important signal is complex-valued, the complex exponential.

s t A 2 f 0 t φ A φ 2 f 0 t
Here, denotes -1 . A φ is known as the signal's complex amplitude . Considering the complex amplitude as a complex numberin polar form, its magnitude is the amplitude A and its angle the signal phase. The complex amplitude is also known as a phasor . The complex exponential cannot be further decomposed into more elemental signals, and is the most important signal in electrical engineering ! Mathematical manipulations at first appear to be more difficult because complex-valued numbers areintroduced. In fact, early in the twentieth century, mathematicians thought engineers would not be sufficientlysophisticated to handle complex exponentials even though they greatly simplified solving circuit problems. Steinmetz introduced complex exponentials to electrical engineering, and demonstrated that "mere" engineers could use them to goodeffect and even obtain right answers! See Complex Numbers for a review of complex numbers and complex arithmetic.

The complex exponential defines the notion of frequency: it is the only signal that contains only one frequency component. The sinusoid consists of two frequencycomponents: one at the frequency f 0 and the other at f 0 .

This decomposition of the sinusoid can be traced to Euler's relation.
2 f t 2 f t 2 f t 2
2 f t 2 f t 2 f t 2
2 f t 2 f t 2 f t
The complex exponential signal can thus be written in terms of its real and imaginary parts using Euler's relation. Thus,sinusoidal signals can be expressed as either the real or the imaginary part of a complex exponential signal, the choicedepending on whether cosine or sine phase is needed, or as the sum of two complex exponentials. These two decompositions aremathematically equivalent to each other.
A 2 f t φ A φ 2 f t
A 2 f t φ A φ 2 f t

Graphically, the complex exponential scribes a circle in the complex plane as time evolves. Its real and imaginary partsare sinusoids. The rate at which the signal goes around the circle is the frequency f and the time taken to go around is the period T . A fundamental relationship is T 1 f .

Using the complex plane, we can envision the complex exponential's temporal variations as seen in the above figure( [link] ). The magnitude of the complex exponential is A , and the initial value of the complex exponential at t 0 has an angle of φ . As time increases, the locus of points traced by the complexexponential is a circle (it has constant magnitude of A ). The number of times per second we go around the circle equals the frequency f . The time taken for the complex exponential to go around the circle once is known asits period T , and equals 1 f . The projections onto the real and imaginary axes of the rotating vector representing the complex exponentialsignal are the cosine and sine signal of Euler's relation ( [link] ).

Real exponentials

As opposed to complex exponentials which oscillate, real exponentials decay.

s t t τ

The real exponential.

The quantity τ is known as the exponential's time constant , and corresponds to the time required for the exponential to decrease by afactor of 1 , which approximately equals 0.368 . A decaying complex exponential is the product of a real and a complex exponential.

s t A φ t τ 2 f t A φ 1 τ 2 f t
In the complex plane, this signal corresponds to an exponential spiral. For such signals, we can define complex frequency as the quantity multiplying t .

Unit step

The unit step function is denoted by u t , and is defined to be

u t 0 t 0 1 t 0

The unit step.

This signal is discontinuous at the origin. Its value at the origin need not be defined, and doesn't matter in signaltheory.
This kind of signal is used to describe signals that "turn on" suddenly. For example, tomathematically represent turning on an oscillator, we can write it as the product of a sinusoid and a step: s t A 2 f t u t .

Pulse

The unit pulse describes turning a unit-amplitude signal on for a duration of Δ seconds, then turning it off.

p Δ t 0 t 0 1 0 t Δ 0 t Δ
The pulse.
We will find that this is the second most important signal in communications.

Square wave

The square wave sq t is a periodic signal like the sinusoid. It too has an amplitude and a period, which must be specified tocharacterize the signal. We find subsequently that the sine wave is a simpler signal than the square wave.

The square wave.

Questions & Answers

show that the set of all natural number form semi group under the composition of addition
Nikhil Reply
explain and give four Example hyperbolic function
Lukman Reply
_3_2_1
felecia
⅗ ⅔½
felecia
_½+⅔-¾
felecia
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
SABAL Reply
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
Pawel
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Ifeanyi
on number 2 question How did you got 2x +2
Ifeanyi
combine like terms. x + x + 2 is same as 2x + 2
Pawel
x*x=2
felecia
2+2x=
felecia
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
mariel Reply
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
Pawel
how do I set up the problem?
Harshika Reply
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
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Shirley Reply
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Abdullahi
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Mark
I need quadratic equation link to Alpa Beta
Abdullahi Reply
find the value of 2x=32
Felix Reply
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
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Mark
use the y -intercept and slope to sketch the graph of the equation y=6x
Only Reply
how do we prove the quadratic formular
Seidu Reply
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Darius
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Shirley Reply
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Seidu
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Opoku
what is math number
Tric Reply
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
Answers please
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Source:  OpenStax, Fundamentals of electrical engineering i. OpenStax CNX. Aug 06, 2008 Download for free at http://legacy.cnx.org/content/col10040/1.9
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