Signals and Systems - Introduction to Signals

8/8/2019 Signals and Systems - Introduction to Signals

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Chapter 1. Introduction to Signals

. Signal Classifications and Properties*

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module will begin our study of signals and systems by laying out some of the fundamentals of signal classi fication. It is essentially an introducmportant definitions and properties that are fundamental to the discussion of signals and systems, with a brief discussion of each.

assifications of Signals

ntinuous-Time vs. Discrete-Time

e names suggest, this classification is determined by whether or not the time axis is discrete (countable) orcontinuous (Figure 1.1). A consignal will contain a value for all real numbers along the time axis. In contrast to this, a discrete-time signal, often created by sampling a conal, will only have values at equally spaced intervals along the time axis.

Figure 1.1.

alog vs. Digital

difference between analog and digital is similar to the difference between continuous-time and discrete-time. However, in this case the diffeves the values of the function. Analog corresponds to a continuous set of possible function values, while digital corresponds to a discrete set ible function values. An common example of a digital signal is a binary sequence, where the values of the function can only be one or zero.

Figure 1.2.

riodic vs. Aperiodic

odic signals repeat with some periodT, while aperiodic, or nonperiodic, signals do not (Figure 1.3). We can define a periodic function throwing mathematical expression, where tcan be any number and Tis a positive constant: f(t) = f(T+ t) The fundamental period of our function

mallest value ofTthat the still allows Equation to be true.


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Figure 1.3.

(a) A periodic signal with periodT0

(b) An aperiodic signal

ite vs. Infinite Length

e name implies, signals can be characterized as to whether they have a finite or infinite length set of values. Most finite length signals are useng with discrete-time signals or a given sequence of values. Mathematically speaking, f(t) is a finite-length signal if it is nonzero over a finval t1 < f(t) < t2 where t1 > and t2 < . An example can be seen in Figure 1.4. Similarly, an infinite-length signal, f(t) , is defined as no

all real numbers: f(t)

Figure 1.4.

Finite-Length Signal. Note that it only has nonzero values on a set, finite interval.

usal vs. Anticausal vs. Noncausal

sal signals are signals that are zero for all negative time, while anticausal are signals that are zero for all positive time. Noncausal signals als that have nonzero values in both positive and negative time (Figure 1.5).


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Figure 1.5.

(a) A causal s ignal

(b) An anticausal s ignal

(c) A noncausal s ignal

en vs. Odd

ven signal is any signal fsuch that f(t) = f( t) . Even signals can be easily spotted as they are symmetric around the vertical axis. An odd se other hand, is a signal fsuch that f(t) = (f( t)) (Figure 1.6).

Figure 1.6.

(a) An even signal

(b) An odd s ignal

g the definitions of even and odd signals, we can show that any signal can be written as a combination of an even and odd signal. That is, eval has an odd-even decomposition. To demonstrate this, we have to look no further than a single equation. B

plying and adding this expression out, it can be shown to be true. Also, it can be shown that f(t) + f( t) fulfills the requirement of an even funct

f(t) f( t) fulfills the requirement of an odd function (Figure 1.7).

ample 1.1.


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Figure 1.7.

(a) The signal we will decompos e using odd-even decomposition

(b) Even part:

(c) Odd part:

(d) Check: e(t) + o(t) = f(t)

terministic vs. Random

terministic signal is a signal in which each value of the signal is fixed and can be determined by a mathematical expression, rule, or table. s the future values of the signal can be calculated from past values with complete confidence. On the other hand, a random signalhas a lot rtainty about its behavior. The future values of a random signal cannot be accurately predicted and can usually only be guessed based on theages of sets of signals (Figure 1.8).

Figure 1.8.

(a) Deterministic Signal

(b) Random Signal


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ample 1.2.

nsider the signal defined for all real tdescribed by

s signal is continuous time, analog, aperiodic, infinite length, causal, neither even nor odd, and, by definition, deterministic.

nal Classifications Summary

module describes just some of the many ways in which signals can be classified. They can be continuous time or discrete time, analog or digdic or aperiodic, finite or infinite, and deterministic or random. We can also divide them based on their causality and symmetry properties. Tr ways to classi fy signals, such as boundedness, handedness, and continuity, that are not discussed here but will be described in subsequenules.

. Signal Size and Norms*

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size" of a signal would involve some notion of i ts strength. We use the mathematical concept of the norm to quantify this concept for both conand discrete-time signals. As there are several types of norms that can be defined for signals, there are several different conceptions of sign

gnal Energy

nite Length, Continuous Time Signals

most commonly encountered notion of the energy of a signal defined on R is the L2 norm defined by the square root of the integral of the squ

ignal, for which the notation

| | f| | 2 = ( | f( t) |

2dt) 1 / 2 .

ever, this idea can be generalized through definition of the Lpnorm, which is given by

| | f| | p= ( | f( t) |

pdt) 1 / p

l 1 p < . Because of the behavior of this expression as p approaches , we furthermore define

h is the least upper bound of |f(t)|. A signal fis said to belong to the vector space Lp(R) if ||f||p< .

ample 1.3.

r example, consider the function defined by

e L1 norm is

e L2 norm is

e L norm is


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ite Length, Continuous Time Signals

most commonly encountered notion of the energy of a signal defined on R[a,b] is the L2 norm defined by the square root of the integral of the

e signal, for which the notation

| | f| | 2 = (b

a| f( t) |2dt) 1 / 2 .

ever, this idea can be generalized through definition of the Lpnorm, which is given by

| | f| | p= (b

a| f( t) |pdt) 1 / p


h is the least upper bound of |f(t)|. A signal fis said to belong to the vector space Lp(R[a,b]) if ||f||p< . The periodic extension of such a sign

infinite energy but finite power.

ample 1.4.

r example, consider the function defined on R[ 5,3] by

e L1 norm is

| | f| | 1 = 3 5 | f( t) |dt=

3 5 | t| dt= 17 .

e L2 norm is

| | f| | 2 = (3 5 | f( t) |

2dt) 1 / 2 = (3 5 | t|2dt) 1 / 2 7 . 12

e L norm is

nite Length, Discrete Time Signals

most commonly encountered notion of the energy of a signal defined on Z is the l2 norm defined by the square root of the sumation of the squ

ignal, for which the notation

ever, this idea can be generalized through definition of the lpnorm, which is given by


h is the least upper bound of |f(n)|. A signal fis said to belong to the vector space lp(Z) if ||f||p< .

ample 1.5.

r example, consider the function defined by

e l1 norm is


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e l2 norm is

e l norm is

ite Length, Discrete Time Signals

most commonly encountered notion of the energy of a signal defined on Z[a,b] is the l2 norm defined by the square root of the sumation of thee signal, for which the notation

ever, this idea can be generalized through definition of the lpnorm, which is given by


h is the least upper bound of |f(n)|. In this case, this least upper bound is simply the maximum value of |f(n)|. A signal fis said to belong to the ve lp(Z[a,b]) if ||f||p< . The periodic extension of such a signal would have infinite energy but finite power.

ample 1.6.

r example, consider the function defined on Z[ 5,3] by

e l1 norm is

e l2 norm is

e l norm is

nal Norms Summary

notion of signal size or energy is formally addressed through the mathematical concept of norms. There are many types of norms that can be gnals, some of the most important of which have been discussed here. For each type norm and each type of signal domain (continuous or difinite or infinite) there are vector spaces defined for signals of finite norm. Finally, while nonzero periodic signals have infinite energy, they haver if their single period units have finite energy.

. Signal Operations*

roduction

module will look at two signal operations affecting the time parameter of the signal, time shifting and time scaling. These operations are verymon components to real-world systems and, as such, should be understood thoroughly when learning about signals and systems.


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nipulating the Time Parameter

me Shifting

shifting is, as the name suggests, the shifting of a signal in time. This is done by adding or subtracting a quantity of the shift to the time variaunction. Subtracting a fixed positive quantity from the time variable will shift the signal to the right (delay) by the subtracted quantity, while addposi tive amount to the time variable will shift the signal to the left (advance) by the added quantity.

Figure 1.9.

f(t T) moves (delays) fto the right byT.

me Scaling

scaling compresses or dilates a signal by multiplying the time variable by some quantity. If that quantity is greater than one, the signal becomower and the operation is called compression, while if the quantity is less than one, the signal becomes wider and is called dilation.

Figure 1.10.

f(at) compresses fbya .

ample 1.7.

ven f(t) we woul like to plot f(at b). The figure below describes a method to accomplish this.

Figure 1.11.

(a) Begin with f(t)

(b) Then replace twith atto get f(at)

(c) Finally, replace twith to get


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me Reversal

tural question to consider when learning about time scaling is: What happens when the time variable is multiplied by a negative number? Thes is time reversal. This operation is the reversal of the time axis, or flipping the signal over the y-axis.

Figure 1.12.

Reverse the time axis

me Scaling and Shifting Demonstration

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nal Operations Summary

e common operations on signals affect the time parameter of the signal. One of these is time shifting in which a quantity is added to the timemeter in order to advance or delay the signal. Another is the time scaling in which the time parameter is multiplied by a quantity in order to dilpress the signal in time. In the event that the quantity involved in the latter operation is negative, time reversal occurs.

. Common Continuous Time Signals*

roduction

re looking at this module, hopefully you have an idea of what a signal is and what basic classifications and properties a signal can have. In real is a function defined with respect to an independent variable. This variable is often time but could represent any number of things. Mathemanuous time analog signals have continuous independent and dependent variables. This module will describe some useful continuous time anals.

portant Continuous Time Signals

nusoids

of the most important elemental signal that you will deal with is the real-valued sinusoid. In its continuous-time form, we write the general exprcos(t+ ) where A is the amplitude, is the frequency, and is the phase. Thus, the period of the sinusoid is

Figure 1.13.

Sinusoid withA = 2 ,w= 2 , and = 0 .

mplex Exponentials

mportant as the general sinusoid, the complex exponential function will become a critical part of your study of signals and systems. Its genenuous form is written as Astwhere s = + j is a complex number in terms of, the attenuation constant, and the angular frequency.

it Impulses


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mportant as the general sinusoid, the complex exponential function will become a critical part of your study of signals and systems. Its geneete form is written as Asnwhere s = + j , is a complex number in terms of, the attenuation constant, and the angular frequency.

discrete time complex exponentials have the following property. jn= j( + 2)nGiven this property, if we have a complex exponential withuency + 2, then this signal "aliases" to a complex exponential with frequency , implying that the equation representations of discrete conentials are not unique.

it Impulses

second-most important discrete-time signal is the unit sample, which is defined as

Figure 1.16. Unit Sample

The unit sample.

e detail is provided in the section on the discrete time impulse function. For now, it suffices to say that this signal is crucially important in the stete signals, as it allows the sifting property to be used in signal representation and signal decomposition.

it Step

her very basic signal is the unit-step function defined as

Figure 1.17.

Discrete-Time Unit-Step Function

step function is a useful tool for testing and for defining other signals. For example, when different shifted versions of the step function are muher signals, one can select a certain portion of the signal and zero out the rest.

mmon Discrete Time Signals Summary

e of the most important and most frequently encountered signals have been discussed in this module. There are, of course, many other signaficant consequence not discussed here. As you will see later, many of the other more complicated signals will be studied in terms of those lis Especially take note of the complex exponentials and unit impulse functions, which will be the key focus of several topics included in this cou

. Continuous Time Impulse Function*

roduction

gineering, we often deal with the idea of an action occurring at a point. Whether it be a force at a point in space or some other signal at a poiit becomes worth while to develop some way of quantitatively defining this. This leads us to the idea of a unit impulse, probably the second mrtant function, next to the complex exponential, in this systems and signals course.

ac Delta Function

Dirac delta function, often referred to as the unit impulse or delta function, is the function that defines the idea of a unit impulse in continuoumally, this function is one that is infinitesimally narrow, infinitely tall, yet integrates to one. Perhaps the simplest way to visualize this is as a rece from to with a height of . As we take the limit of this setup asapproaches 0, we see that the width tends to zero and the heigh

inity as the total area remains constant at one. The impulse function is often written as (t) .


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Figure 1.18.

This is one way to visualize the Dirac Delta Function.

Figure 1.19.

e it is quite difficult to draw something that is infinitely tall, we represent the Dirac wi th an arrow centered at the point it is applied. If we wish to scale it, we m ay write the value it i

next to the point of the arrow. This i s a unit impulse (no scaling).

w is a brief list a few important properties of the unit impulse without going into detail of their proofs.

Impulse Properties

t) = ( t)

, where u(t) is the unit step.

t) ( t) = f( 0 ) ( t)

ast of these is especially important as it gives rise to the sifting property of the dirac delta function, which selects the value of a function at a sand is especially important in studying the relationship of an operation called convolution to time domain analysis of linear time invariant systsifting property is shown and derived below. f( t) ( t) dt=

f( 0 ) ( t) dt= f( 0 ) ( t) dt= f( 0 )

it Imulse Limiting Demonstration


ntinuous Time Unit Impulse Summary

continuous time unit impulse function, also known as the Dirac delta function, is of great importance to the study of signals and systems. Infor

unction with infinite height ant infinitesimal width that integrates to one, which can be viewed as the limiting behavior of a unit area rectangle aows while preserving area. It has several important properties that will appear again when studying systems.

. Discrete Time Impulse Function*

roduction

gineering, we often deal with the idea of an action occurring at a point. Whether it be a force at a point in space or some other signal at a poiit becomes worth while to develop some way of quantitatively defining this. This leads us to the idea of a unit impulse, probably the second mrtant function, next to the complex exponential, in this systems and signals course.

it Sample Function


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unit sample function, often referred to as the unit impulse or delta function, is the function that defines the idea of a unit impulse in discrete e are not nearly as many intricacies involved in its definition as there are in the definition of the Dirac delta function, the continuous time impuon. The unit sample function simply takes a value of one at n=0 and a value of zero elsewhere. The impulse function is often written as (n) .

Figure 1.20. Unit Sample

The unit sample.

w we will briefly list a few important properties of the unit impulse without going into detail of their proofs.

Impulse Properties

n) =( n)

( n ) =u ( n ) u ( n 1 )

n ) ( n ) = f( 0 )( n )

ast of these is especially important as it gives rise to the sifting property of the unit sample function, which selects the value of a function at a and is especially important in studying the relationship of an operation called convolution to time domain analysis of linear time invariant syst

sifting property is shown and derived below.

screte Time Unit Impulse Summary

discrete time unit impulse function, also known as the unit sample function, is of great importance to the study of signals and systems. The funs a value of one at time n=0 and a value of zero elsewhere. It has several important properties that will appear again when studying systems.

. Continuous Time Complex Exponential

*

roduction

plex exponentials are some of the most important functions in our study of signals and systems. Their importance stems from their status asnfunctions of linear time invariant systems. Before proceeding, you should be fami liar with complex numbers.

e Continuous Time Complex Exponential

mplex Exponentials

complex exponential function will become a critical part of your study of signals and systems. Its general continuous form is written as Astw

+ is a complex number in terms of, the attenuation constant, and the angular frequency.

ler's Formula

mathematician Euler proved an important identity relating complex exponentials to trigonometric functions. Specifically, he discovered theymously named identity, Euler's formula, which states that

ejx= cos ( x) + jsin ( x)

h can be proven as follows.

der to prove Euler's formula, we start by evaluating the Taylor series forezabout z= 0, which converges for all complex z, at z= jx. The resu


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use the second expression contains the Taylor series for cos(x) and sin(x) about t= 0, which converge for all real x. Thus, the desired result en.

osing x= tthis gives the result

ejt= cos ( t) + jsin ( t)

h breaks a continuous time complex exponential into its real part and imaginary part. Using this formula, we can also derive the followingonships.

ntinuous Time Phasors

s been shown how the complex exponential with purely imaginary frequency can be broken up into its real part and its imaginary part. Now coral complex frequency s = + jwhere is the attenuation factor and is the frequency. Also consider a phase difference . It follows that

e( + j ) t+ j

= et

(cos ( t+ ) + jsin ( t+ )) .

, the real and imaginary parts ofestappear below.

g the real or imaginary parts of complex exponential to represent sinusoids with a phase delay multiplied by real exponential is often useful ad attenuated phasor notation.

can see that both the real part and the imaginary part have a sinusoid times a real exponential. We also know that sinusoids osci llate betweenegative one. From this it becomes apparent that the real and imaginary parts of the complex exponential will each oscillate within an envelo

ed by the real exponential part.

Figure 1.21.

(a) Ifis negative, we have the case of a decaying exponential window.

(b) Ifis positive, we have the case of a growing exponential window.

(c) Ifis zero, we have the case of a constant window.

The shapes possible for the real part of a complex exponential. Notice that the oscillations are the result of a cosine, as there is a local m aximum att= 0 .

er's Formula Demonstration


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ntinuous Time Complex Exponential Summary

inuous time complex exponentials are signals of great importance to the study of signals and systems. They can be related to sinusoids thror's formula, which identifies the real and imaginary parts of purely imaginary complex exponentials. Eulers formula reveals that, in general, themaginary parts of complex exponentials are sinusoids multiplied by real exponentials. Thus, attenuated phasor notation is often useful in stude signals.

. Discrete Time Complex Exponential*

roduction

plex exponentials are some of the most important functions in our study of signals and systems. Their importance stems from their status asnfunctions of linear time invariant systems. Before proceeding, you should be fami liar with complex numbers.

e Discrete Time Complex Exponential

mplex Exponentials

complex exponential function will become a critical part of your study of signals and systems. Its general discrete form is written as Asnwhe+ , is a complex number in terms of, the attenuation constant, and the angular frequency.

discrete time complex exponentials have the following property, which will become evident through discussion of Euler's formula.n= ( + 2)nGiven this property, if we have a complex exponential with frequency + 2, then this signal "aliases" to a complex exponentiauency , implying that the equation representations of discrete complex exponentials are not unique.

ler's Formula

mathematician Euler proved an important identity relating complex exponentials to trigonometric functions. Specifically, he discovered theymously named identity, Euler's formula, which states that

ejx= cos ( x) + jsin ( x)

h can be proven as follows.

der to prove Euler's formula, we start by evaluating the Taylor series forezabout z= 0, which converges for all complex z, at z= jx. The resu

use the second expression contains the Taylor series for cos(x) and sin(x) about t= 0, which converge for all real x. Thus, the desired result en.

osing x= n this gives the result

ejn= cos ( n ) + jsin ( n )

h breaks a discrete time complex exponential into its real part and imaginary part. Using this formula, we can also derive the following relatio

screte Time Phasors

s been shown how the complex exponential with purely imaginary frequency can be broken up into its real part and its imaginary part. Now coral complex frequency s = + jwhere is the attenuation factor and is the frequency. Also consider a phase difference . It follows that

e( + j ) n + j= en(cos ( n + ) + jsin ( n + )) .


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, the real and imaginary parts ofesnappear below.

g the real or imaginary parts of complex exponential to represent sinusoids with a phase delay multiplied by real exponential is often useful ad attenuated phasor notation.

can see that both the real part and the imaginary part have a sinusoid times a real exponential. We also know that sinusoids osci llate betweenegative one. From this it becomes apparent that the real and imaginary parts of the complex exponential will each oscillate within an enveloed by the real exponential part.

Figure 1.22.

(a) Ifis negative, we have the case of a decaying exponential window.

(b) Ifis positive, we have the case of a growing exponential window.

(c) Ifis zero, we have the case of a constant window.

shapes possible for the real part of a complex exponential. Notice that the oscillations are the result of a cosine, as there is a local m aximum att= 0 . Of course, these drawings

sampled i n a discrete time setting.

er's Formula Demonstration


screte Time Complex Exponential Summary

inuous time complex exponentials are signals of great importance to the study of signals and systems. They can be related to sinusoids thror's formula, which identifies the real and imaginary parts of purely imaginary complex exponentials. Eulers formula reveals that, in general, themaginary parts of complex exponentials are sinusoids multiplied by real exponentials. Thus, attenuated phasor notation is often useful in stude signals.

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Signals and Systems - Introduction to Signals