signal & system

11

Introduction to Mathematical Modeling of Signals and SystemsMathematical Representation of Signals Signals represent or encode information

In communications applications the information is almostalways encoded

In the probing of medical and other physical systems,where signals occur naturally, the information is not pur-posefully encoded

In human speech we create a waveform as a function oftime when we force air across our vocal cords and throughour vocal tract

!"#$%&'()'*+"),-%'*.+&/+0"/)+-'1*0"(&+--1&+2&'#"/)+".'%,3/&,%/"$*/'",*+3+%/&$%,3"-$4*,3/),/".,&$+-"'.+&/$#+5"!

Introduction to Mathematical Modeling of Signals and Systems

1-2

Signals, such as the above speech signal, are continuousfunctions of time, and denoted as a continuous-time signal

The independent variable in this case is time, t, but could beanother variable of interest, e.g., position, depth, temperature,pressure

The mathematical notation for the speech signal recorded bythe microphone might be

In order to process this signal by computer means, we maysample this signal at regular interval , resulting in

(1.2) The signal is known as a discrete-time signal, and is

the sampling period Note that the independent variable of the sampled signal is

the integer sequence Discrete-time signals can only be evaluated at integer val-

ues

% "

&%% '> @ % '&% =

% '> @ &%

' } ! & " & ! } ^ `

13

The speech waveform is an example of a one-dimensionalsignal, but we may have more that one dimension

An image, say a photograph, is an example of a two-dimen-sional signal, being a function of two spatial variables, e.g.

If the image is put into motion, as in a movie or video, wenow have a three-dimensional image, where the third inde-pendent variable is time, Note: movies and videos are shot in frames, so actually

time is discretized, e.g., (often fps) To manipulate an image on a computer we need to sample the

image, and create a two-dimensional discrete-time signal(1.3)

where m and n takes on integer values, and and repre-sent the horizontal and vertical sampling periods respectively

Mathematical Representation of Systems In mathematical modeling terms a system is a function that

transforms or maps the input signal/sequence, to a new out-put signal/sequence

(1.4)

where the subscripts c and d denote continuous and discretesystem operators

( ! )

! ) "

" '&%o & &%e '"=

( * '> @ ( *'! '') =

'! ')

) " &+ ! " ^ `=) '> @ &, ! '> @^ `=


14

Because we are at present viewing the system as a pure math-ematical model, the notion of a system seems abstract anddistant

Consider the microphone as a system which converts soundpressure from the vocal tract into an electrical signal

Once the speech waveform is in an electrical waveform for-mat, we might want to form the square of the signal as a firststep in finding the energy of the signal, i.e.,

(1.5)

The squarer system also exists for discrete-time signals, andin fact is easier to implement, since all we need to do is mul-tiply each signal sample by itself

) " ! " > @!=

) " ! " > @!=!"-61,&+&"-7-/+#

15

(1.6) If we send through a second system known as a digital

filter, we can form an estimate of the signal energy This is a future topic for this course

Thinking About Systems Engineers like to use block diagrams to visualize systems Low level systems are often interconnected to form larger

systems or subsystems Consider the squaring system

The ideal sampling operation, described earlier as a means toconvert a continuous-time signal to a discrete-times signal isrepresented in block diagram form as an ideal C-to-D con-verter

) '> @ ! '> @ ! ! '> @ ! '> @= =) '> @

& (((^ `! " ) "

((( !! " ) "

T"$-","4+*+&$%"-7-/+#

80+,39:/':;

9'*.+&/+&

&%

! " ! '> @ ! '&% =

!"-7-/+#"(,&,#+/+&"/),/-(+%$2$+-"/)+"-,#(3+"-(,%$*4


16

A more complex system, depicted as a collection of subsys-tem blocks, is a system that records and then plays back anaudio source using a compact disk (CD) storage medium

The optical disk reader shown above is actually a high-levelblock, as it is composed of many lower-level subsystems,e.g., Laser, on a sliding carriage, to illuminate the CD An optical detector on the same sliding carriage A servo control system positions the carriage to follow the

track over the disk A servo speed control to maintain a constant linear veloc-

ity as 1/0 data is read from different portions of the disk more ...

The Next Step Basic signals, composed of linear combinations of trigono-

metric functions of time will be studied next We also consider complex number representations as a means

to simplify the combining of more than one sinusoidal signal

21

Sinusoids A general class of signals used for modeling the inter-

action of signals in systems, are based on the trigono-metric functions sine and cosine

The general mathematical form of a single sinusoidal signalis

(2.1)where denotes the amplitude, is the frequency in radi-ans/s (radian frequency), and is the phase in radians The arguments of and are in radians

We will spend considerable time working with sinusoidal sig-nals, and hopefully the various modeling applications pre-sented in this course will make their usefulness clear

Example:

The pattern repeats every This time interval is known as the period of

x t A Z0t I+ cos=

A Z0I

cos sin

x t 10 2S 440 t 0.4S> @cos=

1 440e 0.00227 2.27ms= =x t

Review of Sine and Cosine Functions

22

The text discusses how a tuning fork, used in tuning musicalinstruments, produces a sound wave that closely resembles asingle sinusoid signal In particular the pitch A above middle C has an oscillation

frequency of 440 hertz

Review of Sine and Cosine Functions Trigonometric functions were first encountered in your K12

math courses The typical scenario to explain sine and cosine functions is

depicted below

The right-triangle formed in the first quadrant has sides oflength x and y, and hypotenuse of length r

The angle has cosine defined as x/r and sine defined as y/r

The above graphic also shows how a point of distance rand angle in the first quadrant of the xy plane is related

T

T


23

to the x and y coordinates of the point via sin( ) and cos( ),e.g.,

(2.2) Moving beyond the definitions and geometry interpretations,

we now consider the signal/waveform properties

The function plots are identical in shape, with the sine plotshifted to the right relative to the cosine plot by

This is expected since a well known trig identity states that(2.3)

We also observe that both waveforms repeat every radi-ans; read period =

Additionally the amplitude of each ranges from -1 and 1 A few key function properties and trigonometric identities

x y r T r Tsincos =

S 2e

Tsin T S 2e cos=2S

2S


24

are given in the following tables

For more properties consult a math handbook

Table 2.1: Some sine and cosine properties

Property EquationEquivalence

or

Periodicity , when k is an integer; holds for sine also

Evenness of cosine

Oddness of sine

Table 2.2: Some trigonometric identities

Number Equation1

2

34

56

7

Tsin T S 2e cos=Tcos T S 2e+ sin=T 2Sk cos Tcos=

T cos Tcos=T sin Tsin=

sin2T cos2T+ 1=

2Tcos cos2T sin2T=2Tsin 2 T Tcossin=D Er sin D E D Esincosrcossin=D Er cos D E D Esinsincoscos=

cos2T 12--- 1 2Tcos+ =

sin2T 12--- 1 2Tcos =


25

The relationship between sine and cosine show up in calculustoo, in particular

(2.4)

This says that the slope at any point on the sine curve is thecosine, and the slope at any point on the cosine curve is thenegative of the sine

Example: Prove Identity #6 Using Identities #1 and #2 If we add the left side of 1 to the right side of 2 we get

(2.5)

Example: Find an expression for in terms of ,, and using #5

Let and , then write out #5 under both signchoices

(2.6)

or

(2.7)

d TsindT-------------- T and

d TcosdT---------------cos Tsin= =

2cos2T 1 2Tcos+=

or cos2T 12--- 1 2Tcos+ =

8Tcos 9Tcos7Tcos TcosD 8T= E T=

8T T+ cos 8T T 8T Tsinsincoscos=8T T cos 8T Tcoscos 8T Tsinsin+=

9T 7Tcos+cos 2 8T Tcoscos=+

8Tcos 9T 7Tcos+cos2 Tcos-------------------------------------=

Review of Complex Numbers

26

Review of Complex Numbers See Appendix A of the text for more information A complex number is an ordered pair of real numbers1

denoted The first number, x, is called the real part, while the second

number, y, is called the imaginary part For algebraic manipulation purposes we write

where ; electrical engi-neers typically use j since i is often used to denote current

Note: The rectangular form of a complex number is as defined

above,

The corresponding polar form is

1.Tom M. Apostle, Mathematical Analysis, second edition, Addison Wesley, p. 15, 1974.

z x y =

x y x iy+= x jy+= i j 1= =

1 1u 1= j ju 1=

z x y x jy+= =

z rejT r T z e j zarg= = =


27

We can plot a complex number as a vector

Example:: , , ,

x y

z 2 j5+= z 4 j3= z 5 j0+= z 3 j3=


ECE 2610 Signals and Systems 28

Example:: , , &

For complex numbers and wedefine/calculate

z 2 45q= z 3 150q= z 3 80 q=

z1 x1 jy1+= z2 x2 jy2+=

z1 z2+ x1 x2+ j y1 y2+ (sum)+=z1 z2 x1 x2 j y1 y2 (difference)+=

z1z2 x1x2 y1y2 j x1y2 y1x2+ (product)+=z1z2----

x1x2 y1y2+ j x1y2 y1x2 x22 y2

2+------------------------------------------------------------------------- (quotient)=


29

MATLAB is also consistent with all of the above, startingwith the fact that i and j are predefined to be

To convert from polar to rectangular we can use simple trigo-nometry to show that

(2.24)

Similarly we can show that rectangular to polar conversion is

(2.25)

z1 x12 y1

2+ (magnitude)=

z1 tan1 y1 x1e (angle)=

z1* x1 jy1 (complex conjugate)=

1

polar

rectangular

x r Tcos=y r Tsin=

r x2 y2+=

T tan 1 y xe note add S outside Q1 & Q4r=


210

Example: Rect to Polar and Polar to Rect Consider

In MATLAB we simply enter the numbers directly and thenneed to use the functions abs() and angle() to convert

>> z1 = 2 + j*5

z1 = 2.0000e+00 + 5.0000e+00i

>> [abs(z1) angle(z1)]

ans = 5.3852e+00 1.1903e+00 % mag & phase in rad Using say a TI-89 calculator is similar

Consider In MATLAB we simply enter the numbers directly as a

complex exponential

>> z2 = 2*exp(j*45*pi/180)

z2 = 1.4142e+00 + 1.4142e+00i

z1 2 j5+=

z2 2 45q=


211

Using the TI-89 we can directly enter the polar form usingthe angle notation or using a complex exponential

Example: Complex Arithmetic Consider and Find

>> z1 = 1+j*7;>> z2 = -4-j*9;>> z1+z2

ans = -3.0000e+00 - 2.0000e+00i Using the TI-89 we obtain

z1 1 j7+= z2 4 j9=

z1 z2+


212

Find >> z1*z2

ans = 5.9000e+01 - 3.7000e+01i Using the TI-89 we obtain

Find >> z1/z2

ans =-6.9072e-01 - 1.9588e-01i

Eulers Formula: A special mathematical result, of specialimportance to electrical engineers, is the fact that

(2.26)

z1z2

z1 z2e

TI-89Results

ejT T j Tsin+cos=

Sinusoidal Signals

213

Turning (2.26) around yields (inverse Euler formulas)

and (2.27)

It also follows that(2.28)

Sinusoidal Signals A general sinusoidal function of time is written as

(2.29)where in the second form

Since it follows that swings between , sothe amplitude of is A

The phase shift in radians is , so if we are given a sine sig-nal (instead of the cosine version), we see via the equivalenceproperty that

(2.30)which implies that

Engineers often prefer the second form of (2.8) where isthe oscillation frequency in cycles/s.

Tsin ejT e jT

2j----------------------= Tcos e

jT e jT+2

----------------------=

z x jy+ r Tcos jr Tsin+= =

x t A Z0t I+ cos A 2Sf0t I+ cos= =

Z0 2Sf0=Tcos 1d x t Ar

x t I

x t A Z0t Ic+ sin A Z0t Ic S 2e+ cos= =

I Ic S 2e=f0

Z02S------

rad/srad

----------- f0 sec

1 =

Sinusoidal Signals

214

Example: Clearly, , cycles/s, and rad

Since this signal is periodic, the time interval between max-ima, minima, and zero crossings, for example, are identical

Relation of Frequency to Period A signal is periodic if we can write

(2.31)where the smallest satisfying (2.10) is the period

For a single sinusoid we can relate to by considering

(2.32)

From the periodicity property of cosine, equality is main-tained if , so we need to have

x t 20 2S 40 t 0.4S> @cos=A 20= f0 40= I 0.4S=

140------ 0.025s=

25ms=

MaximaInterval(period)

x t T0+ x t =

T0T0 f0

x t T0+ x t =A Z0 t T0+ I+ cos A Z0t I+ cos=

Z0t I Z0T0+ + cos Z0t I+ cos=

T 2Skr cos T cos=

Sinusoidal Signals

215

(2.33)

So we see that and are reciprocals, with the units of being time and the units of inverse time or cycles per sec-ond, as stated earlier In honor of Heinrich Hertz, who first demonstrated the

existence of radio waves, cycles per second is replacedwith Hertz (Hz)

Z0T0 2S T02SZ0------= =

or 2Sf0 T0 T01f0----=

T0 f0 T0f0

Sinusoidal Signals

216

Example: with , 100, and 0 Hz

The inverse relationship between time and frequency will beexplored through out this course

Period doubles asfrequency halves

A constant signalas the oscillationfrequency is zero

5 2Sf0t cos f0 200=

Sinusoidal Signals

217

Phase Shift and Time Shift We know that the phase shift parameter in the sinusoid moves

the waveform left or right on the time axis To formally understand why this is, we will first form an

understanding of time-shifting in general Consider a triangularly shaped signal having piece wise con-

tinuous definition

(2.34)

Now we wish to consider the signal As a starting point we note that is active over just the

interval , so with we have(2.35)

which means that is active over The piece wise definition of can be obtained by direct

substitution of everywhere appears in (2.34)

s t 2t, 0 t 1 2ed d13--- 4 2t , 1 2e t 2d d

0, otherwise

=

0 1 2 3t

-1

1s t

2t13--- 4 2t

12---

x1 t s t 2 =s t

0 td 2d t t 2o0 t 2 2d d 2 t 4d d

x1 t 2 t 4d dx1 t

t 2 t

Sinusoidal Signals

218

(2.36)

In summary we see that the original signal is moved tothe right by 2 s

Example: Plot With we expect that the signal will shift to the left

by one second

x1 t 2 t 2 , 0 t 2 1 2ed d13--- 4 2 t 2 , 1 2e t 2 2d d

0, otherwise

=

2t 4, 2 t 5 2ed d13--- 8 2t , 5 2e t 4d d

0, otherwise

=

1 2 3 4t

0

1x1 t s t 2 =

52---

2 t 2 13--- 8 2t

s t

s t 1+ t t 1+o

0 1 2 3t

-1

1s t 1+

12---

Sinusoidal Signals

219

The new equations are obtained as before(2.37)

so

(2.38)

Modeling time shifted signals shows up frequently In general terms we say that

(2.39)is delayed in time relative to if , and advanced intime relative to if

A cosine signal has positive peak located at If this signal is delayed by the peak shifts to the right and

the corresponding phase shift is negative Consider

0 t 1+ 2d d 1 t 1d d

s t 1+ 2 t 1+ , 0 t 1+ 1 2ed d13--- 4 2 t 1+ , 1 2e t 1+ 2d d

0, otherwise

=

2t 2,+ 1 t 1 2ed d13--- 2 2t , 1 2e t 1d d

0, otherwise

=

x1 t s t t1 =

s t t1 0!s t t1 0

t 0=

t1

x0 t A Z0t cos=

Sinusoidal Signals

220

(2.40)

which implies that in terms of phase shift we have

For a given phase shift we can turn the above analysis aroundand solve for the time delay via

(2.41)

Since , we can also write the phase shift in termsof the period

(2.42)

An important point to note here is that both cosine and sineare mod functions, meaning that phase is only uniqueon a interval, say or

Example: Suppose ms and ms Direct substitution into (2.21) results in

(2.43)

We need to reduce this value modulo to the interval by adding (or subtracting as needed) multiples of

The result is the reduced phase value

x0 t t1 A Z0 t t1 > @cos=A Z0t Z0t1> @cos=

I Z0t1=

t1IZ0------

I2Sf0-----------= =

T0 1 f0e=

I 2Sf0t1 2St1T0-----

= =

2S2S ( S S] (0 2S]

t1 10= T0 3=

I 2S 103------

203------S 6.6667S= = =

2S( S S] 2S

Sampling and Plotting Sinusoids

221

(2.44)

Does this result make sense? A time delay of 10 ms with a period of 3 ms means that we

have delayed the sinusoid three full periods plus 1 ms A 1 ms delay is 1/3 of a period, with half of a period corre-

sponding to rad, so a delay of 1/3 period is a phase shift of; agrees with the above analysis

The value of phase shift that lies on the interval isknown as the principle value

Sampling and Plotting Sinusoids When plotting sinusoidal signals using computer tools, we

are also faced with the fact that only a discrete-time version

I 203------S 6S+20 18+

3------------------S 23---S 0.6667S= = = =

S2 3e S 0.6667S=

t (ms)

Actual Delay of 10 msModulo the perioddelay of 1 ms

Blue = no delayRed = 10 ms Delay

S I Sd


222

of

may be generated and plotted This fact holds true whether we are using MATLAB, C, Math-

ematica, Excel, or any other computational tool When we need to realize that sample spacing needs

to be small enough relative to the frequency such thatwhen plotted by connecting the dots (linear interpolation),the waveform picture is not too distorted In Chapter 4 we will discuss sampling theory, which will

tell us the maximum sample spacing (minimum samplingrate which is ), such that the sequence

can be used to perfectly reconstruct from

For now we are more concerned with having a good plotappearance relative to the expected sinusoidal shape

A reasonable plot can be created with about 10 samples perperiod, that is with

We will now consider several MATLAB example plots >> t = 0:1/(5*3):1; x = 15*cos(2*pi*3*t-.5*pi);>> subplot(311)>> plot(t,x,'.-'); grid>> xlabel('Time in seconds')>> ylabel('Amplitude')>> t = 0:1/(10*3):1; x = 15*cos(2*pi*3*t-.5*pi);>> subplot(312)

x t A 2Sf0t I+ cos=

t nTsof0

1 Tsex n> @ x nTs = x t

x n> @

Ts 1 10f0 e| T0 10e=


223

>> plot(t,x,'.-'); grid>> xlabel('Time in seconds')>> ylabel('Amplitude')>> t = 0:1/(50*3):1; x = 15*cos(2*pi*3*t-.5*pi);>> subplot(313)>> plot(t,x,'.-'); grid>> xlabel('Time in seconds')>> ylabel('Amplitude')>> print -depsc -tiff sampled_cosine.eps

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 120

10

0

10

20

Time in seconds

Ampl

itude

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 120

10

0

10

20

Time in seconds

Ampl

itude

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 120

10

0

10

20

Time in seconds

Ampl

itude

f0 = 3 Hz, A = 15, I = -S/2Ts

T05

------=

Ts1T010

---------=

TsT050------=

5 Samplesper period

10 Samplesper period

50 Samplesper period

!"#$%&'()'$"*&*+,-%.(-*/(01-."2.

3:3;

Complex Exponentials and PhasorsModeling signals as pure sinusoids is not that common. We typi-cally have more that one sinusoid present. Manipulating multiplesinusoids is actually easier when we form a complex exponentialrepresentation.

Complex Exponential Signals Motivated by Eulers formula above, and the earlier defini-

tion of a cosine signal, we define the complex exponentialsignal as

(2.44)where and

Note that using Eulers formula

(2.45)

We see that the complex sinusoid has amplitude A, phaseshift , and frequency rad/s Note in particular that

(2.46)

! " #$% Z!" I+ =! " #= ! " ! " ^ `"#$ Z!" I+= =

! " #$% Z!" I+ =# Z!" I+ %&' %# Z!" I+ '()+=

I Z!

*+ ! " ^ ` # Z!" I+ %&'=,- ! " ^ ` # Z!" I+ '()=

!"#$%&'()'$"*&*+,-%.(-*/(01-."2.

3:3

2 1 0 1 21

0

1Time = 0.0000 s

2 1 0 1 21

0

1Time = 0.1250 s

2 0 21

0

1Time = 0.2500 s

2 0 21

0

1Time = 0.3750 s

2 0 21

0

1Time = 0.5000 s

2 0 21

0

1Time = 0.6250 s

2 0 21

0

1Time = 0.7500 s

2 0 21

0

1Time = 0.8750 s

$ % S 1e $%!

$%S 1e

01-."2(?//,+,"*

3:3@

The inverse Euler formulas can be used to see that a cosinesignal is composed of positive and negative frequency expo-nentials

(2.51)

Phasor AdditionWe often have to deal with multiple sinusoids. When the sinu-soids are at the same frequency, we can derive a formula of theform

(2.52)

At present we have only the trig identities to aid us, and thisapproach becomes very messy for large N.

Phasor Addition Rule We know that when complex numbers are added we must add

real and imaginary parts separately Consider the sum

# Z!" I+ %&' #$% Z!" I+ $

% Z!" I+ +0

-----------------------------------------------------

=

/0---($

%Z!" /0---(4$

%Z!"+=

/0---! " /

0---!4 " +=

*+ ! " ^ `=

#, Z!" I,+ %&', /=

-

# Z!" I+ %&'=

01-."2(?//,+,"*

3:A6

(2.53)

The above is valid since the real and imaginary parts addindependently, that is

(2.54)

and the same holds for the imaginary part Secondly, a real sinusoid can always be written in terms of a

complex sinusoid via

(2.55)Proof:

#,$jI,

, /=

-

(,, /=

-

({ #$%I= =

*+ (,, /=

-

*+ (,^ `, /=

-

=

# Z!" I+ %&' *+ #$% Z!" I+ ^ `=

#, Z!" I,+ %&', /=

-

*+ #,$% Z!" I,+ ^ `, /=

-

=

*+ #,$%I,

, /=

-

$%Z!"

=

*+ #$%I $%Z!"^ `=

*+ #$% Z!" I+ ^ `=

# Z!" I+ %&'=

+&55&6'78,&$79:;

01-."2(?//,+,"*

3:A5

Example: Phasor Addition Rule in Action Consider the sum

(2.56)

The frequency of the sinusoids is 15 Hz Using phasor notation we can write that

(2.57)

so in the phasor addition rule

(2.58) We perform the complex addition and conversion back to

polar form using the TI-89

so

(2.59)

& " &/ " &0 " +=156= .!S" .6S /7!e+ %&'

8 950 .!S" 7!S /7!e+ %&'+

&/ " *+ 156$%.6S /7!e $%.!S"^ `=

&0 " *+ 950$%7!S /7!e $%.!S"^ `=

(/ 156$%.6S /7!e

= (0 950$%7!S /7!e

=

(

I8()8#":(")'

( (/ (0+ 15;.

01-."2(?//,+,"*

3:A3

Finally,(2.60)

We can check this by directly plotting the waveform in MAT-LABXX#*#D#:K?"AY:N?YBK:@9CXX#Q?#D#F@YN&.%AZ:N)(N*MZYN)("?H:BCXX#Q9#D#[@9N&.%AZ:N)(N*MH:N)("?H:BCXX#Q#D#Q?MQ9C#!),.*#3%(0/#4.,=#10=#,(0-#%*\,-%

The measured amplitude, 10.822, is close to the expectedvalue

& " /!57< .!S" &/ "

&0 "

& "

01-."2(?//,+,"*

3:AA

The location of the peak can be converted to phase via

(2.61)

Summary of Phasor Addition When we need to form the sum of sinusoids at the same fre-

quency, we obtain the final amplitude A and phase via

(2.62)where and

(2.63)

Example:

Find From the given we observe that

I 0 S //5

019.,B.("C(+1&(DE*,*8(F"2G

3:A;

To perform the complex addition we will work step-by-step To add complex numbers we convert to rectangular form

Now,

For use in the phasor sum formula we likely need the answerin polar form

Physics of the Tuning ForkThe tuning fork signal generation example discussed earlier wasimportant because it is an example of a physical system thatwhen struck, produces nearly a pure sinusoidal signal.

By pure we mean a signal composed of a single frequencysinusoid, no other sinusoids at other frequencies, say har-monics (multiples of ) are present

(/ .S0---%&' %. S

0---'()+ %.= =

(0 6S1---

%&' %6 S1---

'()+ .56.66 %.56.66= =

(. . %0+=

( %. .56.66 %.56.66 . %0+ + +=

019.,B.("C(+1&(DE*,*8(F"2G

3:A

signal & system

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