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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"!
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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
% "
&%% '> @ % '&% =
% '> @ &%
' } ! & " & ! } ^ `
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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 '"=
( * '> @ ( *'! '') =
'! ')
) " &+ ! " ^ `=) '> @ &, ! '> @^ `=
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Introduction to Mathematical Modeling of Signals and Systems
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
-
Introduction to Mathematical Modeling of Signals and Systems
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
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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
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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
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Review of Sine and Cosine Functions
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
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Review of Sine and Cosine Functions
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 =
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Review of Sine and Cosine Functions
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-------------------------------------=
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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= = =
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Review of Complex Numbers
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=
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Review of Complex Numbers
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)=
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Review of Complex Numbers
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=
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Review of Complex Numbers
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=
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Review of Complex Numbers
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+
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Review of Complex Numbers
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=
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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 =
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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=
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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
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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=
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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
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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---
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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=
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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
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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
-
Sampling and Plotting Sinusoids
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=
-
Sampling and Plotting Sinusoids
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