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BioE 1410 - Homework 1 - Answers
1. (10 pts.) Express each of the following complex numbers in Cartesian form x + jy ,
drawing a picture of each on the complex plane.
A.
1 j 2 1 j 0 1 0 1
e = e = e =
2
2
2
2
1
x=
y=0
2
(2 j )
= e 2e j = e 2 ( 1) = e 2
B. e
x = e2
C.
2 j
e
5
1
x=
5
y=0
( )= 15 (1 + j )
4
y=
1
5
0j
0
D. 3e = 3e = 3
x=3
E.
y=0
( )
j e j 7 = je j e j 6 = je j = j
x=0
y = 1
5
2 of
j 3
j 2
j
F. e = e e = e = ( 1) = 1
x =1
y=0
G.
2e
H.
1+ j 2
j = 2e e 2 = 2e( j )
x=0
y = 2e
2 (e j
4
x =
)= 1 j 2
= (1 j )
2 y = j
2. (10 pts.) Specify r and in the polar form re for following complex numbers,
drawing a picture of each on the complex plane.
( - < , r 0 ) <--- note range of variables
A.
(1 j ) (1 + j ) = (1 j ) (1 j ) = 1 2 j + j
(1 + j ) (1 j )
2
=
2
B. 3(1 + j )
=
4
r =1
1
r = 3(1 + 1)2 = 3 2
j
C. 17 = 17e
=
r = 17
2
=j
5
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D.
4 = 2j
r=2
=
2
E. 2 ( j 2 )
3
= 4
r=2
F. 2 = 2 (e )
=
G.
H.
r= 2
1 1
= j
4 2
1
=
r=
2
2
2j
( 3 j )= ( 32 j j )(( 33 ++ jj ))= (3 2+ 1)(j
2
= 3
r =1
) 12 (j
3 1 =
)
3 1
5
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3. (15 pts.) For each signal, draw the signal with correct scales for both ordinate and abscissa.
A.
f (t ) = cos(4t )
2
1
0.5
0
-0.5
-1
-1
t
B. f (t ) = e
-0.5
0
0.5
1
0
-0.5
-1
-1.5
-2
-2.5
-3
-1
1
(1+ j )
4. Given z =
2
-0.5
0
0.5
1
First in Cartesian coordinates, and then using phasors, solve for zk , k = 0,1,2, 3,....,8
Draw a diagram of the nine vectors on the complex plane.
1
(1 + j ) = e 4 . Then raising it to each successive
2
power z k rotates the unit phasor by another = 45 .
4
First, convert it to a phasor. z =
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
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5. Using the phasor representations, prove the following trigonometric identity:
sin (2t ) = 2 sin (t )cos(t ) =
e jt e jt e jt + e jt 2
=
2
2 j e 2 jt e 2 jt
= sin (2t )
2j
6. Fill in the blanks.
The sine and cosine functions are both examples of a general class of waveform known as
___sinusoids_, and differ only in their _phase__. Any two sinusoids with the same
frequency, when added together, create another sinusoid. Systems obeying Hooke’s Law
produce sinusoidal motion because the _(negative) second derivative_ of a sinusoid
equals a constant times that sinusoid. Taking the derivative of a sinusoid shifts the phase
by _-90_ degrees, which can also be expressed as __-PI/2_ radians. The only function
whose first derivative equals itself is __e^x___ . A phasor (complex exponential) has
special properities, including that multiplying one phasor by another shifts the phase of
one by the __phase__ of the other.
7. True / False (enter “T” or “F” in each blank).
_T__ Every complex number can be represented by a phasor, and visa versa.
_T__ Any sinusoid can be expressed as the sum of two complex exponentials.
_F__ The sum of a complex number and its conjugate can have a non-zero imaginary
component.
_T__ The square root of a phasor always has half the angle.