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Unformatted text preview: ECE 300
Signals and Systems
Homework 5 Due Date: Tuesday October 9, 2007 at the beginning of class Problems: 1. Determine the complex Fourier series coefﬁcients for the following periodic signals: a) x(t) = —1 + cos(2t) + 3 cos[4t + b) x(t) = 0052(21) (Hint: Use Euler’s identify) 0) x0)=200ﬂ30+4sm[6hmgj 2. Find the Fourier series representation for the signal indicated using hand analysis.
Clearly indicate the values of coo and the ck. Hints: (1) Draw the signal, and then use the sifting property to calculate the ck . (2) If you understand how to do this, there is very
little work involved. xm=2ﬂcw)
p=—oo
3. For the periodic square wave x(t) with period To = 0.5 and 1 OSt<025
x0)
4 025$t<05 show that the Fourier series coefﬁcients are given by :z—J—k odd ck: k7:
0 k even where x(t) = 2k ckejk4’" Fall 2007 . 4. Simplify each of the following into the form ck = a(k)e'jﬂ‘“sinc(/ik) ej7kﬂ _e—j2k7r
3) ck : .
kn}
e—j27rk _e—j57rk
b) ck z .
jk
ej5k _ej2k
C) Ck =
k 1E tl+£ 2”
ScrambledAnswers c =372'e I 2 sine 33—15 , c =3ej<2k 2)sinc % ,c =9e12k sine k—9—
" 2 " 2n " 2 5. For the periodic signal shown below, with period T = 4 a) Determine the fundamental frequency (00. b) Determine the average value. 0) Determine the average power in the DC component of the signal. d) Determine an expression for the expansion coefﬁcients, ck. You must write your
expression in terms of the sine function, and possibly a leading phase term. Fall 2007 . 6. Assume x(t) has the spectrum shown below (all angles are multiples of 45 degrees): Amplitude and Phase Spectrum l l 1
_ _ _ _ _ ~ e _ _ _ _ _ _ . . ~..L——_}‘‘_»_l_~—4—A~‘—Al‘~~
l g l I
l l I
m ‘ ' ‘ ‘ ‘ L ' ‘ ' ' ' “ ‘ ‘ ’ '4?”“‘§ ‘ ‘ ‘ ‘ “it ' ‘ ‘ ‘ ' ‘ ' ‘ ‘ “I"“"
3 i l g .
i I ’ ’ ’ ’ ’ ’ ‘ ’ ’ ' ’ ’ “ “J” ’ ' ’ f ’i’
E z z l l
< _ — _ — 2— e _ A A 1 7 _ ~ . . e . . — . ~f_rv———§ — — — A —_¢+_~—I__V_
: E 1
H‘l. . . _ . _ _ _ _ _ _ _ P _ ‘ _ AA: _ _ A ‘ _..;y . A . v . _ . V v _ ___..“_‘
I l E
, l _ 7, a a .4; a We)
1 0 1 2 3 4
Index
200 I I
«f. l l
l I ‘5” l I
150 e e 7 , 77L 7 e 7 e ,4 r . 7 , r . . my , a e , ,1 7 , t e e iii 7 e 7 7 7 7 7 7 7 , t ,,I r 7 r , ,7
i i I l l
A a j  I
g: 100  "1 ~~~~  :  “l AAA g1 I l I I I
o, 50 ~~~~~~~~~~ ~~g ———— "%"“‘r‘““‘: ~~~~ ~«3—————y———4»
CD 5 I l ; I
E 0» We J , a er
'3 l I I l I
50 .7 L I}: r, I ,I, , l
l l I g z 
I I l l ’p i I
.100 i L I, , i
4 3 2 1 0 1 2 3 4
index
. a) What is the average value ofx(t)? b) What is the average gower in x(t) ? 0) Write an expression for x(t) in terms of cosines (Assume (00 =1). Your expression
must be real. Special Note: We will be using the code you write in the next part for the next few
homeworks and labs, so be sure you do this and understand what is going on! 7. (Matlab/PreLab Problem) Read the Appendix and then do the following: a) Copy the ﬁle Trigonometric_Fourier_Series.m (you wrote this for homework 4) to
ﬁle Complex_Fourier_Series.m. b) Modify Complex_Fourier_Series.m so it computes the average value ea 0) Modify Complex_Fourier_Series.m so it also computes ck for k =1 to k = N Fall 2007 d) Modify Complex_Fourier_Series.m so it also computes the Fourier series estimate
using the formula N
x(t) z CD + Z 2 l ck  cos(ka)at + lick)
k:l You will probably need to use the Matlab functions abs and angle for this. e) Using the code you wrote in part d, ﬁnd the complex Fourier series representation
for the following functions (deﬁned over a single period) ﬁ0)=ah40 OSt<3
t OSZ<2
fxn= 3 23t<3
0 3St<4
0 —2$t<—1
1 —13t<2
ZSt<3 t : f3() 3
0 3St<4 These are the same functions you used for the trigonometric Fourier series. Use N = 10 and turn in your plots for each of these functions. Also, turn in your Matlab program for one of these. Note that the values of low and high will be different for each of these
functions! Fall 2007 Appendix In the majority of this course we will be using the complex (or exponential) form of the
Fourier series, since it is really easier to do various mathematical things with it once you get used to it. Exponential Fourier Series If x(t) is a periodic function with fundamental period T,
then we can represent x(t) as a Fourier series 00 00
.k ,
x(t): E cke’ “’“’= E )(ke’W
k——oo k=—ao where ma :2;— is the fundamental period, c0 is the average (or DC, i.e. zero frequency) value, and
l T
c0 = 51— 6[x(t)dt T
ck :% Ix(t)e‘jkw"'dt
0 fx(t) is a real function, then we have the relationships [ck =i c_k  (the magnitude is even)
and do, = —z(ck (the phase is odd). Using these relationships we can then write x(t) = c, + 2 2 c, lcos(ka)0t + ac, )
k~1 This is usually a much easier form to deal with, since it lends itself easily to thinking of a
phasor representation of x(t). This will be particularly useful when we starting ﬁltering periodic signals. Fall 2007 :4/ FE/ CK ﬁfﬂié/ﬂwéﬂu 7,
6/) W5) 5 4* “SK—2;“) fng/Vz‘r‘ CD 77:9) : QwS/gg) 7: 794/“, m a) W24) 7 ’/%wS‘/2é) + 3 MEI/yZQ—QB . . Q If
” .u J. 5% l “119 L )3 L
’ /"'(2€ +36 4’3 26, f >6 mﬁmxw gm 3;sz % 2 33$ 9: SEN Qwrﬁwww wkmmmm aw Selma 50 SHEETS 22142 100 SHEETS 23‘} $HEE§TS§ 22441 5E} SHEETS 22442 ‘IQO SHEETS
2M) 3HEETS 22441
«144 22 «657 . ‘ 6. Assume x(t) has the spectrum shown below (all angles are multiples of 45 degrees): Amplitude and Phase Spectrum a) What is the average value Ofx(t) ? b) What is the average Qwer in x(t) ? c) Write an expression for x(t) in terms of cosines (Assume mo =1). Your expression
must be real. a) 32:340/0" r *3 32‘ I») a: Mom Woman/z meg/3— 21m : 21+2'22 +2./2: 9+9f2:im (91(17): 00‘!’ Z {CK}
KZi \
: 34— 2.199030“: ~‘?o°)+ 2. \ w:(?'<*+‘i50)
MW,/\ A W“ \ (We) = '3+ Rw§LKH¢200> r1w§<QKE NS") ...
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 Fall '07
 Throne
 Fourier Series, Periodic function

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