Communication Theory I
Midterm Exam.
November 18, 2022
1. (4%) Sketch the double-sided amplitude and phase spectra of 6 sin(8⇡t + ⇡/6) + 8 sin(6⇡t + ⇡/4).
2. (6%) Classify each of the following signals as an energy signal or a power
signal
by calculating
⇣
⌘
P1
t 4n
3t
the energy or the power. (a) 4 cos(6⇡t + 2⇡/3) (b) e u( t) (c) n= 1 ⇤ 2 .
3. (4%) If x(t) has the Fourier series x(t) =
1
X
Xn ej2⇡nf0 t and y(t) = x(t + t0 ) =
n= 1
express Yn in terms of Xn .
4. (5%) Find the Fourier transform of ⇧(t
1
X
Yn ej2⇡nf0 t ,
n= 1
3) exp[j6⇡(t
3)] + ⇧(t + 3) exp[j6⇡(t + 3)].
5. (6%) Find the energy spectral density of the following signals. (a) 5e
6t
u(t) (b) 2⇧(4t).
6. (4%) Without actually computing the Fourier transforms of the following time functions, tell
which ones should be real and even functions of f and which ones should be imaginary and odd
functions of f . Explain your reasons. (a) ⇧(t/4) ⇤(t/2) (b) ⇤(2t)sgn(t).
7. (4%) Find the power spectral density and the autocorrelation function of 4 cos(30⇡t + ⇡/5).
Q
Q
f
f
8.
(4%) A filter has amplitude response |H(f )| = 2 ( 100
) + ( 200
) and phase response ✓(f ) =
(
⇡
f, |f | < 150
300
. Find the output for each of the inputs given below. For which cases is
0,
elsewhere
the transmission distortion? Is it amplitude or phase distortion? (a) cos(120⇡t) + 2 cos(180⇡t) (b)
2 cos(80⇡t) + cos(140⇡t).
P
9. (5%) For a signal x(t), the sampled waveform is x (t) = x(t) ⇥ 1
nTs ). Consider the
n= 1 (t
Q
filter with the impulse response function h(t) = [(t 12 ⌧ )/⌧ ] (⌧ << Ts ) whose input signal is x (t)
and output signal is y(t). (a) Find the spectrum of the output, Y (f ), in terms of the spectrum of
the input, X(f ). (3%) (b) Determine the relationship between ⌧ and the bandwidth of X(f ), W ,
required to minimize distortion in the recovered waveform. (2%)
2.5B
10. (6%) Given the bandpass signal spectrum X(f ) = ⇧( f B
) + ⇧( f +2.5B
), sketch spectra for the
B
following sampling rates fs and indicate which ones are suitable: (a) 2B (b) 3B (c) 5B.
Q
11. (4%) Assume that the Fourier transform of x(t) is X(f ) = A (f /2W ). Determine the spectrum
of the signal [ 34 x(t) 14 j x̂(t)]ej⇡W t .
12. (4%) Assume that a DSB signal xc (t) = m(t) cos(2⇡fc t + ) is demodulated using the demodulation carrier 2 cos(2⇡fc t + ✓). Derive the demodulated output yD (t).
P
2(t
13. (5%) The periodic signal 2K 1
n= 1 ⇤(
Determine the modulation efficiency for a = 0.7.
0.5Ts nTs )
)
Ts
K is applied to an AM modulator.
14. (4%) An AM modulator has output xc (t) = 20 cos(300⇡t) + 4 cos(340⇡t) + 4 cos(260⇡t). Determine the modulation index.
15. (6%) Consider the demodulation of an SSB signal by carrier reinsertion. (a) Plot the block
diagram. (b) Should the locally generated carrier be phase coherent with the original modulation
carrier? Explain the reason.
16. (4%) The bandwidth of the message signal m(t) is W and the signal m(t) cos(2⇡fc t) is sent
8
>
<
2,
|f | > fc + a
into a filter whose transfer function is H(f ) = > a1 (|f | fc + a), fc a  |f |  fc + a where
:
0,
|f | < fc a
0 < a < W . The transmitted signal is the output of this filter. (a) What is the bandwidth of
the transmitted signal? (b) The coherent DSB demodulator is used at the receiver. What is the
demodulated signal?
17. (2%) In a superheterodyne receiver, the frequency of the local oscillator is 1400 kHz. If the
image frequency is 950 kHz, what is the carrier frequency?
18. (5%) Consider AM signal [Ac + m(t)] cos !c t with m(t) = Am cos !m t. There is an interference
signal Ai cos(!c + !i )t. Derive the demodulated output (by approximation) if Ai >> Ac .
19. (3%) Consider the PAM signal with the width ⌧ and the period Ts . What is the restriction if
the receiver is only an ideal LPF?
h
R
i
20. (10%) An FM modulator has output xc (t) = 80 cos 2⇡fc t + 2⇡fd t m(↵)d↵ where fd = 16
Q
Hz/V. Assume that m(t) is the rectangular pulse 2 [ 16 (t 3)]. (a) Sketch the phase deviation in
radians. (b) Sketch the frequency deviation in hertz. (c) Determine the peak frequency deviation
in hertz. (d) Determine the peak phase deviation in radians. (e) Determine the power at the
modulator output.
21. (5%) An FM modulator has fc = 1500 Hz and fd = 20 Hz/V. The modulator has input
m(t) = 10 cos 20⇡t. (a) What is the modulation index? (3%) (b) Is this narrowband FM? Why?
(2%)
6叫批到 8叫比⼗到
amplitude
osjeijlfyijéslfyje外引tyé外引
i 洲 3ei外州 4éntyeigy
肶n
ii
3
3 y
a p⼆ 点
f
81可直接看1
Ehinx
ii
t
⼆
in
fi
iilièiydtfjdtiieit
ii
power signal
o
0
Energy Signal
⼩
Filiidtiiiiildtiliidtiliiiitylii
E
PEN
斯
i power signal
法Il
制裁yinft
⼆
Xlttyejhejznnfltty
点然竺
E it
⼆
1法到
hiifyyeitu
if Ntitoleid
ifxliliy
Yihe
令
得證
tii
dtu
ihxlileidi.eiihe.info 得證
I Hènsinclfsl
iltsleinsinlfslei
⼆
ihleitiluyei
nsinlfsleitsindfyeōisinclflasl6
or
啪
5ěuHis 侧
in 州
⼆
滅 磩
5
州州 i
N
⼆
Gihiiilfly
h
i
z
n
i
i
z
⼆
7evensignal
ÉH
Gdtlxdli
seven and purelyreal
ioddsignd is oddandpurely imaginary
R.li8 叫 可直接寫答案
絒⼆ Yblf151 4外州
Nil
州州圳dt
1 4州了啦 1.4叫了叫圳 副dt
16位 ilmlboīvtintiyihoinlldt
8oshoinl
州
HM
to
50
qasliht
so too
i
g
2以118州
乬
ǐ
f
冷
以1120比 ⼀到 ⼗ 2 叫 批
咧
oslnonltiltzcosllfoiltj
上
7
可直接寫
䲚
yzoslbiltoslyibosl8 iltashi.it
6以18班 制 taslltiy
⼆
7可直接寫
amplitude distortion
州 州州 1Hlfkiéglfhltienftolteihfqteijlftqo
nyoslnōltl 以1 批 褚
州州 州 州
3é 外州 the 外州 tehfteinf 州
州圤6叫秕 訓 㖄 呲 踏
川
州 州 Hi 就外剁 油 歐 州
州 丌 1與1 Hlfl isinkflei
⼆
焮 洲洲
⼆
iimlfléxlf ⼀點
YHHfssinchfleiidlfmH.fi
NH
n
Tlìffwhftn
fzzw
知 nihil
if
Mㄟ
焩
3BǛB
㐺
巧 巧 2513313
a
n
suitable
狐⼼ 物 㤈 猴⼼ 你 ⼼
如 j niin
bio.BZ Bz奶奶
bl
suitable
如
n
似必
j nb
H
仙洲
焮州
bibzBz.is
n
如
㗀
j nb
bi.bz
㗀
Bz奶奶
iili.si
lixH
Nfǐliiljsgnlf州州到
⼆
ixlflisgnlf 1Nf則
inliil
州
w
州
q
w
⼀
回
sq
ii
yi
州 1佣畫圖
Zcoslwiy
ykmltla州 州 iasltt01
㤰
mkllosl烌坤州 州
州 mltlan
⼆
n寫到這扣分
Mǖii
⼆
些
ftzkn信仙洲
州
⼆
2M信ltltzkn 信怡川川火
iǚynkidt
ǛǛ1點
⼀
步
is
till
塔 1就_it9唯 ī
⼆
Effing
Ǜǜj fgi.MN
⼆
Iooskoilt 4叫了0化州州 4叫了蚍 洲
⼆
Iocoshoilt8叫洲 州州
州 Hioslosliáof
b.cn Wta
⼩
州
n
i
o
is
is
火
hi NoHE
hw
fni950kHz
idfilhol.lt
㖄asw.tt 㳤回 北 佣畫圖
1
liashitwilt
iihaslwdltlmcoslwmylwiytlioslw北州
tilwdlthcoslwmycoslwdltlioslw川川州
fisinhhinNi ilhihilwmytfiaslwitHNNcy
f.imNitlsinhiy
⻔
lih
Xdtlylilmaslwmytlicoslwiykliilwitl
⼆
⼆
llftlmoslwmithlmoslwmyfioslwiytliailwtliiy
Hndktlmhhoshntllctlmoslwmyj
fi
HX
xnpni
ilitllihhhoslwif
fi
T
I Is
i
州 jmndx
nii
t
Hit oil
t
炒
lh
6
bl 姒 爍 絒
⼆
AM
32
6
gylmax 冮 HE
州
州mi384ivlrdl
ftp.mowy
it
t
a
b
管 If
No
0
⼆
i a 1
0
20
⼆
年