TELECOMMUNICATIONS

TELECOMMUNICATIONS

Dr. Hugh Blanton

ENTC 4307/ENTC 5307

Solutions to Test 1

Dr. Blanton - ENTC 4307 - Solution_Test_1 3

1. Definitions1. Definitions

a. RF—Implies the frequencies where the wavelength, , becomes equal or smaller than the physical components in the circuits.

b. Continuous—A signal which can take an infinite number of different values between a minimum and a maximum.


c. Discrete—A signal which can take only a finite number of different value.

d. Wireless—A term used to describe telecommunications in which electromagnetic waves (rather than some form of wire) carry the signal over part or all of the communication path.


e. Random—A rule that assigns one and only one numerical value to each sample point in a random experiment.

f. Deterministic—A model where a set inputs always produce the same outputs.


2. Complex Numbers2. Complex Numbers

24

23

2

1

jz

jz

a. Express z1 and z2 in polar form.

6933605631 .. z

4349153472142 .. zb. Find | z1 | and | z2 | and calculate the product z1 z2 and the ratio z1 / z2 in polar form.

605631 .z

472142 .z

744511912451621 .. zz

87501728062021 ../ zz


3. RF3. RF

The presence of the transmission line can be ignored when:

010.l

a. l = 0.1 m, f = 1 GHz transmission line cannot be ignored!

0103333030

1030

101

1039

8

...

..

lm

f

c

b. l = 1 m, f = 1.8 GHz transmission line cannot be ignored!

0105999016670

1016670

1081

1039

8

...

..

.

lm

f

c


3. RF3. RF

c. l = 0.01 m, f = 0.9 GHz transmission line cannot be ignored!

0103000033330

1033330

1090

1039

8

...

..

.

lm

f

c

d. l = 5 cm, f = 5.4 GHz transmission line cannot be ignored!

0108993005560

05005560

1045

1039

8

...

..

.

lm

f

c

mmc

mmcl 050

100

15 .*


4. Smith Chart4. Smith Chart50oZ

2550 jZL

a. Reflection coefficient ():

97752425025100

25

502550

502550..

j

j

j

j

ZZ

ZZ

oL

oL

b. Standing wave ratio (SWR):

641242501

242501

1

1.

.

.

SWR

c. Input impedance (ZIN):

21530024061050240610 ...... jjZjz ININ


4. Smith Chart4. Smith Chart

d. Input admittance (YIN):

siemensjjYjy ININ )..(.... 001400324007064150

1070641

e. Shortest line length


4. Smith Chart4. Smith Chart501 .jzL 0.144

0.494

240610 .. jzIN 070641 .. jyIN


077905301 ..~ ljyINstub

50oZ 2075 jZL

4051

50

2075..~ j

jzL

166062204051

11..

..~~ j

jzy

LL

53011 .~ jyIN

53012 .~ jyIN

42205302 ..~ ljyINstub

1130035014801 ... d

3140035034902 ... d

5. Stub Tuning5. Stub Tuning


6. Uniform Distribution6. Uniform Distribution

21

2

4

4

42

1 22

0

22

0

xdxxxE )(

3

4

6

8

62

1 232

0

32

0

22

x

dxxxE )(

022

12

0

2

0

sin

cos)(cos dxE

333

2232

0

22

2

0

22

3

8

2

1

32

12

2

1

2

1

xmmxx

dxmxmx

dxmxmxE

)(

)()(


7. PDF7. PDFpX(x)

x

½

⅜

⅛

1 2 3

8

9

16

18

16

5

16

9

4

1

16

49

16

312

4

1

1616

3

4

8

1

8

3

2

1

3

2

22

1

21

0

2

3

2

2

1

1

0

3

0

xxx

dxxdxxdxxdxxxfxE x )()(


8. Signal8. Signal

r(t) = s(t) + n(t)

s(t) = 5 cos(2 x 1000t )+ 10cos (2 x 1100t)

noise n(t) is white noise with power No = 0.05 watt/Hz

passband between 990 Hz and 1100 Hz

wattsfdfHzwattPn 60500501100

990

1100

990 ./.

wattsSn 5622

125

2

10

2

5 22

.

dBwatts

SNR 17104106

562..

.


9. RC9. RC• The output power spectral density is the

input density multiplied by the square of the magnitude of the transfer function:

x(t)

R

C

2)()()( fHfGfG xy

22221

2

RCf

NfG o

y)(

/)(

22 4

2

fa

ae ta

Note the Fourier transform pair:

2

2222

222

21

2

21

2

)(

/

)(

/

fCR

CR

N

RCf

N oo


2

222

2222 2

1

12

21

2

)(

/

)(

/

fCR

RCRC

N

fCR

CR

N oo

2222

222

41

2

2

2

21

12

)(

/

)(

/

fCR

RCRC

N

fCR

RCRC

N oo


222

222

42

2

2

2

21

12

)(

/

)(

/

fa

a

RC

N

fCR

RCRC

N oo

RCawhere

1

)(/

)(

/

yRCoo Ge

RC

N

fa

a

RC

N

2

2

42

2

2

2222


10. Ideal Filter10. Ideal Filter

H(f)

ffm-fm

1

otherwise

ffNfHfGfG mo

xy0

22 /)()()(

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