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Solutions for lecture assignments

S-72.1140 Transmission Methods in Telecommunication Systems (5 cr)

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Assignment

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Solution

(i) example: IEEE 802 specifies a large number on local area networking standard. Relating technologies include for instance TDMA, FDMA and CDMA and several technologies of integrated circuit are applied in realization of practical devices (as routers and terminals). IETF standardizes Internet.

(ii) Standardization refers to recommendations to assure compatibility among manufacturers, regulation is must that need to be followed as in the way how electrical appliances must be constructed for being safe

(iii) Effect of market can be quantified (evaluated) for instance in terms of potential or expanding market size, expected revenues and investment costs required to act or even dominate in the market

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Assignment

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Assignment

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Assignment

solution:

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Assignment

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Conclusions: Nonlinearity is reflected as extra harmonics at the output of the circuit

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Assignment

ans: 20 dB

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Assignment

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Solution

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Assignment

2

2

0

1( ) , ( ) cos

1cos ( ) cos( )sin( ) , 2 /

2 2

1 1 2 2 1 1cos sin

2 2 2 2

T

T

P v t dt v t tT

xkx dx kx kx C k T

k

x x x TP

T T T T

Solution

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Assignment

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Solution

a)

3

00

3

00

( ) cos2 4

cos (2 1) / 4

bpn

n

nV f t

t n

b)

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Assignment

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Solutionu(t)

u(-t)

u(t-t1)u(-t+t1)

u(t+t1), t1>0

a)

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Solutionb)

( ) ( ) ( )lp bp C CV f V f f u f f

( )bp CV f f

( )bpV f

cfcf

2 cf

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Assignment

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Assignment

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Solutions

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Assignment

Solution: FM can utilize bandwidth expansion to increase post detection SNR. Therefore, information transfer capacity of FM accommodates to channel better than in AM.

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Assignment

( )( )

d tt

dt

( ) ( )t

t d

Solution:

2 2( ) ( ) 2 , (2) 2 2 2 /(2 ) turns = 2.6 turns

( ) (t)=d ( ) / 2 2 , (2) 2 4

(t)=d ( ) / 2 (constant)

i t t t

ii t dt t

t dt

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Assignment

(i) Briefly summarize what is the main difference between FM and PM ?

(ii) How would you generate FM by using a PM modulator?

Solution(i) In PM instantaneous phase is directly proportional to modulating signal amplitude, in FM instantaneous frequency is directly proportional to modulating signal amplitude.(ii) One need to integrate the modulating signal before applying it to the PM modulator

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Assignment

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Example

1sinc2

2 2

sin(2 ) 1expand sinc( )

2 2 2

2 sin(2 )swap , regroup 2

2 2

2 sinc(2 ) denote pulse width in time domain by 22

sinc( )

fWt

W W

Wt fx

Wt W W

W Wf tf t W

Wf W

tW Wf W

W

tf

1

2W

1

2W

WWtf

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Assignment

Show that for radio transmission following simplification applies:

2 24 4d fd

Lc

10 1092.4 20log 20logdB GHz kmL f d

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Assignment

Solution:

2 24 4d fd

Lc

10 1092.4 20log 20logdB GHz kmL f d

24 4

10log 20logdB

fd fdL

c c

9 34 10 1020log GHz kmf d

c

9 3

8

~92.4

4 10 1020log 20log 20log

3 10 GHz kmf d

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Assignment

How many dB:s channel SNR should be increased in order to accommodate 16-level signaling?

Therefore SNR should be increased 6 dB

T2B log(16) 8Mb/sbr

2log (1 )

ln(1 )

ln2exp( ln2 / ) 1

exp( ln2 / ) 1 255 24dB

C B SNR

C SNR

BC B SNR

SNR C B

10log(63) 18dB

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Assignment

SolutionM() is determined from the condition |Jn()| > , therefore the curve on the previous page can be obtained for instance by fixing a value for and then adjusting n until the condition is met. Note: n = 1 equals the case with two sidebands (magnitude spectra the same as for AM)

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Assignment

SolutionDefinition of voltage controlled oscillator (VCO) indicates that it is an FM modulator because its instantaneous output frequency is directly proportional to modulating signal amplitude. Therefore, the signal after VCO is an FM wave. The inherent integration of FM in output phase must be cancelled for PM signal, that is then realized by setting the differentiator to the input of VCO.

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Assignment

Consider the following RC-filter transfer function

Inspect the relating group delay and state what is a frequency range of low linear distortion

3dB

1 1( )

1 2 1 ( / )

H fj fRC j f B

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Frequency range of low linear distortion?

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Assignment

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Solution

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Assignment

(i) How would you know if a channel has in general non-some linear characteristics?

(ii) How would you measure a linear channel?

(iii) Suggest how to measure a non-linear channel!

(i) Channel output has frequency components that can’t be found in its input(ii) For instance by sending a known signal (frequency spectra) to its input and by monitoring the output. Note the frequency range of the input signal. The result is system transfer function.(iii) The same method as in (ii). However, now both frequency and amplitude of the signal should be changed. Often a single sinusoidal is applied. The result is system transfer characteristics. For more discussions about (ii) and (iii) see few slides ahead.

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Assignment

(i) Determine channel input impedance for Vg=1 V, Rg = 50 ohms and Vi=0.5 V

(ii) Assume load impedance is Zl= 50+j10 ohms and Zg= 50 ohms. How much is the dissipation angle theta?

gV

iV g g gZ R X

LL LZ R X

, cosg g LL i i

i L

V Z ZP V I

V Z

2 2cos / / , ,tot tot tot tot tot tot g L tot L gR Z R R X X X X R R R

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Solution

(i) Voltage divides evenly between the resistive components. Therefore the load equals internal impedance and matching is perfect.(ii) Dissipation angle follows directly from given equations:

2 2

2

cos / / ,

,

arccos 100 / 100 100

tot tot tot tot tot

tot g L tot L g

R Z R R X

X X X R R R

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Assignment

Channel input has two frequency components at f1=1 kHz and f2=2 kHz. The channel frequency response follows H(f)=exp(-jf). How much is the time difference of the frequency components after this channel? Is this a linear or non-linear channel?

Solution:

d

( )

Y(f)= y(t) =Kexp -j t ( )

( ) ,arg ( ) 2

2 1 1/ 2d

d d

H f

X f

H f K H f ft f

t t

F

Both frequency components have the same delayafter the channel!

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Assignment

How would you measure in practice 3rd order intercept point?

2 31 2 3( ) ( ) ( ) ( )y t a x t a x t a x t

( ) cos cos( )c c dx t t A t

1 34 / 3A a a

Ans: it would be required to apply two frequencies to the circuit and adjust the other one by simultaneously monitoring the output by a spectrum analyzer

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Assignment

How the formulas below explain the change of pitch for a train when it passes a station in full speed?

2 2cos

1cos

2d

v tl

vf

t

Doppler effect [7]

ans: the angle ofsignal source changes

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Assignment

How would you measure interference level in a channel?

Solution: By monitoring channel output by a spectrum analyzer. If the interference results from channel nonlinearity, for instance 3rd order intercept point measurement could be taken to quantify relative magnitude of the interference.

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Assignment

( ) E[ ( )]v t v t 2 2( ) E[ ( )]v t v t

( ) ( ) E[ ( ) ( )]v t v t v t v t

/ 2

/ 2

1( ) lim ( )T

Ti iTv t v t dt

T

/ 2

/ 2

1( ) ( )T

Ti iv t v t dt

T

2

1

2solution: ( )t

t

y t dt w

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Assignment

/ /R R

R R T

T T

S S WS N W B

B W B

/ /

2 (DSB,AM) / / 2

10 (FM) / /10

R R T

T R R

T R R

S N W B

B W S N

B W S N

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Assignment

Solution: The respective probability must be integrated from Gaussian pdf, yielding for instance in terms of Q-function 2Q(VT/) where is the square root ofX.

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Assignment

( )sin ( )( )

( )

2

n n

v

c

q

R

A t tt

A

n t

S

2

2 2

/ 2

sinR c

q n n

i q R LP

S A

n A

n n N B

( )sin[ ( )]( ) arctan

( )cos[ ( )]n n

v

c n n

A t tt

A A t t

Explain how the results shown connects to the phasor diagram!

48 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen

Assignment

SolutionM() is determined from the condition |Jn()| > , therefore the curve on the previous page can be obtained for instance by fixing a value for and then adjusting n until the condition is met. Note: n = 1 equals the case with two sidebands (magnitude spectra the same as for AM)

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Assignment

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Solution

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Assignment Determine average power for the following signals

T

T

A

-A

A

-A

A/2

-A/2

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Solution

T

A

-A

A

-A

A/2

-A/2

2 2 21 1( ) ( )

2 2RS A A A

T

2 2 21 1 5( ) ( / 2)

2 2 8RS A A A

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Assignment

What is the impulse response of the matched filter for the following signaling waveform?

How would you determine the respective output signal (after the matched filter)?

T

A ans: Impulse response is the time-reversed version of the signaling pulse

ans: The output is the convolution of input pulse and the impulse response of the matched filter

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Assignment

(i) Determine probability that all bits are inverted by channel noise in a 4-bit code word when bit-error prob. is p

(ii) What are the cases when error is introduced in a 5-bit code word in repetition decoding?

4( )

( ) - all bits are inverted

- 1 bit is inverted

- 2 bits are inverted

i P p

ii

55 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen

Assignment

A channel coding system applies codes 00010 and 11100 for encoding ’1’ and ’0’, respectively.

(i) What is the minimum Hamming distance?(ii) How many code bit errors can be detected by the code?(iii) Received code is 00000. Can it be corrected by the decoder?

Solution: The minimum Hamming distance is the number of different bits (in this case dmin = 4) required to swap a code word to another code word. Error detection is always possible if the number of errors in dmin-1. If the number of errors equals dmin or is larger Error detection is not always possible

For received code 00000 receiver will assume one bit error that can be detected and corrected. However, if the transmitted code

was 11100 a three-bit error has happened that can not be corrected

56 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen

Assignment

(i) How to localize data-biterrors?

(ii) How to localize parity-bit errors? m1 m2

m3

m4

c1

c3c2

57 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen

Solution

(i) How to localize data-bit errors?

m1 by determining parity of circles c1 & c2m2 by determining parity of circles c1 & c3m3 by determining parity of circles c2 & c3m4 by determining parity of circles c1-c3

(ii) How to localize parity-bit errors?

c1: determining parity of c1 (c2 & c3 ok)c2: determining parity of c2 (c1 & c3 ok)c3: determining parity of c3 (c1 & c2 ok)

m1 m2

m3

m4

c1

c3c2

58 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen

Example

Assume received code is [0 1 1 0 0 0 0] and odd parity, is thisa correct code word? Assume odd parity.

m1 m2

m3

m4

c1

c3c2

59 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen

Solution

Assume received code is [0 1 1 0 0 0 0] and odd parity, is thisa correct code word?

Ans: error must be in p3 circle,its parity is even. Because bothp1 & p2 are odd, p3 – bit must be inerror

0

1

0

1

0 0

0

m1 m2

m3

m4

c1

c3c2