Digital Communication 601_tutorial

7/31/2019 Digital Communication 601_tutorial

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DIGITAL COMMUNICATIONS ENGINEERING 303 (11330) Tutorial 1

1. Are the following two signals periodic? If so, find their period.(a)

( )

+

= tttx

3

1

cos25

1

sin

(b) ( ) ) ( )tttx sin22sin +=

2. Sketch the following signals, and show whether the signals are energy or poweror neither.

(a) )()( tAutx =

(b) )()( ttutx =

where

= 00

01

)( t

t

tu

andA is a constant.

3. The output of a nonlinear device is given by).()( 2 txty =

Given ( ) ( ) ( )cos 600 2 cos 6800x t t t = + .

(a) Determine ( ).y t (b)Sketch the amplitude spectrum of (i) the input signal ( )x t , and (ii) the output

signal ( ).y t

(c) Comment on the spectral components observed at the output of the nonlineardevice.

4. A channel has the following amplitude response ( )H f , and phase response ( )f :10( ) 100 4000

( ) 20 100 4000 .

H f f Hzf

f f f Hz

=

=

The signal at the input of the channel is given by

( ) ( ) ( )cos 600 2 cos 6800x t t t = + .

(a) Determine the signal, ( )y t , at the output of the channel.(b)Comment on the influence of the channel on the input signal.


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5. A periodic signal ( )tx of period oT is expressed in terms of its complex exponentialFourier series as

( )tjnctx on

n exp)(

=

=

whereo

oT

2= is the fundamental angular frequency, n is an integer, and nc

correspond to the Fourier coefficients, given by

dttjntxT

coT

oo

n )exp()(1

0 = .

Given that ( )tx is a rectangular pulse train of pulse duration

4

oT= and

amplitude A Volts.

(c) Derive an expression for nc , and express it in terms of

)sin(.

(d)Tabulate the values of nc for |n| up to 8.(e) Calculate the power Ptof )(tx in the time domain.(f) Calculate the power Pf of )(tx in the frequency domain using the values of nc

for |n| up to 8.

(g)Comment on the values ofPt and Pfobtained.


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1. Given the Fourier transforms of the following two functions:( )fTT

T

trect sinc

, and

++

T

fT

fA

T

tA

2

1

2

1

2cos

.

(a) Determine the Fourier transforms of the two half-cosine pulses g1(t) and g2(t), asshown in Fig. 1a and Fig. 1b, respectively.

(b) Show that the above two pulses have the same amplitude response, and energyspectral density given by

.)14(

)(cos4)(

2222

222

=

fT

TfTAfe

2 A trapezoidal pulse is shown in Fig. 2.

(a) Use the property ( ) ( 2 ) ( )n nn

d g tj f G f

dt , and show that the Fourier

transform of the trapezoidal pulse g(t) is given by

)].(sin[)](sin[)(

)(22 abab

ab

ttfttfttf

AfG +

=

(b) Express ( )G f in terms of sinc( ) function, where sin( )sinc( ) AAA

= .

(c) Find the Fourier transform of ( )g t if a bt t= .(d) Given that 0at = , derive the Fourier transform of ( )g t .

g1(t)

-T/2 0 T/2

A

t

Fig.1a

g2(t)

A

t0 T

Fig.1b

-tb -ta 0 ta tbt

A

g(t)

Fig. 2


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3 Evaluate the transfer function of the following linear system.

4 A signal )(ts consists of three components ),(1 ts )(2 ts , and )(3 ts , such that

).5000cos(5)1500cos(3)200cos(2)()()()( 321 ttttstststs ++=++=

The signal )(ts is transmitted through a linear time invariant network with the transfer

function given by)](exp[|)(|)( fjfHfH = .

(a) The resultant output signal )(tso remains distortionless. Determine the amplituderesponse |)(| fH and phase response )( f , given that the output signal

component due to )(1 ts is given by

).100200cos(3)(1 = ttso

(b) Write down an expression for the output signal, )(tso .

+

Delay

t

dt

+

-

x(t y(t


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1

DIGTAL COMMUNICATIONS ENGINEERING 601 (11330) Tutorial 3

1. Given the signal).1608000sin()402000cos(2)(

+=

tttx

(i) What is the minimum sampling rate required to avoid aliasing in the sampledsignal?

(ii) Determine the average power of )(tx .

2. A signal )(ty is given by)320sin(2)200cos()( ttty += .

)(ty is ideally sampled at 300=sf Hz, and the sampled signal )(tz passes through

an ideal lowpass filter with a cutoff frequency of 250 Hz.

Determine the frequency components that appear at the output of the lowpass filter.

3. An analogue signal, )4000cos(5.0)4000sin(2)( tttx += , is sampled at a rate of 8000samples/second. Show that the average power )(tx is given by

21

0

][1

=

=

N

n

nxN

P , whereNis the number of samples in one period of )(tx .

4. A bandlimited signal has a maximum frequency component atB Hz, and an averagepower of S watts. Gaussian noise with a uniform spectral density of S 6102

W/Hz from 0 to 4B Hz is added to this signal. The resultant signal plus noise is then

sampled using an impulse train at a rate of3B Hz.

Calculate the signal-to-noise power ratio (SNR) of the sampled signal in a bandwidth

from 0 toB Hz.

5. An analogue signal, ( ) sin(600 ) 2 cos(2000 ) 2 cos(6800 ),x t t t t = + + is sampledusing a sample-and-hold device at a rate of8000 samples/second.

Determine the spectral components at the output of the sample-and-hold device for

frequencies up to half the Nyquist sampling frequency.


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1

DIGITAL COMMUNICATIONS ENGINEERING 601 (11330) ___ Tutorial 4

1. Design a uniform quantiser for an input signal with a dynamic range of 10 V. Its

output is represented by four binary bits. Find the quantiser output value and the

quantisation error for an input signal of amplitude 1.2 Vif the quantiser is of the (i)midrise type, and (ii) midtread type. The quantisation error is taken to be the

difference between the output and input of the quantiser.

Hint: Draw the transfer function of the quantiser and calculate the relevant step size.

2. To demonstrate the advantage of using logarithmic companding for low-level signals,we use the same linear midrise quantiser of (1). This time the linear quantiser is

preceded by a = 255 compandor. The -law compression )(xF

and its inverse

)(1xF

characteristics are given by

11for)1ln(

|)|1ln()sgn()(

+

+= x

xxxF

and )sgn(1)1(1

)(1 xxF

x

+=

where


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2

4. A signal )(tv bandlimited to 3 kHz is sampled at a rate %3

133 higher than the

Nyquist sampling rate. The maximum acceptable quantisation error is 0.5 % of the

peak amplitude pv of )(tv . The quantised samples are then binary encoded.

Find the transmission rate of the resulting binary coded signal. What is the theoreticalminimum transmission bandwidth for this signal?

5. An analog signal with frequency components ranging from 300 Hz to 3400 Hz issampled, quantised, and encoded into a PCMsignal. The input signal is assumed to be

sampled at a rate 25% above the Nyquist sampling rate, and that the sampled signal is

quantised into 256levels.

The PCMsignal is then transmitted using discrete pulse amplitude modulation (i.e.,

M-ary PCM) over a baseband transmission link. Determine the minimumtransmission rate if each signalling pulse is allowed to take on the following number

of amplitude levels: (i) 3, and (ii) 8.

Supplementary Note:

A basic relationship in communication states that a maximum of2B independent pieces

of information per second can be transmitted, error free, over a noiseless channel of

bandwidthB Hz.

This statement is the result of sampling theorem. A noise-free channel of bandwidthB Hz

can support an error-free transmission of a signal of bandwidth B Hz. Now, this B Hzsignal can be exactly reconstructed from its Nyquist samples at a rate of2B samples/sec

(orHz). This suggests that theB Hz signal can be completely specified by 2B independent

pieces of information per second. As the channel is able to transmit this B Hz signalerror-free, then it should also be able to transmit, error-free, 2B independent pieces of

information per second.

This relationship leads us to conclude that the theoretical minimum bandwidth of the

channel for the binary PCMsystem of 1(a) is equal to 342

68=

kHz.


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1. Additive white Gaussian noise (AWGN) has a probability density function given by

=

2

2

2exp

2

1)(

vvp

where is the rms value of theAWGN.

What is the probability that the noise is larger than A Volts? You may express your result in

terms of the complementary error function given by

=x

z dzexerfc22

)(

.

2. In a baseband ternary transmission system, the received signal at the decision instant is either0 orA or A. The threshold levels used in the decision device are set at

2

A . The noise at the

receiver input is white and Gaussian with a probability density function given by

=

2

2

2exp

2

1)(

nnp

where is the rms value of the noise at the input of the decision device.

Assuming that the three signaling levels are equally likely, and the symbols transmitted in

adjacent time slots are statistically independent, derive the average probability of symbol

errors, eP ,of this system.


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1

DIGITAL COMMUNICATIONS ENGINEERING 601 (11330) __ Tutorial 6

1. The Fourier transformR() of a basic pulse r(t) used in a binary transmission system is givenby

=

elsewhere0

102||for1|)(|

4

R

and constantaisandallfor)( =

where |R()| and () are the amplitude response and phase response of R(),respectively.

(a)Show that this pulse r(t) satisfies the Nyquist criterion for zero intersymbol interference.(b)What is the transmission rate (in bits per second) that could be supported by this basic

pulse r(t) to achieve zero intersymbol interference at the decision instant?

2. A binary data source is transmitted over a channel of bandwidth 60 kHz using a 4-levelsignaling in a system with raised cosine spectrum. The roll-off factor is 0.5.

(a) Calculate the signaling rate.(b) What is the data rate of the binary source?

3. The received signal at the detection node of a regenerator is represented by)()( nTtgbtr

n

n =

=

where 1=nb is thethn binary data bit, and T is the bit period. The isolated waveform

)(tg is given by

=

2

302.2exp)(T

ttg .

(i) Determine the signal levels at each decision instant. You may neglect any value of001.0|)(| tg .

(ii)

Specify the data patterns that give rise to the worst case eye opening.(iii) The received signal is corrupted by additive white Gaussian noise with the probabilityof its amplitude exceeding A Volts given by

=>

22

1)(

AerfcAnp

Assume the input binary data is random, derive an expression for the average bit-error

rate, eP .


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2

4. In a binary communication system, the received waveform r(t) of an isolated pulse has thefollowing values:

(0) 1( ) 0.3( ) 0.1( ) 0 for 2, 3, ... .

b

b

b

r

r T

r T

r nT n

=

=

=

= =

where Tb is the signalling bit period, and 0=t corresponds to the decision instant.

(i) Assuming that unipolar signalling is used, determine all the possible signal values at adecision instant.

(ii) Specify the data bit patterns that contribute to the worst case eye-opening?(iii) The received signal is corrupted by additive white Gaussian noise with the probability

of its amplitude exceedingA Volts given by

1( ) .

2 2

Ap n A erfc

> =

where is the rms value of the noise at the input of a threshold detector.

Assume random input binary data, derive an expression for the probability of bit errors, Pe,based on the worst-case eye-opening.


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1. Determine the information capacity of a 1 Mb/s random binary data stream. Thisbinary sequence is then encoded into a 4-level code by transforming a 4-bit binary

word into a 2-symbol 4-level word. Assuming that all four levels are equal, determinethe transmission efficiency of this 4-level code.

2. Consider the binary sequence 1010110001. Draw the waveforms for the followingsignaling formats.

(a) Unipolar NRZ signaling(b)Polar RZ signaling(c) Manchester signaling(d)AMI NRZ signalingFor the AMI NRZ signalling of (d), an error occurs at the 5

thpulse position during

transmission. Comment on the detection process caused by this single transmissionerror.

3. Determine theHDB3 encodings of the following two binary sequences:(i) 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1, .(ii) 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1, .You may use the + and

signs to denote the positive and negative pulses,

respectively.

4. A binary data stream is encoded using B6ZS, the resulting coded sequence is givenby

(i) + 0 0 0 + 0 + 0 0 + + 0 + 0 + 0 + 0(ii) 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 Determine the two original binary data sequences.


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1. A pulse s(t) is shown in Fig. 1.(a) Determine and sketch the impulse response of a filter matched to s(t).(b) Plot the output of the matched filter as a function of time.(c) What is the peak value of the matched filter output?

2. The amplitude of the pulse s(t) in Fig. 1 is halved. Determine how the duration of the

pulse has to be changed, so that a filter matched to this new pulse has the same

performance as the matched filter in question (1). You may assume that both cases

operate with the same additive white Gaussian noise.

3. A 2 kb/s input binary data is transmitted using an AMI-RZ signaling format. An

integrate-and-dump device is used for matched filter detection. It is assumed that the

signaling pulse is rectangular with a unity mark-space ratio. The amplitude of the

signaling pulse is 2 Volts.

(a) Determine and sketch the output of the integrate-and-dump device for the inputbinary data sequence:

1, 1, 0, 0, 1, 0, 1, 1, 1.

(b) Comment on the output of the integrate-and-dump device if the signaling pulseis non return-to-zero.

s(t)

t0

A

-A

T/2

T

Fig. 1.

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Digital Communication 601_tutorial