Transcribed Text
Question (01): Block diagram analysis [35 marks]
Consider the block diagrams for cascade, parallel and feedback connections in Fig.
A
communication system consists of many interconnected subsystems. Let us assume that
these subsystems exhibit linear and limeinvariant (LTI) properties and the corresponding
inputoutput relationship is described by their individual transfer functions H1(5) and H2(f).
Whenever each subsystem is described by individual transfer functions, it is possible and
desirable to obtain the equivalent transfer function of the overall system.
X(f)
H1(f)
H2(f)
Y(f)
(a) Cascade connection
H1(f)
+
X(f)
€
Y (f)
+
A
H2(f)
(b) Parallel connection
X(f)
H1(f)
Y
(f)
H2(f)
(c) Feedback connection
Figure 1: Basic block diagrams of communication systems
(a). Find the overall transfer functions of cascade, parallel and feedback connections in
terms of H1(5) and H2(f).
1
(b.3) Show that can be approximated as follows:

[hint: Use binomial expansion assuming a small echo, so that K2 << 1. Then
take the first three terms by dropping all other higher order terms. Note that
L=1+++P++++++ for x<1.]
(b.4) Draw a block diagram for representing this equalizer in timedomain by using
delay and scaler multiplication blocks [hint: see Table
Question (03): Lowpass, highpass, bandpass and bandrejection filtering [30 marks]
Filters are important buildingblocks of communication systems.
(a) Recall that the transfer function of an ideal lowpass filter can be written as
1/1 < B.
HLP(f)
=
0,
Ifl > B.
(a.1) Find and sketch the amplitude and phase response of this ideal lowpass filter.
[hint: sketch the phase response for I/l < B as it can be arbitrary for I/I>B.]
(a.2) Show that the impulse response of the lowpass filer is given by
hLp(t)=2KB sinec(2B(tta)) for 00<<<00
(a.3) Sketch the impulse response her(t).
(b) Let the input to this ideal lowpass be a rectangular pulse given by I(t) = A rect (+).
x(t)
HLP(f)
y(t)
Figure 5: Lowpass filtering
(b.1) Let the time delay be zero (i.e., td = 0) in the aforementioned impulse response
of the ideal lowpass filter. Show that the output signal of the lowpass filtering
in Fig.
can be written as
 
where Si(x) is defined as the Sine Integral and given by
Si(x) = 1. l
4
(c) The impulse response of the ideal lowpass filter can be truncated to obtain a practically
realizable causal function. The corresponding truncated impulse response can written
as follows:
h(t) = 0, 2K B sinc (2B(t  ta)), elsewhere 0 < t < 2td
(c.1) Sketch this truncated impulse response h(t).
(c.2) Show that the transfer function of this truncated impulse response is given by
K
= + 
(d) The amplitude spectrum of an ideal highpass filter is given by Fig.
(H())
K
f
 fi
fi
Figure 6: Highpass filtering
(d.1) Show that the transfer function of an ideal highpass filter can be written as
Hup(f)
(d.2) Find and sketch the impulse response hup(t) of this ideal highpass filter.
(e) The amplitude spectrum of an ideal bandpass filter is given by Fig.
7
H(f)l
K
f

fo

fi
fi
fu
Figure 7: Highpass filtering
The transfer function of an ideal highpass filter can be written as follows:
HBP(f) = 0, elsewhere
(e.1) Find and sketch the impulse response hBP(t) of this ideal bandpass filter.
5
(b). Any LTI operation has an equivalent transfer function. Table lists four transfer func
tions obtained by applying transform properties of Fourier transform to four primitive
timedomain operations.
Table 1: Transfer functions of four primitive timedomain operations
Timedomain operation
Description
Transfer function
Scalar multiplication
H(f)=K
Time delay
y(+)==(tts)
= exp(j2xftd
Differentiation
y(t)= =
of
Integration
H(1)121
Consider the zeroorder hold system in Fig. It has several applications in communi
cation systems. For instance, if a signal has discrete sample points, then a zeroorder
hold can be used to interpolate between the points.
r(t)
floodt
y(t)
Delay
T
Figure 2: Zeroorder hold system
(b.1) Draw a block diagram to represent the zeroorder hold in frequency domain.
(b.2) Find the overall transfer function of the zeroorder hold.
(b.3) Find and sketch the impulse response of the zeroorder hold.
(b.4) Let x(t) = A rect () be applied to the zeroorder hold. Find y(t) for the following
three cases; (i) T << T, (ii) T = T and (iii) T >> T.
(c) Show that the integrated value of the signal r(t) over the interval T (i.e., y(t) =
can be obtained by passing r(t) through the zeroorder hold system.
(hint: Express r(t) in terms of its inverse Fourier transform and then use the transfer
function of the zeroorder hold system evaluated in part (b).]
(d) The zeroorder hold is cascaded with another subsystem with a transfer function H1(f)
as shown in Fig.
I(t)
I'mo dt
H1(5)
y(t)
Delay
T
Figure 3: Cascaded zeroorder hold system
2
(d.1) Let the impulse response of the second block in Fig. be = u(t)  u(t  To),
where n(t) is the unit step function. Find and sketch the overall impulse response
of this cascaded system when T > To.
(d.2) Let the transfer function of the second block in Fig. be =
Find and sketch the overall impulse response of this cascaded system when T >>
1/B.
Question (02): Equalization [35 marks]
Recall that the transfer function of a distortionless transmission channel can be written as
=
where K and td are constants (see lecture 5 for more information). Theoretically, linear
distortion (in both amplitude and phase) in transmission channels can be mitigated by
using equalization. Consider the transmission channel with the cascaded equalizer for linear
distortion in Fig.
I(t)
Hc(f)
Heq(f)
y(t)
Channel
Equalizer
Figure 4: Channel with equalizer for linear distortion
(a) Find the transfer function of the equalizer Heq(f) which transforms the overall fre
quency response of the cascaded system to be a distortionless one in terms Hc(f), K
and t.
(b) Typically, wireless transmission systems suffer from multipath distortion caused by
multiple propagation paths between the transmitter and receiver. The channel output
of such a system can be written as follows:
y(t) = Kir(t  t1) + K2z(t  t2),
where t2>t1. This case ensures that the second term corresponds to an echo of the
first term.
(b.1) Find the impulse response he(t) of the aforementioned multipath channel.
(b.2) Show that the transfer function of the multipath channel is given by
where k: = K2/K1 and to = to  t1.
(b.3) If K  K1 and ts = t1, show that the transfer function of the equalizer Heq(f)
which makes the the overall frequency response of the system a distortionless one
can be written as
1
Heq(f) = 1+kexp(~j2nfto)
3
(f) The amplitude spectrum of an ideal bandrejection filter is given by Fig.
H(f)l
K
f
fu

fi
fi
fu
Figure 8: Highpass filtering
(f.1) Show that the transfer function of an ideal bandrejection filter can be written as
= 
(f.2) Find and sketch the impulse response hBR(t) of this ideal bandrejection filter.
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