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Question 1

The auto-correlation function for an energy signal, f(t), is given by

R(tau) = 1 exp(-tau2/8).

Calculate the energy of this signal.

Question 2

In a binary communications system with bit period 11, in order to send bit O, we send the following signal

s0(t) = 2t, for 0<t<11/2 + 3, and, zero, elsewhere,

and, in order to send bit 1, we send the following signal

s1(t) = 2t, for 11/2 - 3<t<11, and, zero, elsewhere.

If we represent signals s0(t) and s1(t) by their vector representation, what would be the distance between the corresponding vectors?

You should round your final answer to two decimal digits. That is, if the answer is 12.456, you can leave it as 12.46.

Question 3

In a binary communications system, with a bit rate of 36 Hz, we use a correlation receiver for an additive white Gaussian noise channel with two-sided power spectral density NO/2 = 1 E -2 (in Joules). Bits O and 1 are equally likely and are, respectively. represented by the following signals:

so(t) = 10 sin(3t), for o<t < T, and zero elsewhere.

s1(t) =10 cos(3t), for 0 <t <T, and zero elsewhere,

where T is the bit period of the above communications system, and the amplitudes are in  unit.

Specify the value of x, if the BER for this system is given by Q(x).

You should round your final answer to two decimal digits. That is, if the answer is 12.456, you can leave it as 12.46.

Question 4

We have a signal generator that can generate pulses of a fix duration of 2.3 micro second and a fix amplitude 1,806 (in ) at a carrier frequency of 1 GHz. We can, however, change the phase of the carrier as desired. Using this generator, we need to send 1 Gb of data through a free space channel of 10 km length. Our transmitter antenna has 20 dB gain, and the receiver antenna has 1 m2 of area.

We consider two options:

1- Using BPSK modulation: and

2- Using QPSK modulation.

In both cases, we can use the corresponding correlation receiver for an AWGN channel with N0/2 = 10-8J.

In the answer box, specify x-y, if the BER for the above BPSK system is given by Q(x), and the BER for the QPSK system is given by Q(y).

You should round your final answer to two decimal digits, that is, if the answer is 12.456, you can leave it as 12.46.

Question 5

What is small-scale fading? Briefly explain by giving an example.

Your answer should fit into one short paragraph. You may highlight the relevant keywords.

Question 6

In a DWDM optical-fibre communications system, there are 40 channels per optical fibre, each supporting 2.5 Gbps of data transmission using OOK modulation.

1. Explain how wavelength division multiplexing helps us better utilise the available bandwidth in an optical fibre.

2. The above DWDM system is being used to exchange data between two data centres 100 km apart from each other. If the typical data traffic between the two centres is 10 Tbps, how many strands of fibre are required to connect these two centres?

3. For all channels in the above DWDM system, the channel loss is known to be roughly 0.2 dB/km and the transmitted intensity for bit 1 is 0.1mw. Find the bit error rate of the system, for a channel with central wavelength 1.5 um, assuming that the receiver is an ideal photon-counter. (No thermal noise or dark current is assumed; photodetectors have unity quantum efficiencies.)