ELEC/XJEL3430
Digital Communications
PROBLEM SET 5
Problem 1: Bandpass Binary Communication
Consider a binary communication system with bit rate 1/T in which equi-probable databits 0 and 1 are, respectively, encoded by the following pulse functions:
s0 (t) = A cos(wt), 0 < t < 2T / 3
, and zero elsewhere.
s1(t) = A cos(wt), T / 3 < t < T
Suppose that the communication channel is an AWGN one with two-sided spectral density N0 / 2 with no other distortions.
Suppose wT>>1.
a) Find the energy associated with the above pulse shapes. What is the average energy per bit, Eb ?
b) Find the inner product between s0 (t) and s1(t) .
c) Draw s0 (t) and s1(t) as vectors in a plane. Specify the length of each vector as well as the angle between the two vectors.
d) Obtain and draw the optimal matched-filter receiver for this signaling.
e) Find the bit error rate for this system and write it in terms of Eb / N0 .
f) Using MATLAB, plot the bit error rate in part (e) as well as in part (g) of Problem 1 (on the same plot) versus Eb / N0 in dB. Use the log scale for the vertical axis and find the value of Eb / N0 at which the BER = 1E-5. Label each graph properly.
g) Now, let us consider the loss in the channel. Suppose our communication channel is a free- space channel of 15 km length. At the transmitter, we use a directed antenna of 14 dB gain, and at the receiver we use a receiver antenna with 2 m2 area. Can you write the bit error rate in part (e) in terms of Eb / N0 at the transmitter.
h) For N0 = 10- 15 Joules and T = 1 ns, plot BER vs transmitted power. How much transmission power do you need to achieve BER = 1 E -5?
Problem 2: Sending a Digitised Signal
In order to transmit a band-limited analogue signal, with a bandwidth of 10 kHz, we sample the signal at twice its Nyquist rate and quantise each sample using an 8-bit uniform quantiser. The resulted signal will then be sent using a digital communications system.
(a) How many bits per second are being generated by the above source-sampler-quantiser?
(b) Suppose we use QPSK modulation to transmit, in real time, the above generated data. If no channel or source coding is in place, how many symbols per second do we need to send?
(c) Now consider a BPSK modulator, with a bit rate as specified by part (a) and with a carrier frequency of 1 GHz . At the receiver, we use a correlation receiver. Assume that the channel is distortion-free, but the receiver has additive white Gaussian noise with two-sided power spectral density of 1E-16 Joules. Find the minimum required energy per bit, at the receiver, as well as its corresponding power, to achieve a bit error rate less than or equal to 1E-5. Assume that Q(4.25) = 1E-5.
(d) Next, let us assume that, instead of the distortion-free channel in part (c), the channel
between the transmitter and receiver is a 40-km-long free-space channel with no fading; see Fig. 1. If we use a transmitter antenna with a 10 dB gain and a receiver antenna with an area of 1 m2, how much power is required for the transmitter in part (c)? (Hint: if the transmitter antenna gain is 1, then, at any distance L, the transmitted power would be spread equally over a sphere with a radius of L.)
Fig. 1. The setup in part (d) and (e).
(e) Suppose the free-space channel in part (d), suffers from small-scale fading effects.
Explain what small-scale fading is and suggest one technique to mitigate it.
(f) Repeat parts (c) and (d) for a real-time communications system that used QPSK modulation.
Problem 3: Fading Channels
Consider a binary communication system with bit rate 1/T in which equi-probable databits 0 and 1 are, respectively, encoded by the following signals:
s0 (t) = A cos(wt), 0 < t < T
s1(t) = - A cos(wt), 0 < t < T .
The above signal will go through a fading channel with a random amplitude gain X. That is the received amplitude for bit 0 will be AX, and that of bit 1 would be -AX. Suppose X is uniformly distributed between 0 and 1.
a) Calculate the average energy per bit at the transmitter.
b) If X = x, for a fixed value of x, what would be the average energy per bit at the receiver?
c) Average over X in your answer in part (b) to find the average energy per bit at the receiver.
d) Suppose the receiver has an additive white Gaussian noise with double-sided power spectral density of N0 / 2 . Draw the correlation receiver for this system.
e) Calculate the bit error rate for your correlation receiver. Hint: You can first find the
conditional error rate when X or thermal noise is known, and then average over that variable.