Discrete Communications

Objective:

Revision of fundamental concepts and demonstration of necessary foundation skills.

Assessment:

The assignment will comprise 20% of your final mark and all ten problems will be of equal value.

The problems

1. Complete the problem below AND use MATLAB to confirm your sketches, ensure you submit your MATLAB code and the plots it generates.

The signals in Fig. P3,3-6 are modulated signals with carrier cos 10t. Find the Fourier transforms of these signals by using the appropriate properties of the Fourier transform and Table 3.1. Sketch the amplitude and phase spectra for Fig. P3.3-6a and b.

Hint: These functions can be expressed in the form g(t) cos 2Πf_ot.

2. The distortion caused by multipath transmission can be partly corrected by a tapped delay-line equalizer. Show that if α << 1, the distortion in the multipath system in Fig. 3.31a can be approximately corrected if the received signal in Fig. 3.31a is passed through the tapped delay-line equalizer shown in Fig. P3.6-2.

Hint: From Eq. (3.64a), it is clear that the equalizer filter transfer function should be H_eq(f) = 1/ (1+ αe^-jΠfΔt). Use the fact that 1/(1 - x) = 1 + x + x² + x³ + ...... if x << 1 to show what should be the tap parameters ai to make the resulting transfer function

H (f)H_eq(f) ≈ e^-jΠft_d

3. The random binary signal x(t) shown in Fig. P3.8-2 transmits one digit every T_b seconds. A binary 1 is transmitted by a pulse p(t) of width T_b/2 and amplitude A., a binary 0 is transmitted by no pulse. The digits 1 and 0 are equally likely and occur randomly. Determine the autocorrelation function R_x(τ) and the PSD S_x(f).

4. The American Standard Code for Information Interchange (ASCII) has 128 characters, which are binary-coded. If a certain computer generates 100,000 characters per second, determine the following:

(a) The =Tiber of bits (binary digits) required per character.

(b) The number of bits per second required to transmit the computer output, and the minimum bandwidth required to transmit this signal.

(c) For single error deiection capability, an additional bit (panty bit) is added to the code of each character. Modify your answers in parts (a) and (b) in view of this information.

5. Five telemetry signals, each of bandwidth 240 Hz, are to be transmitted simultaneously by binary KM. The signals must be sampled at least 20% above the Nyquist rate. Framing and synchroniz¬ing requires an additional 0.5% extra bits. A PCM encoder is used to convert these signals before they are time-multiplexed into a single data stream. Determine the minimum possible data rate (bits per second) that must be transmitted, and the minimum bandwidth required to transmit the multiplex signal.

6. In a binary data transmission using duobinary pulses, sample values of the received pulses were read as follows:

12000 -200 -20200 -20220 -2

(a) Explain if there is any error.

(b) Can you guess the correct transmitted digit sequence? There is more than one possible correct sequence. Give as many correct sequences as passible, assuming that more than crie detection error is extremely unlikely.

7.  Using a computer-based tool of your choice (MATLAB, MS-Excel, or anything else you consider appropriate) construct a program which can automatically discover errors in duobinary pulse sequences and decode them to recover the original bit sequence in the case where the sequence is error-free. Test your solution out on the pulse sequences presented in 7.3-10 below (a previous tutorial problem)

120 - 2 - 200 - 202002000 - 2

and 7.3-11 below.

12000 - 200 - 20200 - 20220 - 2

12000 -200 -20200 -20220 -2

8. In a certain binary communication system that uses Nyquist's criterion pulses, a received pulse P_r(t) (see Fig. 7.22a) has the following nonzero sample values:

p_r(0) = 1
P_r(T_b) = 0.1 p_r(-T_b) = 0.3
p_r(24) 0.02 p_r(-24)= -0.07

(a) Determine the tap settings of a three-tap, zero-forcing equalizer.

(b) Using the equalizer in part (a), find the residual nonzero ISI,

9 Take the MATLAB program binary_eye. m from the end from the end of chapter 7 (available on UTSOnline) and modify it so that you can add noise of various forms to the pulses. Examine the resulting eye diagrams, print some of them out and add your own comments and explanations.

10 Review Example 7.4 again below which we studied in lecture 5 ...

Example 7.4 Determine the PSD of the quaternary (4-ary) baseband signaling in Fig. 7.28 when the message bits 1 and 0 are eqUally likely.
The 4-ary line code has four distinct symbols corresponding to the four different combinations of two message bits. One such mapping is

                 -3 message bits 00
                 -1 message bits 01
ak =
                 +1 message bits 10
                 +3 message bits 11 (7.56)

Therefore, all four values of a_k are equally likely, each with a chance of 1 in 4. Recall that

R_o = lim_N→∞ 1/NΣa²_k

Within the summation, 1/4 of the a_k will be ±1, and ±3 Thus,

R_o = lim_N→∞ 1/N[N/4(-3)² + N/4 (-1)² + + N/4(1)² + + N/4(3)²] = 5

On the other hand, for n > 0, we need to determine

R_n = lim_N→∞ 1/N ∑_k a_ka_k+n

To find this average value, we build a table with all the possible values of the product a_ka_k+n:

From the foregoing table listing all the possible products of a_ka_k+n, we see that each product in the summation a_ka_k+n, can take on any of the following six values ±1, ±3, ±9. First, (±1, ±9) are equally likely (1 in 8). On the other hand, 13 are equally likely (1 in 4).

Thus, we can show that

R_n = lim_N→∞1/N[N/8(-9) + N/8(+9) + N/8(-1) + N/8(+1) + N/4(-3) + N/4(+3)] = 0

As a result,

S_x(f) = 5/T_s ⇒ S_y(f) = 5/T_s|P(f)|²

Thus, the M-ary line code generates the same PSD. shape as binary polar signaling. The only difference is that it utilizes 5 times the original signal power.

Notice there is a suggestion that R_n = 0 for all n > 0. Using MATLAB or MS-Excel, demonstrate that this is the case for 8-ary coding. Then use your results to develop an argument which proves that it must be true for all M-ary schemes.

Note: a rigorous proof isn't required in this instance.