DIGITAL COMMUNICATIONS (QUESTIONS AND ANSWERS)
1. Assume that a source produces the English letters (from A to Z, including space), and all these symbols will be produced with equal probability. Determine the entropy. Solution: Letters A to Z plus space = 27 characters/symbols P(i) = equal probability to all symbols H = log 2N bits per symbols H = log 2(27) H = 4.7549 bits per symbol
2. Consider a source that produces four symbols with probabilities of ½, ¼, 1/8, and 1/8, and all symbols are independent of each other. Determine the entropy. Solution: P(symbol1) = ½ P(symbol2) = ¼ P(symbol3) = 1/8 P(symbol4) = 1/8 H = - ∑P(i)log2P(i)
H = -(
H = - (-1.75) H = 1.75 bits per symbol
3. A telephone touch-tone keypad has the digits 0 to 9, plus the t he * and # keys. Assume the probability of sending * or # is 0.005 and the probability of sending 0 to 9 is 0.099 each. I f the keys are pressed at a rate of 2 keys/s, compute the entropy and data rate for this source. Solution: 0 to 9 = 10 symbols -> P(i) = 0.099 * / # = 2 symbols -> P(i) = 0 .005 Rate = 2 keys/sec H = - ∑P(i)log2P(i) H = - [ 10*(0.099log 20.099) + 2*(0.005log 20.005) ] H = - (-3.3795) = 3.3795 bits per keys Data rate = 3.3795 bits/keys * 2 keys/sec Data rate = 7.7590 bps
4. Consider a voice-grade line for which W = 3100 Hz, SNR = 30 dB (i.e., the signal-to-noise ratio is 1000:1). Determine the channel capacity. Solution: W = 3100 Hz SNR = 30 dB C = W log2(1+S/N) C = 3.1kHz*log2(1+1030/10) C = 30.8984 kbps
5. Consider an experiment in which two coins are thrown. What is the probability of getting one head and one tail? Solution: Possible results in tossing two coins = {HH, HT, TH, TT} Probability of getting one head and one tail = 2/4 = ½
6. A coin is flipped twice. Four different events are defined. A is the event of getting a head on the first flip. B is the event of getting a tail on the second flip. C is the event of a match between the two flips. D is the elementary event of a head on both flips. Find Pr{A}, Pr{B}, Pr{C}, Pr{D}, Pr{A|B}, and Pr{C|D}. Are A and B i ndependent? Are C and D independent? Solution: The events are defined by the following combination of outcomes. A = HH, HT B = HT, TT C = HH, TT D = HH Therefore, Pr{A} = Pr{B} = Pr{C} = 1/2 and Pr{D} = 1/4 Pr{A|B} = 0.5 and Pr{C|D} = 1 Since Pr{A|B} = Pr{A} , the event of a head on the first flip is independent of that of a tail on the second flip. Since Pr{C|D} ≠ Pr{C} , the event of a match and that of two heads are not independent
7. Which of the following codes are uniquely decipherable? For those that are uniquely decipherable, determine whether they are instantaneous. a. 0, 01, 001, 0011, 101 b. 110, 111, 101, 01 c. 0, 01, 011, 0110111
8. Find the minimum average length of a code with four messages with probabilities 1/8, 1/8, 1/4, and 1/2, respectively. Solution: H = - ∑P(i)log2P(i) H = - (1/8log21/8 + 1/8log21/8 + 1/4log21/4 + 1/2log21/2) = - (-1.75) H ≥ 1.75
9. Find the Huffman code for the following seven messages with probabilities as indicated: S1 S2 S3 S4 S5 S6 S7 0.05 0.15 0.2 0.05 0.15 0.3 0.1
10. Suppose that we wish to code five words, s 1, s2, s3, s4, and s5 with probabilities 1/16, 1/8, 1/4, 1/16, and 1/2, respectively.
11. Find the Shannon-Fano code for the following seven messages with probabilities as indicated:
S1 0.05
S2 S3 0.15 0.2
S4 0.05
S5 S6 0.15 0.3
S7 0.1
12. Calculate the number of levels if the number of bits per sample is: a. 8 (as in telephony) b. 16 (as in compact disc audio systems) Solution: N a. 2 = M 8 M = 2 = 256 levels N b. 2 = M 16 M = 2 = 65,536 levels
13. A telephone line has a bandwidth of 3.2 kHz and a signal-to-noise ratio of 35 dB. A signal is transmitted down this line using a four-level code. What is the maximum theoretical data rate? Solution: BW = 3.2kHz SNR = 35dB Level = 4 I = 2Blog2M I = 2(3.2kHz)log2(4) I = 12.8 kbps
14. An attempt is made to transmit a baseband frequency of 30 kHz using a digital audio system with a sampling rate of 44.1 kHz. What audible frequency would result? Solution: fm = 30Khz fs = 44.1 kHz fa = fs – fm fa = 44.1kHz – 30kHz fa = 14.1kHz
15. Find the maximum dynamic range for a linear PCM system using 16-bit quantizing. Solution: DR = 1.76 + 6.02M dB DR = 1.76 + 6.02(6) DR = 98.08 dB
16. Calculate the minimum data rate needed to transmit audio with a sampling rate of 40 kHz and 14 bits per sample. Solution: fs = 40kHz M = 14 bits DR = fsM = (40kHz)(14bits) DR = 560kbps
17. For a PCM system with the following parameters, determine (a) minimum sample rate, (b) minimum number of bits used in the PCM code, (c) resolution, and (d) quantization error. Maximum analog input frequency = 4 kHz Maximum decoded voltage at the receiver = ±2.55 V Minimum dynamic range = 46 dB Solution: a. fs = 2fm = 2(4kHz) = 8kHz N
b. DR = 2 -1 46/10 N DR = 10 = 2 -1 N 46/10 2 = 10 +1 N = 15.2809 = 16 bits c.
Resolution = Vmax / DR Resolution =
d. Qe = resolution/2 = 64.0531/2 Qe = 32.0255uV
18. A signal at the input to a mu-law compressor is positive with its voltage one-half the maximum value. What proportion of the maximum output voltage is produced? Solution: Vin = 0.5Vmax
⁄ ⁄
19. Code a positive-going signal with amplitude 30% of the maximum allowed as a PCM sample. 20. Convert the 12-bit sample 100110100100 into an 8-bit compressed code. 21. Suppose an input signal to a µ-law compressor has a positive voltage and amplitude 25% of the maximum possible. Calculate the output voltage as a percentage of the maximum output. 22. How would a signal with 50% of the maximum input voltage be coded in 8-bit PCM, using digital compression? 23. Convert a sample coded (using mu-law compression) as 11001100 to a voltage with the maximum sample voltage normalized at 1 V. 24. Convert the 12-bit PCM sample 110011001100 to an 8-bit compressed sample.
25. Suppose a composite video signal with a baseband frequency range from dc to 4 MHz is transmitted by linear PCM, using eight bits per sample and a sampling rate of 10 MHz. a. How many quantization levels are there? b. Calculate the bit rate, ignoring overhead.
c. What would be the maximum signal-to-noise ratio, in decibels? d. What type of noise determines the answer to part (c)? 26. The compact disc system of digital audio uses two channels with TDM. Each channel is sampled at 44.1 kHz and coded using linear PCM with sixteen bits per sample. Find: a. the maximum audio frequency that can be r ecorded (assuming ideal filters) b. the maximum dynamic range in decibels c. the bit rate, ignoring error correction and framing bits d. the number of quantizing levels
27. A modulator transmits symbols, each of which has sixty-four different possible states, 10, 000 times per second. Calculate the baud rate and bit rate. Solution: M = 64 Baud = 10 000 times per second Baud rate = 10 000 baud or 10 kbaud f b = baud x N = 10 000 x log2 64 = 60 kbps
28. Determine the baud and minimum bandwidth necessary to pass a 10 kbps binary signal using amplitude-shift keying. Solution: f b = 10 000 bps N = 1 (for ASK) B = f b / N = 10 000 / 1 = 10 000 Hz Baud = f b / N = 10 000 / 1 = 10 000 baud per second
29. Determine (a) the peak frequency deviation, (b) minimum bandwidth, and (c) baud for a binary FSK signal with a mark frequency of 49 kHz, a space frequency of 51 kHz, and an input bit rate of 2 kbps. Solution: fm = 49khz fs = 51kHz fb = 2kbps a. b. c.
30. The GSM cellular radio system uses GMSK in a 200-kHz channel, with a channel data rate of 270.833 kb/s. Calculate: (a) the frequency shift between mark and space (b) the transmitted frequencies if the carrier (center) frequency is exactly 880 MHz (c) the bandwidth efficiency of the scheme in b/s/Hz Solution: (a) f m – f s = 0.5 f b = 0.5 x 270.833 kb/s = 135.4165 kHz (b) f max = f c + 0.25f b = 880 MHz + 0.25 x 270.833 kHz = 880.0677 MHz f min = f c – 0.25f b = 880 MHz – 0.25 x 270.833 kHz = 879.93229 MHz
(c) The GSM system has a bandwidth efficiency of 270.833 / 200 = 1.35 b/s/Hz, comfortably under the theoretical maximum of 2 b/s/Hz for a two-level code.
31. Write the ASCII codes for the characters below. - B - b 32. For the following sequence of bits, identify the ASCII-encoded character, the start and stop bits, and the parity bits (assume even parity and two stop bits). 11111101000001011110001000 33. For the following string of ASCII-encoded characters, identify each character (assume odd parity): 01001111010101000001011011 34. A digital communication system has a symbol alphabet composed of four entries, and a transition matrix given by the following:
a. Find the probability of a single transmitted symbol being in error assuming that all four input symbols are equally probable at any time b. Find the probability of a correct symbol transmission. c. If the symbols are denoted as A, B, C, and D, find the probability that the transmitted sequence BADCAB will be received as DADDAB. Solution: a. Pe | 0 sent = P10 + P20 + P30 = ¼ + ¼ + ¼ = ¾ Pe | 1 sent = P01 + P21 + P31 = ½ + 1/6 + 1/6 = 5/6 Pe | 2 sent = P02 + P12 + P32 = 1/6 + ½ + 1/6 = 5/6 Pe | 3 sent = P03 + P13 + P23 = 1/6 + 1/6 + 1/3 = 2/3
∑ ()
b. c.
35. Consider a binary symmetric channel for which the conditional probability of error p = 10 -4 , and symbols 0 and 1 occur with equal probability. Calculate the following probabilities: a. The probability of receiving symbol 0 b. The probability of receiving symbol 1 c. The probability that symbol 0 was sent, given that symbol 0 is received d. The probability that symbol 1 was sent, given that symbol 0 is received Solution: a. P(B0) = ½ b. P(B1) = ½ -4 c. P(A0|B0)=1-10 -4 d. P(A0|B1) = 10
36. Find the minimum distance for the following code consisting of four code words: 0111001, 1100101, 0010111, 1011100 How many bit errors can be detected? How many bit errors can be corrected?
37.
38.
39. Determine the BCS for the following data and CRC generating sequence: Data, G = 10110111 CRC P = 110011 40. Determine the VRCs and LRC for the following ASCII-encoded message: THE CAT. Use odd parity for the VRCs and even parity for the LRC 41. For a 12-bit data string of 101100010010, determine the number of Hamming bits required, arbitrarily place the Hamming bits into the data string, determine the logic condition of each Hamming bit, assume an arbitrary single-bit transmission error, and prove that the Hamming code will successfully detect the error.
42. Determine the Hamming bits for the ASCII character “B”. Insert the Hamming bits into every other bit location starting from the left. 43. Determine the Hamming bits for the ASCII character “C” (use odd parity and two stop bits). Insert the Hamming bits into every other location starting from at the right
