# Digital modulation techniques

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1 Outline Introduction Signal, random variable, random process and spectra Analog modulation Analog to digital conversion Digital transmission through baseband channels Signal space representation Optimal receivers Digital modulation techniques Channel coding Synchronization Information theory Communications Engineering 1

2 Digital modulation techniques Binary digital modulation M-ary digital modulation Comparison study Chapter 8.2,8.3.3, , ,9.7 Communications Engineering 2

3 Digital modulation techniques In digital communications, the modulation process corresponds to switching or keying the amplitude, frequency, or phase of a sinusoidal carrier wave corresponding to incoming digital data Three basic digital modulation techniques 1. Amplitude-shift keying (ASK) - special case of AM 2. Frequency-shift keying (FSK) - special case of FM 3. Phase-shift keying (PSK) - special case of PM We use signal space approach in receiver design and performance analysis Communications Engineering 3

4 Binary digital modulation In binary signaling, the modulator produces one of two distinct signals in response to one bit of source data at a time. Binary modulation type Communications Engineering 4

5 Binary digital modulation Binary Phase-Shift Keying (BPSK) Modulation bit duration : carrier frequency, chosen to be for some fixed integer or : transmitted signal energy per bit, i.e., The pair of signals differ only in a 180-degree phase shift Communications Engineering 5

6 Binary digital modulation Binary Phase-Shift Keying (BPSK) Signal space representation: So and A binary PSK system is characterized by a signal space that is one-dimensional (N=1), and has two message points (M=2) Assume that the two signals are equally likely, i.e., Communications Engineering 6

7 Binary digital modulation Binary Phase-Shift Keying (BPSK) The optimal decision boundary is the midpoint of the line joining these two message points Decision rule: 1. Guess signal s1(t) (or binary 1) was transmitted if the received signal point falls in region 2. Guess signal s2(t) (or binary 0) was transmitted otherwise (r 0) Communications Engineering 7

8 Binary digital modulation Binary Phase-Shift Keying (BPSK) Probability of error analysis. The conditional probability of the receiver deciding in favor of s2(t) given that s1(t) was transmitted is Due to symmetry P( e s 2 ) = P( r > 0 s 2 ) = Q 2E N b 0 Communications Engineering 8

9 Binary digital modulation Binary Phase-Shift Keying (BPSK) Probability of error analysis. Since the signals s1(t) and s2(t) are equally likely to be transmitted, the average probability of error is This ratio is normally called bit energy to noise density ratio (SNR/bit) Communications Engineering 9

10 Binary digital modulation Binary Phase-Shift Keying (BPSK) Transmitter. Communications Engineering 10

11 Binary digital modulation Binary Phase-Shift Keying (BPSK) Receiver. θ is the carrier-phase offset, due to propagation delay or oscillators at the transmitter and receiver are not synchronous The detection is coherent in the sense of phase synchronization and timing synchronization Communications Engineering 11

12 Binary digital modulation Binary Frequency-Shift Keying (BFSK) Modulation : transmitted signal energy per bit : transmitted frequency with separation is selected so that s1(t) and s2(t) are orthogonal, i.e., (Example?) Communications Engineering 12

13 Binary digital modulation Binary Frequency-Shift Keying (BFSK) Signal space representation: Communications Engineering 13

14 Binary digital modulation Binary Frequency-Shift Keying (BFSK) Decision regions: 1. Guess signal s1(t) (or binary 1) was transmitted if the received signal point falls in region (r2>r1) 2. Guess signal s2(t) (or binary 0) was transmitted otherwise (r 0) Communications Engineering 14

15 Binary digital modulation Binary Frequency-Shift Keying (BFSK) Probability of error analysis. Given that s1 is transmitted Since the condition r2>r1 corresponds to the receiver making a decision in favor of symbol s2, the conditional probability of error when s1 is transmitted is given by n1 and n2 are i.i.d. Gaussian with Then is Gaussian with Communications Engineering 15

16 Binary digital modulation Binary Frequency-Shift Keying (BFSK) Probability of error analysis. By symmetry, we also have Since the two signals are equally likely to be transmitted, the average probability of error for coherent binary FSK is Communications Engineering 16

17 Binary digital modulation Binary Frequency-Shift Keying (BFSK) Transmitter. Communications Engineering 17

18 Binary digital modulation Binary Frequency-Shift Keying (BFSK) Receiver. Communications Engineering 18

19 Binary digital modulation Binary Amplitude-Shift Keying (BASK) Modulation. Average energy per bit Decision region Communications Engineering 19

20 Binary digital modulation Binary Amplitude-Shift Keying (BASK) Probability of error analysis. Average probability of error Prove it! Communications Engineering 20

21 Binary digital modulation Comparison In general, Communications Engineering 21

22 Binary digital modulation Comparison Communications Engineering 22

23 Binary digital modulation Example Binary data are transmitted over a microwave link at the rate of 10 6 bits/sec and the PSD of the noise at the receiver input is watts/hz. Find the average carrier power required to maintain an average probability of error for coherent binary FSK. What if noncoherent binary FSK? Communications Engineering 23

24 Binary digital modulation Update We have discussed coherent modulation schemes, e.g., BPSK, BFSK, BASK, which need coherent detection assuming that the receiver is able to detect and track the carrier wave s phase In many practical situations, strict phase synchronization is not possible. In these situations, noncoherent reception is required. We now consider non-coherent detection on binary FSK and differential phase-shift keying (DPSK) Communications Engineering 24

25 Binary digital modulation Non-coherent scheme: BFSK Consider a binary FSK system, the two signals are : unknown random phases with uniform distribution Communications Engineering 25

26 Binary digital modulation Non-coherent scheme: BFSK Since Choose four basis functions as Signal space representation Communications Engineering 26

27 Binary digital modulation Non-coherent scheme: BFSK The vector representation of the received signal Communications Engineering 27

28 Binary digital modulation Non-coherent scheme: BFSK Decision rule: Conditional pdf ML Similarly Communications Engineering 28

29 Binary digital modulation Non-coherent scheme: BFSK For ML decision, we need to evaluate Removing the constant terms We have Communications Engineering 29

30 Binary digital modulation Non-coherent scheme: BFSK By definition where I0() is a modified Bessel function of the zero-th order Thus, the decision rule becomes: choose s1 if Communications Engineering 30

31 Binary digital modulation Non-coherent scheme: BFSK Note that this Bessel function is monotonically increasing. Therefore, we choose s1 if 1. Useful insight: we just compare the energy in the two frequencies and pick the larger (envelope detector) 2. Carrier phase is irrelevant in decision making Communications Engineering 31

32 Binary digital modulation Non-coherent scheme: BFSK Structure. See Section Communications Engineering 32

33 Binary digital modulation Comparison Communications Engineering 33

34 Binary digital modulation Differential PSK (DPSK) Non-coherent version of PSK Phase synchronization is eliminated using differential encoding 1. Encode the information in phase difference between successive signal transmission. 2. Send 0, advance the phase of the current signal by 180 o 3. Send 1, leave the phase unchanged Provided that the unknown phase θ contained in the received wave varies slowly (constant over two bit intervals), the phase difference between waveforms received in two successive bit intervals will be independent of θ Communications Engineering 34

35 Binary digital modulation Differential PSK (DPSK) Generate DPSK signals in two steps 1. Differential encoding of the information binary bits. 2. Phase shift keying Differential encoding starts with an arbitrary reference bit Communications Engineering 35

36 Binary digital modulation Differential PSK (DPSK) Structure. Communications Engineering 36

37 Binary digital modulation Differential PSK (DPSK) Differential detection. Output of integrator (assume noise free) The unknown phase θ becomes irrelevant. The decision becomes: if (bit 1), then y>0; if (bit 0), then y<0 Communications Engineering 37

38 Binary digital modulation Comparison Communications Engineering 38

39 Binary digital modulation Comparison Communications Engineering 39

40 M-ary digital modulation Why? Communications Engineering 40

41 M-ary digital modulation In binary data transmission, send only one of two possible signals during each bit interval Tb In M-ary data transmission, send one of M possible signals during each signaling interval T In almost all applications, M=2 n and T=nTb, where n is an integer Each of the M signals is called a symbol These signals are generated by changing the amplitude, phase, frequency, or combined forms of a carrier in M discrete steps. Thus, we have MASK, MPSK, MFSK, and MQAM Communications Engineering 41

42 M-ary digital modulation M-ary Phase-shift Keying (MPSK) Modulation: The phase of the carrier takes on M possible values Signal set Es=Energy per symbol, Basis functions Communications Engineering 42

43 M-ary digital modulation M-ary Phase-shift Keying (MPSK) Signal space representation. Communications Engineering 43

44 M-ary digital modulation M-ary Phase-shift Keying (MPSK) Signal constellations. Communications Engineering 44

45 M-ary digital modulation M-ary Phase-shift Keying (MPSK) Euclidean distance The minimum Euclidean plays an important role in determining error performance as discussed previously (union bound) In the case of PSK modulation, the error probability is dominated by the erroneous selection of either one of the two signal points adjacent to the transmitted signal point Consequently, an approximation to the symbol error probability is Communications Engineering 45

46 M-ary digital modulation M-ary Phase-shift Keying (MPSK) Exercise: Consider the M=2, 4, 8 PSK signal constellations. All have the same transmitted signal energy Es. Determine the minimum distance between adjacent signal points For M=8, determine by how many db the transmitted signal energy Es must be increased to achieve the same as M=4. Communications Engineering 46

47 M-ary digital modulation M-ary Phase-shift Keying (MPSK) For large M, doubling the number of phases requires an additional 6 db/bit to achieve the same performance Communications Engineering 47

48 M-ary digital modulation M-ary Quadrature Amplitude Modulation (MQAM) In MPSK, in-phase and quadrature components are interrelated in such a way that the envelope is constant (circular constellation) If we relax this constraint, we get M-ary QAM Communications Engineering 48

49 M-ary digital modulation M-ary Quadrature Amplitude Modulation (MQAM) Modulation: is the energy of the signal with the lowest amplitude are a pair of independent integers Basis functions Signal space representation Communications Engineering 49

50 M-ary digital modulation M-ary Quadrature Amplitude Modulation (MQAM) Signal constellation. Communications Engineering 50

51 M-ary digital modulation M-ary Quadrature Amplitude Modulation (MQAM) Probability of error analysis. Upper bound of the symbol error probability Think about the increase in Eb required to maintain the same error performance if the number of bits per symbol is increased from k to k+1, where k is large. Communications Engineering 51

52 M-ary digital modulation M-ary Frequency-shift Keying (MFSK) (Multitone Signaling) Signal set: where Correlation between two symbols Communications Engineering 52

53 M-ary digital modulation M-ary Frequency-shift Keying (MFSK) (Multitone Signaling) For orthogonality, the minimum frequency separation is Communications Engineering 53

54 M-ary digital modulation M-ary Frequency-shift Keying (MFSK) (Multitone Signaling) Geometrical representation. Basis functions. Communications Engineering 54

55 M-ary digital modulation M-ary Frequency-shift Keying (MFSK) (Multitone Signaling) Probability of error. Communications Engineering 55

56 M-ary digital modulation Notes Pe is found by integrating conditional probability of error over the decision region, which is difficult to compute but can be simplified using union bound Pe depends only on the distance profile of the signal constellation Communications Engineering 56

57 M-ary digital modulation Gray Code Symbol errors are different from bit errors When a symbol error occurs, all k bits could be in error In general, we can find BER using nij the number of different bits between si and sj Gray coding is a bit-to-symbol mapping, where two adjacent symbols differ in only one bit out of the k bits An error between adjacent symbol pairs results in one and only one bit error Communications Engineering 57

58 M-ary digital modulation Gray Code Communications Engineering 58

59 M-ary digital modulation Example The 16-QAM signal constellation shown right is an international standard for telephone-line modems (called V.29) Determine the optimum decision boundaries for the detector Derive the union bound of the probability of symbol error assuming that the SNR is sufficiently high so that errors only occur between adjacent points Specify a Gray code for this 16- QAM V.29 signal constellation Communications Engineering 59

60 M-ary digital modulation Gray Code For MPSK with Gray coding, we know that an error between adjacent symbols will most likely occur. Thus, bit error probability can be approximated by For MFSK, when an error occurs, anyone of the other symbols may result equally likely. Thus, k/2 bits every k bits will on average be in error when there is a symbol error. The bit error rate is approximately half of the symbol error rate Think about why MQAM is more preferrable? Communications Engineering 60

61 M-ary digital modulation Channel bandwidth and transmit power are two primary communication resources and have to be used as efficient as possible Power utilization efficiency (energy efficiency): measured by the required Eb/N0 to achieve a certain bit error probability Spectrum utilization efficiency (bandwidth efficiency): measured by the achievable data rate per unit bandwidth Rb/B It is always desired to maximize bandwidth efficiency at a minimal required Eb/N0 Communications Engineering 61

62 M-ary digital modulation Consider for example you are a system engineer in Huawei/ZTE, designing a part of the communication systems. You are required to design a modulation scheme for three systems using MFSK, MPSK or MQAM only. State the modulation level M to be low, medium or high Communications Engineering 62

63 M-ary digital modulation Energy efficiency comparison Communications Engineering 63

64 M-ary digital modulation Energy efficiency comparison MFSK: At fixed Eb/N0, increasing M can provide an improvement on Pb; At fixed Pb, increasing M can provide a reduction in the Eb/N0 MPSK: BPSK and QPSK have the same energy efficiency. At fixed Eb/N0, increasing M degrades Pb; At fxied Pb, increasing M increases the Eb/N0 requirement MFSK is more energy efficient than MPSK Communications Engineering 64

65 M-ary digital modulation Bandwidth efficiency comparison To compare bandwidth efficiency, we need to know the power spectral density (power spectra) of a given modulation scheme MPSK/MQAM If is rectangular, the bandwidth of main-lobe is If it has a raised cosine spectrum, the bandwidth is Communications Engineering 65

66 M-ary digital modulation Bandwidth efficiency comparison In general, bandwidth required to pass MPSK/MQAM signal is approximately given by The bit rate is So the bandwidth efficiency may be expressed as But for MFSK, bandwidth required to transmit MSFK signal is Adjacent frequencies need Bandwidth efficiency to be separated by 1/2T to maintain orthogonality Communications Engineering 66

67 M-ary digital modulation Bandwidth efficiency comparison In general, bandwidth required to pass MPSK/MQAM signal is approximately given by The bit rate is So the bandwidth efficiency may be expressed as MPSK/MQAM is more bandwidth efficient than MFSK But for MFSK, bandwidth required to transmit MSFK signal is Adjacent frequencies need Bandwidth efficiency to be separated by 1/2T to maintain orthogonality Communications Engineering 67

68 M-ary digital modulation Fundamental tradeoff: Bandwidth Efficiency vs. Energy Efficiency To see the ultimate power-bandwidth tradeoff, we need to use Shannon s channel capacity theorem: Channel capacity is the theoretical upperbound for the maximum rate at which information could be transmitted without error (Shannon 1948) Specifically, for a bandlimited channel corrupted by AWGN, the maximum achievable rate is given by Note that Thus, Communications Engineering 68

69 M-ary digital modulation Fundamental tradeoff: Bandwidth Efficiency vs. Energy Efficiency Communications Engineering 69

70 M-ary digital modulation Fundamental tradeoff: Bandwidth Efficiency vs. Energy Efficiency In the limits as R/B goes to 0, we get This value is called the Shannon limit. Received Eb/N0 must be >-1.59 db to ensure reliable communication BPSK and QPSK require the same Eb/N0 of 9.6 db to achieve Pe=10-5. However, QPSK has a better bandwidth efficiency. MQAM is superior to MPSK MPSK/MQAM increases bandwidth efficiency at the cost of energy efficiency MFSK trades energy efficiency at reduced bandwiidth efficiency Communications Engineering 70

71 M-ary digital modulation Fundamental tradeoff: Bandwidth Efficiency vs. Energy Efficiency Which modulation to use? Communications Engineering 71

72 M-ary digital modulation Consider for example you are a system engineer in Huawei/ZTE, designing a part of the communication systems. You are required to design a modulation scheme for three systems using MFSK, MPSK or MQAM only. State the modulation level M to be low, medium or high Communications Engineering 72

73 M-ary digital modulation Practical applications BPSK: WLAN IEEE b (1 Mbps) QPSK: QAM: FSK: 1. WLAN IEEE b (2 Mbps, 5.5 Mbps, 11 Mbps) 2. 3G WCDMA 3. DVB-T (with OFDM) 1. Telephone modem (16-QAM) 2. Downstream of Cable modem (64-QAM, 256-QAM) 3. WLAN IEEE a/g (16-QAM for 24 Mbps, 36 Mbps; 64-QAM for 38 Mbps and 54 Mbps) 4. LTE cellular Systems 5. 5G 1. Cordless telephone 2. Paging system Communications Engineering 73

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