30. Signal Space Analysis of BASK, BFSK, BPSK, and QAM

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1 Signal Space Analysis of BASK, BFSK, BPSK, an QAM on Mac 3. Signal Space Analysis of BASK, BFSK, BPSK, an QAM The vector-space representation of signals an the optimum etection process which chooses the signal closest to the receive signal is particularly useful in signal esign an in probability-of-error calculations. The material presente herein inclues the error performance of binary amplitue-shift keying (BASK, binary frequency-shift keying (BFSK, binary phase-shift keying (BPSK, an M-ary quarature amplitue moulation (QAM signals. rror Performance of Binary ASK A binary amplitue-shift keying (BASK signal can be efine by s(t Acos( πf c t, t T, elsewhere (3. where A is a constant, f c is the carrier frequency, an T is the bit uration. It has a power P A /, so that A P. Thus equation (3. can be written as s(t Pcos( π f c t, t T, elsewhere PT cos( π f c t, t T T, elsewhere cos( π f c t, t T T, elsewhere (3. where PT is the energy containe in a bit uration. Figure 3. shows the signal constellation iagram of BASK signals an the conitional probability ensity functions associate with the signals. Figure 3. BASK signal constellation iagram an the conitional probability ensity functions associate with the signals. The energy containe in a bit uration is an so (3.3 3.

2 Signal Space Analysis of BASK, BFSK, BPSK, an QAM on Mac (3.4 It is assume that the noise is aitive white Gaussian noise (AWGN with a two-sie power spectral ensity of n /, zero mean an fixe variance σ n /. In the presence of AWGN, the conitional probability ensity function of assuming that s is transmitte is f( / πσ e σ an the probability of error given that s is transmitte is P e / f( / / πσ e σ (3.5 Similarly, the probability of error given that s is transmitte is P e / / f( / πσ ( e σ P e where f( / πσ ( e σ (3.6 Let p be the probability of sening s an p be the probability of sening s. For equally likely transmission of binary signals, we have p p.5. The average probability of error is given by P e p P e + p P e P e / e σ πσ (3.7 3.

3 Signal Space Analysis of BASK, BFSK, BPSK, an QAM on Mac Let y. Then y σ σ an y σ equation (3.7, we get. Substituting y an y into P e e π -y y y σ [ π e-y y ] y erfc( σ erfc( n erfc( (3.8 4n where the complementary error function is erfc (x π e-α α (3.9 x σ n / for AWGN, an. rror Performance of Binary PSK A binary phase-shift keying (BPSK signal can be efine by s(t +A cos πf c t, < t < T (3. where A is a constant, f c is the carrier frequency, an T is the bit uration. It has a power P A /, so that A P. Thus equation (3. can be written as s(t + P cos πf c t + PT T cos πf c t + T cos πf c t (3. 3.3

4 Signal Space Analysis of BASK, BFSK, BPSK, an QAM on Mac where PT is the energy containe in a bit uration. Figure 3. shows the signal constellation iagram of BPSK signals an the conitional probability ensity functions associate with the signals. Figure 3. BPSK signal constellation iagram an the conitional probability ensity functions associate with the signals. The energy containe in a bit uration is ( (3. an so 4 (3.3 In the presence of AWGN, the conitional probability ensity function of assuming that s is sent is f( / πσ ( + e σ (3.4 an the probability of error given that s is transmitte is P e f( / πσ ( + e σ (3.5 Similarly, the probability of error given that s is sent is P e f( / πσ ( e σ P e (3.6 where 3.4

5 Signal Space Analysis of BASK, BFSK, BPSK, an QAM on Mac f( / πσ ( e σ (3.7 Let p be the probability of sening s an p be the probability of sening s. For equally likely transmission of binary signals, we have p p.5. The average probability of error is P e p P e + p P e P e ( + e σ πσ (3.8 + Let y. Then y σ ( + σ an y σ equation (3.8, we get. Substituting y an y into P e + y σ π π + y σ erfc ( + σ erfc ( σ erfc( n e -y y e -y y erfc( n (3.9 where is the threshol value, the complementary error function is 3.5

6 Signal Space Analysis of BASK, BFSK, BPSK, an QAM on Mac erfc (x π e-α α (3. x σ n / for AWGN, an 4. rror Performance of Orthogonal Binary FSK A binary frequency-shift keying (BFSK signal can be efine by s(t Acos( π f c t, t T Acos( π f c t, elsewhere (3. where A is a constant, f c an f c are the carrier frequencies, an T is the bit uration. It has a power P A /, so that A P. Thus equation (3. can be written as s(t Pcos( π f c t, t T Pcos( π f c t, elsewhere PT cos( π f c t T, t T PT cos( π f c t T, elsewhere cos( π f c t T, t T cos( π f c t, T elsewhere (3. where PT is the energy containe in a bit uration. Figure 3.3 shows the signal constellation iagram of orthogonal BFSK signals an the conitional probability ensity functions associate with the signals. Figure 3.3 Orthogonal BFSK signal constellation iagram an the conitional probability ensity functions associate with the signals. The energy in a bit uration is ( (3.3 an 3.6

7 Signal Space Analysis of BASK, BFSK, BPSK, an QAM on Mac (3.4 If we rotate the two orthonormal axes an to u an u, we en up with a signal constellation iagram for BPSK signals with a separation istance of. Thus, the average probability of error is P e erfc( σ erfc( n erfc( (3.5 n σ n / for AWGN, an. Table 3. summarises the performance of BPSK, orthogonal BFSK, an BASK signals in the presenec of AWGN. Moulation P e erfc( σ Relative (B erfc( n BPSK 4 erfc( n Orthogonal BFSK erfc( n -3 BASK erfc( 4n -6 Table 3. Performance of various moulation systems. Also, we can express the performance of these moulation systems in terms of the average signal energy, where / (3.6 for BASK signals, 3.7

8 Signal Space Analysis of BASK, BFSK, BPSK, an QAM on Mac (3.7 for orthogonal BFSK signals, an (3.8 for BPSK signals. This is summarise in Table 3.. Moulation P e Relative (B BPSK erfc( n erfc( n Orthogonal BFSK erfc( n erfc( n -3 BASK / erfc( 4n erfc( n -3 Table 3. Performance of various binary moulation systems in terms of the average signal energy. rror Performance of M-ary QAM Consier the (M6-ary quarature amplitue moulation (QAM signal constellation iagram of Figure 3.4 as an example. Figure ary QAM signal constellation iagram. The average signal energy of an M-ary QAM signal is M i (3.9 M i where i is the energy of s i. For M 6, we have 6 [ ] s 569,,, s 43478,,,,,,, s,, 3, 5 3.8

9 Signal Space Analysis of BASK, BFSK, BPSK, an QAM on Mac 5 (3.3 an so 5 (3.3 To fin the probability of error, it is simpler to first fin the probability P c of correct etection. It is apparent from Figure 3.4 that we can group signal points into 3 classes an compute P c base on the 3 ecision regions as shown in Figure 3.5. Figure 3.5 Decision regions. Let n an n be aitive white Gaussian noise samples with two-sie power spectral ensity of n /, zero mean an fixe variance σ n / along the -axis an - axis, respectively. Case - four inner signal points (s 5,6,9, : In the presence of AWGN, the probability of correct etection given that s i, i 5, 6, 9,, is transmitte is P(C/s i P(-/ < n < / x P(-/ < n < / P(-/ < n < / x P(-/ < n < / p x p (3.3 where p P(-/ < n < / / e σ / πσ / e σ πσ (

10 Signal Space Analysis of BASK, BFSK, BPSK, an QAM on Mac Let y. Then y σ σ an y σ equation (3.33, we get. Substituting y an y into p σ erf( σ erf( where the error function is π e -y y (3.34 n erf (x π x e-α α (3.35 an σ n / for AWGN. Case - four corner signal points (s,3,,5 : In the presence of AWGN, the probability of correct etection given that s i, i, 3,, 5, is transmitte is where P(C/s i P( < n < / x P(-/ < n < (3.36 P(-/ < n < x P(-/ < n < P(-/ < n < x P(-/ < n < r x r r P(-/ < n < e σ / πσ 3.

11 Signal Space Analysis of BASK, BFSK, BPSK, an QAM on Mac e σ / πσ + e σ πσ / e σ πσ + e σ πσ.5p +.5 (3.37 Case 3 - eight ege signal points (s,,4,7,8,,3,4 : In the presence of AWGN, the probability of correct etection given that s i, i,, 4, 7, 8,, 3, 4, is transmitte is P(C/s i P(-/ < n < / x P(-/ < n < P(-/ < n < / x P(-/ < n < p x r (3.38 Let P(s i be the probability of sening s i, for i,,..., 5. Thus, the total probability of correct etection is P c M 5 i Ps ( i P(C/s 3 i (3.39 a priori probability With equally likely transmission of s i, we have P c 4 ( 6 p x p + 4 ( 6 r x r + 8( 6 p x r 6 [ 4 p + 4( p + + 8p ( p + ] (3p+ 6 (3.4 The probability of error is P e ( - P c - (3p+ 4 (3p ( p(5 + 3p 6 3.

12 Signal Space Analysis of BASK, BFSK, BPSK, an QAM on Mac For normal operation, p approaches an P e <<, an then P e 3( p8 6 3 ( - p (3.4 Substituting p erf ( n an into equation (3.4, we have 5 P e 3 [ erf ( n ] 3 erfc ( n (3.4 xample 3. Consier the 4-ary QAM signal constellation iagram as shown in Figure 3.6. The average signal energy is an Figure ary QAM signal constellation iagram. 4 [ ] 5. From our previous analysis of 6-ary QAM signals, the probability of correct etection in 4-ary QAM signals given that s i, i,,, 3, is transmitte is where P(C/s i P( < n < / x P(-/ < n < P(-/ < n < x P(-/ < n < P(-/ < n < x P(-/ < n < r x r 3.

13 Signal Space Analysis of BASK, BFSK, BPSK, an QAM on Mac r.5p +.5 p erf( σ erf( n an σ n / for AWGN. The total probability of correct etection is P c M 3 i Ps ( i P(C/s 3 i a priori probability With equally likely transmission of s i, we have P c 4 ( 4 r x r ( p + ( + p 4 The probability of error is P e ( - P c - ( + p 4 ( p ( 3 + p 4 For normal operation, p approaches an P e <<, an then P e 4 ( p ( - p ( Substituting p erf ( n an into equation (3.43, we have P e erf ( erfc(. n n 3.3

14 Signal Space Analysis of BASK, BFSK, BPSK, an QAM on Mac References [] M. Schwartz, Information Transmission, Moulation, an Noise, 4/e, McGraw- Hill, 99. [] H. Taub an D. L. Schilling, Principles of Communication Systems, /e, MaGraw-Hill,

15 Signal Space Analysis of BASK, BFSK, BPSK, an QAM on Mac π σ e - σ Figure 3. Probability ensity function f ( / T sin π f c t f ( / ( π σ e - σ Pe Pe s s / T cos π f c t BASK signal constellation iagram an the conitional probability ensity functions associate with the signals. ( + π σ e - σ f ( / Pe s - T sin π f ( c t - f ( / π σ e - σ Pe s T cos π f t c Figure 3. BPSK signal constellation iagram an the conitional probability ensity functions associate with the signals. (u + π σ e - f (u / σ T cos π f t c s s u f (u / (u - π σ e - σ T cos π f c t u Figure 3.3 Orthogonal BFSK signal constellation iagram an the conitional probability ensity functions associate with the signals. 3.5

16 Signal Space Analysis of BASK, BFSK, BPSK, an QAM on Mac s T sin π f c t Decision bounary s s s 3 / / / / / s 7 s 6 s 5 s 4 s 8 s s 9 s T cos π f c t s 5 s 4 s 3 s Figure ary QAM signal constellation iagram. / s s -/ / / -/ s / -/ -/ -/ (a (b Figure 3.5 Decision regions. (c 3.6

17 Signal Space Analysis of BASK, BFSK, BPSK, an QAM on Mac T sin π f c t Decision bounary s s / / T cos π f c t s s 3 / / Figure ary QAM signal constellation iagram. 3.7

Thus there are three basic modulation techniques: 1) AMPLITUDE SHIFT KEYING 2) FREQUENCY SHIFT KEYING 3) PHASE SHIFT KEYING

Thus there are three basic modulation techniques: 1) AMPLITUDE SHIFT KEYING 2) FREQUENCY SHIFT KEYING 3) PHASE SHIFT KEYING CHAPTER 5 Syllabus 1) Digital modulation formats 2) Coherent binary modulation techniques 3) Coherent Quadrature modulation techniques 4) Non coherent binary modulation techniques. Digital modulation formats: