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Chapter 8: M -ary Signaling Techniques A First Course in Digital Communications Ha H. Nguyen and E. Shwedyk February 2009 A First Course in Digital Communications 1/46

Chapter 8: M -ary Signaling Techniques Introduction There are benefits to be gained when M -ary ( M = 4 ) signaling methods are used rather than straightforward binary signaling. In general, M -ary communication is used when one needs to design a communication system that is bandwidth efficient. Unlike QPSK and its variations, the gain in bandwidth is accomplished at the expense of error performance. To use M -ary modulation, the bit stream is blocked into groups of λ bits ⇒ the number of bit patterns is M = 2 λ . The symbol transmission rate is r s = 1 /T s = 1 / ( λT b ) = r b /λ symbols/sec ⇒ there is a bandwidth saving of 1 /λ compared to binary modulation. Shall consider M -ary ASK, PSK, QAM (quadrature amplitude modulation) and FSK. A First Course in Digital Communications 2/46

Chapter 8: M -ary Signaling Techniques Optimum Receiver for M -ary Signaling ✞ r ✒ ✞ m ✡ s i ( t ( t ✡ m ✁ ✂ ✟ ✠ ✁ ✄ ✆ ✎ ✁ ✂ ✟ ✠ ✁ ✄ ) ˆ ) � ✁ ✂ ✄ ☎ ✆ i ✝ i ☛ ☞ ☛ ✓ ✡ ✡ ✄ ✠ ✌ ✍ ✎ ✏ ✆ ✄ ✑ ✆ ☎ ✆ ✏ ✔ ✆ ✄ ✑ w ( t ) w ( t ) is zero-mean white Gaussian noise with power spectral density of N 0 2 (watts/Hz). Receiver needs to make the decision on the transmitted signal based on the received signal r ( t ) = s i ( t ) + w ( t ) . The determination of the optimum receiver (with minimum error) proceeds in a manner analogous to that for the binary case. A First Course in Digital Communications 3/46

Chapter 8: M -ary Signaling Techniques Represent M signals by an orthonormal basis set, { φ n ( t ) } N n =1 , N ≤ M : s i ( t ) = s i 1 φ 1 ( t ) + s i 2 φ 2 ( t ) + · · · + s iN φ N ( t ) , � T s s ik = s i ( t ) φ k ( t )d t. 0 Expand the received signal r ( t ) into the series r ( t ) = s i ( t ) + w ( t ) = r 1 φ 1 ( t ) + r 2 φ 2 ( t ) + · · · + r N φ N ( t ) + r N +1 φ N +1 ( t ) + · · · For k > N , the coefficients r k can be discarded. Need to partition the N -dimensional space formed by � r = ( r 1 , r 2 , . . . , r N ) into M regions so that the message error probability is minimized. A First Course in Digital Communications 4/46

Chapter 8: M -ary Signaling Techniques N − dimensiona l observatio n space ✗ = r r r ✖ r ( , , , ) M 1 2 ℜ 1 s t m Choose ( ) or 1 1 ℜ ℜ ✕ 2 M s t m Choose ( ) or s t m 2 2 Choose ( ) or M M The optimum receiver is also the minimum-distance receiver : Choose m i if � N k =1 ( r k − s ik ) 2 < � N k =1 ( r k − s jk ) 2 ; j = 1 , 2 , . . . , M ; j � = i. A First Course in Digital Communications 5/46

Chapter 8: M -ary Signaling Techniques M -ary Coherent Amplitude-Shift Keying ( M -ASK) � 2 s i ( t ) = V i cos(2 πf c t ) , 0 ≤ t ≤ T s T s � 2 = [( i − 1)∆] φ 1 ( t ) , φ 1 ( t ) = cos(2 πf c t ) , 0 ≤ t ≤ T s , T s i = 1 , 2 , . . . , M. s 1 t s 3 t s k ( t s 2 t ✙ ✘ s M − 1 t s M ( t ( ) ) ( ) ( ) ( ) ) φ 1 t ( ) ∆ − ) ∆ − ) ∆ ∆ ( k − ) ∆ ( M ( M 0 2 1 2 1 t = kT s r s i ( t ( t r ✣ ✤ ✣ ✥ ✚ m ) ) kT ✦ ✛ ✜ ✢ ˆ s ( ) i 1 • t ✣ d ✧ ✛ ✜ ✢ ✜ − k T ( 1 ) s w ( t φ 1 t ) ( ) N 0 WGN, strength watts/Hz 2 A First Course in Digital Communications 6/46

Chapter 8: M -ary Signaling Techniques Minimum-Distance Decision Rule for M -ASK  � � � � k − 3 k − 1 s k ( t ) , if ∆ < r 1 < ∆ , k = 2 , 3 , . . . , M − 1  2 2 r 1 < ∆ Choose s 1 ( t ) , if . 2 � �  M − 3 s M ( t ) , if r 1 > ∆ 2 ( ) f r s t ( ) k 1 r 1 ∆ ( k − ) ∆ 0 1 ★ s t ⇐ s M t Choose ( ) Choose ( ) 1 s k t Choose ( ) A First Course in Digital Communications 7/46

Chapter 8: M -ary Signaling Techniques Error Performance of M -ASK ( ) f r s t ( ) k 1 r 1 ∆ ∆ 2 2 M � P [ error ] = P [ s i ( t )] P [ error | s i ( t )] . i =1 � � � P [ error | s i ( t )] = 2 Q ∆ / 2 N 0 , i = 2 , 3 , . . . , M − 1 . � � � P [ error | s i ( t )] = Q ∆ / 2 N 0 , i = 1 , M. P [ error ] = 2( M − 1) � � � Q ∆ / 2 N 0 . M A First Course in Digital Communications 8/46

Chapter 8: M -ary Signaling Techniques Modified M -ASK Constellation The maximum and average transmitted energies can be reduced, without any sacrifice in error probability, by changing the signal set to one which includes the negative version of each signal. � s i ( t ) = (2 i − 1 − M )∆ 2 cos(2 πf c t ) , 0 ≤ t ≤ T s , i = 1 , 2 , . . ., M. 2 T s � �� � V i ✬ ✭ ✮ φ 1 t ( ) 3 ∆ ∆ ∆ 3 ∆ 0 ✩ − − ✩ 2 2 2 2 ✬ ✯ ✮ φ 1 t ( ) ✫ ✪ − 2 ∆ − ∆ ∆ ∆ 2 0 � M M (2 i − 1 − M ) 2 = ( M 2 − 1)∆ 2 = ∆ 2 i =1 E i � E s = . M 4 M 12 i =1 � log 2 M = ( M 2 − 1)∆ 2 E s (12 log 2 M ) E b E b = ⇒ ∆ = M 2 − 1 12 log 2 M A First Course in Digital Communications 9/46

Chapter 8: M -ary Signaling Techniques Probability of Symbol Error for M -ASK �� �� � � P [ error ] = 2( M − 1) 6 E s = 2( M − 1) 6 log 2 M E b Q Q . ( M 2 − 1) N 0 M 2 − 1 M M N 0 �� � P [ bit error ] = 1 λP [ symbol error ] = 2( M − 1) 6 log 2 M E b M log 2 M Q (with Gray mapping) M 2 − 1 N 0 −1 10 −2 10 M =16 ( W =1/4 T b ) −3 10 P [symbol error] M =8 ( W =1/3 T b ) −4 10 M =4 ( W =1/2 T b ) −5 10 M =2 ( W =1/ T b ) −6 10 −7 10 0 5 10 15 20 25 E b / N 0 (dB) W is obtained by using the W T s = 1 rule-of-thumb. Here 1 /T b is the bit rate (bits/s). A First Course in Digital Communications 10/46

Chapter 8: M -ary Signaling Techniques Example of 2-ASK (BPSK) and 4-ASK Signals Baseband information signal 1 0 −1 0 Tb 2Tb 3Tb 4Tb 5Tb 6Tb 7Tb 8Tb 9Tb 10Tb BPSK Signalling 2 0 −2 0 Tb 2Tb 3Tb 4Tb 5Tb 6Tb 7Tb 8Tb 9Tb 10Tb 4−ASK Signalling 2 0 −2 0 2Tb 4Tb 6Tb 8Tb 10Tb A First Course in Digital Communications 11/46

Chapter 8: M -ary Signaling Techniques M -ary Phase-Shift Keying ( M -PSK) � � 2 πf c t − ( i − 1)2 π s i ( t ) = V cos , 0 ≤ t ≤ T s , M i = 1 , 2 , . . . , M ; f c = k/T s , k integer ; E s = V 2 T s / 2 joules � ( i − 1)2 π � � ( i − 1)2 π � s i ( t ) = V cos cos(2 πf c t )+ V sin sin(2 πf c t ) . M M φ 1 ( t ) = V cos(2 πf c t ) , φ 2 ( t ) = V sin(2 πf c t ) √ E s √ E s . � ( i − 1)2 π � � ( i − 1)2 π � � � s i 1 = E s cos , s i 2 = E s sin . M M The signals lie on a circle of radius √ E s , and are spaced every 2 π/M radians around the circle. A First Course in Digital Communications 12/46

Chapter 8: M -ary Signaling Techniques Signal Space Plot of 8-PSK � � 2 πf c t − ( i − 1)2 π s i ( t ) = V cos , 0 ≤ t ≤ T s , M i = 1 , 2 , . . . , M ; f c = k/T s , k integer ; E s = V 2 T s / 2 joules φ 2 t ( ) s t ↔ ( ) 011 3 ↔ s 4 t s t ↔ 010 ( ) ( ) 001 2 E s π ↔ s 5 t 4 ↔ s t 110 ( ) ( ) 000 1 0 φ 1 t ( ) ↔ ↔ s 6 t s t 111 ( ) ( ) 100 8 s t ↔ ( ) 101 7 A First Course in Digital Communications 13/46

Chapter 8: M -ary Signaling Techniques Signal Space Plot of General M -PSK � � 2 πf c t − ( i − 1)2 π s i ( t ) = V cos , 0 ≤ t ≤ T s , M i = 1 , 2 , . . . , M ; f c = k/T s , k integer ; E s = V 2 T s / 2 joules φ 2 t ( ) s 2 t ( ) E s π s 1 t M ( ) 2 φ 1 t ( ) 0 − π M 2 s M ( t ) A First Course in Digital Communications 14/46

Chapter 8: M -ary Signaling Techniques Optimum Receiver for M -PSK t = T s ✱ r T s ( ) 1 • t d Compute 0 ( ) ( ) r − s 2 + r − s 2 r ✰ ( t m ) ˆ i i 1 1 2 2 i i = M for 1 , 2 , , φ 1 t ( ) t = T s and choose ✲ r T s ( ) 2 • the smallest t d 0 φ 2 t ( ) P [ error ] = P [ error | s 1 ( t )] � � r 2 = 1 − f ( r 1 , r 2 | s 1 ( t ))d r 1 d r 2 . Region 2 s 2 t Choose ( ) r 1 ,r 2 ∈ Region 1 s 2 t ( ) E s s 1 t π M ( ) r 1 0 Region 1 s 1 t Choose ( ) A First Course in Digital Communications 15/46

Chapter 8: M -ary Signaling Techniques Lower Bound of P [ error ] of M -PSK r 2 Region 2 s 2 t Choose ( ) r 2 s 2 t s 2 t ( ) ( ) ℜ 1 ( ) E π M sin E s s s 1 t s 1 t π M ( ) π ( ) M r r 1 ( ) 0 1 0 E Region 1 ,0 s s 1 t Choose ( ) P [ error | s 1 ( t )] > P [ r 1 , r 2 fall in ℜ 1 | s 1 ( t )] , or � π � � � � P [ error | s 1 ( t )] > Q sin 2 E s /N 0 . M A First Course in Digital Communications 16/46

Chapter 8: M -ary Signaling Techniques Upper Bound of P [ error ] of M -PSK r 2 s 2 t ( ) ℜ 1 ( ) π E M sin s r s 1 t ( ) π M 2 Region 2 r ( ) 0 1 s 2 t E Choose ( ) ,0 s s 2 t ( ) r E 2 s s 1 t π M ( ) r ( ) 1 0 E ,0 Region 1 s r s 1 t − π Choose ( ) M 1 0 s 1 t ( ) ( ) E π M sin s ℜ s M ( t 2 ) P [ error] < P [ r 1 , r 2 fall in ℜ 1 | s 1 ( t )] + P [ r 1 , r 2 fall in ℜ 2 | s 1 ( t )] , or � π � � � � P [ error] < 2 Q sin 2 E s /N 0 , M A First Course in Digital Communications 17/46