Chapter 8: M -ary Signaling Techniques EE456 – Digital Communications Professor Ha Nguyen September 2016 EE456 – Digital Communications 1
Chapter 8: M -ary Signaling Techniques Chapter 8: M -ary Signaling Techniques EE456 – Digital Communications 2
Chapter 8: M -ary Signaling Techniques Introduction to M -ary Signaling Techniques There are benefits to be gained when QPSK is used rather than the straightforward BPSK signaling. In general, an M -ary communication system can be designed for either bandwidth efficient or power efficient applications. Whether being bandwidth efficient or power efficient depends on how the M -ary signal set is constructed. Unlike QPSK ( M = 4 ), when designed for bandwidth-efficient applications with M > 4 , 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. EE456 – Digital Communications 3
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. EE456 – Digital Communications 4
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. EE456 – Digital Communications 5
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. EE456 – Digital Communications 6
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 2 t s 3 t s k ( t s M − 1 t s M ( t ( ) ( ) ( ) ) ( ) ) φ 1 t ✙ ✘ ( ) ∆ ∆ − ) ∆ ( M − ) ∆ ( M − ) ∆ ( k 2 2 1 0 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 WGN, strength 0 watts/Hz 2 EE456 – Digital Communications 7
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 � ∆ r 1 > � M − 3 s M ( t ) , if 2 ( ) f r s ( t ) 1 k r 1 ∆ − ) ∆ 0 ( k 1 ⇐ ⇒ Choose ( ) s t Choose s M ( t ) 1 Choose s k ( t ) 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 2( M − 1) � � � P [ error ] = Q ∆ / 2 N 0 . M For a given M , P [ error ] depends on the noise power ( N 0 ) and the minimum distance ∆ . This means that moving the origin of the signal constellation does not affect the performance! EE456 – Digital Communications 8
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 moving the origin to the center of the constellation, i.e., allowing both positive and negative values of the signal component. � 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 EE456 – Digital Communications 9
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). EE456 – Digital Communications 10
Chapter 8: M -ary Signaling Techniques Equivalent Implementation Structures of Optimum M -ASK Receiver t = kT s ∆ ( t ) r kT r s ( ) ∫ 1 • d t r 1 0 − ( k 1 ) T s φ = π ( t ) 2 cos( 2 f t ) 1 T s c t = kT s h ( t ) ∆ ~ ( t ) r 1 r 1 ~ r 1 t 0 T s π V cos( 2 f t ) c In practice, the thresholds (i.e., decision boundaries) for the Slicer are computed (estimated) directly from the output of the matched filter. The minimum distance ˜ ∆ between closest signal points is related to the “average” of | ˜ r 1 | as: M/ 2 M/ 2 � ˜ � � ˜ � = M ˜ 1 ∆ 1 ∆ ∆ ∆ = 4 | ˜ r 1 | � � � � ˜ | ˜ r 1 | ≈ � (2 k − 1) 2 + ˜ w 1 � ≈ (2 k − 1) 2 + ˜ w 1 ⇒ � � � � M/ 2 M/ 2 4 M k =1 k =1 The minimum distance ˜ r 2 ∆ can also be computed from the “average” of ˜ 1 as: � M/ 2 � 2 = ( M 2 − 1) ˜ � ˜ ∆ 2 12 r 2 1 ∆ � ˜ r 2 1 1 ≈ (2 k − 1) ⇒ ∆ = M 2 − 1 . M/ 2 2 12 k =1 EE456 – Digital Communications 11
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 EE456 – Digital Communications 12
Chapter 8: M -ary Signaling Techniques M -ary Phase-Shift Keying ( M -PSK) � � 2 πf c t − ( i − 1)2 π , 0 ≤ t ≤ T s ; i = 1 , 2 , . . . , M ; E s = V 2 T s / 2 s i ( t ) = V cos M � ( i − 1)2 π � ( i − 1)2 π � � = V cos cos(2 πf c t ) + V sin sin(2 πf c t ) . M M � ( i − 1)2 π � � ( i − 1)2 π � � � s i 1 = E s cos , s i 2 = E s sin . M M 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 The signals lie on a circle of radius √ E s , and are spaced every 2 π/M radians around the circle. EE456 – Digital Communications 13
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 ✱ • t the smallest d 0 φ 2 t ( ) P [ error ] = P [ error | s 1 ( t )] = 1 − P [ correct | s 1 ( t )] �� = 1 − f ( r 1 , r 2 | s 1 ( t ))d r 1 d r 2 . r 2 Region 2 r 1 ,r 2 ∈ Region 1 s 2 t � �� � Choose ( ) P [ correct | s 1 ( t )] s 2 t ( ) E s s 1 t π ( ) M r 1 0 Region 1 s 1 t Choose ( ) EE456 – Digital Communications 14
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 ( ) E π M sin s s 1 t s π M ( ) s 1 t ( ) r π M 1 r 0 Region 1 ( ) 1 0 E ,0 s 1 t s Choose ( ) P [ error | s 1 ( t )] = P [( r 1 , r 2 ) fall outside Region 1 | s 1 ( t )] > P [( r 1 , r 2 ) fall in ℜ 1 | s 1 ( t )] , � π � � � � P [ error | s 1 ( t )] > Q sin 2 E s /N 0 . M EE456 – Digital Communications 15
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