ML Performance of M -ary Signaling Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay October 10, 2012 1 / 43
Performance of ML Decision Rule for M -ary signaling
ML Decision Rule for M -ary Signaling The ML decision rule for M -ary signaling in a real AWGN channel is � y , s i � − � s i � 2 � � 1 ≤ i ≤ M � y − s i � 2 = arg max δ ML ( y ) = arg min 2 1 ≤ i ≤ M The ML decision rule for M -ary signaling in a complex AWGN channel is Re ( � y , s i � ) − � s i � 2 � � 1 ≤ i ≤ M � y − s i � 2 = arg max δ ML ( y ) = arg min 2 1 ≤ i ≤ M In both cases, the rule can be represented as δ ML ( y ) = arg max 1 ≤ i ≤ M Z i where Z i is the decision statistic 3 / 43
ML Decision Rule for Binary Signaling ML decision rule � y , s i � − � s i � 2 � � δ ML ( y ) = arg max 1 ≤ i ≤ 2 Z i = arg max 2 1 ≤ i ≤ 2 Probability of error � � s 0 − s 1 � 2 � � s 0 − s 1 � � P e = Q = Q 2 σ 2 N 0 � s 0 � 2 + � s 1 � 2 � Let E b = 1 � . For antipodal signaling, 2 �� � 2 E b P e = Q N 0 4 / 43
ML Decision Rule for Binary Signaling For on-off keying, s 1 ( t ) = s ( t ) and s 0 ( t ) = 0 and �� � E b P e = Q N 0 For orthogonal signaling, s 1 ( t ) and s 2 ( t ) are orthogonal �� � E b P e = Q N 0 5 / 43
Performance Comparison of Antipodal and Orthogonal Signaling 10 − 1 Orthogonal Antipodal 10 − 3 10 − 5 P e 10 − 7 10 − 9 0 2 4 6 8 10 12 14 16 18 20 E b N 0 (dB) 6 / 43
ML Decision Rule for QPSK Y s ( −√ E b , √ E b ) ( √ E b , √ E b ) Y c ( −√ E b , −√ E b ) ( √ E b , −√ E b ) � � � � � � P e | 1 = Pr Y c < 0 or Y s < 0 � ( E b , E b ) was sent � 7 / 43
ML Decision Rule for QPSK � � � � � � P e | 1 = Pr Y c < 0 or Y s < 0 � ( E b , E b ) was sent � �� �� � � 2 E b 2 E b − Q 2 = 2 Q N 0 N 0 By symmetry, P e | 1 = P e | 2 = P e | 3 = P e | 4 Since the four constellation points are equally likely, the probability of error is given by �� �� 4 � � P e = 1 2 E b 2 E b � − Q 2 P e | i = P e | 1 = 2 Q 4 N 0 N 0 i = 1 8 / 43
ML Decision Rule for 16-QAM 16-QAM 3 A A − 3 A − A A 3 A − A − 3 A Exact analysis is tedious. Approximate analysis is sufficient. 9 / 43
Revisiting the Q function
Revisiting the Q function X ∼ N ( 0 , 1 ) � ∞ � − t 2 � 1 Q ( x ) = P [ X > x ] = √ exp dt 2 2 π x p ( t ) Q ( x ) Φ( x ) x t 11 / 43
Bounds on Q ( x ) for Large Arguments Q ( x ) 10 − 1 UB in (1) LB in (1) 10 − 3 10 − 5 10 − 7 5 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 x � e − x 2 ≤ Q ( x ) ≤ e − x 2 � 1 − 1 2 2 √ √ (1) x 2 x 2 π x 2 π 12 / 43
Bounds on Q ( x ) for Small Arguments Q ( x ) 10 − 1 UB in (1) UB in (2) 10 − 3 10 − 5 10 − 7 5 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 x Q ( x ) ≤ 1 2 e − x 2 (2) 2 13 / 43
Bounds on Q ( x ) for Small Arguments 1 Q ( x ) UB in (1) 0 . 8 UB in (2) 0 . 6 0 . 4 0 . 2 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 x 14 / 43
Q Functions with Smallest Arguments Dominate 10 0 Q ( x ) Q ( x ) + Q ( 2 x ) Q ( x ) + Q ( 2 x ) + Q ( 3 x ) 10 − 1 10 − 2 10 − 3 0 0 . 5 1 1 . 5 2 2 . 5 3 x • P e in AWGN channels can typically be bounded by a sum of Q functions • The Q function with the smallest argument is used to approximate P e 15 / 43
Union Bound Analysis
Union Bound for M -ary Signaling in AWGN The conditional error probability given H i is true is � � � � � � P e | i = Pr ∪ j � = i Z i < Z j � H i � Since P ( A ∪ B ) ≤ P ( A ) + P ( B ) , we have � � � � � s j − s i � � � � � P e | i ≤ Pr Z i < Z j � H i = Q � 2 σ j � = i j � = i The error probability for prior probabilities π i is given by � � s j − s i � � � � � P e = π i P e | i ≤ Q π i 2 σ i i j � = i 17 / 43
Union Bound for QPSK Y s s 2 s 1 Y c s 3 s 4 � � � � � � � � � � � P e | 1 = Pr ∪ j � = 1 Z 1 < Z j � H 1 ≤ Pr Z 1 < Z j � H 1 � � j � = 1 � � s 2 − s 1 � � � � s 3 − s 1 � � � � s 4 − s 1 � � P e | 1 ≤ Q + Q + Q 2 σ 2 σ 2 σ �� �� � � 2 E b 4 E b = 2 Q + Q N 0 N 0 18 / 43
Union Bound for QPSK Union bound on error probability of ML rule �� �� � � 2 E b 4 E b P e ≤ 2 Q + Q N 0 N 0 Exact error probability of ML rule �� �� � � 2 E b 2 E b − Q 2 P e = 2 Q N 0 N 0 19 / 43
QPSK Error Events E 1 = [ Z 2 > Z 1 ] ∪ [ Z 3 > Z 1 ] ∪ [ Z 4 > Z 1 ] = [ Z 2 > Z 1 ] ∪ [ Z 4 > Z 1 ] Z 2 > Z 1 Z 3 > Z 1 Z 4 > Z 1 Y s Y s Y s s 2 s 1 s 2 s 1 s 2 s 1 Y c Y c Y c s 3 s 4 s 3 s 4 s 3 s 4 E 1 Y s s 2 s 1 Y c s 3 s 4 20 / 43
Intelligent Union Bound for QPSK � � � � P e | 1 = Pr ( Z 2 > Z 1 ) ∪ ( Z 4 > Z 1 ) � H 1 � � � � � � � � � ≤ Pr Z 2 < Z 1 � H 1 + Pr Z 2 < Z 1 � H 1 � � � � s 2 − s 1 � � � � s 4 − s 1 � � = Q + Q 2 σ 2 σ �� � 2 E b = 2 Q N 0 By symmetry P e | 1 = P e | 2 = P e | 3 = P e | 4 and �� � 2 E b P e ≤ 2 Q N 0 21 / 43
Summary of results for QPSK Exact error probability of ML rule �� �� � � 2 E b 2 E b − Q 2 P e = 2 Q N 0 N 0 Union bound on error probability of ML rule �� �� � � 2 E b 4 E b P e ≤ 2 Q + Q N 0 N 0 Intelligent union bound on error probability of ML rule �� � 2 E b P e ≤ 2 Q N 0 22 / 43
Intelligent Union Bound for 16-QAM 3 A A − 3 A − A A 3 A − A − 3 A Assignment 4 23 / 43
Nearest Neighbors Approximation Let d min be the minimum distance between constellation points d min = min i � = j � s i − s j � Let N d min ( i ) denote the number of nearest neighbors of s i � d min � P e | i ≈ N d min ( i ) Q 2 σ Averaging over i we get � d min � P e ≈ ¯ N d min Q 2 σ where ¯ N d min denotes the average number of nearest neighbors 24 / 43
Nearest Neighbors Approximation for 16-QAM 3 A A − 3 A − A A 3 A − A − 3 A Assignment 4 25 / 43
Bit Error Probability of ML Rules
Bit Error Probability of ML Decision Rule • Probability of bit error is also termed bit error rate (BER) • For fixed SNR, symbol error probability depends only on constellation geometry • For fixed SNR, BER depends on both constellation geometry and the bits to signal mapping Gray coded bitmap for QPSK Other bitmap for QPSK Y s Y s 10 00 11 00 Y c Y c 11 01 10 01 • For an M -ary constellation, number of possible bitmaps is M ! = M ( M − 1 ) · · · 3 · 2 · 1 27 / 43
Bit Error Rate for QPSK using Gray Bitmap Gray coded bitmap for QPSK Y s 10 00 Y c 11 01 Conditional BER when b [ 1 ] b [ 2 ] = 00 is � � � 1 b [ 1 ]ˆ ˆ � P b | 00 = 2 Pr b [ 2 ] = 01 � b [ 1 ] b [ 2 ] = 00 � � � � + 1 b [ 1 ]ˆ ˆ � 2 Pr b [ 2 ] = 10 � b [ 1 ] b [ 2 ] = 00 � � � � b [ 1 ]ˆ ˆ � + Pr b [ 2 ] = 11 � b [ 1 ] b [ 2 ] = 00 � 28 / 43
Bit Error Rate for QPSK using Gray Bitmap Gray coded bitmap for QPSK Y s 10 00 Y c 11 01 � 2 E b Let α = N 0 � � � b [ 1 ]ˆ ˆ � Pr b [ 2 ] = 01 � b [ 1 ] b [ 2 ] = 00 = Q ( α ) [ 1 − Q ( α )] � � � � ˆ b [ 1 ]ˆ � Pr b [ 2 ] = 10 � b [ 1 ] b [ 2 ] = 00 = Q ( α ) [ 1 − Q ( α )] � � � � Q 2 ( α ) ˆ b [ 1 ]ˆ � Pr b [ 2 ] = 11 � b [ 1 ] b [ 2 ] = 00 = � 29 / 43
Bit Error Rate for QPSK using Gray Bitmap Gray coded bitmap for QPSK Y s 10 00 Y c 01 11 Conditional BER when b [ 1 ] b [ 2 ] = 00 is 1 2 Q ( α ) [ 1 − Q ( α )] + 1 2 Q ( α ) [ 1 − Q ( α )] + Q 2 ( α ) = P b | 00 �� � 2 E b = Q ( α ) = Q N 0 �� � 1 2 E b � � P b = P b | 00 + P b | 01 + P b | 10 + P b | 11 = Q 4 N 0 30 / 43
Bit Error Rate for QPSK using Other Bitmap Other bitmap for QPSK Y s 11 00 Y c 10 01 Conditional BER when b [ 1 ] b [ 2 ] = 00 is � � � 1 b [ 1 ]ˆ ˆ � P b | 00 = 2 Pr b [ 2 ] = 01 � b [ 1 ] b [ 2 ] = 00 � � � � + 1 b [ 1 ]ˆ ˆ � 2 Pr b [ 2 ] = 10 � b [ 1 ] b [ 2 ] = 00 � � � � b [ 1 ]ˆ ˆ � + Pr b [ 2 ] = 11 � b [ 1 ] b [ 2 ] = 00 � 31 / 43
Bit Error Rate for QPSK using Other Bitmap Other bitmap for QPSK Y s 11 00 Y c 10 01 � 2 E b Let α = N 0 � � � b [ 1 ]ˆ ˆ � Pr b [ 2 ] = 01 � b [ 1 ] b [ 2 ] = 00 = Q ( α ) [ 1 − Q ( α )] � � � � Q 2 ( α ) ˆ b [ 1 ]ˆ � Pr b [ 2 ] = 10 � b [ 1 ] b [ 2 ] = 00 = � � � � ˆ b [ 1 ]ˆ � Pr b [ 2 ] = 11 � b [ 1 ] b [ 2 ] = 00 = Q ( α ) [ 1 − Q ( α )] � 32 / 43
Bit Error Rate for QPSK using Other Bitmap Other bitmap for QPSK Y s 11 00 Y c 10 01 Conditional BER when b [ 1 ] b [ 2 ] = 00 is 2 Q ( α ) [ 1 − Q ( α )] + 1 1 2 Q 2 ( α ) + Q ( α ) [ 1 − Q ( α )] = P b | 00 �� � 3 2 Q ( α ) − Q 2 ( α ) ≈ 3 2 Q ( α ) = 3 2 E b = 2 Q N 0 �� � 1 ≈ 3 2 E b � � P b = P b | 00 + P b | 01 + P b | 10 + P b | 11 2 Q 4 N 0 33 / 43
Comparison of Modulation Schemes
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