Digital Modulation Saravanan Vijayakumaran sarva@ee.iitb.ac.in - PowerPoint PPT Presentation

Digital Modulation Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay August 19, 2013 1 / 21

Digital Modulation Definition The process of mapping a bit sequence to signals for transmission over a channel. Information Source Channel Modulator Source Encoder Encoder Channel Information Source Channel Demodulator Destination Decoder Decoder 2 / 21

Digital Modulation Example (Binary Baseband PAM) 1 → p ( t ) and 0 → − p ( t ) p ( t ) − p ( t ) A t t − A 3 / 21

Classification of Modulation Schemes • Memoryless • Divide bit sequence into k -bit blocks • Map each block to a signal s m ( t ) , 1 ≤ m ≤ 2 k • Mapping depends only on current k -bit block • Having Memory • Mapping depends on current k -bit block and L − 1 previous blocks • L is called the constraint length • Linear • Complex baseband representation of transmitted signal has the form � u ( t ) = b n g ( t − nT ) n where b n ’s are the transmitted symbols and g is a fixed baseband waveform • Nonlinear 4 / 21

Signal Space Representation

Signal Space Representation of Waveforms • Given M finite energy waveforms, construct an orthonormal basis s 1 ( t ) , . . . , s M ( t ) → φ 1 ( t ) , . . . , φ N ( t ) � �� � Orthonormal basis � 1 � ∞ if i = j � φ i , φ j � = φ i ( t ) φ ∗ j ( t ) dt = 0 otherwise −∞ • Each s i ( t ) is a linear combination of the basis vectors N � s i ( t ) = s i , n φ n ( t ) , i = 1 , . . . , M n = 1 � � T • s i ( t ) is represented by the vector s i = s i , 1 · · · s i , N • The set { s i : 1 ≤ i ≤ M } is called the signal space representation or constellation 6 / 21

Constellation Point to Waveform φ 1 ( t ) s i , 1 × φ 2 ( t ) s i , 2 × . . . . . . + s i ( t ) φ N − 1 ( t ) s i , N − 1 × φ N ( t ) s i , N × 7 / 21

Waveform to Constellation Point φ ∗ 1 ( t ) � s i , 1 × φ ∗ 2 ( t ) � × s i , 2 . . . . . . . . . s i ( t ) φ ∗ N − 1 ( t ) � s i , N − 1 × φ ∗ N ( t ) � × s i , N 8 / 21

Gram-Schmidt Orthogonalization Procedure • Algorithm for calculating orthonormal basis for s 1 ( t ) , . . . , s M ( t ) • Consider M = 1 φ 1 ( t ) = s 1 ( t ) � s 1 � where � s 1 � 2 = � s 1 , s 1 � • Consider M = 2 φ 1 ( t ) = s 1 ( t ) φ 2 ( t ) = γ ( t ) � s 1 � , � γ � where γ ( t ) = s 2 ( t ) − � s 2 , φ 1 � φ 1 ( t ) • Consider M = 3 φ 1 ( t ) = s 1 ( t ) φ 2 ( t ) = γ 1 ( t ) φ 3 ( t ) = γ 2 ( t ) � s 1 � , � γ 1 � , � γ 2 � where γ 1 ( t ) = s 2 ( t ) − � s 2 , φ 1 � φ 1 ( t ) γ 2 ( t ) = s 3 ( t ) − � s 3 , φ 1 � φ 1 ( t ) − � s 3 , φ 2 � φ 2 ( t ) 9 / 21

Gram-Schmidt Orthogonalization Procedure • In general, given s 1 ( t ) , . . . , s M ( t ) the k th basis function is φ k ( t ) = γ k ( t ) � γ k � where k − 1 � γ k ( t ) = s k ( t ) − � s k , φ i � φ i ( t ) i = 1 is not the zero function • If γ k ( t ) is zero, s k ( t ) is a linear combination of φ 1 ( t ) , . . . , φ k − 1 ( t ) . It does not contribute to the basis. 10 / 21

Gram-Schmidt Procedure Example s 1 ( t ) s 3 ( t ) 1 1 3 t 2 t -1 s 2 ( t ) s 4 ( t ) 1 1 2 t 3 t -1 -1 11 / 21

Gram-Schmidt Procedure Example φ 1 ( t ) φ 3 ( t ) 1 1 √ 2 2 3 t 2 t -1 � √ � T = 2 0 0 φ 2 ( t ) s 1 √ � � T = 0 2 0 s 2 1 � √ √ 2 � T = 2 0 1 s 3 2 t √ � � T = − 2 0 1 s 4 − 1 √ 2 12 / 21

Properties of Signal Space Representation • Energy � ∞ N | s m ( t ) | 2 dt = � | s m , n | 2 = � s m � 2 E m = −∞ n = 1 • Inner product � s i ( t ) , s j ( t ) � = � s i , s j � 13 / 21

Modulation Schemes

Pulse Amplitude Modulation • Signal Waveforms s m ( t ) = A m p ( t ) , 1 ≤ m ≤ M where p ( t ) is a pulse of duration T and A m ’s denote the M possible amplitudes. • Example M = 2, p ( t ) is a real pulse A 1 = − A , A 2 = A for a real number A -A A 15 / 21

Pulse Amplitude Modulation • Example M = 4, p ( t ) is a real pulse A 1 = − 3 A , A 2 = − A , A 3 = A , A 4 = 3 A -3A -A A 3A 16 / 21

Phase Modulation • Complex envelope of phase modulated signals s m ( t ) = p ( t ) e j π ( 2 m − 1 ) 1 ≤ m ≤ M , M where p ( t ) is a real baseband pulse of duration T • Corresponding passband signals � √ 2 s m ( t ) e j 2 π f c t � s p m ( t ) = Re � π ( 2 m − 1 ) � √ = 2 p ( t ) cos cos 2 π f c t M � π ( 2 m − 1 ) � √ − 2 p ( t ) sin sin 2 π f c t M 17 / 21

Constellation for PSK QPSK, M = 4 01 11 00 10 Octal PSK, M = 8 011 001 010 000 110 100 111 101 18 / 21

Quadrature Amplitude Modulation • Complex envelope of QAM signals s m ( t ) = ( A m , i + jA m , q ) p ( t ) , 1 ≤ m ≤ M where p ( t ) is a real baseband pulse of duration T • Corresponding passband signals � √ 2 s m ( t ) e j 2 π f c t � s p m ( t ) = Re √ √ = 2 A m , i p ( t ) cos 2 π f c t − 2 A m , q p ( t ) sin 2 π f c t 19 / 21

Constellation for QAM 16-QAM 20 / 21

Thanks for your attention 21 / 21