Digital Modulation Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay August 8, 2012 1 / 45
Digital Modulation Definition The process of mapping a bit sequence to signals for transmission over a channel. Example (Binary Baseband PAM) 1 → p ( t ) and 0 → − p ( t ) p ( t ) − p ( t ) A t t − A 2 / 45
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 • Modulated 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 waveform • Nonlinear 3 / 45
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 • 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 5 / 45
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 × 6 / 45
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 × 7 / 45
Gram-Schmidt Orthogonalization Procedure • Algorithm for calculating orthonormal basis • Given s 1 ( t ) , . . . , s M ( t ) the k th basis function is φ k ( t ) = γ k ( t ) √ E k where � ∞ | γ k ( t ) | 2 dt E k = −∞ k − 1 � γ k ( t ) = s k ( t ) − c k , i φ i ( t ) i = 1 c k , i = � s k ( t ) , φ i ( t ) � , i = 1 , 2 , . . . , k − 1 8 / 45
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 9 / 45
Gram-Schmidt Procedure Example φ 1 ( t ) φ 3 ( t ) 1 1 √ 2 2 3 t 2 t -1 � √ � T φ 2 ( t ) = 2 0 0 s 1 √ � � T = 0 2 0 1 s 2 √ 2 � √ 2 t � T = 2 0 1 s 3 √ − 1 � T � √ 2 = − 2 0 1 s 4 10 / 45
Properties of Signal Space Representation • Energy � ∞ N � | s m , n | 2 = � s m � 2 | s m ( t ) | 2 dt = E m = −∞ n = 1 • Inner product � s i ( t ) , s j ( t ) � = � s i , s j � 11 / 45
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. • Usually, M = 2 k and amplitudes A m take the values A m = 2 m − 1 − M , 1 ≤ m ≤ M Example (M=4) A 1 = − 3 , A 2 = − 1 , A 3 = + 1 , A 4 = + 3 • Baseband PAM: p ( t ) is a baseband signal • Passband PAM: p ( t ) = g ( t ) cos 2 π f c t where g ( t ) is baseband 13 / 45
Constellation for PAM M = 2 0 1 M = 4 00 01 11 10 14 / 45
Phase Modulation • Complex Envelope of 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 • 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 15 / 45
Constellation for PSK QPSK, M = 4 01 11 00 10 Octal PSK, M = 8 011 001 010 000 110 100 111 101 16 / 45
Quadrature Amplitude Modulation • Complex Envelope of 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 • 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 17 / 45
Constellation for QAM 16-QAM 18 / 45
Power Spectral Density of Digitally Modulated Signals
PSD Definition for Digitally Modulated Signals • Consider a real binary PAM signal ∞ � u ( t ) = b n g ( t − nT ) n = −∞ where b n = ± 1 with equal probability and g ( t ) is a baseband pulse of duration T • PSD = F [ R u ( τ )] Not stationary or WSS 20 / 45
Cyclostationary Random Process Definition (Cyclostationary RP) A random process X ( t ) is cyclostationary with respect to time interval T if it is statistically indistinguishable from X ( t − kT ) for any integer k . Definition (Wide Sense Cyclostationary RP) A random process X ( t ) is wide sense cyclostationary with respect to time interval T if the mean and autocorrelation functions satisfy m X ( t ) = m X ( t − T ) for all t , R X ( t 1 , t 2 ) = R X ( t 1 − T , t 2 − T ) for all t 1 , t 2 . 21 / 45
Stationarizing a Cyclostationary Random Process Theorem Let S ( t ) be a cyclostationary random process with respect to the time interval T. Suppose D ∼ U [ 0 , T ] and independent of S ( t ) . Then S ( t − D ) is a stationary random process. Proof Sketch Let V ( t ) = S ( t − D ) . We prove that V ( t 1 ) ∼ V ( t 1 + τ ) . � T 1 P [ V ( t 1 + τ ) = v ] = P [ S ( t 1 + τ − x ) = v ] dx T 0 � T − τ 1 = P [ S ( t 1 − y ) = v ] dy T − τ � T 1 = P [ S ( t 1 − y ) = v ] dy T 0 = P [ V ( t 1 ) = v ] 22 / 45
Stationarizing a Cyclostationary Random Process Proof Sketch (Contd) We prove that V ( t 1 ) , V ( t 2 ) ∼ V ( t 1 + τ ) , V ( t 2 + τ ) . P [ V ( t 1 + τ ) = v 1 , V ( t 2 + τ ) = v 2 ] � T 1 = P [ S ( t 1 + τ − x ) = v 1 , S ( t 2 + τ − x ) = v 2 ] dx T 0 � T − τ 1 = P [ S ( t 1 − y ) = v 1 , S ( t 2 − y ) = v 2 ] dy T − τ � T 1 = P [ S ( t 1 − y ) = v 1 , S ( t 2 − y ) = v 2 ] dy T 0 = P [ V ( t 1 ) = v 1 , V ( t 2 ) = v 2 ] 23 / 45
Stationarizing a Wide Sense Cyclostationary RP Theorem Let S ( t ) be a wide sense cyclostationary RP with respect to the time interval T. Suppose D ∼ U [ 0 , T ] and independent of S ( t ) . Then S ( t − D ) is a wide sense stationary RP . Proof Sketch Let V ( t ) = S ( t − D ) . We prove that m V ( t ) is a constant function. m V ( t ) = E [ V ( t )] = E [ S ( t − D )] = E [ E [ S ( t − D ) | D ]] E [ S ( t − D ) | D = x ] = E [ S ( t − x )] = m S ( t − x ) � T � T E [ E [ S ( t − D ) | D ]] = 1 m S ( t − x ) dx = 1 m S ( y ) dy T T 0 0 24 / 45
Stationarizing a Wide Sense Cyclostationary RP Proof Sketch (Contd) We prove that R V ( t 1 , t 2 ) is a function of t 1 − t 2 = kT + ǫ R V ( t 1 , t 2 ) = E [ V ( t 1 ) V ∗ ( t 2 )] = E [ S ( t 1 − D ) S ∗ ( t 2 − D )] � T 1 = R S ( t 1 − x , t 2 − x ) dx T 0 � T 1 = R S ( t 1 − kT − x , t 2 − kT − x ) dx T 0 � T − ǫ 1 = R S ( t 1 − kT − ǫ − y , t 2 − kT − ǫ − y ) dy T − ǫ � T − ǫ 1 = R S ( t 1 − t 2 − y , − y ) dy T − ǫ � T 1 = R S ( t 1 − t 2 − y , − y ) dy T 0 25 / 45
Power Spectral Density of a Realization Time windowed realizations have finite energy x T o ( t ) = x ( t ) I [ − To 2 ] ( t ) 2 , To S T o ( f ) = F ( x T o ( t )) | S T o ( f ) | 2 ˆ S x ( f ) = (PSD Estimate) T o PSD of a realization | S T o ( f ) | 2 ¯ S x ( f ) = lim T o T o →∞ To � | S T o ( f ) | 2 − 1 2 T o ( u − τ ) du = ˆ − ⇀ x T o ( u ) x ∗ R s ( τ ) ↽ T o T o − To 2 26 / 45
Power Spectral Density of a Cyclostationary Process S ( t ) S ∗ ( t − τ ) ∼ S ( t + T ) S ∗ ( t + T − τ ) for cyclostationary S ( t ) To � 1 2 ˆ s ( t ) s ∗ ( t − τ ) dt R s ( τ ) = T o − To 2 � KT 1 2 s ( t ) s ∗ ( t − τ ) dt = for T o = KT KT − KT 2 K � T 2 1 1 � = s ( t + kT ) s ∗ ( t + kT − τ ) dt T K 0 k = − K 2 � T 1 → E [ S ( t ) S ∗ ( t − τ )] dt T 0 � T 1 = R S ( t , t − τ ) dt = R V ( τ ) T 0 PSD of a cyclostationary process = F [ R V ( τ )] 27 / 45
Power Spectral Density of a Cyclostationary Process To obtain the PSD of a cyclostationary process • Stationarize it • Calculate autocorrelation function of stationarized process • Calculate Fourier transform of autocorrelation or • Calculate autocorrelation of cyclostationary process R S ( t , t − τ ) • Average autocorrelation between 0 and T , � T R S ( τ ) = 1 0 R S ( t , t − τ ) dt T • Calculate Fourier transform of averaged autocorrelation R S ( τ ) 28 / 45
Power Spectral Density of Linearly Modulated Signals
PSD of a Linearly Modulated Signal • Consider ∞ � u ( t ) = b n p ( t − nT ) n = −∞ • u ( t ) is cyclostationary wrt to T if { b n } is stationary • u ( t ) is wide sense cyclostationary wrt to T if { b n } is WSS • Suppose R b [ k ] = E [ b n b ∗ n − k ] • Let S b ( z ) = � ∞ k = −∞ R b [ k ] z − k • The PSD of u ( t ) is given by e j 2 π fT � | P ( f ) | 2 � S u ( f ) = S b T 30 / 45
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