PSD of Digitally Modulated Signals Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay 1 / 30
Digital Modulation
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 3 / 30
Digital Modulation Example (Binary Baseband PAM) 1 → p ( t ) and 0 → − p ( t ) p ( t ) − p ( t ) A t t − A 4 / 30
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 5 / 30
PSD Definition for Linearly Modulated Signals
PSD Definition for Linearly 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 )] Neither SSS nor WSS 7 / 30
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 . 8 / 30
Power Spectral Density of a Cyclostationary Process To obtain the PSD of a cyclostationary process with period T • Calculate autocorrelation of cyclostationary process R X ( t , t − τ ) • Average autocorrelation between 0 and T , � T R X ( τ ) = 1 0 R X ( t , t − τ ) dt T • Calculate Fourier transform of averaged autocorrelation R X ( τ ) 9 / 30
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 x ( τ ) T o T o − To 2 10 / 30
Power Spectral Density of a Cyclostationary Process X ( t ) X ∗ ( t − τ ) ∼ X ( t + T ) X ∗ ( t + T − τ ) for cyclostationary X ( t ) To 1 � 2 ˆ x ( t ) x ∗ ( t − τ ) dt R x ( τ ) = T o − To 2 KT 1 � 2 x ( t ) x ∗ ( t − τ ) dt = (for T o = KT ) KT − KT 2 K 2 − 1 � T 1 1 � x ( t + kT ) x ∗ ( t + kT − τ ) dt = (for even K ) T K 0 k = − K 2 � T 1 − → E [ X ( t ) X ∗ ( t − τ )] dt T K →∞ 0 � T 1 = R X ( t , t − τ ) dt = R X ( τ ) T 0 PSD of a cyclostationary process = F [ R X ( τ )] 11 / 30
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 ] k = −∞ R b [ k ] z − k • Let S b ( z ) = � ∞ • The PSD of u ( t ) is given by e j 2 π fT � | P ( f ) | 2 � S u ( f ) = S b T 12 / 30
PSD of a Linearly Modulated Signal R u ( τ ) � T 1 = R u ( t + τ, t ) dt T 0 � T ∞ ∞ 1 � � = E [ b n b ∗ m p ( t − nT + τ ) p ∗ ( t − mT )] dt T 0 n = −∞ m = −∞ � − ( m − 1 ) T ∞ ∞ 1 � � = E [ b m + k b ∗ m p ( λ − kT + τ ) p ∗ ( λ )] d λ T − mT k = −∞ m = −∞ � ∞ ∞ 1 � E [ b m + k b ∗ m p ( λ − kT + τ ) p ∗ ( λ )] d λ = T −∞ k = −∞ � ∞ ∞ 1 � p ( λ − kT + τ ) p ∗ ( λ ) d λ = R b [ k ] T −∞ k = −∞ 13 / 30
PSD of a Linearly Modulated Signal � ∞ ∞ R u ( τ ) = 1 � p ( λ − kT + τ ) p ∗ ( λ ) d λ R b [ k ] T −∞ k = −∞ � ∞ | P ( f ) | 2 p ( λ + τ ) p ∗ ( λ ) d λ ↔ −∞ � ∞ | P ( f ) | 2 e − j 2 π fkT p ( λ − kT + τ ) p ∗ ( λ ) d λ ↔ −∞ | P ( f ) | 2 ∞ � R b [ k ] e − j 2 π fkT S u ( f ) = F [ R u ( τ )] = T k = −∞ e j 2 π fT � | P ( f ) | 2 � = S b T k = −∞ R b [ k ] z − k . where S b ( z ) = � ∞ 14 / 30
PSD of Line Codes
Line Codes 0 1 1 0 1 1 1 0 1 0 1 Unipolar NRZ Polar NRZ Bipolar NRZ Manchester Further reading: Digital Communications , Simon Haykin, Chapter 6 16 / 30
Unipolar NRZ • Symbols independent and equally likely to be 0 or A P ( b [ n ] = 0 ) = P ( b [ n ] = A ) = 1 2 • Autocorrelation of b [ n ] sequence A 2 k = 0 2 R b [ k ] = A 2 k � = 0 4 • p ( t ) = I [ 0 , T ) ( t ) ⇒ P ( f ) = T sinc ( fT ) e − j π fT • Power Spectral Density S u ( f ) = | P ( f ) | 2 ∞ � R b [ k ] e − j 2 π kfT T k = −∞ 17 / 30
Unipolar NRZ A 2 T 4 sinc 2 ( fT ) + A 2 T ∞ 4 sinc 2 ( fT ) � e − j 2 π kfT S u ( f ) = k = −∞ A 2 T 4 sinc 2 ( fT ) + A 2 ∞ f − n � � 4 sinc 2 ( fT ) � = δ T n = −∞ A 2 T 4 sinc 2 ( fT ) + A 2 = 4 δ ( f ) 18 / 30
Normalized PSD plot 1 Unipolar NRZ S u ( f ) 0 . 5 A 2 T 0 0 . 5 1 1 . 5 2 fT 19 / 30
Polar NRZ • Symbols independent and equally likely to be − A or A P ( b [ n ] = − A ) = P ( b [ n ] = A ) = 1 2 • Autocorrelation of b [ n ] sequence A 2 k = 0 R b [ k ] = 0 k � = 0 • P ( f ) = T sinc ( fT ) e − j π fT • Power Spectral Density S u ( f ) = A 2 T sinc 2 ( fT ) 20 / 30
Normalized PSD plots 1 Unipolar NRZ Polar NRZ S u ( f ) 0 . 5 A 2 T 0 0 . 5 1 1 . 5 2 fT 21 / 30
Manchester • Symbols independent and equally likely to be − A or A P ( b [ n ] = − A ) = P ( b [ n ] = A ) = 1 2 • Autocorrelation of b [ n ] sequence A 2 k = 0 R b [ k ] = 0 k � = 0 � fT � π fT • P ( f ) = jT sinc � � e − j π fT sin 2 2 • Power Spectral Density � fT � � π fT � S u ( f ) = A 2 T sinc 2 sin 2 2 2 22 / 30
Normalized PSD plots 1 Unipolar NRZ Polar NRZ Manchester S u ( f ) 0 . 5 A 2 T 0 0 . 5 1 1 . 5 2 fT 23 / 30
Bipolar NRZ • Successive 1’s have alternating polarity 0 → Zero amplitude 1 → + A or − A • Probability mass function of b [ n ] 1 P ( b [ n ] = 0 ) = 2 1 P ( b [ n ] = − A ) = 4 1 P ( b [ n ] = A ) = 4 • Symbols are identically distributed but they are not independent 24 / 30
Bipolar NRZ • Autocorrelation of b [ n ] sequence A 2 / 2 k = 0 − A 2 / 4 R b [ k ] = k = ± 1 0 otherwise • Power Spectral Density � A 2 2 − A 2 e j 2 π fT + e − j 2 π fT �� � T sinc 2 ( fT ) S u ( f ) = 4 A 2 T 2 sinc 2 ( fT ) [ 1 − cos ( 2 π fT )] = A 2 T sinc 2 ( fT ) sin 2 ( π fT ) = 25 / 30
Normalized PSD plots 1 Unipolar NRZ Polar NRZ Manchester Bipolar NRZ S u ( f ) 0 . 5 A 2 T 0 0 . 5 1 1 . 5 2 fT 26 / 30
PSD of Passband Modulated Signals
Relating the PSDs of a Passband Modulated Signal and its Complex Envelope • Definitions • s p ( t ) is a passband signal realization with complex envelope s ( t ) • For observation interval T o , ˆ s p ( t ) = s p ( t ) I � � ( t ) − To 2 , To 2 • ˆ s p ( t ) has complex envelope ˆ s ( t ) s p ( t ) ↔ ˆ s ( t ) ↔ ˆ • ˆ S p ( f ) and ˆ S ( f ) • PSD approximations for s p ( t ) and s ( t ) 2 2 � � � � � ˆ � ˆ S p ( f ) S ( f ) � � � � � � S s p ( f ) ≈ , S s ( f ) ≈ T o T o • From the relationship between the deterministic signals 1 � � ˆ S ( f − f c ) + ˆ ˆ S ∗ ( − f − f c ) S p ( f ) = √ 2 • Since ˆ S ( f − f c ) and ˆ S ∗ ( − f − f c ) do not overlap, we have �� 2 � 2 = 1 2 � � � � � � ˆ � ˆ � ˆ S p ( f ) S ( f − f c ) + S ∗ ( − f − f c ) � � � � � � 2 � � � 28 / 30
Relating the PSDs of a Passband Modulated Signal and its Complex Envelope • Dividing by T o 2 2 2 � � � � � � � ˆ � ˆ � ˆ S p ( f ) S ( f − f c ) S ∗ ( − f − f c ) � � � � � � = 1 � � � + T o 2 T o T o • As the observation interval T o → ∞ , we get S s p ( f ) = 1 2 [ S s ( f − f c ) + S s ( − f − f c )] • By a similar argument, we get S s ( f ) = 2 S s p ( f + f c ) u ( f + f c ) 29 / 30
References • Section 2.5, Fundamentals of Digital Communication , Upamanyu Madhow, 2008 • Section 2.3.1, Fundamentals of Digital Communication , Upamanyu Madhow, 2008 • Chapter 6, Digital Communications , Simon Haykin, 2006 30 / 30
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