Phase and Timing Synchronization Saravanan Vijayakumaran - PowerPoint PPT Presentation

Phase and Timing Synchronization Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay 1 / 14

The System Model • Consider the following complex baseband signal s ( t ) K − 1 � s ( t ) = b i p ( t − iT ) i = 0 where b i ’s are complex symbols • Suppose the LO frequency at the transmitter is f c � √ 2 s ( t ) e j 2 π f c t � s p ( t ) = Re . • Suppose that the LO frequency at the receiver is f c − ∆ f • The received passband signal is y p ( t ) = As p ( t − τ ) + n p ( t ) • The complex baseband representation of the received signal is then y ( t ) = Ae j ( 2 π ∆ ft + θ ) s ( t − τ ) + n ( t ) 2 / 14

The System Model K − 1 y ( t ) = Ae j ( 2 π ∆ ft + θ ) � b i p ( t − iT − τ ) + n ( t ) i = 0 • The unknown parameters are A , τ , θ and ∆ f Timing Synchronization Estimation of τ Carrier Synchronization Estimation of θ and ∆ f • The preamble of a packet contains known symbols called the training sequence • The b i ’s are known during the preamble 3 / 14

Carrier Phase Estimation • The change in phase due to the carrier offset ∆ f is 2 π ∆ fT in a symbol interval T • The phase can be assumed to be constant over multiple symbol intervals • Assume that the phase θ is the only unknown parameter • Assume that s ( t ) is a known signal in the following y ( t ) = s ( t ) e j θ + n ( t ) • The likelihood function for this scenario is given by � 1 Re ( � y , se j θ � ) − � se j θ � 2 � �� L ( y | θ ) = exp σ 2 2 • Let � y , s � = Z = | Z | e j φ = Z c + jZ s e − j θ Z = | Z | e j ( φ − θ ) � y , se j θ � = Re ( � y , se j θ � ) = | Z | cos ( φ − θ ) � se j θ � 2 � s � 2 = 4 / 14

Carrier Phase Estimation • The likelihood function for this scenario is given by � 1 | Z | cos ( φ − θ ) − � s � 2 � �� L ( y | s θ ) = exp σ 2 2 • The ML estimate of θ is given by θ ML = φ = arg ( � y , s � ) = tan − 1 Z s ˆ Z c s c ( − t ) Sample at t = 0 × y c ( t ) + LPF s s ( − t ) Sample at t = 0 √ 2 cos 2 π f c t tan − 1 Z s ˆ y p ( t ) θ ML Z c √ − 2 sin 2 π f c t s c ( − t ) Sample at t = 0 − × LPF y s ( t ) + s s ( − t ) Sample at t = 0 5 / 14

Phase Locked Loop • The carrier offset will cause the phase to change slowly • A tracking mechanism is required to track the changes in phase • For simplicity, consider an unmodulated carrier √ y p ( t ) = 2 cos ( 2 π f c t + θ ( t )) + n p ( t ) • The complex baseband representation is y ( t ) = e j θ ( t ) + n ( t ) • For an observation interval T o , the log likelihood function is given by � � ln L ( y | θ ) = 1 − T o � � � y , e j θ ( t ) � Re σ 2 2 • We get ˆ θ ML by maximizing � T o � � � y , e j θ ( t ) � J [ θ ( t )] = Re = [ y c ( t ) cos θ ( t ) + y s ( t ) sin θ ( t )] dt 0 6 / 14

Phase Locked Loop • A necessary condition for a maximum at ˆ θ ML is � T o � ∂ � � − y c ( t ) sin ˆ θ ML + y s ( t ) cos ˆ � ∂θ J [ θ ( t )] = 0 = ⇒ θ ML dt = 0 � � ˆ 0 θ ML � � y , je j ˆ � θ ML � = ⇒ Re = 0 � y p , − sin ( 2 π f c t + ˆ = ⇒ θ ML ) � = 0 � y p ( t ) sin ( 2 π f c t + ˆ = ⇒ − θ ML ) dt = 0 T o � T o () dt y p ( t ) × VCO sin ( 2 π f c t + ˆ θ ) 7 / 14

Symbol Timing Estimation • Consider the complex baseband received signal y ( t ) = As ( t − τ ) e j θ + n ( t ) where A , τ and θ are unknown and s ( t ) is known • For γ = [ τ, θ, A ] and s γ ( t ) = As ( t − τ ) e j θ , the likelihood function is � 1 Re ( � y , s γ � ) − � s γ � 2 � �� L ( y | γ ) = exp σ 2 2 • For a large enough observation interval, the signal energy does not depend on τ and � s γ � 2 = A 2 � s � 2 • For s MF ( t ) = s ∗ ( − t ) we have � Ae − j θ � y , s γ � = y ( t ) s ∗ ( t − τ ) dt � Ae − j θ = y ( t ) s MF ( τ − t ) dt Ae − j θ ( y ⋆ s MF )( τ ) = 8 / 14

Symbol Timing Estimation • Maximizing the likelihood function is equivalent to maximizing the following cost function − A 2 � s � 2 � � Ae − j θ ( y ⋆ s MF )( τ ) J ( τ, A , θ ) = Re 2 • For ( y ⋆ s MF )( τ ) = Z ( τ ) = | Z ( τ ) | e j φ ( τ ) we have � � Ae − j θ ( y ⋆ s MF )( τ ) Re = A | Z ( τ ) | cos ( φ ( τ ) − θ ) • The maximizing value of θ is equal to φ ( τ ) • Substituting this value of θ gives us the following cost function J ( τ, A , θ ) = A | ( y ⋆ s MF )( τ ) | − A 2 � s � 2 J ( τ, A ) = argmax 2 θ 9 / 14

Symbol Timing Estimation • The ML estimator of the delay picks the peak of the matched filter output ˆ τ ML = argmax | ( y ⋆ s MF )( τ ) | τ s c ( − t ) × LPF y c ( t ) + Squarer s s ( − t ) √ 2 cos 2 π f c t Pick y p ( t ) + ˆ τ ML the peak √ − 2 sin 2 π f c t s c ( − t ) − × LPF y s ( t ) + Squarer s s ( − t ) 10 / 14

Early-Late Gate Synchronizer • Timing tracker which exploits symmetry in matched filter output p ( t ) Matched Filter Output 1 1 0 T t T 2 T t 11 / 14

Early-Late Gate Synchronizer Matched Filter Output Optimum Sample 1 Early Sample Late Sample T − δ T T + δ 2 T t • The values of the early and late samples are equal 12 / 14

Early-Late Gate Synchronizer � × T () dt Sampler Magnitude Advance by δ + Symbol Loop r ( t ) waveform + VCC Filter generator − Delay by δ � × T () dt Sampler Magnitude • The motivation for this structure can be seen from the following approximation dJ ( τ ) ≈ J ( τ + δ ) − J ( τ − δ ) d τ 2 δ 13 / 14

References • Section 4.3, Fundamentals of Digital Communication , Upamanyu Madhow, 2008 14 / 14