Envelope/phase delays correction in an EER radio architecture - PowerPoint PPT Presentation

Envelope/phase delays correction in an EER radio architecture Envelope/phase delays correction in an EER radio architecture Jean-François Bercher and Corinne Berland Esiee-Paris, France Nice, december 11, 2006 1/ 17

+ + + + + + Envelope/phase delays correction in an EER radio architecture Background Efficient modulations present significant envelope variations 2/ 17

+ + + + + + Envelope/phase delays correction in an EER radio architecture Background Efficient modulations present significant envelope variations Non linearities in RF transmitter cause severe distorsions which would impact both EVM and output spectrum 2/ 17

Envelope/phase delays correction in an EER radio architecture Background Efficient modulations present significant envelope variations Non linearities in RF transmitter cause severe distorsions which would impact both EVM and output spectrum Possible solution: Envelope Elimination and Restoration (EER) — Polar components are amplified separately Class D PA Class D PA + + 2 2 2 2 I I Q Q PWM PWM I I + + 2 2 2 2 I I Q Q 90° 90° 90° +/- +/- Q Q Class E PA Class E PA + + 2 2 2 2 I I Q Q Frequency Frequency synthesizer synthesizer 2/ 17

Envelope/phase delays correction in an EER radio architecture Background BUT time mismatch ∆ between envelope and phase signals at the recombination has especially a great impact on EVM and spectral re-growths. This is caused by different operations on each of the two paths. 3/ 17

Envelope/phase delays correction in an EER radio architecture Background BUT time mismatch ∆ between envelope and phase signals at the recombination has especially a great impact on EVM and spectral re-growths. This is caused by different operations on each of the two paths. Maximum of spectral regrowths 20 20 20 20 0.45 0.4 0 0 0 0 0.35 Normalized log10 of spectral regrowths 30% EVM 30% EVM Spectrum Spectrum Spectrum Spectrum 10ns 10ns 10ns 10ns -20 -20 -20 -20 0.3 4ns 4ns 4ns 4ns 0.25 -40 -40 -40 -40 2ns 2ns 2ns 2ns 6.5% EVM 6.5% EVM 0.2 -60 -60 -60 -60 0.15 0.1 -80 -80 -80 -80 0.05 4940 4940 4940 4940 4960 4960 4960 4960 4980 4980 4980 4980 5000 5000 5000 5000 5020 5020 5020 5020 5040 5040 5040 5040 5060 5060 5060 5060 0 Mega_Hertz Mega_Hertz Mega_Hertz Mega_Hertz 0 5 10 15 20 25 30 35 40 ∆ in % of symbol period Need of a synchronization procedure... 3/ 17

Envelope/phase delays correction in an EER radio architecture Proposed procedure Proposed procedure Initial situation: distorded output ρ (n) ρ (t) ρ (t- ∆ 1) RF Transmitter φ (n) φ (t) φ (t- ∆ 2) ρ (t- ∆ 1)cos( φ (t- ∆ 2)) ρ (t)cos( φ (t)) 4/ 17

Envelope/phase delays correction in an EER radio architecture Proposed procedure Proposed procedure If envelope and phase are delayed by τ 1 and τ 2 ρ (n) ρ (t+ τ 1) ρ (t+ τ 1- ∆ 1) RF Transmitter φ (n) φ (t+ τ 2) φ (t+ τ 2- ∆ 2) ρ (t+ τ 1- ∆ 1)cos( φ (t+t2- ∆ 2)) ρ (t)cos( φ (t)) ' ρ (t)cos( φ (t)) 4/ 17

Envelope/phase delays correction in an EER radio architecture Proposed procedure Proposed procedure If envelope and phase are delayed by τ 1 and τ 2 ρ (n) ρ (t+ τ 1) ρ (t+ τ 1- ∆ 1) RF Transmitter φ (n) φ (t+ τ 2) φ (t+ τ 2- ∆ 2) ρ (t+ τ 1- ∆ 1)cos( φ (t+t2- ∆ 2)) ρ (t)cos( φ (t)) ' ρ (t)cos( φ (t)) Procedure: adjusts delays τ 1 and τ 2 such that the output be- comes synchronous with the (unmodified) input 4/ 17

Envelope/phase delays correction in an EER radio architecture Proposed procedure Proposed procedure ρ (n) ρ (t+ τ 1) ρ (t+ τ 1- ∆ 1) RF Transmitter φ (n) φ (t+ τ 2) φ (t+ τ 2- ∆ 2) ρ (t+ τ 1- ∆ 1)cos( φ (t+t2- ∆ 2)) ρ (t)cos( φ (t)) ' ρ (t)cos( φ (t)) 5/ 17

Envelope/phase delays correction in an EER radio architecture Proposed procedure Proposed procedure ρ (n) ρ (t+ τ 1) ρ (t+ τ 1- ∆ 1) RF Transmitter φ (n) φ (t+ τ 2) φ (t+ τ 2- ∆ 2) τ1,τ2 + − ρ (t+ τ 1- ∆ 1)cos( φ (t+ τ 2- ∆ 2)) ρ (t)cos( φ (t)) 5/ 17

Envelope/phase delays correction in an EER radio architecture Criterion Criterion � ρ ( t ) cos ( φ ( t )) � cos ( φ ( t )) � � Let x ( t ) = and cos ( φ ( t )) = ρ ( t ) sin ( φ ( t )) sin ( φ ( t )) . Let t 1 = t + τ 1 − ∆ 1 and t 2 = t + τ 2 − ∆ 2 . The criterion is J ( τ 1 , τ 2 ) = E [ | x ( t ) − ρ ( t 1 ) cos ( φ ( t 2 )) | q ] With q = 2 , | x | 2 = x t x , the criterion is simply ✞ ☎ J ( τ 1 , τ 2 ) = 4 R ( 0 , 0 ) − 4 R ( τ 1 − ∆ 1 , τ 2 − ∆ 2 ) ✝ ✆ with R ( µ 1 , µ 2 ) = E [ ρ ( t ) cos ( φ ( t )) ρ ( t − µ 1 ) cos ( φ ( t − µ 2 ))] 6/ 17

Envelope/phase delays correction in an EER radio architecture Criterion Criterion 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 4 0 2 4 3 2 0 1 0 −1 −2 −2 −3 −4 −4 Envelope delay Phase delay 7/ 17

Envelope/phase delays correction in an EER radio architecture Criterion Criterion 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 1.5 1.5 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 −1.5 −1.5 Phase delay Envelope delay 7/ 17

Envelope/phase delays correction in an EER radio architecture Criterion Criterion 8/ 17

Envelope/phase delays correction in an EER radio architecture Criterion Criterion There exist local minima 8/ 17

Envelope/phase delays correction in an EER radio architecture Criterion Criterion There exist local minima A descent algorithm will avoid local minima for delays ∆ 1 , ∆ 2 ≤ T s , with T s the symbol period. 8/ 17

Envelope/phase delays correction in an EER radio architecture Criterion Criterion There exist local minima A descent algorithm will avoid local minima for delays ∆ 1 , ∆ 2 ≤ T s , with T s the symbol period. J ( τ 1 , τ 2 ) is not factorizable as J ( τ 1 , τ 2 ) = K ( τ 1 ) L ( τ 2 ) , because R ( µ 1 , µ 2 ) � = R ρ ( µ 1 ) R φ ( µ 2 ) , but not so far. . . 8/ 17

Envelope/phase delays correction in an EER radio architecture Criterion Criterion There exist local minima A descent algorithm will avoid local minima for delays ∆ 1 , ∆ 2 ≤ T s , with T s the symbol period. J ( τ 1 , τ 2 ) is not factorizable as J ( τ 1 , τ 2 ) = K ( τ 1 ) L ( τ 2 ) , because R ( µ 1 , µ 2 ) � = R ρ ( µ 1 ) R φ ( µ 2 ) , but not so far. . . R ( τ 1 , τ 2 ) reduces to the autocorrelation function R ( τ ) for τ = τ 1 = τ 2 : the behaviour of the error function is linked to the shaping filter h . 8/ 17

Envelope/phase delays correction in an EER radio architecture Algorithm Descent algorithms Descent algorithms Gradient algorithm � τ 1 � τ 1 � � ∂ J � � ∂τ 1 ( n + 1 ) = ( n ) − γ ∂ J τ 2 τ 2 ∂τ 2 τ 1 = τ 1 ( n ) , τ 2 = τ 2 ( n ) Newton algorithm τ ( n + 1 ) = τ ( n ) − γ H ( n ) − 1 G ( n ) with H hessian matrix, G gradient vector. 9/ 17

Envelope/phase delays correction in an EER radio architecture Algorithm Descent algorithms The gradient is  � � � cos ( φ ( t 2 )) t e ( t ) ∂ J ( τ 1 , τ 2 ) d ρ ( u ) = − 2 E  �  ∂τ 1 du  � u = t 1 � � � ρ ( t 1 ) dcos ( φ ( t 2 )) t e ( t ) ∂ J ( τ 1 , τ 2 ) d φ ( u ) = − 2 E  �  ∂τ 2 du  � u = t 2 with dcos ( . ) = [ − sin ( . ) cos ( . )] and e ( t ) = x ( t ) − ρ ( t + τ 1 − ∆ 1 ) cos ( φ ( t + τ 2 − ∆ 2 )) . 10/ 17

Envelope/phase delays correction in an EER radio architecture Algorithm Descent algorithms The gradient is  � � � cos ( φ ( t 2 )) t e ( t ) ∂ J ( τ 1 , τ 2 ) d ρ ( u ) = − 2 E  �  ∂τ 1 du  � u = t 1 � � � ρ ( t 1 ) dcos ( φ ( t 2 )) t e ( t ) ∂ J ( τ 1 , τ 2 ) d φ ( u ) = − 2 E  �  ∂τ 2 du  � u = t 2 with dcos ( . ) = [ − sin ( . ) cos ( . )] and e ( t ) = x ( t ) − ρ ( t + τ 1 − ∆ 1 ) cos ( φ ( t + τ 2 − ∆ 2 )) . Of course, we do not have analytical expressions of the statistical expectations involved, and have to resort to “blocks estimates”, or iterative/adaptive versions. 10/ 17

Envelope/phase delays correction in an EER radio architecture Algorithm LMS-like algorithms LMS-like algorithm The LMS algorithm consists in using the instantaneous gradient rather than the (correct) statistical average, then update the equations at each new sample.  � � � t 1 ( n ) cos ( φ ( t 2 ( n ))) t e ( t ) d ρ ( u ) τ 1 ( n + 1 ) = τ 1 ( n )+ γ 1 ( n ) E  �  du  � � � � t 2 ( n ) ρ ( t 1 ( n )) dcos ( φ ( t 2 ( n ))) t e ( t ) d φ ( u ) τ 2 ( n + 1 ) = τ 2 ( n )+ γ 2 ( n ) E  �  du  � 11/ 17

Envelope/phase delays correction in an EER radio architecture Algorithm LMS-like algorithms LMS-like algorithm The LMS algorithm consists in using the instantaneous gradient rather than the (correct) statistical average, then update the equations at each new sample.  � � � t 1 ( n ) cos ( φ ( t 2 ( n ))) t e ( t ) d ρ ( u ) τ 1 ( n )+ γ 1 ( n ) ✓ ❙ τ 1 ( n + 1 ) = E  �  du  � � � � t 2 ( n ) ρ ( t 1 ( n )) dcos ( φ ( t 2 ( n ))) t e ( t ) d φ ( u ) τ 2 ( n )+ γ 2 ( n ) ✓ ❙ τ 2 ( n + 1 ) = E  �  du  � 11/ 17