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

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Envelope/phase delays correction in an EER radio architecture

Jean-François Bercher and Corinne Berland

Esiee-Paris, France

Nice, december 11, 2006

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Envelope/phase delays correction in an EER radio architecture Background

Efficient modulations present significant envelope variations

+ + + + + +

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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

+ + + + + +

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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

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I Q + PWM Class D PA Class E PA

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I I Q +

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Q I Q + Frequency synthesizer +/- 90°

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I Q + PWM Class D PA Class E PA

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I I Q +

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Q I Q + Frequency synthesizer +/- 90° 90°

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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.

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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.

10ns 2ns 4ns

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Mega_Hertz Spectrum

30% EVM 6.5% EVM

10ns 2ns 4ns

4960 4980 5000 5020 5040 4940 5060

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Mega_Hertz Spectrum

10ns 2ns 4ns

4960 4980 5000 5020 5040 4940 5060

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10ns 2ns 4ns

4960 4980 5000 5020 5040 4940 5060

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Mega_Hertz Spectrum Mega_Hertz Spectrum

30% EVM 6.5% EVM

5 10 15 20 25 30 35 40 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Maximum of spectral regrowths ∆ in % of symbol period Normalized log10 of spectral regrowths

Need of a synchronization procedure...

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Envelope/phase delays correction in an EER radio architecture Proposed procedure

Proposed procedure

Initial situation: distorded output

ρ(t-∆1) φ(t-∆2) RF Transmitter ρ(t) φ(t) ρ(t)cos(φ(t)) ρ(t-∆1)cos(φ(t-∆2)) ρ(n) φ(n) 4/ 17

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Envelope/phase delays correction in an EER radio architecture Proposed procedure

Proposed procedure

If envelope and phase are delayed by τ1 and τ2

ρ(t+τ1-∆1) φ(t+τ2-∆2) RF Transmitter ρ(t+τ1) φ(t+τ2) ρ(t)cos(φ(t)) ρ(t+τ1-∆1)cos(φ(t+t2-∆2)) ρ(n) φ(n) ' ρ(t)cos(φ(t)) 4/ 17

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Envelope/phase delays correction in an EER radio architecture Proposed procedure

Proposed procedure

If envelope and phase are delayed by τ1 and τ2

ρ(t+τ1-∆1) φ(t+τ2-∆2) RF Transmitter ρ(t+τ1) φ(t+τ2) ρ(t)cos(φ(t)) ρ(t+τ1-∆1)cos(φ(t+t2-∆2)) ρ(n) φ(n) ' ρ(t)cos(φ(t))

Procedure: adjusts delays τ1 and τ2 such that the output be- comes synchronous with the (unmodified) input

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Envelope/phase delays correction in an EER radio architecture Proposed procedure

Proposed procedure

ρ(t+τ1-∆1) φ(t+τ2-∆2) RF Transmitter ρ(t+τ1) φ(t+τ2) ρ(t)cos(φ(t)) ρ(t+τ1-∆1)cos(φ(t+t2-∆2)) ρ(n) φ(n) ' ρ(t)cos(φ(t)) 5/ 17

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Envelope/phase delays correction in an EER radio architecture Proposed procedure

Proposed procedure

ρ(t+τ1-∆1) φ(t+τ2-∆2) RF Transmitter ρ(t+τ1) φ(t+τ2) ρ(t)cos(φ(t)) ρ(t+τ1-∆1)cos(φ(t+τ2-∆2)) ρ(n) φ(n) + − τ1,τ2 5/ 17

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Envelope/phase delays correction in an EER radio architecture Criterion

Criterion

Let x(t) = ρ(t)cos(φ(t)) ρ(t)sin(φ(t))

and cos(φ(t)) =

cos(φ(t)) sin(φ(t))

.

Let t1 = t +τ1 −∆1 and t2 = t +τ2 −∆2. The criterion is J(τ1,τ2) = E[|x(t)−ρ(t1)cos(φ(t2))|q] With q = 2, |x|2 = xtx, the criterion is simply

✞ ✝ ☎ ✆

J(τ1,τ2) = 4R(0,0)−4R(τ1 −∆1,τ2 −∆2) with R(µ1,µ2) = E[ρ(t)cos(φ(t))ρ(t − µ1)cos(φ(t − µ2))]

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Envelope/phase delays correction in an EER radio architecture Criterion

Criterion

−4 −2 2 4 −4 −3 −2 −1 1 2 3 4 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Envelope delay Phase delay

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Envelope/phase delays correction in an EER radio architecture Criterion

Criterion

−1.5 −1 −0.5 0.5 1 1.5 −1.5 −1 −0.5 0.5 1 1.5 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Envelope delay Phase delay

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Envelope/phase delays correction in an EER radio architecture Criterion

Criterion

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Envelope/phase delays correction in an EER radio architecture Criterion

Criterion

There exist local minima

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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 ≤ Ts, with Ts the symbol period.

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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 ≤ Ts, with Ts the symbol period. J(τ1,τ2) is not factorizable as J(τ1,τ2) = K(τ1)L(τ2), because R(µ1,µ2) = Rρ(µ1)Rφ(µ2), but not so far. . .

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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 ≤ Ts, with Ts the symbol period. J(τ1,τ2) is not factorizable as J(τ1,τ2) = K(τ1)L(τ2), because R(µ1,µ2) = Rρ(µ1)Rφ(µ2), but not so far. . . R(τ1,τ2) reduces to the autocorrelation function R(τ) for τ = τ1 = τ2: the behaviour of the error function is linked to the shaping filter h.

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Envelope/phase delays correction in an EER radio architecture Algorithm Descent algorithms

Descent algorithms

Gradient algorithm τ1 τ2

(n+1) =

τ1 τ2

(n)−γ
∂J

∂τ1 ∂J ∂τ2

τ1=τ1(n),τ2=τ2(n)

Newton algorithm τ(n+1) = τ(n)−γH(n)−1G(n) with H hessian matrix, G gradient vector.

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Envelope/phase delays correction in an EER radio architecture Algorithm Descent algorithms

The gradient is       

∂J(τ1,τ2) ∂τ1

= −2E

dρ(u)

du

u=t1

cos(φ(t2))t e(t)

∂J(τ1,τ2)

∂τ2

= −2E

dφ(u)

du

u=t2

ρ(t1)dcos(φ(t2))t e(t)

with dcos(.) = [−sin(.) cos(.)] and

e(t) = x(t)−ρ(t +τ1 −∆1)cos(φ(t +τ2 −∆2)).

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Envelope/phase delays correction in an EER radio architecture Algorithm Descent algorithms

The gradient is       

∂J(τ1,τ2) ∂τ1

= −2E

dρ(u)

du

u=t1

cos(φ(t2))t e(t)

∂J(τ1,τ2)

∂τ2

= −2E

dφ(u)

du

u=t2

ρ(t1)dcos(φ(t2))t e(t)

with dcos(.) = [−sin(.) cos(.)] and

e(t) = x(t)−ρ(t +τ1 −∆1)cos(φ(t +τ2 −∆2)). Of course, we do not have analytical expressions of the statistical expectations involved, and have to resort to “blocks estimates”,

r iterative/adaptive versions.

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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.        τ1(n+1) = τ1(n)+γ1(n)E

dρ(u)

du

t1(n) cos(φ(t2(n)))t e(t)
τ2(n+1)

= τ2(n)+γ2(n)E

dφ(u)

du

t2(n) ρ(t1(n))dcos(φ(t2(n)))t e(t)
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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.        τ1(n+1) = τ1(n)+γ1(n)✓

❙

E

dρ(u)

du

t1(n) cos(φ(t2(n)))t e(t)
τ2(n+1)

= τ2(n)+γ2(n)✓

❙

E

dφ(u)

du

t2(n) ρ(t1(n))dcos(φ(t2(n)))t e(t)
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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.      τ1(n+1) = τ1(n)+γ1(n) dρ(u)

du

t1(n) cos(φ(t2(n)))t e(t)

τ2(n+1) = τ2(n)+γ2(n) dφ(u)

du

t2(n) ρ(t1(n))dcos(φ(t2(n)))t e(t)

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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.      τ1(n+1) = τ1(n)+γ1(n) dρ(u)

du

t1(n) cos(φ(t2(n)))t e(t)

τ2(n+1) = τ2(n)+γ2(n) dφ(u)

du

t2(n) ρ(t1(n))dcos(φ(t2(n)))t e(t)

What do we need?

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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.      τ1(n+1) = τ1(n)+γ1(n) dρ(u)

du

t1(n) cos(φ(t2(n)))t e(t)

τ2(n+1) = τ2(n)+γ2(n) dφ(u)

du

t2(n) ρ(t1(n))dcos(φ(t2(n)))t e(t)

What do we need? the error e(t)

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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.      τ1(n+1) = τ1(n)+γ1(n) dρ(u)

du

t1(n) cos(φ(t2(n)))t e(t)

τ2(n+1) = τ2(n)+γ2(n) dφ(u)

du

t2(n) ρ(t1(n))dcos(φ(t2(n)))t e(t)

What do we need? the error e(t) → easy to compute

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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.      τ1(n+1) = τ1(n)+γ1(n) dρ(u)

du

t1(n) cos(φ(t2(n)))t e(t)

τ2(n+1) = τ2(n)+γ2(n) dφ(u)

du

t2(n) ρ(t1(n))dcos(φ(t2(n)))t e(t)

What do we need? the error e(t) → easy to compute the derivatives dρ(u)

du

u=t1(n) and dφ(u)

du

u=t2(n) dcos(φ(t2(n))

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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.      τ1(n+1) = τ1(n)+γ1(n) dρ(u)

du

t1(n) cos(φ(t2(n)))t e(t)

τ2(n+1) = τ2(n)+γ2(n) dφ(u)

du

t2(n) ρ(t1(n))dcos(φ(t2(n)))t e(t)

What do we need? the error e(t) → easy to compute the derivatives dρ(u)

du

u=t1(n) and dφ(u)

du

u=t2(n) dcos(φ(t2(n))

→ approximation by finite differences of ρ(t) and cos(φ(t))

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Envelope/phase delays correction in an EER radio architecture Algorithm Interpolation

Interpolation

BUT the actual inputs of the system ρ(t +τ1(n)) and cos(φ(t +τ2(n)) are not known!

ρ(t+τ1-∆1) φ(t+τ2-∆2) RF Transmitter ρ(t+τ1) φ(t+τ2) ρ(t)cos(φ(t)) ρ(t+τ1-∆1)cos(φ(t+τ2-∆2)) ρ(n) φ(n) + − τ1,τ2 12/ 17

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Envelope/phase delays correction in an EER radio architecture Algorithm Interpolation

Interpolation

BUT the actual inputs of the system ρ(t +τ1(n)) and cos(φ(t +τ2(n)) are not known! What is only available are sequences ρ(n) and cos(φ(n)). One needs to interpolate so as to generate delayed inputs. Possibilities : Linear interpolation Newton interpolation Bessel interpolation Stirling interpolation

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Envelope/phase delays correction in an EER radio architecture Algorithm Results

Example of results – ∆1 = 0.36Ts,∆2 = 0.12Ts

200 400 600 800 1000 −0.05 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Estimated envelope delay Symbols Estimated delay γ=0.06 True delay Estimated delay γ=0.3 200 400 600 800 1000 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Estimated phase delay Symbols Estimated delay γ=0.06 True delay Estimated delay γ=0.3

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Envelope/phase delays correction in an EER radio architecture Algorithm Results

Example of results – ∆1 = 0.36Ts,∆2 = 0.12Ts

100 200 300 400 500 600 700 800 900 1000 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 Instantaneous error between the true and restored signals

586 588 590 592 594 596 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 Signals comparison Symbols true signal restored signal signal without delays compensation

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Envelope/phase delays correction in an EER radio architecture Algorithm Results

Results – ∆1 = 0.36Ts,∆2 = 0.12Ts – EVM=0.62%

−0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 −100 −80 −60 −40 −20 20 Normalized frequency Ideal Uncorrected Corrected 14/ 17

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Envelope/phase delays correction in an EER radio architecture Algorithm Results

Results – ∆1 = 0.76Ts,∆2 = 0.32Ts – EVM=1.2%

−0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 −100 −80 −60 −40 −20 20 Normalized frequency Ideal Uncorrected Corrected 15/ 17

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Envelope/phase delays correction in an EER radio architecture Algorithm Results

Behaviour with adaptation step γ

10

−2

10

−1

10 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Error vector Magnitude (%) γ

Bessel Linear Newton

EVM

10

−2

10

−1

10 100 200 300 400 500 Settling time γ Symbols bessel linear newton

Convergence

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Envelope/phase delays correction in an EER radio architecture Summary

Summary

proposed a new algorithm for delays correction in EER architecture, characterized and validated by simulations

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Envelope/phase delays correction in an EER radio architecture Summary

Summary

proposed a new algorithm for delays correction in EER architecture, characterized and validated by simulations Omitted sequence of decreasing steps, analytic derivation of J(τ1,τ2)

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Envelope/phase delays correction in an EER radio architecture Summary

Summary

proposed a new algorithm for delays correction in EER architecture, characterized and validated by simulations Omitted sequence of decreasing steps, analytic derivation of J(τ1,τ2) Todo block estimates - Newton algorithm Best strategy of minimization better/other interpolation - Farrow structure account for additive noise/robustness to noise dsp implementation - finite precision effects

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