A Refined Skew Matrix Model of the CIM3 in the Up-Mixer Extending the Duality of I/Q Imbalance (RFIT-2022)
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Presented at RFIT-2022 Session info : Advanced Circuit and System Design Date : Aug 30, 2022 Place : Busan, Korea Cite with the shortened URL if you like https://lnkd.in/gBhJmSRa [ URL/DOI of the paper/preprint ] DOI) 10.1109/RFIT54256.2022.9882474 URL) https://ieeexplore.ieee.org/document/9882474 https://www.researchgate.net/publication/363510202_A_Refined_Skew_Matrix_Model_of_the_CIM3_in_the_Up-Mixer_Extending_the_Duality_of_IQ_Imbalance Following(corrective) work of https://lnkd.in/gwJhqgb8
![1/16
GCT Semiconductor, Inc.
RFIT 2022
Table of Contents
β Introduction
β Meaning of the study on the CIM3-only DPD model.
β Review of the previous work of joint I/Q-imbalance and CIM3 model.
β Duality between the components of I/Q gain/phase mismatch
β Review of the duality in I/Q imbalance model.
β Extension and application to CIM3 model with conjugate signal representation
β Correction/Enhancement to the CIM3 models introduced in 5 years ago
β Identification of missing terms in prior works.
β Evaluation of the improvement after the correction.
β LMS adaptation revisited and its simplification
β Frequency domain => Time domain : Parsevalβs theorem
β Link to other works already established for I/Q imbalance : circularity
β Conclusion
β Refined version of joint CIM3 + I/Q imbalance model
[ pp. 2 ~ 5 ]
[ pp. 6 ~ 8 ]
[ pp. 9 ~ 10 ]
[ pp. 11 ~ 15 ]
[ p. 16 ]](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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A Refined Skew Matrix Model of the CIM3 in the Up-Mixer Extending the Duality of I/Q Imbalance (RFIT-2022)
- 1. GCT Semiconductor, Inc. RFIT 2022 A Refined Skew Matrix Model of the CIM3 in the Up-Mixer Extending the Duality of I/Q Imbalance Ealwan Lee GCT Semiconductor, Inc. Aug 30, 2022 Session T3B.4 (2:30 pm ~ 2: 50 pm) Advanced Circuit and System Designs
- 2. 1/16 GCT Semiconductor, Inc. RFIT 2022 Table of Contents β Introduction β Meaning of the study on the CIM3-only DPD model. β Review of the previous work of joint I/Q-imbalance and CIM3 model. β Duality between the components of I/Q gain/phase mismatch β Review of the duality in I/Q imbalance model. β Extension and application to CIM3 model with conjugate signal representation β Correction/Enhancement to the CIM3 models introduced in 5 years ago β Identification of missing terms in prior works. β Evaluation of the improvement after the correction. β LMS adaptation revisited and its simplification β Frequency domain => Time domain : Parsevalβs theorem β Link to other works already established for I/Q imbalance : circularity β Conclusion β Refined version of joint CIM3 + I/Q imbalance model [ pp. 2 ~ 5 ] [ pp. 6 ~ 8 ] [ pp. 9 ~ 10 ] [ pp. 11 ~ 15 ] [ p. 16 ]
- 3. 2/16 GCT Semiconductor, Inc. RFIT 2022 Introduction β What is CIM3 and Why it became nuisance ? β CIM3 = 3rd order Counter Inter-Modulation β Up-conversion mixer β One of the key factor in the SAW-less Tx implementation of 4G/5G/6G RF IC solution ο Violating the out-of-band emission spec from UL band to the DL band of specialty network. β Lowering the CIM3 inside the channel & band helps still in many ways. β Straightforward and Simple Approach β Lowering the signal level solves CIM3 problem at least but in trade-off with other metrics. β Reduction of signal by x1 dB => reduction of CIM3 by x3 dB. β A study of simple but plausible/consistent mathematical model of CIM3 helps β Characterizing, pushing to the limit of the analog circuitry in a systematic way. β Can be combined with other CIM3 reduction method. DL of other bands affected without TX SAW filter UL in operation fc fc+fm fc-3fm No effects to DL of other bands UL in operation fc IMD3 CIM3 * UL+DL in XDD or Full-duplex CIM3
- 4. 3/16 GCT Semiconductor, Inc. RFIT 2022 Review of Prior Works in RFIT2017 β Mathematical models of CIM3 + DPD up-mixer β Same model in cascade with complementary(typically negative) parameters. β 1st order cancellation as in typical I/Q imbalance compensator β Joint compensation with a single skew matrix + X X X X + zi zq yi yq π + πΉπ + ππ β ππ π β πΉπ β ππ β ππ πΉπ + ππ β ππ + X X X X + xi xq zi zq π β πΉπ β ππ β ππ π + πΉπ + ππ β ππ βπΉπ β ππ β ππ CIM3 distortion model (analog circuit) Digital Pre-Distortion model (digital processing) ππ = ππ β ππ ππ = ππ β ππ D/A D/A w/o DPD w/o DPD ππ + π β ππ ππ + π β ππ Image @ -fm CIM3 @ -3fm counter 3rd order intermodulation
- 5. 4/16 GCT Semiconductor, Inc. RFIT 2022 CIM3 in spectrum and phasor diagram β Despite perfect synchronization in digital domain, β Delay in feed-back path(οt), non-coherency between RF and BB(οο), Tx and Rx(οο±) matters. β No change in the spectrum of the up-mixer output. β Any distortion/compensation model should explain the effect of οt. reference vector (π¦+1) = y+1,i + jοy+1,q @ +f CIM3 vector @ -3f conjugate of reference vector (ΰ΄€ π¦+1) = y+1,i - jοy+1,q Re Im Joint I/Q imb + CIM3 compensator tone generator accumulator de-rotator +f -3f, -f, +f duration = N/f Synchronized with 1/f Up mixer {ο€g, ο€p ;Ο΅g, Ο΅p} RF-PLL Report * y-3 = CIM3 * y-1 = Image * y+1 = Desired eg, ep dg, dp Rx I/Q imb compensator D/A οt Variation of the delay in feed-back path +ππ πβπ βππ πβπ βππ πβπ +f -f -3f A/D οο¦ οο±
- 6. 5/16 GCT Semiconductor, Inc. RFIT 2022 Just Scribbling to Figure Out Something Else β Only combination of two terms tried to make (-3 * f) component in previous work. β 4 terms were able to be combined becoming insensitive to the phase shift. β cos 2ππ‘ term missed and can complement the missing part of sin 2ππ‘ . β But, should the number of parameters be increased from 2 to 4, then ? π¦π = π₯π + 0π₯π 3 + π2π₯π 2 π₯π 1 + π1π₯π 1 π₯π 2 + 0π₯π 3 π¦π = π₯π + 0π₯π 3 β π2π₯π 2 π₯π 1 + π1π₯π 1 π₯π 2 + 0π₯π 3 π¦π = π₯π + ΰ· π=0 3 πππ₯π π π₯π 3βπ π¦π = π₯π + ΰ· π=0 3 ππ π₯π 3βπ π₯π π Enforcing 0 oversight in prior work π₯π + 1π β π₯π = cos ππ‘ + 1π β sin ππ‘ π₯π 2 π₯π 1 β 1π β π₯π 1 π₯π 2 = 1 2 β sin 2ππ‘ β cos ππ‘ β 1π β sin ππ‘ Enforcing 0 oversight in prior work π₯π 1 π₯π 2 + 1π β π₯π 2 π₯π 1 = 1 2 π β sin 2ππ‘ β cos ππ‘ β 1π β sin ππ‘ π₯π 3 β π₯π 1 π₯π 2 β 1π β π₯π 2 π₯π 1 + 1π β π₯π 3 = cos 2ππ‘ β cos ππ‘ β 1π β sin ππ‘ βπ₯π 3 + 1π β π₯π 1 π₯π 2 + π₯π 2 π₯π 1 β 1π β π₯π 3 = π β cos 2ππ‘ β cos ππ‘ β 1π β sin ππ‘ π3 ? π0 ? fIF -3 fIF
- 7. 6/16 GCT Semiconductor, Inc. RFIT 2022 Duality of I/Q imbalance model in the (down)-mixer β gain mismatch(Ο΅g) and phase mismatch(Ο΅p) are exchangeable under signal rotation. β explaining the consistency of image signal and IRR in spectrum against co-ordinate rotation. Another proof by (2nd) geometric interpretation Down-mixer [2018] 1. L1-norm based LMS calibration 2. Completeness of symmetric skew matrix Applied to up-mixer in this paper. (Ο΅g/2, Ο΅g/2) => (ο€g,ο€g) Proof by (1st) simple arithmetic
- 8. 7/16 GCT Semiconductor, Inc. RFIT 2022 Duality of I/Q imbalance in conjugate signal representation β Real number matrix representation (used in two previous works) β intuitive and straightforward β Conjugate signal representation β More compact form is available. π = π + πΉ β ΰ΄₯ π π β ππ Ξ€ π π = π β ππ Ξ€ π π + πΉ β ΰ΄₯ π β ππ Ξ€ π π = π β ππ Ξ€ π π + πΉ β ππ Ξ€ π π β π β ππ Ξ€ π π ΰ·₯ π = ΰ·₯ π + πΉ β ππ Ξ€ π π β ΰ΄₯ ΰ·₯ π = ΰ·₯ π + ΰ·© πΉ β ΰ΄₯ ΰ·₯ π ΰ·₯ π β π β ππ Ξ€ π π ΰ·₯ π β π β ππ Ξ€ π π ΰ·© πΉ β πΉ β ππ Ξ€ π π π¦π π¦π = 1 + πΏπ +πΏπ +πΏπ 1 β πΏπ β π₯π π₯π = 1 0 0 1 β π₯π π₯π + +πΏπ +πΏπ +πΏπ βπΏπ β π₯π π₯π π = π¦π + π β π¦π π = π₯π + π β π₯π ΰ΄₯ π = π₯π β π β π₯π πΉ = πΏπ + π β πΏπ ο Rotation by ο°/4 ΰ·© πΉ = βπΏπ + π β πΏπ ο α πΏπ = βπΏπ α πΏπ = +πΏπ Conjugate signal representation (in complex number) Another proof (3rd) in conjugate representation
- 9. 8/16 GCT Semiconductor, Inc. RFIT 2022 Application of the Duality to CIM3 of up-mixer β Conjugate signal representation β desired signal (+fm from carrier frequency) : π β Image signal (-fm from carrier frequency) : ΰ΄₯ π β CIM3 (-3fm from carrier frequency) : ΰ΄₯ ππ π = π + πΉ β ΰ΄₯ π + π β ΰ΄₯ ππ = π + πΉ β ΰ΄₯ π + π β ΰ΄₯ ππ β ΰ΄₯ π π β ππ Ξ€ π π = π + πΉ β ΰ΄₯ π + π β ΰ΄₯ ππ β ΰ΄₯ π β ππ Ξ€ π π = π β ππ Ξ€ π π + πΉ β ππ Ξ€ π π β π β ππ Ξ€ π π + π β πππ β π β ππ Ξ€ π π π β π β ππ Ξ€ π π ΰ·₯ π = ΰ·₯ π + ΰ·© πΉ β ΰ΄₯ ΰ·₯ π + ΰ·€ π β ΰ΄₯ ΰ·₯ ππ β ΰ΄₯ ΰ·₯ π = ΰ·₯ π + ΰ·© πΉ β ΰ΄₯ ΰ·₯ π + ΰ·€ π β π β ΰ΄₯ ΰ·₯ π ΰ·₯ π β π β ππ Ξ€ π π ΰ·₯ π β π β ππ Ξ€ π π ΰ·© πΉ β πΉ β ππ Ξ€ π π ΰ·€ π β π β πππ Η ππ = βππ Η ππ = βππ Rotation by ο°/4 as in I/Q imbalance ο π = ππ + π β ππ π β ΰ΄₯ ΰ·₯ ππ π β ππ + π β ππ = 2 β π₯π β π₯π + π β π₯π 2 β π₯π 2 ππ = 2 β π₯π β π₯π ππ = π₯π 2 β π₯π 2
- 10. 9/16 GCT Semiconductor, Inc. RFIT 2022 Donβt be confused with (Ο΅g, Ο΅g) Identification of the component missed in our prior work β Comparison with prior works in matrix representation β degree of freedom for CIM3 is kept to 2 in spite of added term : (Ο΅g, Ο΅g) π¦π π¦π = 1 0 0 1 + πΏπ + ππ β ππ + ππ β ππ β +1 0 0 β1 + πΏπ + ππ β ππ β ππ β ππ β 0 +1 +1 0 β π₯π π₯π ππ = 2 β π₯π β π₯π ππ = π₯π 2 β π₯π 2 π = π₯π β π₯π π¦π π¦π = 1 0 0 1 + +πΏπ πΏπ πΏπ βπΏπ + ππ β ππ +ππ +ππ βππ β π₯π π₯π + ππ β ππ βππ βππ βππ β π₯π π₯π Terms missed in previous work and complemented in this work Terms simplified and merged in previous work with matrix arithmetic. Skew matrix for I/Q imbalance ΰ·€ ππ = 2 β ΰ·€ π₯π β ΰ·€ π₯π = π₯π 2 β π₯π 2 ΰ·€ ππ = ΰ·€ π₯π 2 β ΰ·€ π₯π 2 = 2 β π₯π β π₯π re-scaled and re-named for generalization Rotation by ο°/4
- 11. 10/16 GCT Semiconductor, Inc. RFIT 2022 Comparison of two models side-by-side β Re-plot of previous works by flipping the direction of before/after the DPD for comparison. β Reflection of the sequential estimation of ο€ and ο β Scope of the problem has extended. β All the harmonics at -f, +3f, +5f not just -3f should be suppressed at its best of the given model. β Cascade of a single skew matrix has its limitation leaving artifact on +3f, +5f. Before DPD After DPD Image @ -9 MHz -26 => -56 (dBc) CIM3 @ -27 MHz -38 => -64 (dBc) Image @ -9 MHz -26 => -71 (dBc) CIM3 @ -27 MHz -38 => -72 (dBc) Prior model in RFIT2017 @ Seoul - Single run applied from for ο€ ,then ο. Refined model in RFIT2022 @ Busan - Optimal ο€, ο obtained with iterative joint LMS adaptation.
- 12. 11/16 GCT Semiconductor, Inc. RFIT 2022 Derivation of LMS Adaptation Formulae β Frequency domain β Obtain the Fourier-transform at each desired frequency, π +π , π(β3π) β Process only π +π to obtain π +π 3 β Derived measure is insensitive to the rotation caused by the delay of the feed-back path. +fm +3fm -3fm -fm 0 π(βπππ) β πβπππ ππβπ π(+π) β π+πππ ππβπ +fm +3fm -3fm -fm 0 π(+π)π β π+πππ ππβπ π(βπππ) β πβπππ πππ βπ β Ξ€ π βπππ π +ππ π β π(βπππ) β π(+ππ)π π +ππ π β π(+ππ)π π = π + πΉ β ΰ΄₯ π + π β ΰ΄₯ ππ Norm of the signal that can be absorbed into update factor.
- 13. 12/16 GCT Semiconductor, Inc. RFIT 2022 Derivation of LMS Adaptation Formulae (contβd) β Frequency domain β How to align the desired and distorted component with the same offset from carrier(DC) ? β Squaring in time-domain => Convolution in freq-domain. β Previous method did not pre-processed time-domain signal before DFT. β This measure is also insensitive to the rotation caused by the delay of the feed-back path. β Frequency domain => Time domain : next page π[π]π +fm +2fm -3fm -fm 0 π(+ππ)π β π+πππ πππ -2fm π(βπππ)βπ(+ππ)βπβπππ πππ -6fm +fm +2fm -3fm -fm 0 π(βπππ) β πβπππ πππ π(+ππ)βπ+πππ πππ -2fm π[π] βπ β π βπππ β π ππ π β π(βπππ) β π(ππ) β π(ππ)π +π β ππ βπ β ππ +π β ππ βπ β ππ
- 14. 13/16 GCT Semiconductor, Inc. RFIT 2022 Time domain : Circularity in I/Q imbalance => Circularity in CIM3 β Real and Imaginary part of higher order statistics should be zero, respectively. β Time-domain formula from the frequency-domain relation is derived using Parsevalβs theorem. β Circularity is preserved under the signal rotation including +ο°/4. ΰ·‘ πΉ π β ΰ·‘ πΉ π β π + ππΉ β π(βπ) β π(+π) ΰ·‘ πΉ π β ΰ·‘ πΉ π β π + ππΉ β π[π]π ΰ· π π β ΰ· π π β π + ππ β π βππ β π +π βπ β ΰ· π π β π + ππ β π βππ β π +π +π = ΰ· π π β π + ππ β π βππ β π +π β π +π +π π¬ ππ[π] β ππ[π] β π π¬ ππ π π β ππ π π β π π¬ ππ π + ππ π β ππ π β ππ π β π π¬ ππ[π] β ππ[π] β ππ π + ππ[π] β ππ[π] β ππ[π] β π π¬ ππ π + ππ[π] π β ππ[π] β ππ[π] π β π β ππ[π] β ππ[π] π β π ΰ· π π β ΰ· π π β π + ππ β π[π]π π[π]π βπ[π] βπΉ[π] π[π] CIM3 I/Q imbalance(Image) π[π]π Parsevalβs Theorem Parsevalβs Theorem @ Β±π β ππ @ Β±π β ππ
- 15. 14/16 GCT Semiconductor, Inc. RFIT 2022 Joint Compensation Model => Joint LMS adaptation β Lead-in stage : Only πΏπ, πΏπ is updated. β After the images falls below pre-defined threshold, steps into main stage. β Main stage : Both πΏπ, πΏπ and ππ, ππ are jointly updated. 0 2 4 6 8 10 12 -120 -110 -100 -90 -80 -70 -60 -50 -40 -30 Time (us) @ 160Msps rate for fm p = 10 MHz CIM3, Image (dBc) CIM3 @ time-domain CIM3 @ freq-domain Image @ time-domain Image @ freq-domain Window size and Update interval of freq-domain ππ, ππ πΏπ, πΏπ
- 16. 15/16 GCT Semiconductor, Inc. RFIT 2022 Trivia and Tips for the Sake of Reference β Parsevalβs theorem β Well-known form in Electrical Engineering β Energy measured in time domain or frequency domain(Fourier-transform) are same. β Generalized form with two functions β Inner product of two functions in two spaces under unitary transforms are same. β Dependency between signal domain and update method. β Frequency domain update can be done at sampling rate if DFT is done with sliding window. β Currently only a single value is obtained per non-overlapping block for efficient implementation. β Time domain update can also be done in block-wise manner as with Frequency domain. β However, update at sampling rate in time domain requires less hardware than block-wise one. β Especially, π¦4 can be obtained from π¦2 already obtained for the I/Q imbalance calibration. ΰ· π=ββ +β π[π] π = ΰΆ± βπ +π π¨(π) β π¨(π)π π ΰ· π=ββ +β π[π] β π[π] = ΰΆ± βπ +π π¨(π) β π©(π)π π π[π] ππ[π] π/π π[π] ππ[π] π/π
- 17. 16/16 GCT Semiconductor, Inc. RFIT 2022 Conclusion β A refined model for joint CIM3 and I/Q imbalance model has been proposed. β Duality similar to that of I/Q imbalance has been also applied to CIM3. β Missing components identified and complemented. β Framework for the proposed model unified to get the help from β Conjugate signal representation β Circularity of the signal. β Derivation of the joint LMS adaptation formulae. β Optimization => LMS β Frequency domain => Time domain β Block-wise => Sample-wise β URL of the slide/presentation β [4] Duality of the I/Q imbalance : https://lnkd.in/gy24S4kF β [5] CIM3 in RFIT2017 : https://lnkd.in/gwJhqgb8