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![1/16
GCT Semiconductor, Inc.
RFIT 2022
Table of Contents
❑ Introduction
◆ Meaning of the study on the CIM3-only DPD model.
◆ Review of the previous work of joint I/Q-imbalance and CIM3 model.
❑ Duality between the components of I/Q gain/phase mismatch
◆ Review of the duality in I/Q imbalance model.
◆ Extension and application to CIM3 model with conjugate signal representation
❑ Correction/Enhancement to the CIM3 models introduced in 5 years ago
◆ Identification of missing terms in prior works.
◆ Evaluation of the improvement after the correction.
❑ LMS adaptation revisited and its simplification
◆ Frequency domain => Time domain : Parseval’s theorem
◆ Link to other works already established for I/Q imbalance : circularity
❑ Conclusion
◆ Refined version of joint CIM3 + I/Q imbalance model
[ pp. 2 ~ 5 ]
[ pp. 6 ~ 8 ]
[ pp. 9 ~ 10 ]
[ pp. 11 ~ 15 ]
[ p. 16 ]](https://image.slidesharecdn.com/rfit2022-ealwanleet3b4-220828055639-285914d3/75/A-Refined-Skew-Matrix-Model-of-the-CIM3-in-the-Up-Mixer-Extending-the-Duality-of-I-Q-Imbalance-RFIT-2022-2-2048.jpg)
![6/16
GCT Semiconductor, Inc.
RFIT 2022
Duality of I/Q imbalance model in the (down)-mixer
❑ gain mismatch(ϵg) and phase mismatch(ϵp) are exchangeable under signal rotation.
◆ 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.
(ϵg/2, ϵg/2) => (g,g)
Proof by (1st)
simple arithmetic](https://image.slidesharecdn.com/rfit2022-ealwanleet3b4-220828055639-285914d3/75/A-Refined-Skew-Matrix-Model-of-the-CIM3-in-the-Up-Mixer-Extending-the-Duality-of-I-Q-Imbalance-RFIT-2022-7-2048.jpg)
![12/16
GCT Semiconductor, Inc.
RFIT 2022
Derivation of LMS Adaptation Formulae (cont’d)
❑ Frequency domain
◆ How to align the desired and distorted component with the same offset from carrier(DC) ?
◆ Squaring in time-domain => Convolution in freq-domain.
✓ Previous method did not pre-processed time-domain signal before DFT.
✓ This measure is also insensitive to the rotation caused by the delay of the feed-back path.
❑ Frequency domain => Time domain : next page
𝒚[𝒏]𝟐
+fm +2fm
-3fm -fm 0
𝒀(+𝒇𝒎)𝟐
∙ 𝒆+𝒋𝟒𝝅𝒇𝒎𝒕
-2fm
𝒀(−𝟑𝒇𝒎)∙𝒀(+𝒇𝒎)∙𝒆−𝒋𝟒𝝅𝒇𝒎𝒕
-6fm
+fm +2fm
-3fm -fm 0
𝒀(−𝟑𝒇𝒎) ∙ 𝒆−𝒋𝟔𝝅𝒇𝒎𝒕
𝒀(+𝒇𝒎)∙𝒆+𝒋𝟐𝝅𝒇𝒎𝒕
-2fm
𝒚[𝒏]
∆𝝐 ∝ 𝒀 −𝟑𝒇𝒎 ⋅ 𝒀 𝒇𝒎
𝟑
∝ 𝒀(−𝟑𝒇𝒎) ⋅ 𝒀(𝒇𝒎) ⋅ 𝒀(𝒇𝒎)𝟐
+𝟐 ⋅ 𝒇𝒎
−𝟐 ⋅ 𝒇𝒎
+𝟐 ⋅ 𝒇𝒎
−𝟐 ⋅ 𝒇𝒎](https://image.slidesharecdn.com/rfit2022-ealwanleet3b4-220828055639-285914d3/75/A-Refined-Skew-Matrix-Model-of-the-CIM3-in-the-Up-Mixer-Extending-the-Duality-of-I-Q-Imbalance-RFIT-2022-13-2048.jpg)
![13/16
GCT Semiconductor, Inc.
RFIT 2022
Time domain : Circularity in I/Q imbalance => Circularity in CIM3
❑ Real and Imaginary part of higher order statistics should be zero, respectively.
◆ Time-domain formula from the frequency-domain relation is derived using Parseval’s theorem.
◆ Circularity is preserved under the signal rotation including +/4.
𝜹 𝒏 ≔
𝜹 𝒏 − 𝟏 + 𝝁𝜹 ⋅ 𝒀(−𝒇) ⋅ 𝒀(+𝒇)
𝜹 𝒏 ≔
𝜹 𝒏 − 𝟏 + 𝝁𝜹 ⋅ 𝒚[𝒏]𝟐
ො
𝝐 𝒏 ≔ ො
𝝐 𝒏 − 𝟏 + 𝝁𝝐 ⋅ 𝒀 −𝟑𝒇 ⋅ 𝒀 +𝒇 −𝟑
≔ ො
𝝐 𝒏 − 𝟏 + 𝝁𝝐 ⋅ 𝒀 −𝟑𝒇 ⋅ 𝒀 +𝒇 +𝟑
= ො
𝝐 𝒏 − 𝟏 + 𝝁𝝐 ⋅ 𝒀 −𝟑𝒇 ⋅ 𝒀 +𝒇 ⋅ 𝒀 +𝒇 +𝟐
𝑬 𝒚𝒊[𝒏] ⋅ 𝒚𝒒[𝒏] → 𝟎
𝑬 𝒚𝒊
𝟐
𝒏 − 𝒚𝒒
𝟐
𝒏 → 𝟎
𝑬 𝒚𝒊 𝒏 + 𝒚𝒒 𝒏 ⋅ 𝒚𝒊 𝒏 − 𝒚𝒒 𝒏 → 𝟎
𝑬 𝒚𝒊[𝒏] ⋅ 𝒚𝒒[𝒏] ⋅ 𝒚𝒊 𝒏 + 𝒚𝒒[𝒏] ⋅ 𝒚𝒊[𝒏] − 𝒚𝒒[𝒏] → 𝟎
𝑬 𝒚𝒊 𝒏 + 𝒚𝒒[𝒏]
𝟐
⋅ 𝒚𝒊[𝒏] − 𝒚𝒒[𝒏]
𝟐
− 𝟐 ⋅ 𝒚𝒊[𝒏] ⋅ 𝒚𝒒[𝒏]
𝟐
→ 𝟎
ො
𝝐 𝒏 ≔ ො
𝝐 𝒏 − 𝟏 + 𝝁𝝐 ⋅ 𝒚[𝒏]𝟒
𝒚[𝒏]𝟒
∆𝝐[𝒏]
∆𝜹[𝒏]
𝒚[𝒏]
CIM3
I/Q imbalance(Image)
𝒚[𝒏]𝟐
Parseval’s
Theorem
Parseval’s
Theorem
@ ±𝟐 ⋅ 𝒇𝒎
@ ±𝟏 ⋅ 𝒇𝒎](https://image.slidesharecdn.com/rfit2022-ealwanleet3b4-220828055639-285914d3/75/A-Refined-Skew-Matrix-Model-of-the-CIM3-in-the-Up-Mixer-Extending-the-Duality-of-I-Q-Imbalance-RFIT-2022-14-2048.jpg)
![15/16
GCT Semiconductor, Inc.
RFIT 2022
Trivia and Tips for the Sake of Reference
❑ Parseval’s theorem
◆ Well-known form in Electrical Engineering
✓ Energy measured in time domain or frequency domain(Fourier-transform) are same.
◆ Generalized form with two functions
✓ Inner product of two functions in two spaces under unitary transforms are same.
❑ Dependency between signal domain and update method.
◆ Frequency domain update can be done at sampling rate if DFT is done with sliding window.
✓ Currently only a single value is obtained per non-overlapping block for efficient implementation.
◆ Time domain update can also be done in block-wise manner as with Frequency domain.
✓ However, update at sampling rate in time domain requires less hardware than block-wise one.
✓ Especially, 𝑦4
can be obtained from 𝑦2
already obtained for the I/Q imbalance calibration.
𝒏=−∞
+∞
𝒂[𝒏] 𝟐
= න
−𝝅
+𝝅
𝑨(𝒇) ⋅ 𝑨(𝒇)𝒅𝒇
𝒏=−∞
+∞
𝒂[𝒏] ⋅ 𝒃[𝒏] = න
−𝝅
+𝝅
𝑨(𝒇) ⋅ 𝑩(𝒇)𝒅𝒇
𝒚[𝒏]
𝒀𝒇[𝒌]
𝟏/𝒇
𝒚[𝒏]
𝒀𝒇[𝒌]
𝟏/𝒇](https://image.slidesharecdn.com/rfit2022-ealwanleet3b4-220828055639-285914d3/75/A-Refined-Skew-Matrix-Model-of-the-CIM3-in-the-Up-Mixer-Extending-the-Duality-of-I-Q-Imbalance-RFIT-2022-16-2048.jpg)
![16/16
GCT Semiconductor, Inc.
RFIT 2022
Conclusion
❑ A refined model for joint CIM3 and I/Q imbalance model has been proposed.
◆ Duality similar to that of I/Q imbalance has been also applied to CIM3.
◆ Missing components identified and complemented.
❑ Framework for the proposed model unified to get the help from
◆ Conjugate signal representation
◆ Circularity of the signal.
❑ Derivation of the joint LMS adaptation formulae.
◆ Optimization => LMS
◆ Frequency domain => Time domain
◆ Block-wise => Sample-wise
❑ URL of the slide/presentation
◆ [4] Duality of the I/Q imbalance : https://lnkd.in/gy24S4kF
◆ [5] CIM3 in RFIT2017 : https://lnkd.in/gwJhqgb8](https://image.slidesharecdn.com/rfit2022-ealwanleet3b4-220828055639-285914d3/75/A-Refined-Skew-Matrix-Model-of-the-CIM3-in-the-Up-Mixer-Extending-the-Duality-of-I-Q-Imbalance-RFIT-2022-17-2048.jpg)
Uploaded byEalwan Lee
A Refined Skew Matrix Model of the CIM3 in the Up-Mixer Extending the Duality of I/Q Imbalance (RFIT-2022)
The document presents a refined skew matrix model of a third-order counter inter-modulation (CIM3) in an up-mixer, highlighting its relevance in 4G/5G/6G RF IC solutions. It reviews previous works on joint I/Q imbalance and the CIM3 model while introducing corrections and enhancements to these models. The conclusion emphasizes the development of an improved joint CIM3 and I/Q imbalance model.
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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. RFIT2022 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. RFIT2022 Table of Contents ❑ Introduction ◆ Meaning of the study on the CIM3-only DPD model. ◆ Review of the previous work of joint I/Q-imbalance and CIM3 model. ❑ Duality between the components of I/Q gain/phase mismatch ◆ Review of the duality in I/Q imbalance model. ◆ Extension and application to CIM3 model with conjugate signal representation ❑ Correction/Enhancement to the CIM3 models introduced in 5 years ago ◆ Identification of missing terms in prior works. ◆ Evaluation of the improvement after the correction. ❑ LMS adaptation revisited and its simplification ◆ Frequency domain => Time domain : Parseval’s theorem ◆ Link to other works already established for I/Q imbalance : circularity ❑ Conclusion ◆ 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. RFIT2022 Introduction ❑ What is CIM3 and Why it became nuisance ? ◆ CIM3 = 3rd order Counter Inter-Modulation ◆ Up-conversion mixer ✓ One of the key factor in the SAW-less Tx implementation of 4G/5G/6G RF IC solution Violating the out-of-band emission spec from UL band to the DL band of specialty network. ✓ Lowering the CIM3 inside the channel & band helps still in many ways. ❑ Straightforward and Simple Approach ◆ Lowering the signal level solves CIM3 problem at least but in trade-off with other metrics. ✓ Reduction of signal by x1 dB => reduction of CIM3 by x3 dB. ❑ A study of simple but plausible/consistent mathematical model of CIM3 helps ◆ Characterizing, pushing to the limit of the analog circuitry in a systematic way. ◆ 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. RFIT2022 Review of Prior Works in RFIT2017 ❑ Mathematical models of CIM3 + DPD up-mixer ◆ Same model in cascade with complementary(typically negative) parameters. ✓ 1st order cancellation as in typical I/Q imbalance compensator ✓ Joint compensation with a single skew matrix + X X X X + zi zq yi yq 𝟏 + 𝜹𝒈 + 𝝆𝒛 ⋅ 𝝐𝒈 𝟏 − 𝜹𝒈 − 𝝆𝒛 ⋅ 𝝐𝒈 𝜹𝒑 + 𝝆𝒛 ⋅ 𝝐𝒑 + X X X X + xi xq zi zq 𝟏 − 𝜹𝒈 − 𝝆𝒙 ⋅ 𝝐𝒈 𝟏 + 𝜹𝒈 + 𝝆𝒙 ⋅ 𝝐𝒈 −𝜹𝒑 − 𝝆𝒙 ⋅ 𝝐𝒑 CIM3 distortion model (analog circuit) Digital Pre-Distortion model (digital processing) 𝝆𝒙 = 𝒙𝒊 ⋅ 𝒙𝒒 𝝆𝒛 = 𝒛𝒊 ⋅ 𝒛𝒒 D/A D/A w/o DPD w/o DPD 𝒛𝒊 + 𝒋 ⋅ 𝒛𝒒 𝒚𝒊 + 𝒋 ⋅ 𝒚𝒒 Image @ -fm CIM3 @ -3fm counter 3rd order intermodulation
- 5. 4/16 GCT Semiconductor, Inc. RFIT2022 CIM3 in spectrum and phasor diagram ❑ Despite perfect synchronization in digital domain, ◆ Delay in feed-back path(t), non-coherency between RF and BB(), Tx and Rx() matters. ◆ No change in the spectrum of the up-mixer output. ❑ Any distortion/compensation model should explain the effect of t. reference vector (𝑦+1) = y+1,i + jy+1,q @ +f CIM3 vector @ -3f conjugate of reference vector (ത 𝑦+1) = y+1,i - jy+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 {g, p ;ϵg, ϵp} RF-PLL Report * y-3 = CIM3 * y-1 = Image * y+1 = Desired eg, ep dg, dp Rx I/Q imb compensator D/A t Variation of the delay in feed-back path +𝟐𝝅𝒇∆𝒕 −𝟐𝝅𝒇∆𝒕 −𝟔𝝅𝒇∆𝒕 +f -f -3f A/D
- 6. 5/16 GCT Semiconductor, Inc. RFIT2022 Just Scribbling to Figure Out Something Else ❑ Only combination of two terms tried to make (-3 * f) component in previous work. ❑ 4 terms were able to be combined becoming insensitive to the phase shift. ◆ cos 2𝜔𝑡 term missed and can complement the missing part of sin 2𝜔𝑡 . ◆ But, should the number of parameters be increased from 2 to 4, then ? 𝑦𝑖 = 𝑥𝑖 + 0𝑥𝑖 3 + 𝑏2𝑥𝑖 2 𝑥𝑞 1 + 𝑏1𝑥𝑖 1 𝑥𝑞 2 + 0𝑥𝑞 3 𝑦𝑞 = 𝑥𝑞 + 0𝑥𝑞 3 − 𝑏2𝑥𝑞 2 𝑥𝑖 1 + 𝑏1𝑥𝑞 1 𝑥𝑖 2 + 0𝑥𝑖 3 𝑦𝑖 = 𝑥𝑖 + 𝑛=0 3 𝑏𝑛𝑥𝑖 𝑛 𝑥𝑞 3−𝑛 𝑦𝑞 = 𝑥𝑞 + 𝑛=0 3 𝑐𝑛 𝑥𝑖 3−𝑛 𝑥𝑞 𝑛 Enforcing 0 oversight in prior work 𝑥𝑖 + 1𝑗 ∙ 𝑥𝑞 = cos 𝜔𝑡 + 1𝑗 ∙ sin 𝜔𝑡 𝑥𝑖 2 𝑥𝑞 1 − 1𝑗 ∙ 𝑥𝑖 1 𝑥𝑞 2 = 1 2 ∙ sin 2𝜔𝑡 ∙ cos 𝜔𝑡 − 1𝑗 ∙ sin 𝜔𝑡 Enforcing 0 oversight in prior work 𝑥𝑖 1 𝑥𝑞 2 + 1𝑗 ∙ 𝑥𝑖 2 𝑥𝑞 1 = 1 2 𝑗 ∙ sin 2𝜔𝑡 ∙ cos 𝜔𝑡 − 1𝑗 ∙ sin 𝜔𝑡 𝑥𝑖 3 − 𝑥𝑖 1 𝑥𝑞 2 − 1𝑗 ∙ 𝑥𝑖 2 𝑥𝑞 1 + 1𝑗 ∙ 𝑥𝑞 3 = cos 2𝜔𝑡 ∙ cos 𝜔𝑡 − 1𝑗 ∙ sin 𝜔𝑡 −𝑥𝑞 3 + 1𝑗 ∙ 𝑥𝑖 1 𝑥𝑞 2 + 𝑥𝑖 2 𝑥𝑞 1 − 1𝑗 ∙ 𝑥𝑖 3 = 𝑗 ∙ cos 2𝜔𝑡 ∙ cos 𝜔𝑡 − 1𝑗 ∙ sin 𝜔𝑡 𝑏3 ? 𝑏0 ? fIF -3 fIF
- 7. 6/16 GCT Semiconductor, Inc. RFIT2022 Duality of I/Q imbalance model in the (down)-mixer ❑ gain mismatch(ϵg) and phase mismatch(ϵp) are exchangeable under signal rotation. ◆ 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. (ϵg/2, ϵg/2) => (g,g) Proof by (1st) simple arithmetic
- 8. 7/16 GCT Semiconductor, Inc. RFIT2022 Duality of I/Q imbalance in conjugate signal representation ❑ Real number matrix representation (used in two previous works) ◆ intuitive and straightforward ❑ Conjugate signal representation ◆ More compact form is available. 𝒚 = 𝒙 + 𝜹 ⋅ ഥ 𝒙 𝒚 ⋅ 𝒆𝒋 Τ 𝝅 𝟒 = 𝒙 ⋅ 𝒆𝒋 Τ 𝝅 𝟒 + 𝜹 ⋅ ഥ 𝒙 ⋅ 𝒆𝒋 Τ 𝝅 𝟒 = 𝒙 ⋅ 𝒆𝒋 Τ 𝝅 𝟒 + 𝜹 ⋅ 𝒆𝒋 Τ 𝝅 𝟐 ⋅ 𝒙 ⋅ 𝒆𝒋 Τ 𝝅 𝟒 𝒚 = 𝒙 + 𝜹 ⋅ 𝒆𝒋 Τ 𝝅 𝟐 ⋅ ഥ 𝒙 = 𝒙 + ෩ 𝜹 ⋅ ഥ 𝒙 𝒚 ≜ 𝒚 ⋅ 𝒆𝒋 Τ 𝝅 𝟒 𝒙 ≜ 𝒙 ⋅ 𝒆𝒋 Τ 𝝅 𝟒 ෩ 𝜹 ≜ 𝜹 ⋅ 𝒆𝒋 Τ 𝝅 𝟐 𝑦𝑖 𝑦𝑞 = 1 + 𝛿𝑔 +𝛿𝑝 +𝛿𝑝 1 − 𝛿𝑔 ⋅ 𝑥𝑖 𝑥𝑞 = 1 0 0 1 ⋅ 𝑥𝑖 𝑥𝑞 + +𝛿𝑔 +𝛿𝑝 +𝛿𝑝 −𝛿𝑔 ⋅ 𝑥𝑖 𝑥𝑞 𝒚 = 𝑦𝑖 + 𝑗 ⋅ 𝑦𝑞 𝒙 = 𝑥𝑖 + 𝑗 ⋅ 𝑥𝑞 ഥ 𝒙 = 𝑥𝑖 − 𝑗 ⋅ 𝑥𝑞 𝜹 = 𝛿𝑔 + 𝑗 ⋅ 𝛿𝑝 Rotation by /4 ෩ 𝜹 = −𝛿𝑝 + 𝑗 ⋅ 𝛿𝑔 ሚ 𝛿𝑔 = −𝛿𝑝 ሚ 𝛿𝑝 = +𝛿𝑔 Conjugate signal representation (in complex number) Another proof (3rd) in conjugate representation
- 9. 8/16 GCT Semiconductor, Inc. RFIT2022 Application of the Duality to CIM3 of up-mixer ❑ Conjugate signal representation ◆ desired signal (+fm from carrier frequency) : 𝒙 ◆ Image signal (-fm from carrier frequency) : ഥ 𝒙 ◆ CIM3 (-3fm from carrier frequency) : ഥ 𝒙𝟑 𝒚 = 𝒙 + 𝜹 ⋅ ഥ 𝒙 + 𝝐 ⋅ ഥ 𝒙𝟑 = 𝒙 + 𝜹 ⋅ ഥ 𝒙 + 𝝐 ⋅ ഥ 𝒙𝟐 ⋅ ഥ 𝒙 𝒚 ⋅ 𝒆𝒋 Τ 𝝅 𝟒 = 𝒙 + 𝜹 ⋅ ഥ 𝒙 + 𝝐 ⋅ ഥ 𝒙𝟐 ⋅ ഥ 𝒙 ⋅ 𝒆𝒋 Τ 𝝅 𝟒 = 𝒙 ⋅ 𝒆𝒋 Τ 𝝅 𝟒 + 𝜹 ⋅ 𝒆𝒋 Τ 𝝅 𝟐 ⋅ 𝒙 ⋅ 𝒆𝒋 Τ 𝝅 𝟒 + 𝝐 ⋅ 𝒆𝒋𝝅 ⋅ 𝒙 ⋅ 𝒆𝒋 Τ 𝝅 𝟒 𝟐 ⋅ 𝒙 ⋅ 𝒆𝒋 Τ 𝝅 𝟒 𝒚 = 𝒙 + ෩ 𝜹 ⋅ ഥ 𝒙 + 𝝐 ⋅ ഥ 𝒙𝟐 ⋅ ഥ 𝒙 = 𝒙 + ෩ 𝜹 ⋅ ഥ 𝒙 + 𝝐 ⋅ 𝝆 ⋅ ഥ 𝒙 𝒚 ≜ 𝒚 ⋅ 𝒆𝒋 Τ 𝝅 𝟒 𝒙 ≜ 𝒙 ⋅ 𝒆𝒋 Τ 𝝅 𝟒 ෩ 𝜹 ≜ 𝜹 ⋅ 𝒆𝒋 Τ 𝝅 𝟐 𝝐 ≜ 𝝐 ⋅ 𝒆𝒋𝝅 ǁ 𝜖𝑔 = −𝜖𝑔 ǁ 𝜖𝑝 = −𝜖𝑝 Rotation by /4 as in I/Q imbalance 𝝐 = 𝜖𝑔 + 𝑗 ⋅ 𝜖𝑝 𝝆 ≜ ഥ 𝒙𝟐 𝝆 ≜ 𝜌𝑔 + 𝑗 ⋅ 𝜌𝑝 = 2 ⋅ 𝑥𝑖 ⋅ 𝑥𝑞 + 𝑗 ⋅ 𝑥𝑞 2 − 𝑥𝑖 2 𝜌𝑔 = 2 ⋅ 𝑥𝑖 ⋅ 𝑥𝑞 𝜌𝑝 = 𝑥𝑞 2 − 𝑥𝑖 2
- 10. 9/16 GCT Semiconductor, Inc. RFIT2022 Don’t be confused with (ϵg, ϵg) Identification of the component missed in our prior work ❑ Comparison with prior works in matrix representation ◆ degree of freedom for CIM3 is kept to 2 in spite of added term : (ϵg, ϵg) 𝑦𝑖 𝑦𝑞 = 1 0 0 1 + 𝛿𝑔 + 𝜌𝑔 ⋅ 𝜖𝑔 + 𝜌𝑝 ⋅ 𝜖𝑝 ⋅ +1 0 0 −1 + 𝛿𝑝 + 𝜌𝑔 ⋅ 𝜖𝑝 − 𝜌𝑝 ⋅ 𝜖𝑔 ⋅ 0 +1 +1 0 ⋅ 𝑥𝑖 𝑥𝑞 𝜌𝑔 = 2 ⋅ 𝑥𝑖 ⋅ 𝑥𝑞 𝜌𝑝 = 𝑥𝑞 2 − 𝑥𝑖 2 𝜌 = 𝑥𝑖 ⋅ 𝑥𝑞 𝑦𝑖 𝑦𝑞 = 1 0 0 1 + +𝛿𝑔 𝛿𝑝 𝛿𝑝 −𝛿𝑔 + 𝜌𝑔 ⋅ 𝜖𝑔 +𝜖𝑝 +𝜖𝑝 −𝜖𝑔 ⋅ 𝑥𝑖 𝑥𝑞 + 𝜌𝑝 ⋅ 𝜖𝑝 −𝜖𝑝 −𝜖𝑝 −𝜖𝑝 ⋅ 𝑥𝑖 𝑥𝑞 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 𝜌𝑔 = 2 ⋅ 𝑥𝑖 ⋅ 𝑥𝑞 = 𝑥𝑞 2 − 𝑥𝑖 2 𝜌𝑝 = 𝑥𝑞 2 − 𝑥𝑖 2 = 2 ⋅ 𝑥𝑖 ⋅ 𝑥𝑞 re-scaled and re-named for generalization Rotation by /4
- 11. 10/16 GCT Semiconductor, Inc. RFIT2022 Comparison of two models side-by-side ❑ Re-plot of previous works by flipping the direction of before/after the DPD for comparison. ◆ Reflection of the sequential estimation of and ❑ Scope of the problem has extended. ◆ All the harmonics at -f, +3f, +5f not just -3f should be suppressed at its best of the given model. ✓ 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 ,then . Refined model in RFIT2022 @ Busan - Optimal , obtained with iterative joint LMS adaptation.
- 12. 11/16 GCT Semiconductor, Inc. RFIT2022 Derivation of LMS Adaptation Formulae ❑ Frequency domain ◆ Obtain the Fourier-transform at each desired frequency, 𝑌 +𝑓 , 𝑌(−3𝑓) ◆ Process only 𝑌 +𝑓 to obtain 𝑌 +𝑓 3 ◆ Derived measure is insensitive to the rotation caused by the delay of the feed-back path. +fm +3fm -3fm -fm 0 𝒀(−𝟑𝒇𝒎) ∙ 𝒆−𝒋𝟔𝝅𝒇𝒎∆𝒕 𝒀(+𝒇) ∙ 𝒆+𝒋𝟐𝝅𝒇𝒎∆𝒕 +fm +3fm -3fm -fm 0 𝒀(+𝒇)𝟑 ∙ 𝒆+𝒋𝟔𝝅𝒇𝒎∆𝒕 𝒀(−𝟑𝒇𝒎) ∙ 𝒆−𝒋𝟔𝝅𝒇𝒎𝒕 ∆𝝐 ∝ Τ 𝒀 −𝟑𝒇𝒎 𝒀 +𝒇𝒎 𝟑 ∝ 𝒀(−𝟑𝒇𝒎) ⋅ 𝒀(+𝒇𝒎)𝟑 𝒀 +𝒇𝒎 𝟑 ⋅ 𝒀(+𝒇𝒎)𝟑 𝒚 = 𝒙 + 𝜹 ⋅ ഥ 𝒙 + 𝝐 ⋅ ഥ 𝒙𝟑 Norm of the signal that can be absorbed into update factor.
- 13. 12/16 GCT Semiconductor, Inc. RFIT2022 Derivation of LMS Adaptation Formulae (cont’d) ❑ Frequency domain ◆ How to align the desired and distorted component with the same offset from carrier(DC) ? ◆ Squaring in time-domain => Convolution in freq-domain. ✓ Previous method did not pre-processed time-domain signal before DFT. ✓ This measure is also insensitive to the rotation caused by the delay of the feed-back path. ❑ Frequency domain => Time domain : next page 𝒚[𝒏]𝟐 +fm +2fm -3fm -fm 0 𝒀(+𝒇𝒎)𝟐 ∙ 𝒆+𝒋𝟒𝝅𝒇𝒎𝒕 -2fm 𝒀(−𝟑𝒇𝒎)∙𝒀(+𝒇𝒎)∙𝒆−𝒋𝟒𝝅𝒇𝒎𝒕 -6fm +fm +2fm -3fm -fm 0 𝒀(−𝟑𝒇𝒎) ∙ 𝒆−𝒋𝟔𝝅𝒇𝒎𝒕 𝒀(+𝒇𝒎)∙𝒆+𝒋𝟐𝝅𝒇𝒎𝒕 -2fm 𝒚[𝒏] ∆𝝐 ∝ 𝒀 −𝟑𝒇𝒎 ⋅ 𝒀 𝒇𝒎 𝟑 ∝ 𝒀(−𝟑𝒇𝒎) ⋅ 𝒀(𝒇𝒎) ⋅ 𝒀(𝒇𝒎)𝟐 +𝟐 ⋅ 𝒇𝒎 −𝟐 ⋅ 𝒇𝒎 +𝟐 ⋅ 𝒇𝒎 −𝟐 ⋅ 𝒇𝒎
- 14. 13/16 GCT Semiconductor, Inc. RFIT2022 Time domain : Circularity in I/Q imbalance => Circularity in CIM3 ❑ Real and Imaginary part of higher order statistics should be zero, respectively. ◆ Time-domain formula from the frequency-domain relation is derived using Parseval’s theorem. ◆ Circularity is preserved under the signal rotation including +/4. 𝜹 𝒏 ≔ 𝜹 𝒏 − 𝟏 + 𝝁𝜹 ⋅ 𝒀(−𝒇) ⋅ 𝒀(+𝒇) 𝜹 𝒏 ≔ 𝜹 𝒏 − 𝟏 + 𝝁𝜹 ⋅ 𝒚[𝒏]𝟐 ො 𝝐 𝒏 ≔ ො 𝝐 𝒏 − 𝟏 + 𝝁𝝐 ⋅ 𝒀 −𝟑𝒇 ⋅ 𝒀 +𝒇 −𝟑 ≔ ො 𝝐 𝒏 − 𝟏 + 𝝁𝝐 ⋅ 𝒀 −𝟑𝒇 ⋅ 𝒀 +𝒇 +𝟑 = ො 𝝐 𝒏 − 𝟏 + 𝝁𝝐 ⋅ 𝒀 −𝟑𝒇 ⋅ 𝒀 +𝒇 ⋅ 𝒀 +𝒇 +𝟐 𝑬 𝒚𝒊[𝒏] ⋅ 𝒚𝒒[𝒏] → 𝟎 𝑬 𝒚𝒊 𝟐 𝒏 − 𝒚𝒒 𝟐 𝒏 → 𝟎 𝑬 𝒚𝒊 𝒏 + 𝒚𝒒 𝒏 ⋅ 𝒚𝒊 𝒏 − 𝒚𝒒 𝒏 → 𝟎 𝑬 𝒚𝒊[𝒏] ⋅ 𝒚𝒒[𝒏] ⋅ 𝒚𝒊 𝒏 + 𝒚𝒒[𝒏] ⋅ 𝒚𝒊[𝒏] − 𝒚𝒒[𝒏] → 𝟎 𝑬 𝒚𝒊 𝒏 + 𝒚𝒒[𝒏] 𝟐 ⋅ 𝒚𝒊[𝒏] − 𝒚𝒒[𝒏] 𝟐 − 𝟐 ⋅ 𝒚𝒊[𝒏] ⋅ 𝒚𝒒[𝒏] 𝟐 → 𝟎 ො 𝝐 𝒏 ≔ ො 𝝐 𝒏 − 𝟏 + 𝝁𝝐 ⋅ 𝒚[𝒏]𝟒 𝒚[𝒏]𝟒 ∆𝝐[𝒏] ∆𝜹[𝒏] 𝒚[𝒏] CIM3 I/Q imbalance(Image) 𝒚[𝒏]𝟐 Parseval’s Theorem Parseval’s Theorem @ ±𝟐 ⋅ 𝒇𝒎 @ ±𝟏 ⋅ 𝒇𝒎
- 15. 14/16 GCT Semiconductor, Inc. RFIT2022 Joint Compensation Model => Joint LMS adaptation ❑ Lead-in stage : Only 𝛿𝑔, 𝛿𝑝 is updated. ◆ After the images falls below pre-defined threshold, steps into main stage. ❑ Main stage : Both 𝛿𝑔, 𝛿𝑝 and 𝜖𝑔, 𝜖𝑝 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 𝜖𝑔, 𝜖𝑝 𝛿𝑔, 𝛿𝑝
- 16. 15/16 GCT Semiconductor, Inc. RFIT2022 Trivia and Tips for the Sake of Reference ❑ Parseval’s theorem ◆ Well-known form in Electrical Engineering ✓ Energy measured in time domain or frequency domain(Fourier-transform) are same. ◆ Generalized form with two functions ✓ Inner product of two functions in two spaces under unitary transforms are same. ❑ Dependency between signal domain and update method. ◆ Frequency domain update can be done at sampling rate if DFT is done with sliding window. ✓ Currently only a single value is obtained per non-overlapping block for efficient implementation. ◆ Time domain update can also be done in block-wise manner as with Frequency domain. ✓ However, update at sampling rate in time domain requires less hardware than block-wise one. ✓ Especially, 𝑦4 can be obtained from 𝑦2 already obtained for the I/Q imbalance calibration. 𝒏=−∞ +∞ 𝒂[𝒏] 𝟐 = න −𝝅 +𝝅 𝑨(𝒇) ⋅ 𝑨(𝒇)𝒅𝒇 𝒏=−∞ +∞ 𝒂[𝒏] ⋅ 𝒃[𝒏] = න −𝝅 +𝝅 𝑨(𝒇) ⋅ 𝑩(𝒇)𝒅𝒇 𝒚[𝒏] 𝒀𝒇[𝒌] 𝟏/𝒇 𝒚[𝒏] 𝒀𝒇[𝒌] 𝟏/𝒇
- 17. 16/16 GCT Semiconductor, Inc. RFIT2022 Conclusion ❑ A refined model for joint CIM3 and I/Q imbalance model has been proposed. ◆ Duality similar to that of I/Q imbalance has been also applied to CIM3. ◆ Missing components identified and complemented. ❑ Framework for the proposed model unified to get the help from ◆ Conjugate signal representation ◆ Circularity of the signal. ❑ Derivation of the joint LMS adaptation formulae. ◆ Optimization => LMS ◆ Frequency domain => Time domain ◆ Block-wise => Sample-wise ❑ URL of the slide/presentation ◆ [4] Duality of the I/Q imbalance : https://lnkd.in/gy24S4kF ◆ [5] CIM3 in RFIT2017 : https://lnkd.in/gwJhqgb8