A Beamforming Method for Blind Calibration of Time-Interleaved A/D Converters

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1 A Beamforming Method for Blind Calibration of Time-nterleaved A/D Converters Bernard C. Levy University of California, Davis Joint wor with Steve Huang September 30, 200

2 Outline 1 Motivation Problem Formulation Blind Calibration Method Convergence Simulations and Conclusions

3 Motivation 2 Applications requiring ADCs operating at high data rates Time-interleaved ADCs. Constituent ADCs have gain, offset, timing mismatches that need to be estimated and corrected in the digital domain. Correction achieved by digital filter bans operating on ADCs outputs (Johansson and Lowenborg, 2002 Prendergast et al. 2004). Requires 10 to 20 % excess samples. Timing offset estimation can be performed either with test signals or blindly. Blind methods do not lower ADC throughput and can adjust to changes online.

4 Outline 3 Motivation Problem Formulation Blind Calibration Method Convergence Simulations and Conclusions

5 Problem Formulation 4 f is a CT bandlimited signal with bandwidth, can be recovered from its samples if. nstead of using a single fast ADC, we employ at. slow ADCs operating

6 Problem Formulation (cont d) PSfrag replacements!!#... Analog Mux Digital Mux

7 for 8 + $ + % Problem Formulation (cont d) 6 Due to timing offsets, quantization errors and thermal noise, output of $-th ADC )( * =>?A@ * B C ( 0, where,.- / 21 WGN and $43 )(!6 where = sampling period of slow ADCs,!6 timing offset of $-th ADC measured wrt 1st ADC f 94: 94: <!6, analysis filter ban model for D:

8 Q V Q W FHS!TAU JLK M W FHO E W F O P W F O FHSLTAU JLK M V FHO E V F O P V F O Q R FHS!TAU JLK M R FHO E R F O P R F O E FHG JLK M N FHO E N F O P N F O Problem Formulation (cont d)

9 [ g \ \ X C j \ h i C f X Y \ \ \ a b [e\ c 9d: 94: 94: g < 6hf i C 94: < f g 6 94: g < 6f a b []\ 9Z: 9: 9: 9: ` 9: a b Problem Formulation (cont d) 8 For D, let = DTFT of exact ADC outputs, and vector of alias components of fast sampled sequence, and = vector of timing mismatches.

10 ~ y } z z x z z x x o x z z x w qr t s / + $ + D v g f l X Y z 9 6 : h i < =>? C z z z { n 9Z: / p u 9Z: <!6 =>?@ CB 9d: / n 9d: / o 9Z: Dml 9Z: / X c 9Z: / Problem Formulation (cont d) 9 We have where and = Vandermonde matrix with

11 o Problem Formulation (cont d) 10 By inverting \, we can find synthesis filters 9: such that 9d: 9d: 9d: \ For small representation. s, filters admit a closed-form 1st-order Farrow Synthesis filter ban for D:

12 ƒ M W F O LK W FHS!TAU M V F O LK V FHSLTAU M R F O LK R FHS!TAU M N F O LK N FHS!TAU E F G Problem Formulation (cont d) 11

13 Outline 12 Motivation Problem Formulation Blind Calibration Method Convergence Simulations and Conclusions

14 DŠ y } x z x x o x z x w withl } \ X Y x \ g x f w g i C D Let z 6 z 6 z 6 z { 9: / n 9: / o 9Z: D l 9Z: / y \ 94: < f g 6h i C 9 : { 94: < f = h Blind Calibration 13 D, if is ran deficient for ˆ :, where is a reduced Vandermonde matrix. 3 = % of excess samples, the alias matrix

15 D 6 j 6 j j j 0 6 Œ C Œ i C Œg C i C Œ C Œ h f 9Z: / X Y 9Z: - ` j ` j ` j 6 j j 6 j j ` j < Ž f g h < Ž f g h <! Ž < Ž f <! Ž f Œ` B <! ˆŽ f C Blind Calibration (cont d) 14 Can find a nulling filter ban 9 : / such that for ˆ.

16 D 1 D a D 6 ` š 3 ( 3 ( 3 š 3 3 š 3 ( ( 3 b o - ~ b [ ` 9Z: / b n 9Z: / Blind Calibration (cont d) 1 Structure of nulling filter ban: for ˆ, =0 otherwise, where satisfies Set. Then, for small s:

17 % / % where D š 9 œcb ` : 6 p 0 p ~ Ÿž 9 : ideal lowpass filter of bandwidth 0 9Z: )(!6 9Z: / 9Z: /!6 <!6 =>?A@ Ÿž 9Z: Blind Calibration (cont d) 16 Consider the 1st-order Farrow approximation. Let Consider the adaptive null-steering structure

18 ± ³ «µ ³ ± ³ «µ ³ ± ³ «µ ³ µ ³!ª ² ª! ²!ª!ª ² ² ª ² ª ª! ²!!ª ² Blind Calibration (cont d) 1

19 ` 3 D š x x x x w w ` ¹» ¾À Á 9 / º y 3 { } ( y 3 )( ` ` { } 3 ` )( 1 9 / º º )( º º ¼ 9 / º ½21 Blind Calibration (cont d) 18 Consider the objective function where = nulling filter output. We have

20 º Blind Calibration (cont d) 19 Â 9 Á 9 3 / º ¾Ã / º obtained by stochastic gradient algorithm ( º º where - Â step size, with initial condition Ä ~ º

21 Outline 20 Motivation Problem Formulation Blind Calibration Method Convergence Simulations and Conclusions

22 º Â Convergence 21 Use ODE/stochastic averaging method. Assume WSS. small, zero-mean Write adaptive algorithm as ( ÂÆÅ º Ç b / Ç ( º where of the algorithm, ¼4È b É ½ b. Due to the stochastic gradient structure º º ½ ¾Ã º ¹ Á / / Ç» ¼ ¹ 3 so º Êº Ê / = Lyapunov function for ODE ¹ so ODE trajectories converge to a minimum of º.

23 Ò 1 1 Ô Õ i h 6 Ï Ô Õ Ð Ñ Î Ó ( ( Ò º º ¹ š Ï º º º 1 0 h i 6 j 9Z: Ê = j j 1 9Z: Ê j ÐÍ 3 Ñ 3 º 3 3 ( º 1ÌË º Í 3 Î 3 3 º 3 3 Convergence (cont d) 22 For small and with where we neglect cubic terms.

24 º ' Ø Ù ½ ½ ( 3 Ï º Convergence (cont d) 23 For noiseless case, unique minimum of ¹for small offset and offset estimates is if -and -, so and. ¼ 3 / ¼ Ö 21 / Ö 21 must have power in band Ensures that as, ÂÆÚ / whereú= positive definite matrix.

25 Outline 24 Motivation Problem Formulation Blind Calibration Method Convergence Simulations and Conclusions

26 ~ 1-6 ` - -~ -~ 3 ½ - ~ ÜÛ ~ ÜÛ Bandlimited WGN input Simulation parameters timing estimation ch2 ch3 ch4 Signal bandwidth: Â -1~, -. ¼ 3, / design Ü # of Samples x 10 4

27 - - ~ - 6 -~ - ~ ÜÛ - ~ - ÜD~ Multitone sinusoidal input Simulation parameters timing estimation ch2 ch3 ch4 nput frequencies:. -1~, -,,, 0.00 Â design Ü x 10 4

28 Calibrated ADC output Power Spectrum Power Spectrum Frequency in Radian (x π) Frequency in Radian (x π) Uncalibrated ADC, 0dB SNR Calibrated ADC, 0dB SNR

29 Conclusion 28 Blind calibration of time-interleaved ADCs presented, requires 10 to 20 % oversampling and intermittent excitation of certain frequency bands. Simulated for up to 16 channels, but 2 or 4 channels primary interest for today s ADC technology. Postprocessing of analog circuits with mismatched components source of interesting adaptive signal processing problems.

30 29 Than you!!

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