Blind Digital Tuning for an Analog World - Radio Self ...meeting.xidian.edu.cn/workshop/miis2015/uploads/files/Yingbo Hua.pdf · Blind Digital Tuning for an Analog World-Radio Self-Interference

Blind Digital Tuning for an Analog World-

Radio Self-Interference Cancellation

Yingbo HuaUniversity of California at Riverside

July 2015

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Motivation

I Global mobile data traffic grew 69 percent in 2014... (Cisco).

I Demand for radio spectrum continues to increase.

I Spectrum sharing between communications and passive radarbecomes necessary.

I The best radio transceivers in the future must be able totransmit and receive using the same frequency at the sametime (full-duplex radio).

I Radio self-interference cancellation will enable full-duplexradio and facilitate spectrum sharing between communicationsand radar.

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Radio Components

I In addition to antenna/circuit isolation, radio self-interferencecancellation must be done first at radio frontend.

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RF Frontend Cancellation

I The interference channel’s impulse response in baseband:

hInt(t) =I∑

i=1

aie−2πfcτi sinc(W (t − τi ))

I How do we choose an analog cancellation channel to matchthe above by using only passive components with minimumnoise? 4 / 32

An Analog Cancellation Channel

I The RF waveform only passes through passive components.I The phase shifters and time delays are constant with possible

constant errors.I The only variable components are step-attenuators which are

digitally controlled.5 / 32

Attenuation in polar form, i.e., −20 log |G |e−jarg(G )

Figure: 1 attenuator per c-tap Figure: 2 attenuators per c-tap

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Attenuation in polar form, i.e., −20 log |G |e−jarg(G )

Figure: 3 attenuators per c-tap Figure: 4 attenuators per c-tap

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An Alternative Form

I Using 90-degree power splitters to implement desired phases.

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Comparison of Ideal Channel Models:

hInt(t) =I∑

i=1

aie−2πfcτi sinc(W (t − τi ))

hCan(t) = e−j2πfcT0

N−1∑n=0

Gnsinc(W (t − nT ))

I |Gn|.

= |gn,1 + jgn,2 − gn,3 − jgn,4| < 1

I We want hRes(t).

= hInt(t)− hCan(t) to be as small aspossible by adjusting all gn,m.

I If maxi ,j |τi − τj | � 1W , N = 1 would be sufficient.

I Otherwise, N > 1 is needed.

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Minimized Residual Self-Interference Channel

Figure: The distribution of theresidual interference over f , i.e.,E2(fm). T = 1

10W . αp = 2.

Figure: The CDF of the residualinterference, i.e., the CDF ofE

(r)1 . T = 1

10W . αp = 2.

I E2(f ) = E{ |HRes(f )|2|HInt(f )|2

} and E(r)1 is the r th run of∫W /2

−W /2|HRes(f )|2|HInt(f )|2

df

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Practical Constraints

I hInt(t) is unknown except for some poor estimates.

I hCan(t) is also an unknown function of the gains/attenuationsof the step attenuators due to analog interface and hardwareimperfection.

I There is no precise access to the input and/or output of eitherchannel.

I A step attenuator has a limited dynamic range and a fixedstep size.

I To tune a practical cancellation channel, we must toleratemany unknown factors - blind tuning.

I Step attenuators allow blind digital tuning.

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A C-Tap on PCB

Figure: A c-tap on custom madePCB.

Figure: 900 out of 324 points ofattenuations at 2.4GHz.

I Each c-tap has four independent step-attenuators. The complexattenuation of a c-tap densely covers four quadrants of a disk.

I Each step-attenuator has a step size 1dB and 1-32dB dynamic range.There are 324N choices for a cancellation channel with N c-taps - tooslow to do brute force blind digital tuning.

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Methods to Speed Up Blind Digital Tuning:

I Using Additional Down-Converters - Method 1

I Using No Additional Down-Converter - Method 2

I Using Quadratic Model - Method 3

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Using Additional Down-Converters - Method 1

I Assuming that we can introduce a down-converter to tap theinput RF waveform xi (k) of each step-attenuator, thenideally we can write the residual interference xres(k)measurable in baseband as

xres(k) = x0(k)− T{[x1(k), · · · , xNA(k)][g1, · · · , gNA

]T}

I Here T is some unknown linear operator caused by unknownRF couplings.

I If T is known and xi (k) can be measured accurately enough,the classic LS, RLS and LMS algorithms can be applied.

I But in practice, T is unknown and xi (k) is contaminated bynoise. The down-converters are also costly.

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Using No Additional Down-Converter - Method 2

I Without any additional down-converter, we can write theresidual interference measurable in baseband as

xres(k) = x0(k)− X(k)g

where x0(k) and X(k) are unknown.

I The affine/linear relationship holds strongly (in case of IQimbalances) if a modification into real-valued representation isused.

I With unknown x0(k) and X(k), how to find the optimal g tominimize xres(k)?

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A Solution

I Define a sequence of training vectors of g: i.e., g1, · · · , gNT.

I For each training vector gi , Tx transmits the same datamultiple times, and Rx measures the averaged residualxres,i (k), which can be written as

xres,i (k) = x0(k)− X(k)gi + wi (k), i = 1, · · · ,NT

where x0(k) and X(k) can be treated as independent of i .

I Provided that

[1 · · · 1g1 · · · gNT

]has a full row rank, there is

a unique LS solution for x0(k) and X(k).

I Then, the optimal g follows.

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Using Quadratic Model - Method 3

I The previous affine model is sensitive to phase noise.

I One way to remove the phase noise is to choose the power ofxres(k) as the measurement, which leads to

pres = gTAg + gTb + c

where A, b and c are unknown parameters.

I How to find the optimal g to minimize pres?

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A Solution

I If the estimates A and b are given, the optimal g is

g = −1

2A+b

I To find A and b, we can first define a sequence of trainingvectors of g: i.e., g1, · · · , gNT

(not the same as before).

I For each gi , Tx transmits a stream of data, and Rx measuresthe residual power pres,i . We can then write pres,1· · ·

pres,NT

=

1 gT1 gT

1 ⊗ gT1

· · · · · · · · ·1 gT

NTgTNT⊗ gT

NT

cb

vec{A}

= Gθ

I If the training matrix G had a full column rank, there wouldbe a unique LS solution for θ.

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The Quadratic Term gTAg

I With g being real, the unknown matrix A in gTAg can alwaysbe chosen to be real symmetric. Hence, we can write

gTAg = (gT ⊗ gT )vec{A} = (gT ⊗ gT )STDSvec{A}

where, assuming g being 4× 1 and JM×N being the last Mrows of IN×N ,

S = diag [J4×4, J3×4, J2×4, J1×4]

D = diag [1, 2, 2, 2, 1, 2, 2, 1, 2, 1]

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Reduced Model

I Hence, we can rewrite Gθ as pres,1· · ·

pres,NT

=

1 gT1 (gT

1 ⊗ gT1 )ST

· · · · · · · · ·1 gT

NT(gT

NT⊗ gT

NT)ST

cb

DSvec{A}

or equivalently

p = GTθT

I Here, GT and θT of NT × (12NA(NA + 1) + NA + 1) and(12NA(NA + 1) + NA + 1)× 1 are smaller than G and θ.

I For a unique LS solution of A and b, we only need GT to beof full column rank.

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Example of GT of Full Column Rank

I Assume g1 = 0 and the following:I For i = 2, · · · ,NA + 1, gi = αeNA,i−1 with 0 < α ≤ 1 and

eNA,i being the ith column of INA×NA

I For i = NA + 1, · · · , 2NA + 1, gi = βeNA,i−NA−1 with0 < β < 1 and β < α

I For i = 2NA + 2, · · · , 12 (NA + 1)NA + NA + 1,gi = αeNA,l + αeNA,k with 1 ≤ l < k ≤ NA

I Then, we can show that

det{GT} = αN2AβNA(α− β)NA 6= 0

I The above matrix GT is very sparse, and a closed-formexpression of G−1T can also be found.

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Example of GT with NA = 3

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Example of Solution to p = GTθT

Hp = PθT

Figure: The matrix H with∆ = αβ2 − α2β, ∆1 = α2 − β2

and ∆2 = β − α.Figure: The permutation matrixP.

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Basic Facts of Radio Hardware

I Phase Noise

I IQ Imbalance

I Real Linear Model vs Widely Linear (Complex) Model

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Phase Noise

I Tx phase noise: for a baseband signal xBB(t) = xr (t) + jxi (t),its RF signal generated by a practical transmitter is

xRF (t) = xr (t) cos(2πfct + φT (t))− xi (t) sin(2πfct + φT (t))

where φT (t) is the Tx phase noise.

I Rx phase noise: for the above RF signal, its baseband signalgenerated by a practical receiver is

xBB(t) = xBB(t)e jφT (t)+jφR(t)

where φR(t) is the Rx phase noise.

I Phase noise is non-additive noise. If approximated via Taylor’sseries, the noise term is correlated with the signal.

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IQ Imbalances

I Tx IQ imbalances: for a baseband signalxBB(t) = xr (t) + jxi (t), the RF signal generated by a practicaltransmitter is

xRF (t) = (1+δT )xr (t) cos(2πfct+θT )−(1−δT )xi (t) sin(2πfct−θT )

where δT and θT are the Tx amplitude and phase imbalances.I Rx IQ imbalances: for the above RF signal, the baseband

signal generated by a practical receiver isxBB(t) = xr (t) + j xi (t) and

[xr (t)xi (t)

]=

[(1 + δR ) cos θR (1 + δR ) sin θR(1 − δR ) sin θR (1 − δR ) cos θR

]︸ ︷︷ ︸

Rx IQ imbalance

[(1 + δT ) cos θT (1 − δT ) sin θT(1 + δT ) sin θT (1 − δT ) cos θT

]︸ ︷︷ ︸

Tx IQ imbalance

[xr (t)xi (t)

]

I The real-valued linear model holds but the complex-valuedlinear model does not.

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Real Linear Model vs Widely Linear (Complex) Model

I With IQ imbalances, a MIMO radio channel can be describedby

yre(k) =∑l

Hre(l)xre(k − l) + wre(k)︸ ︷︷ ︸Real Linear Model ∈R2nr×1

or equivalently by

yco(k) =∑l

Aco(l)xco(k − l) +∑l

Bco(l)x∗co(k − l) + wco(k)︸ ︷︷ ︸Widely Linear Model ∈Cnr×1

I The real linear model is clearly more straightforward to use,e.g., transmit beamforming designing.

I Why is the widely linear model “more popular”?

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Example: Time-Domain Tx Beamformer

I Assume that we want to design a MIMO zero-forcingbeamformer for nt transmitters, which produces nulls at nr(< nt) receivers.

I The Tx waveform should be xre(k) = Pre(k) ∗ sre(k) and

Hre(k)∗Pre(k) = [Hre,a(k)︸ ︷︷ ︸2nr×2nr

, Hre,b(k)︸ ︷︷ ︸2nr×2(nt−nr )

]∗

Pre,a(k)︸ ︷︷ ︸

2nr×2(nt−nr )Pre,b(k)︸ ︷︷ ︸

2(nt−nr )×2(nt−nr )

= 0

I A solution is Pre,a = Adj{Hre,a(k)} ∗Hre,b(k) andPre,b(k) = −Det{Hre,a(k)}I2(nt−nr )×2(nt−nr )

I The beamformer for widely linear model can be found butneeds extra work.

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Example: Channel Estimation

I Real linear model:

[yre (0), · · · , yre (N − 1)

]︸ ︷︷ ︸R2nr×N

=[

Hre (0), · · · ,Hre (L)] xre (0) · · · xre (N − 1)

· · · · · · · · ·xre (−L) · · · xre (N − 1 − L)

︸ ︷︷ ︸

Xre∈R2nt (L+1)×N

+noise

It requires (XreXTre)−1 ∈ R2nt(L+1)×2nt(L+1)

I Widely linear model

[yco (0), · · · , yco (N − 1)

]︸ ︷︷ ︸Cnr×N

=[

Aco (0), · · · ,Aco (L)] xco (0) · · · xco (N − 1)

· · · · · · · · ·xco (−L) · · · xco (N − 1 − L)

︸ ︷︷ ︸

Xco∈Cnt (L+1)×N

+

+[

Bco (0), · · · ,Bco (L)] x∗co (0) · · · x∗co (N − 1)

· · · · · · · · ·x∗co (−L) · · · x∗co (N − 1 − L)

︸ ︷︷ ︸

X∗co∈Cnt (L+1)×N

+noise

It requires

([Xco

X∗co

] [Xco

X∗co

]H)−1∈ C 2nt(L+1)×2nt(L+1)

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Final Remarks

I Radio self-interference cancellation continues to be achallenging problem which requires significant advances inboth hardware and algorithm designs - an interdisciplinarytopic.

I From algorithm point of view, there are several fundamentalresearch issues such as: Given the model

pres = gTAg + gTb + c + noise

I How to optimally design the training vectors of g to estimateA, b and c subject to some prior distributions?

I If we have a dynamic model for A, b and c (or a recursivealgorithm for estimating them), how to design a controlalgorithm that concurrently adjusts g to drive pres to theminimum?

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RF Frontend Cancellation Result Using Two C-Taps

100MHz bandwidth centered at 2.4GHz. The top curve is before tuning. The 3rd curve is after tuning the 1st

c-tap. The 2nd curve is after tuning the 2nd c-tap. The 4th curve is after re-tuning the 1st c-tap. The bottom

curve is the receiver noise.

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References

1. Y. Hua, Y. Li, C. Mauskar, Q. Zhu, “Blind Digital Tuning for InterferenceCancellation in Full-Duplex Radio”, Asilomar Conference on Signals,Systems and Computers, pp. 1691-1695, Pacific Grove, CA, Nov 2014

2. Y. Hua, Y. Ma, A. Gholian, Y. Li, A. Cirik, P. Liang, “RadioSelf-Interference Cancellation by Transmit Beamforming, All-AnalogCancellation and Blind Digital Tuning,” Signal Processing, Vol. 108, pp.322-340, 2015.

3. A. Cirik, Y. Rong, Y. Hua, “Achievable Rates and QoS Considerations ofFull-Duplex MIMO Radios for Fast Fading Channels with ImperfectChannel Estimation,” IEEE Transactions on Signal Processing, Vol. 62,No. 15, pp. 3874-3886, Aug 2014.

4. A. Gholian, Y. Ma, Y. Hua, “A Numerical Investigation of All-AnalogRadio Self-Interference Cancellation,” IEEE Workshop on SPAWC,Toronto, Canada, June 2014.

5. Y. Hua, Y. Ma, P. Liang, A. Cirik, “Breaking the Barrier of TransmissionNoise in Full-Duplex Radio”, MILCOM, San Diego, CA, Nov 2013.

6. Y. Hua, P. Liang, Y. Ma, A. Cirik and Q. Gao, “A method for broadbandfull-duplex MIMO radio,” IEEE Signal Processing Letters, Vol. 19, No.12, pp. 793-796, Dec 2012.

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