Power Optimized PA Clipping for MIMO-OfDM Systems

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 9, SEPTEMBER 2011 2823

Power Optimized PA Clipping for MIMO-OFDM Systems

Hun Seok Kim, Member, IEEE, and Babak Daneshrad, Member, IEEE

AbstractFor a multiple-input multiple-output (MIMO) or-

thogonal frequency division multiplexing (OFDM) system thatis being pushed into power amplifier (PA) saturation, this letterinvestigates power-optimized PA clipping. Our goal is to identifythe optimum clipping level for a MIMO-OFDM system thatdelivers the desired bit error rate (BER) with minimum powerconsumption in the PA. We present a complete theoreticalframework resulting in an analytical expression for the BER ofa MIMO-OFDM system subject to PA clipping. PA power savingis addressed by the total degradation metric, which shows that asmuch as 6dB power reduction can be achieved by proper choiceof the clipping level.

Index TermsMIMO, OFDM, PA clipping, peak to averagepower ratio, total degradation.

I. INTRODUCTION

H IGH peak-to-average power ratio (PAPR) is a majordrawback of OFDM modulation since it results in exces-sive power consumption at the PA. A number of approacheshave been proposed in this area to reduce PAPR for OFDM.

Among them, signal clipping which artificially limits the

amplitude of the signal to a pre-determined clipping level [1]

[9] is one of the simplest and extensively studied methods.

Although PAPR reduction by signal clipping improves the

PA efficiency, it also degrades the BER performance by

introducing distortion to the desired signal. To quantify this

tradeoff space available by various signal clipping strategies,the concept of the total degradation () [3] [6] is widelyused. The is a function of the PA amplitude clipping level, defined by

() = () + () [in dB] (1)

where () is the performance penalty measured bythe signal-to-noise ratio (SNR) difference (in dB) between the

unclipped system and a system with a finite when bothsatisfy the same BER. Typically, aggressive clipping (small

) results in lower PAPR at the cost of higher SNR penalty.A conventional system designed to avoid any signal clipping(high ), on the other hand, cannot minimize because ofexcessive PAPR even if the SNR penalty is 0.

There is a rich body of work dealing with the issue of

clipped OFDM systems [1] [9]. Analysis provided in these

works, however, is focused on single antenna (SISO) OFDM

systems and is not readily extended to a MIMO-OFDMsystem. Although, the work in [9] goes one step further by

Manuscript received October 17, 2010; revised March 21, 2011; acceptedJune 3, 2011. The associate editor coordinating the review of this paper andapproving it for publication was W. Zhang.

H. S. Kim is with the Applications and Systems R&D Center, TexasInstruments Inc., Dallas, TX, USA (e-mail: [email protected]).

B. Daneshrad is with the EE Department, University of California, LosAngeles, CA, USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TWC.2011.072011.101834

Spatial

Stream

Parser

OFDM

modulation

OFDM

modulation

OFDM

demod.

MIMO

Detector

Spatial

Stream

Deparser

Transmitter

Receiver

QAM

Mapper

QAM

Mapper

Signal

Clipping

Signal

Clipping

Bit

stream

Bit

stream

OFDM

modulation

QAM

Mapper

Signal

Clipping

OFDM

demod.

OFDM

demod.

FEC

Encoder

FEC

decoder

Uncoded

BER

measured

here

Fig. 1. MIMO-OFDM transceiver structure.

analyzing an Alamouti space-time coded system, its result is

limited to the single spatial stream case. The total degradation

of a MIMO-OFDM system (with multiple streams) has not

been fully explored in the literature in part due to the lack ofan analytical BER expression for a clipped MIMO system.

In this work, we derive analytical BER performance modelsfor a clipped MIMO-OFDM system and identify the total

degradation (1) as a function of the clipping level, . As aresult, the optimal tradeoff point is revealed for a MIMO-

OFDM system where the combination of the SNR penaltyand PAPR is minimized while a given BER performance

requirement is satisfied.

I I . SYSTEM MODEL

The structure of the MIMOOFDM system considered in

this work is depicted in Fig. 1. Throughout this letter, we usespatial multiplexing transmission with multiple streams. At the

transmitter, the information bit stream is parsed to multiple

spatial streams. Each stream is then mapped to quadrature-

amplitude-modulated (QAM) symbols and OFDM modulated.

At the receiver, the reverse operations are performed as

depicted in Fig. 1.

A. Signal Model of a SISO-OFDM System with Clipping

We first present the modeling result from [1] for a clipped

SISOOFDM system considering an ideal PA with input and

output characteristic given by

Input: () = ()()

Output: () ={

()() when ()

() otherwise

(2)

where () and () are input and output signals of the PA

respectively and is the input signal power. The amplitude1536-1276/11$25.00 c 2011 IEEE


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2824 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 9, SEPTEMBER 2011

and phase of the input signal at time are denoted by ()and () respectively. When the clipping level is set to , thepeak PA output signal power is bounded by 2.

It is shown in [1] that, in the frequency domain, the OFDM

signal distortion due to clipping is well approximated by azero-mean Gaussian random variable when the number of

OFDM subcarriers, , is reasonably large (e.g., 128).With this assumption, the average PA output transmission

power, , after clipping is approximated ([1] [2]) by

= (

1 2)

. (3)

Using (2) and (3), the PAPR after clipping is presented as a

function of by

() = 2/(

1 2)

. (4)

The frequency domain PA output signal for the OFDMsubcarrier, , can be considered as the sum of the scaledoriginal input signal, , and the Guassian distortion noisesignal, , due to clipping [1]. and are uncorrelated

and the relationship

= + (5)

holds where = 12 + 2 is the signal powerscaling factor (see e.q. (27) of [1]).

The distortion signal power for the OFDM subcarrieris denoted by

E ( ) = (6)where the distortion signal power coefficient, , is ob-

tained by calculating the power spectral density (PSD) of

the clipped signal and subtracting the signal power from

it. For the PSD computation, the upsampled discrete timeauto-correlation [] introduced in [1] (see Appendix A of[1]) is adopted. The PSD of the clipped signal (i.e., discrete

Fourier transform of []) consists of the desired signal

power 2(

1 2)1

and the distortion signal power

. Therefore, is obtained by subtracting the signalpower from the PSD as given in

=1

(1) [], =0,1, ,1

}

2(

1 2)1

(7)

where is the over-sampling factor ( 4 is recommendedto properly approximate continuous time behavior of the

clipped signal [1]) and {} is the output of the-point discrete Fourier transform. It is well known thatclipping produces not only in-band distortion but also out-

of-band interference, which may degrade the performance of

other systems in adjacent bands. However, in this work, we

investigate the minimum (i.e., lower bound) when spectralgrowth due to the clipping is allowed and the effect of the out-

of-band interference is ignored.

B. Signal Model of a MIMOOFDM System with Clipping

Let and represent the number of receive andtransmit antennas. For the MIMOOFDM system, we denote

s as the 1 transmitted symbol vector for the OFDMsubcarrier after clipping while the clipping level is commonfor all spatial streams. The clipped MIMO signal, s, can berepresented as the summation of the scaled original transmitted

symbol vector, s (without clipping), and Gaussian distortion

vector, n, as in

s = s + n (8)while E (s n) = 0, E (nn) = I andE (s s) = I holds.

Assuming additive white Gaussian noise (AWGN) at the

receiver, we denote the 1 received vector y for the OFDM subcarrier as the product of the channelmatrix H and the transmitted symbol vector s plus the 1 AWGN vector n with covariance matrix ofE (nn

) =

2I as in

y = Hs +Hn + n = Hs + n. (9)The additive noise plus distortion signal at the receiver is

denoted by n (= Hn + n). It is worth noting that n is

spatially correlated because of the impact of the MIMO chan-nel on the originally uncorrelated vector n. Its covariance,

C, is obtained by

C = E (nn) = HH + 2I. (10)

We use a frequency-selective quasi-static fading channel

model [10], whereby the channel is assumed to be static within

the duration of a single packet but is independent from one

packet to the next. We denote by (,) the (, ) component of

the frequency domain channel matrix H, which is the channel

response from the transmit antenna to the receive

antenna for the subcarrier. In our channel model, (,)

has zero mean independent circularly symmetric, complexGaussian distribution with unit variance. It is worth noting

that (,) has statistically identical distribution regardless of

the subcarrier index . It is independent in space and time butcorrelated in frequency.

III. GENERALIZED MIMO DETECTION ALGORITHMS FOR

TH E CLIPPED OFDM SIGNAL

In this section, we derive the generalized ML and MMSE

MIMO detection algorithms which are applicable to both

clipped and unclipped MIMO-OFDM signals.

A. ML-MIMO Detection of Clipped MIMO-OFDM Signal

The ML solution for the clipped MIMO signal is derived

using the conditional probability density function (PDF),

(yH, s, ), of the received signal y given channel H,

transmit signal s and . The ML decision, s,, is the s thatmaximizes the conditional PDF as in

s, = arg mins(yH, s, )

= arg mins

12 (yHs)C1 (yHs)

(2) C

. (11)

Therefore, s,

can be obtained by solving

argmins

LyLHs2 = argmins yHs2(12)


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KIM and DANESHRAD: POWER OPTIMIZED PA CLIPPING FOR MIMO-OFDM SYSTEMS 2825

where L is the Cholesky decomposition ofC(= LL),

y = Ly, and H = LH. The matrix L is essentiallythe whitening filter and y is the whitened signal.

B. MMSE-MIMO Detection of Clipped MIMO-OFDM Signal

The MMSE detection solution for the clipped MIMO-

OFDM signal, s,, is computed by the product of the

received vector y and the MMSE weight matrix W,as in

s, = W, y. (13)The MMSE weight matrix is the W matrix which minimizes

the mean square error satisfying

W, = arg minW

E

(s Wy) (s Wy)

. (14)

By solving (14), W, for the clipped MIMO system isobtained by

W, =H(2+HH+2I)

1.(15)

Applying the non-white noise covariance C (10) to the

post detection SNR computing procedure of the unclippedMIMO systems [11] [12], the MMSE post detection SNR of

the clipped MIMO system can be obtained in a closed-formexpression. We denote by , the MMSE post detection SNRof the clipped MIMO system for the spatial stream of the OFDM subcarrier. The expression for , is given by

, =1[

E

s,s

s,s

}],

1

= 1[I22H

(2+

HH+2I)1H],

1

(16)

where [A], is the diagonal component of the matrix A.

IV. ANALYTICAL BER EXPRESSION FOR THE CLIPPED

MIMO OFDM SIGNAL

An analytical expression for the BER is an important step

in quantifying the clipping penalty and (1). We derivean analytical BER expression of a clipped MIMOOFDM

system with the ML and MMSEMIMO detection algorithms

derived in section III. In this letter, uncoded BER is used asthe indicator of the system performance measured at the input

of the forward error correction (FEC) decoder as depicted

in Fig. 1. The uncoded BER allows the system designer to

independently analyze the performance of a clipped MIMO

OFDM system regardless of a specific FEC scheme. Typically,

there is an one-to-one mapping between uncoded BER and the

after-FEC packet error rate, determined by the characteristic

of each FEC scheme.

Under the assumption of gray -QAM mapping, it isknown that the BER and symbol error rate (SER) are related

via the approximation; 1 where = 2()stands for the number of bits per symbol. To estimate the BER,we consider an error event with a correct vector, s, and an error

vector, s, which satisfies Es {ss} = Esss

= I. We

simplify the BER estimation by assuming that the minimum

distance event dominates the performance. We thus only

consider error events which satisfy the relationship, e =s s = [ 0 0 ], where is the differencebetween adjacent QAM symbols and e = = is theminimum Euclidean distance between QAM symbols. Based

on these assumptions, the system BER given noise power 2and clipping level is lower bounded by

1

1

1=0

EH

Es

{ s s.t. s=s

(s s)

1

1=0

EH

Es

{ s s.t. ss=

(s s)

(17)

where (s s) = (s s

H, s, s

, ,2)is the

probability of detecting s

when s is sent.

A. BER Performance with the ML-MIMO Detector

The BER bound for the ML MIMO detector can be

computed by applying the ML detection expression for the

clipped signal given in (12). To derive the BER bound, we

use the fact that, for each symbol in a square -QAM(e.g., 4QAM, 16QAM, etc.) constellation, there are on average(

4 4/)

adjacent symbols with the minimum distance

. Therefore, the BER bound (17) for the ML MIMOdetector can be computed by

1

1=0

EH

Es

{ s s.t. ss=

yHs2yHs2

H, s, s, ,2}

=4 4

1=0

Eh

E

{

2h2

nLh + hLn2h, , ,

2

}(18)

whereh represents a column of

H that satis

fies

He =

hgiven e.

Since only has a non-zero value in either the real or theimaginary part, nLh +

hLn is a real zero-meanGaussian random variable with variance 22h2. TheBER lower bound given and 2 is thus computed as

4 4

1=0

=0

h,()

2

2

(19)

where () stands for the tail probability function of the nor-malized Gaussian distribution, and h,() is the probabilitydensity function (PDF) of

h2 given and 2. Note that

h2 = 2HC1 H, is independent and identicallydistributed regardless of the spatial stream index .


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TABLE I

THE PD F OF 1/

H

H + 0I

1,

1

0 , 0 PDF of 1/HH + 0I1

,

1 given 0 > 0

22

0 00 (1 + 0 1(+1)2 0+1)

32 0 20

0 (1 +02

12(+1)2

0+12(+1))

42

0 3020 ( 12 + 06 16(+1)2 0+26(+1))

33

0 00 (1 + 20 + 202 + 0+2(+1)3 + 2062(+1)2 20+30+1 )

43

0 200 (1 + 0 + 206 + 0+33(+1)3 + 20+2066(+1)2 20+50+33(+1) )

44

0 00 (1 + 30 + 3202 + 306 20+60+62(+1)4 30+360486(+1)3 + 30+620122(+1)2 30+820+1202(+1) )

We derive the exact PDF, h,(), ofh2 using the resultin [11] where the PDF of1/

(HH+ 0I)1

,1 is studied

for a positive constant 0 and a matrix H with independentcomplex Gaussian components. When H is an

complex Gaussian matrix with unit variance as in our channelmodel, the PDF of 1/

(HH+ 0I)1

, 1 is represented

by the function in TABLE I which was originallyderived for the unclipped MMSE MIMO system [11]. In

order to find h,(), we use the matrix inversion lemma,

(HH + 0I)1

= 10H (HH

+ 0I)

1H, which leads

us to relationship

222 ()2h2 1

= 1/

H

H +

2

I1

,

1.(20)

Since the PDF of the random variable on the right side of (20)

is , the exact PDF of h2 is obtained by randomvariable transformation and is given by

h,() =

1

2

2

2

0 =2

when > 0

(2) 1

2

(1)! otherwise.

(21)

It is obvious that when the clipping distortion is negli-

gible ( 1 or 0), h,() is identical to thePDF of 2 HH, which has the Gamma (Chi-square)distribution (21). The plot of h,() is shown in Fig. 2,which indicates that our expression (21) is well matched to

the Monte-Carlo simulation results for various , andclipping levels.

B. BER performance with the MMSE-MIMO detector

The output of an MMSE-MIMO detector is the equalized

symbol vector, s, along with the additive term, n, satisfying

the equation, W, y = s + n, where n repre-sents the post-detection noise-plus-interference. In general, forthe MMSE-MIMO detector, symbol-by-symbol detection [11]

[12] is performed assuming the spatial channel of the

TABLE IIEXAMPLE MODES (PARAMETER SET S)

Mode Constellation MIMO Detection

Mode1 2 3 64QAM MLMode2 2 4 16QAM MLMode3 3 4 QPSK MLMode4 4 4 64QAM MMSEMode5 2 2 16QAM MMSEMode6 3 3 QPSK MMSE

OFDM subcarrier is an independent AWGN channel whose

SNR is dictated by the post-detection SNR, ,, given in(16). In this case, the BER lower bound (17) for the MMSE

detector can be computed by

4 4

1=0

EH

Es

{ s s.t. ss=

(sW, y2 sW, y2

)

=4 4

1

=0EH

{Ee

{

n2 e n2

=4 4

1=0

EH

{

2{} H, ,2

(22)

where is a component of n that satisfies ne =

given e. Note that given H, 2 and is a zero-mean

complex Gaussian random variable whose variance is dictated

by 1/,.

The MMSE post-detection SNR, ,, (16) has independent,identical distribution regardless of the spatial index . We de-note by ,() the PDF of, given 2 and . Applying thematrix inversion lemma and random variable transformations


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KIM and DANESHRAD: POWER OPTIMIZED PA CLIPPING FOR MIMO-OFDM SYSTEMS 2827

0 20 40 60 80 100 120 140 1600

2

4

6

8

10

x 10-4

Monte-Carlo Simulation

Analysis

Value of (for ML) or (for MMSE)2

kh

,i k

,k

p

x

h

PDF

value

or

,

(

)

k

p

x 23, 4, 0.0316,

, 1.2

T R NN N

ML

22, 2, 0.01,

, 1.3

T R NN N

MMSE

24, 4, 0.0032,

, 1.7

T R NN N

MMSE

22, 3, 0.0032,

, 1.7

T R NN N

ML

Fig. 2. Validation of h,() and ,(). Comparison between Monte-

Carlo simulation and analytical PDF.

5 10 15 20 25 30 35 40 45 50

10-3

10-2

SNR dB

BER

Simulation

Analysis

Mode1

Mode2

Mode3

Mode4

Mode5

1.7

1.2 1.41.3 1.7

Fig. 3. BER analysis vs. simulation results for various clipping.

to (16), we arrive at exact the expression

,() =22/

22 +

(1 + ( + 1) ( 22

2

2

+ ))2

(+1)22/(

22+)

1+(+1)

(22

22+

)0 = 222

+

.

(23)

Fig. 2 shows that our expression (23) for ,() is wellmatched to the Monte-Carlo simulation results for various

clipping levels. Finally, using the expression for ,(), werewrite the lower BER bound (22) for the MMSE detector and

obtain the expression

4 4

1

=0

=0,()

22

.

(24)

0.5 1 1.5 2 2.5 33

4

5

6

7

8

9

10

Clipping level

Mode1

Mode2

Mode3

Mode4

Mode5

Mode6

Clipping Level

TotalDeg

radation(dB)

Target BER = 10-4

Target BER = 10-4

Target BER= 10

-3

Fig. 4. Total degradation to satisfy 103 or 104 BER.

C. BER Analysis Validation

To validate the BER analysis, we compared Monte-Carlo

simulations with the results predicted by (19) and (24). Fig.3 shows a side by side comparison of the BER performance

of a collection of representative parameter sets itemized inTABLE II. For BER simulation, we used a MIMOOFDM

system with =128 subcarriers and 20 OFDM symbols perpacket without any pilot or training tones. The channel model

described in section II.B was applied with perfect channel

estimation and synchronization. Fig. 3 confirms that the BER

lower bound given by (19) and (24) is valid and reasonably

close to the actual BER for various clipping levels. For the MLdetector (Mode 1 3), the gap between the actual performance

and our analytical expressions is mainly because of looseness

in the lower bound (17), where we only consider minimum

distance error events ignoring spatial interference. The bound(17) is much tighter for MMSE cases (Mode 4 6) because the

MMSE detector is designed to minimize spatial interference

before making decisions. It is worth noting that the total

degradation (1) is governed by () (SNR gap betweenthe unclipped and the clipped cases) rather than the exact

position of the BER curve. Since the BER performance penalty

due to clipping is accurately predicted by our analytical model,

we will use (19) and (24) with equality to compute the SNRpenalty of clipping in total degradation (1).

V. PA POWER SAVING VIA TOTAL DEGRADATIONANALYSIS

The tradeoff space of clipping is quantified by the total

degradation ( ) metric [3] [6] as a function of definedin (1), which combines the performance degradation penalty

and the PAPR penalty. The objective of the optimal clipping

is to minimize the metric directly related to the PA powerconsumption to achieve a given target BER performance.

The SNR degradation, (), in (1) can be accuratelyfound by evaluating the inverse function of the BER lower

bound expressions (19) or (24). Since the inverse function of


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(19) or (24) is difficult to obtain, we compute () bynumerically evaluating (19) or (24) for a target BER.

Fig. 4 shows the penalty as a function of the clippinglevel to satisfy a target uncoded BER (at the input of FECdecoder) of 103 or 104 for the parameter sets itemizedin TABLE II. The optimal clipping level for each parameter

set is the which minimizes the on each curve. Fig. 4shows that the values resulting from the optimal clipping

were in the range of 3.2dB (for Mode6) to 7.6dB (for Mode1)depending on the parameter set. Meanwhile, the of theconventional unclipped system with 3.1 is at least 9.4dB(this is solely from the PAPR penalty since, by definition, the

SNR penalty is 0 for unclipped systems). In other words, the

optimal clipping can provide the same BER performance with

1.8 6.2dB less penalty (SNR and PAPR penalty combined)

compared to the unclipped system. Fig. 4 suggests that the

optimal clipping level is significantly dependent on the operat-

ing system parameters such as the MIMO detection algorithm,

QAM size, MIMO antenna configuration, etc. It is also worth

noting that modes with smaller constellations (e.g. Mode6 for

MMSE or Mode3 for ML) can extract more gain from optimalclipping since these modes operate at lower SNRs and thusmore aggressive clipping can be tolerated until the clipping

distortion starts to dominate the system performance.

V I . CONCLUSION

We identified optimal PA clipping for MIMO-OFDM sys-

tems as a means of minimizing PA power consumption for

a target BER. The BER performance of a clipped MIMO-

OFDM signal was successfully analyzed and an accurate

BER approximation was provided for various combinations ofsystem parameters. Our studies show that the total degradation

penalty of the optimized system can be 1.8 6.2dB lower than

an unclipped system satisfying the same BER requirement.

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Power Optimized PA Clipping for MIMO-OfDM Systems

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