Analyzing Spectral Regrowth of QPSK and OQPSK Signals

8/2/2019 Analyzing Spectral Regrowth of QPSK and OQPSK Signals

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ANALYZING SPECTRALREGROWTHOF QPSK AND OQPSK SIGNALSRaviv Raich and G. Tong Zhou

School of Electrical and C ompu ter Engineering, G eorgia Institute of technology, A tlanta, GA 30332-02 50r a v iv , [email protected]

ABSTRACTIn this paper, a comparison is ma de betwe en the spe ctral regrowthof quadrature phase shift keyed (QPS K) and offset QPS K (OQP SK)signals as they go through non-line ar amplifications. Contrary toexisting approaches that assume the power amplifier input is Gaus-sian, our analysis is carried out withou t the Gaussian assumption,by using higher-order statistics. We show that it is possible to as-sess quantitatively, whether and how much OQPSK is beneficialin reducing spectral regrowth. Simp le closed form form ulas areobtained w hen the pulse shap e filter is time-limited. A particularmeasu re of spectral broadening is also provid ed

1. INTRODUCTIONQPSK is a popular modulation format that is used in many appli-cations (e.g., IS-95 CDMA). Let us denote a QPSK symbol by smwhere s m = [+1& j ] with probability 0.25 each. A significantdrawback of QPSK is the fl 80 -d eg ree phase change at the 1+ jH -1 - and the 1- f) -1 + transitions. Such transitionsare undesirable if the waveform is to be filtered and subsequentlyprocessed by a nonlinear power amplifier (PA).Nonlinear PAS are used in communication systems for im-proved efficiency because generally, there is an inverse relation-ship between linearity and efficiency [11. Higher efficiency meansthat a larger percentage o f the dc (e.g., battery) power is deliveredto the load, thus increasing battery life and minim izing heat dissi-pation.Figure 1shows in solid line, the AM/AM (amplitude to am-plitude) conversion of a nonlinear PA. The dashed line shows incomparison, a linear A W AM response. Although Ism I = isconstant modulus, the envelope of a filtered QPSK signal couldfluctuate, thus leading to nonlinear distortions. In Figure 1 , thismeans that a filtered QPSK signal could traverse the A-C region ofthe PA response. In addition to the PA compre ssion at the largeramplitudes, the filtered QP SK sign al also experienc es cut-off whenthe input amplitude is close to zero.A remedy is to employ offset QPSK (OQPSK) modulation. InOQPSK, the I- and Q- symbol streams are offset in time by halfthe symbol period, thus avoiding the fl80-degree phase change.For illustration purpose, we can imagine that in Figure 1, insteadof the A-C region, the filtered OQP SK signal traverses through theB-C region of the PA, thus avoiding cut-off.

Despite of this merit of OQPSK, some concerns were raisedabout the overall effectiveness of employing OQ PSK. First of all,it does not lend itself to differential encoding [2]. Furthe rmore, ar-gument has been made that although the QP SK signal experiencesThis work was supported in part by the National Science Foundationgrant M IP 9703312 and by th e State of Georgias Yamacraw Initiative.

0-7803-70414/01/$10.0002001 IEEE

AMIAM conversion121

Fig. 1 . AM /AM characteristic of a nonlinear PA (solid line).

the cut-off effect, it spends a very small percentage of time in thecut-off region. On the other hand, the region that the OQPSK sig-nal spends its time with (e.g., the B-C region in Figure I ) is morecomp ressed and hence more nonlinear than the A-B region that theQPS K signal frequently visits.This paper attem pts to offer a means of quantitatively analyz-ing sp ectral regrowth of a comm unication signal passing througha nonlinear device. Specifically, we com pare the power spectraof filtered QPSK and OQPSK signals after their nonlinear ampli-fication. Although spectral analysis is routinely carried out forcommunications signals, the nonlinearity present in the PA com-plicates the problem. In [3], the authors analyzed the spectral re-growth pattern when the input signal is Gaussian, but their resultshave limited applicability since many communication signals arenon-Gaussian. In [4],a Volterra system approach w as adopted. Weaddress here spectral regrowth of a memoryles s nonlinear device.

2. PROBLEM FORMULATIONA linearly modu lated signal is expressed in the baseba nd as

r mH~ ( t )- sm h ( t - m T ) ,Jz =- mwhere sm = a , + b, is the mt h symbol transmitted, h ( t ) s thebaseband pulse shape filter, A is a real-valued input scale factor,and T is the symbol period. We assume that a , and b are i.i.d.and are mutually independent. We refer to the resulting s m ascircular complex symmetric.When sm is QPSK , we have a , E (1, -1) with equ al proba-bility 0.5, and similarly forb,.

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A filtered OQPSK signal may be written as:A "x ( t ) = - a ,h ( t -mT)Jz m = - m

2A " t - m T - T ) , (2)

where a,m, h, , h ( t ) , , and T are the same as in the filtered QPSKcase. The only difference is that the imaginary part of (2 ) has aT/ 2 delay relative to that of (1).Next, z ( t ) s input to a PA to yield output y ( t ) . Ideally, wewould like y ( t ) = Q z ( t ) , here Q is a constant with la1 > 1 . Bu tin reality, all PAS are inherently nonlinear. In the case of a m emo-ryless nonlinear PA, we can app roximate its baseband inputloutputrelationship by [5 , p. 7351:

00

~ ( t )~ ( t ) Zn+l 1z(t)1"1 (3 1n= O

from which we infer that the complex gain is(4)

It is seen that the complex gain is a function of the input ampli-tude Iz(t) only. This is consistent with the fact that a memorylessnonlinear PA is often characterized by its A W A M (i.e., I y ( t ) l vs .Iz(t)l) nd AM/PM (i.e., L y ( t ) - L z ( t ) vs . Iz(t)l) haracteris-tics. If (z(t ) ls constant such as the case of ( I ) with a rectangularshaped h( t ) see also Section 3), then z ( t )will not experience anynonlinear distortion since the gain in (4 ) is constant. Our objectivehere is to analyze the power spectral density (PSD) of y ( t ) an dits dependence on the PA parameters {azn+l}, he baseband filterh( t ) , nd the input scale factor A.

3. ANALYSISAlthough our analysis on spectral regrowth can be generalized toaccommodate higher-order nonlinearities, for simplicity, we illus-trate our approach using a 3rd-order nonlinear model:

y ( t ) = a 1 z ( t ) + a 3 (z ( t )12z( t )= alz( t )+ a 3 z z ( t ) z * ( t ) . ( 5 )

Since sm has a symmetric distribution, y ( t ) as zero-mean.Therefore, the auto-correlation and auto-covariance functions ofy ( t ) coincide. We define the auto-correlation function of y(t) attime t and lag T as follows:

C z y ( t ; T * ) = E[Y(t)Y * ( t + T ) ] . ( 6 )In ( 6 ) ,T * indicates that conjugation is applied to the lagged copy,y ( t + T ) . Note that T itself is always a real number.function isSince y(t) is cyclostationary, its time-averaged auto-correlation

C Z ~ ( T )= - czy(t; ') d t . (7 )LTThe power spectrum o f y ( t ) is the Fourier transform of Ez Y ( T ) :SZY(f) = 3 T + f { C Z Y ( T ) }

m(8 )1, z Y ( T ) e - - 3 2 x T f d T .

We would like to examine SzYf) or the PA mo del in ( 5 ) andthe input as in ( I ) or (2). Substituting ( 5 ) into ( 6 ) ,we obtain

Our next step is to expand the above covariance terms usingthe Leonov-Shiryaev formula [ 6 ] . Under the circular symmetryassumption of sm , we infer that z ( t ) f ( I ) is circular symmetricas well. Therefore, we find for filtered QPSK,

Note that the OQPSK signal (2) is no longer circular symmet-ric and hence the corresponding 4 1 3 , 4 3 1 , an d 4 3 3 expressionscontain additional terms.The kth-order cumulant of z ( t ) t time t and lags (TI , . . . ,T k E - l ) is defined as

Note that a conjugated lag in the argumen t of ck z (.); e.g., re', m-plies that the corresponding term in the cumulant; e.g., Z* ( t+T P ) ,has conjugation. For the z ( t ) n (l), we haveC k z ( t ; T 1 , . . . T 6 , . . . T;-i )

h(t - mT + Te-I)h*(t- mT + P ) . . .h*( t- mT + T k - l ) ,an d

yks = c u m { s ( t ) , ( t + T I ) , . . . ,s ( t + T P - I ) ,

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s * ( t + q), . , * ( t + q - I ) } .Next, let us analyze Skt(f), hich is the Fourier transform ofInterestingly, when h( t )= 0, V J t J T/2 , he q 5 k e ( t ; T ) ex-& ! ( T ) , the time-average of Q k t ( t ; 7 ) .pressions can be simplified considerably. As a result, w e obtain

A4TA 4A6T

sI3(f) - - ~ ( - f ) [ ~ ( f )w - f ) o ~ * ( - f ) 1 ( 1 3 )S 3 1 ( f ) = +*(-f) w * ( f )o w - f ) o ~ ( - f ) 1 ( 1 4 )S33(f) = - ff" 0 H(-f) 0 H(-f)12 , (15)

where H ( f ) s the Fourier transform of h ( t ) , nd 0 s the convo-lution operator.When h ( t ) s real valued and sym metric, we obtain a surpris-ingly simple expression for the PSD of ~ ( t ) :

where H3(f) H ( f )0 H ( f )0 H ( f ) .We make the following remarks regarding (16):Remark 1 : Potential spectral regrowth is indicated by the Hs(f)term which generally expands the bandwidth of H (f) hrough thetriple convolution.Remark 2: The severity of spectral regrowth is determined bythe coefficient ( a3 /a1 )A2n (16). If the PA is inherently verynonlinear; i.e., the 03 / a 1 ratio is high, then on e needs to reduce theinput amplitude factor A in order to minimize spectral regrowth ~this is referred to as input back-off. In general, spectral regrowthbecomes more severe as A increases.

Now let us consider two baseband filters often studied in theliterature [7]: ,

and

an d

For the rectangular pulse ( 1 7), we obtain

Substituting ( 1 9)-(20) into ( 1 6) , we infer that there is no spec-tral regrowth when the rectangular pulse is used for the z ( t ) n ( I ) .

This is expected since in this case, the resulting Iz(t)l = A hasconstant envelope.For the sinusoidal pulse (1 8), we have

and

This Hz(f)an be shown to have a wider m ainlobe than H ( f ) .bandwidth whereOne way to quantify spectral regrowth is to use a notion of

and

Note that for a symmetric spectrum S ( f ) , he corresponding ( f )=0. Substitution of (16), (21), and (22) into (23) yields the follow-ing bandwidth formula for the cosine pulse (1 8):1- Re@) + 4 .5 [P I21- 3 Re@) + 2 .5 [ P I 2

When the PA is linear, we have a3 = 0 and hence /? = 0.Th e bandwidth formula (25) yields 0.5f0 as the bandwidth of alinear system ., Therefore the ratio, m / ( O . S f o ) , can be usedas a measure of bandwidth expansion and from (25), it is obviousthat this ratio is > 1 for any /3 # 0.When z ( t ) s OQPSK, the analysis is generally more involved.But with either (17) or (1 8), the OQPSK signal in (2 ) has ( z ( t ) (,4 and hence the corresponding PA output . y ( t )= ( a ,+m A2)z ( t )does not experience any spectral regrowth.

4. SIMULATIONSIn this section, we present a numerical example to verify the accu-racy of the expressions (12)-( 16). 1 , 0 0 0 symbols sm were gener-ated and a filtered QPSK signal z ( t )was obtained from equation( I ) with the cosine pulse (18). Th e scale factor was A = 1 andthe sampling period was AT seconds. The resulting z ( t ) wentthrough nonlinear amplification as described by (5) with ai = 1and a3 = -0 .3exp ( j ; ) .Figure 2 shows the theoretical SI] f ) c.f. (12)) in solid linean d its estimate in dashed line. Th e estima te is nothing but the PS Destimate of z ( t ) .Close agreement between the two is observed.Figure 3 shows the theoretical S33 ( f ) c.f. ( 1 5 ) ) in solid lineand its estimate ~ the PS D estimate of z ( t ) ' z * ( t )n dashed line.Similar agreement is observed. Comparing with Figure 2, we seethat the bandwidth of &(f) has increased from that of SI ,(f).

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Indeed, of S33(f) s times or 34% larger than thatof SI1 f) c.f. (23)). Moreover, evaluation of the (12)-(15) termsreveals that 5 3 3 (f) s the major contributor to spectral regrowth inIn Figure 4,we show a comparison between the output spec-trum Szy( f ) hen the input is QPSK (solid line) or OQPS K (dashedline). The bandwidth of the QPSK signal is indeed larger than thatoft he O QPSK signal. In fact, equation (25) with ,B = 0 .3 exp ( j )tells that the bandwidth increase was 14%.

S Z Y ( f .

5. CONCLUSIONSWe have described in this paper, an analytical approach for evalu-ating the power spectra of filtered QPSK an d OQ PSK signals afternonlinear amplification. A salient feature of ou r analysis is thatwe do not need to assume that the PA input is Gaussian. In theQPSK case, we were able to obtain a simple closed form expres-sion for the output PS D when the PA is cubic nonlinear and thebaseband filter is time-limited. We treated the cosine pulse fil-tered QPSW OQP SK signals in detail and provided a measure o fband width expansion. We are currently workin g on applying ouranalysis to more general scenarios.

6. R E F E R E N C E S[ I ] S. C. Cripps, RF Power Amplijiers fo r Wireless Communica-tions, Artech House, Nonvood, MA, 1999.[2] Y. Akaiwa and Y. Nagata, Highly efficient digital mobilecommunication with a linear modulation method, IEEE J . ofSelected Areas in Comm unications, vol. 5, pp. 890-895, June1987.[3] K. G. Gard, H. M. Gutierrez, and M.B. Steer, Characteri-zation of spectral regrowth in microwave amplifiers based on

the nonlinear transformation of a complex gaussian process,IEEE Trans. on Microwave Theory and Techniques, vol. 47 ,no . 7, pp . 1059-1 069, July 1999.

[4] A. Gusmao, V. GonGalves, and N. Esteves, A novel approachto modeling of OQPSK-type digital transmission over nonlin-ear radio channels, IEEE Journal on Selected Areas in Com-munications, vol. 15 , no. 4, pp . 6 4 7 4 5 5 , M ay 1 99 7.

[5] S. Benedetto and E. Biglieri, Principles of Digital Transmis-sion with Wireless Applications, Kluwer Academic I PlenumPublishers, New York, 1999.

[6] D. R. Brillinger, Erne Series: D ata Analysis and Theory,Holden-day Inc., San Francisco, 1981.[7] J. G. Proakis, Digital Communications, McGraw-Hill, 3rdedition, 1995.

r 7Fig.2. The theoretical S11(f) solid line) and its estimate (dashedline).

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Fig.3. The theoretical S33 (f) solid line) and its estimate (dashedline).

-5 -4 -3 -2 -I 0 I 2 3 4 5Y4Fig. 4. The PA output PSD Szy( f ) hen the input is QPSK (solidline) or OQPSK (dashed line).

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Analyzing Spectral Regrowth of QPSK and OQPSK Signals

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