Design of PLL for Carrier Tracking in Uncertain OFDM Systems

Copyright@ IFAC Robust Control Design. Prague. Czech Republic. 2000

DESIGN OF PLL FOR CARRIER TRACKING IN UNCERTAIN OFDM SYSTEMS Yaniv O. and Rapbaeli D. Tel-A t!it! Unit! . ,EE-Sll!tem, e-mail: gerrhonOeng.tau.ac.il

Abstract: In this paper we develop design procedures for carrier tracking loop for OFDM systems or other systems of blocked data. We show that in many cases tracking loop is better than AFC+pilot to combat the phase noise problem. The derivation treats the general case of loop design in situations where phase error measurements are made infrequent enough to invalidate the traditional loop design methodology which is based on analog loop design. Out solution is general and includes arbitrary phase noise and additive noise spectrums, margins, uncertainty and loop delay. IT the optimal loop does not meet the required margin constraints and uncertainties, we propose the use of the QFT synthesis procedure which we extend for this case. In addition, we give simple low order loop design procedure which may be sufficient in many cases. The case where pilot symbols are used is covered and we combine them for best performance. Finally, we show that a large improvement can be gained also in the phase change over the symbol. Copyright @20001FAC Keywords: PLL, Carrier tracking, OFDM, Gain margin, Phase margin

1. INTRODUCTION

not reliable enough to be used in a PLL, but after decoding, the phase of the block can be estimated using the decoded symbols. In the examples above the continuous time section of the analog/digital PLL is replaced by a high sampling rate digital implementation. Since the sampling rate is high relative to the loop bandwidth, the approximation by continuous waveforms is good for all practical purposes.

The phase locked loop (PLL) principle is being successfully used for decades for tracking the carrier phase and the bit timing. Digital implementation of PLL in most cases is based on sampling frequency which is much higher than the loop bandwidth, and the PLL behavior can be approximated by its analog counterpart. There are situations which invalidate this assumption, and new design method need to be developed. There are many possible variations on how the PLL is sampled. In this paper we are interested in the case where continuous time phase is tracked by mixed analog/digital PLL in which the phase detector output is sampled in low rate (relative to the resulting loop bandwidth) and fed to the loop filter. The samplinf!; of the phase detector is undesired but unavoidable con8cquence of the communication system if the data is blocked and information about the phase can be extracted only at the end uf the block. As an example to such system is Orthogonal Frequency Division Multiplexing (OFDM) receivers. Another example is a system which uses block code, and the uncoded symbols are

A general solution using the QFT technique [4], [8],[9], is used to derive a close to optimal loop filter. The loop filter is designed under the conditions of minimizing its nns phase error at the sampling point, with given phase noise spectrum and additive noise level, under the restriction to satisfy gain and phase margins with some gain uncertainty. After the high order solution (near optimal solution) is described, restricted order solutions are also given. First order solutions have closed form when the delay is large enough, and for second order solutions easy to use graphs are given for extracting the range in which the optimal solution can be searched.

609

Many papers has been devoted to the problem of phase noise effect in OFDM and synchronization loops. Several papers have investigated the effect of phase noise on OFDM [1], [14], [13],[17],[15] and show that the sensitivity of OFDM is orders of magnitude more than single carrier schemes with the same bit rate. However, none of these papers considered carrier tracking loop for decrease the need of very stable local oscillators. The common solution is to use differential detection or phase extracted from pilot signal within the symbol and slow AFC loop [13],[10], [5] . Differential detection is known to loose at least 3dB in performance for QPSK or QAM. Mignone [12], used phase estimation from previous symbol to be used in the current symbol. Being able to get good phase estimate extracted from pilot signals requires that the 'signal to noise is good enough, which is not the case for relatively small FFT sizes (e.g. 64 points). The assumption that the carrier phase is not varying during two symbol period, like used in [12] or in differential detectors, requires a very low level of phase noise.

2. TRACKING LOOP FOR OFDM 2.1 Derivation of Phase Error Indication

During a symbol period T, the complex envelope of the transmitted OFDM signal is [14]

s(t) = ej9 (t)

N-l

L

am ej21r ,¥t, 0::; t ::; T.

m=O

where am is the data symbol of the m frequency bin, and B(t) is the carrier phase, which is common to all sub carriers. For simplicity we assume that the channel is fiat and the same constellation is used for all subcarriers. We assume that at the receiver front end there is a phase rotator which derotate the signal by multiplying by e- j8 (t), where O(t) is the estimated phase by the loop. Let us denote the phase error by e(t) = B(t) - O(t) then the signal at the FFT output of the receiver can be written as N-l

rk

L

= akIo +

amIk- m + Nk

(1)

m=O,m;ek

By using properly designed loop, it is possible to considerably reduce the degradation from phase noise. We show that OFDM fits the model used in this paper, therefore the design techniques can be applied. The phase detector estimates can be obtained either from the data of the previous symbol or from pilot symbols. Better, non-causal, solution to OFDM are also presented by using pilot symbols to correct the phase of the current symbol in addition to the loop operation.

where Ik = ~ foT ej21rfteje(t)dt and Nk is the thermal noise contribution. The first term in (1) is the useful signal, ak, which was rotated by the phase of Io which is the average value of e(t) during the symbol, and will be denoted by ea (n) . The second term is called inter bin interference (IBI) and results from the phase error change over the symbol and can be modeled as uncorrelated Gaussian noise for large N . It is difficult to extract information from the IBI regarding the phase error. The first term, however can be used to extract information about the average value, ea (n). The phase detector output can be expressed as

The degradation of OFDM receiver from phase noise is due to two effects. The first is the average phase error over the symbol, which rotates the constellation of all subchannels. The second is the phase change over the symbol which causes to loss in orthogonality. The latter is similar in its effects to additive Gaussian noise. Though the optimization in this paper is done with respect to the average phase error, we observe in many cases that using the loop significantly decreased the phase change over the symbol. Loop optimization with regard to the phase change is the subject of subsequent research effort .

ea = N~N

N-l

L Im{rma:n}

m=O

Here we assumed that am has been decoded successfully. High error rate in am can be tolerated since N phase estimates are averaged. Alternatively, am after re-encoding can be used [12], or pilot symbols with known am can be used. If pilot symbols are used, the noise level
2.2 Degradation of Performance Due to Phase Errors

~-:BJ

=

'"' ~

am

m=O,m;ek

=~ ~

dt

Jej27f~tejec(t)dt J

am

• F(s) denotes a linear time invariant filter to be designed; and • O(t) and n(t) are mutually uncorrelated zero mean stochastic signal with known spectral densities, modeled by stable minimum phase linear time invariant system driven by white noise.

ei27f?tec(t)dt

0

2: am ~ Jei27f ?tec(t)dt, T

N-I

m=O

0

the last equality is because by definition JoT ec(t) O. Since am are uncorrelated,

F

• A denotes the phase detector gain;

T

m=O,m;ek

= j

t Je . (t)

0

N-I

f 2:

k-",

T

am

m=O,m;ek .

J

T

• T denotes a periodic sampling with sampling period T (in the OFDM case its the length of OFDM symbol); • W denotes a decimating filter. In this paper we choose an integrator along a period T, and its transfer function is W(s) = I-T~'T ;

0

N-I

2:

J·2e 7f-,.- e

W

Fig. 1. The sampled PLL linear approximated feedback system

T

N-I

1 = T

def

1:

-

e

The degradation in the OFDM performance is due to both ea(n) which cause rotation in the constele( t) - ea (n) which causes the lation and e c (t) IB!. We like to analyze the effect of ec(t) on the performance. Let

V

A

1:

A

=

Using the notation [x]' for the sampled representation of the continuous signal x(t) with sampling period T[2, page 266] (that is x(t) multiplied by the train of pulses m(t) = L~-oo c5(t - kT)), the error signal e of the system in Fig. 1 is the solution of the following equation

Assuming that the contribution of ec(t) above the frequency N /2T is negligible, we can use the Parseval relation to get

e(s) = O(s) - F(s) [W(Ae

+ n)]* (s).

(2)

Our problem is to design a loop filter, F(s), which minimizes the phase error signal, e(t), subject to the following performance index, data and constraints: Thus, the Signal to IBI ratio is ~ UJ,I

=

• The power spectral density of the noise, n, and phase noise, 0, are 'l>n(w) and 'l>e(w), respectively, and it is assumed that (J and n are uncorrelated. • The phase detector gain, A, is fixed but only known to belong to an interval A E [AI, A 2 ] where Al and A2 are known. Variations in A are the result of change in symbol energy EN or fading. • The performance index is to minimize the rms T value of ea(n) = ~ e(t)dt, the average phase error during a symbol. It is also useful to minimize the rms phase change during a symbol a c . • The open loop response should have some gain and phase margins in order to guarantee a well damped closed loop response and stability in

~,where Ut;

a~ ~fE (~ JoT e~(t)) .We will neglect the influence of ec(t) on 10 since it is related to the second order of ec(t).

3. STATEMENT OF THE PROBLEM Since the sampling rate of the receiver frontend is much higher than the symbol rate, the OFDM receiver operation can be well approximated by a matched filter operating on a continuous signal. For convenience, the loop filter which operate in the front end sampling rate is approximated by a continuous filter F(s). The linear model of the PLL is presented in Fig. 1 where,

J::+

611

case of some gain uncertainty. These margins are defined by a constant , such that

L*(jW)

I1 +L*

1<- ,;A E [A 1,A2),L* = [AWF)*

0'; =

f f

Ja(z)

+ f3(z)Q(z)J 2 ~z

Izl=1

(3)

+

f/l()(z)f/l()(z-1) - a(z)a(z-l)d;

Izl=1

where f3(z) and a(z) satisfy the following equations

4. THE PROPOSED ALGORITHM

f3(Z)f3(Z-1) = f/ln(Z)f/ln(z-l)

From equation (2) e

+ F [W Ae)* = B -

* [WBr [We) = 1 + [AWF)*

= [WB)* 1 + L*

+ f/l()(z)f/l()(z-I),(8)

a(z-1 )f3(z) = -f/l()(z )f/l() (Z-I).

(9)

F [W n)* , Q(z) has the same zeros inside the unit circle as L(z) and these can only be pure delays due to the process Wand the delay in the feedback loop. We therefore denote

[WFj* [Wn)* 1 + [AWF)*

_ L* [Wn)* 1 1 + L* A'

(10)

We want to minimize the rms value of ea (n)<5(t -

nT)

= [WBj*, which is 0'; = E ([We)*)2

where k represents the delay and Qo(z) is minimum phase. Hence

(4)

O'e2

Let us denote by [Wn)(z) , [WB)(z) and L(z) the z-transform representation of the sampled signals, [Wn)*(s), [WB)*(s) and the impulse response of AWF(s), respectively. Using the equality L(z esT) = L*(s) [2, page 280) we get

f I+

0'2 = e

f/l()(Z) 12 dz 1 L z

Izl=1

+

f f

Iz ka(z) + f3(z)Qo(z)1 2dz -;

Izl=1

+

f/l()(z)f/l()(z-I)-a(z)a(z-1))d;.

Izl=1

After removing terms not depending on Qo(z), it is clear that Qo(z) which minimizes O'~ is the same one which minimizes

fI

Lf/ln(z) 12 dz .(5) 1+L Az

Izl=1

where E ([Wn)(z))2 = f/ln(z)f/ln(z-I), E ([WB)(z))2 = f/l() (z)f/l() (Z-1 ). Since L ex A, the argument of each integral in equation (5) in low frequencies is approximately proportional to 1/A 2 , and since in general the spectral density of «I>() is concentrated in low frequencies and that of «I>n is white, we shall assume

Assumption 4.1. For a given F(s), the maximum of O'~(A) over A E [A 1,A2) is O'~(Al)'

0';1 =

f f

Izka(z)

+ f3(z)Qo(z)1

2

d;

Izl=1

+

f/l() (z )f/l() (z-1) - a(z )a(z-1) d; .

Izl=1

Since 13 (z ) = 13 (z -1) it can be chosen minimum phase, thus 0'~1 can be split into its stable and unstable parts

O'e21

=

f

(Ja_J 2

+ Ja+ + f3QoJ 2)

dz -;'

(11)

Izl=1

This assumption means that a solution that minimize O'e(At} subjected to all other constraints is a solution to our problem. Using the notation

where zka(z) = a+(z)+a_(z) is the partial fraction expansion of zka(z) with terms inside and outside the unit circle, respectively. Therefore the optimal Qo is

L(z) 1 1 + L(z) = Q(z), 1 + L(z) = 1 - Q(z), (6) where Q(z) is a stable rational function with poles inside the unit circle, gives

f

=

(12)

(7)

and the optimal open loop, Lopt(z), is by equations (6,10)

By simple complex arithmetic it can be shown that

(13)

0';

=

(Jf/lnQJ2Jf/l()(z) - f/l()QJ2) d;.

Izl=1

612

The same results, without explicit consideration of the expected delays, can be found in many papers, for example in [16] . The filter, Fopt(s), such that [WFopt(s)]* (s) = Lopt(z ) is the solution we seek for, only if the closed loop satisfies the gain and phase margin constraints, 1', over all A E [AI , A2]' However if the open loop gain interval is large and/or the desired margins are large compared to T, Fopt (s) will not be a satisfactory solution. It might even be an unstable solution for some of the possible open loop gains, Le, high gains. In the next step it is shown how to synthesize an appropriate F(s) by modifying Qopt. The gain and phase margin specifications can be expressed as

I I JI L1 (z) I~1' ; VAE [A ,A ]'(14)

L(z) = --L

I1 +

1+

A

Al

1

Ll

2

where Ll(Z) is the open loop when A = Al . Using the former notation A(Z) (15) Ql = Qopt - (3( z )' where A = Al

tion is using the previously detected symbols am . In many cases pilot signals are added to the OFDM symbol. These are unmodulated subcarriers which can be used for phase estimation. The phase of the pilot signals (without noise) is the average carrier phase. This causal information can be used to improve the average phase estimation. The average value of O(t) during the symbol will be denoted by Oa . Let the pilot estimate of 0 be denoted by Ba . Ba = Oa + nl where nl is Gaussian noise with spectrum (1/ J.L)~ , where J.L is the ratio of pilots energy relative to total symbol energy. The estimate phase which combines ,the sources of information in an optimal way is Oa(t) = o:Oa(t) + (1 - o:)Ba, where (1~:)2 = ~f;~l, which is simply because we assume independence between the estimation error due to the previous symbols to the error induced by the current symbol noise. A more convenient implementation is to measure the pilot phase after the correction of the phase by (), so the pilot is measuring e-.. = 0-;' - O'a. Therefore,

our problem reduces, by expression (11) and assumption 4.1, to designing a stable A(Z) which minimizes 2 a e2

=

f IA(Z)I-;, 2

dz

6. IMPROVEMENT BY NON-CAUSAL FILTERS

(16)

The gain and phase margin constraints , the delay and the uncertainty dictate a suboptimal filter F (s). It is possible to estimate the signal, 0, by the sum of {) and u·(kT) during the time interval [kT, kT+T], see Fig. 2. Then apply an optimal filter, which can even be non-causal, in order to improve the estimation of 0 in the sampling time and/or between them. For example 1 when the estimated A value is Aest = 1 and for k = 0 (no sample delay), the optimal filter (calculated as described 0 .06257z2-0 .03795 z h in section 4 ) is Q = 0 .05129z 2 -O .04z+0 .01334 Furt er

Izl=l

subjected to the constraint (14) which is

I

-

AQ(z) A2 ~ 1'; 'v'A E [1, -A ]. 1 + (A - l)Q 1

I

(17)

Substituting equation (15) in equation (17) gives

AQo(z z ) _ A(Z) z[3(z) A 1 + (A-l)Qo _ A A-I z

z[3(z)

Inequality (18) reduces our optimization problem into another optimization problem which is superior in the sense that it guarantees stability, it is simpler and needs less optimization parameters. Moreover, it has a design framework for designing a suboptimal filter, this framework is the feedback synthesis theory known as QFT [3], [9], [8] . Fig. 2. The improved feedback system improvement can be obtained if a longer delay is assumed. Moreover, the change during the symbol can be improved by replacing Q(z) with Q(s) which can be interpreted as interpolating filter.

5. PILOT ASSISTED PHASE ESTIMATION Until this point we assumed a causal operation of the loop, which means that the phase error estima-

613

7. IMPROVEMENT IN PHASE-CHANGE ERROR

Design Using the Quantitative Feedback Theory', Int., Jou. of Robust and Nonlinear Control, Vol. 3, pp. 47-54. [5] Classen F. and Meyr H., 'Frequency synchronization algorithms for OFDM systems suitable for communication over frequency selective fading channels' Vehicular Technology Conference, 8-10 June 1994, 1655 - 1659 [6] D'Azzo J. and Houpis C.H., Linear Control System Analysis and Design Conventional and Modern, 3rd ed., McGraw-Hill, New York, 1988. [7] Doyle J . C., B. A. Francis and A. R. Tannenbaum, Feedback Control Theory, Macmillan Publishing Company, New York, 1992. [8] Horowitz 1., Quantitative Feedback Design Theory (QFT), QFT Publications, Boulder, Colorado, 1992. [9] Horowitz 1., 'Invited Paper - Survey of Quantitative Feedback Theory (QFT)', International Journal of Control, Vol. 53, no. 2, 1991, pp. 25529l. [10] Luise, M., Reggiannini, R., 'Carrier frequency acquisition and tracking for OFDM systems', IEEE Transactions on Communications, Nov. 1996, Vol. 44, 1590 - 1598. [11] Martin G. H., 'Designing Phase-Locked Loops', R.F Design, 1997, Vol. 20, No. 5, p. 56, 58, 60, 62. [12] Mignone, V., Morello, A., 'CD3-0FDM: a novel demodulation scheme for fixed and mobile receivers', IEEE Transactions on Communications, Sept. 1996, Vol. 44, 1144 - 115l. [13] Muschallik C., 'Influence of RF oscillators on an OFDM signal', IEEE Transactions on Consumer Electronics, Aug. 1995 Vol. 41, 592 - 603. [14] Pollet, T ., Van Bladel, M., Moeneclaey, M., 'BER sensitivity of OFDM systems to carrier frequency offset and Wiener phase noise', IEEE Transactions on Communications, Feb.-MarchApril 1995 Vol. 43, 191 - 193. [15] Robertson, P., Kaiser, S., 'Analysis of the effects of phase-noise in orthogonal frequency division multiplex (OFDM) systems', IEEE International Conference on Communications, ICC '95 Seattle, 18-22 June 1995, 1652 - 1657. [16] Shaked U., 'A General Transfer Function Approach to the Discrete-Time Steady-State Linear Quadratic Gaussian Stochastic Control Problem' Int. J. of Control, 1979, Vol. 29, No. 3,361-386. [17] Tomba L., 'On the effect of Wiener phase noise in OFDM systems', IEEE Transactions on Communications, May 1998, Vol. 46, 580 583.

As derived in Section 2.2 the OFDM receiver performance is dependent on both a e and a c . Although the loop was optimized with respect to a e, we have checked the contribution of the loop to the reduction of a c . If the phase noise process is random walk, we do not expect to have improvement in phase change since the phase change in current symbol is independent on previous ones. Only interpolation as noted in previous section can improve the phase change (dramatically). However, for phase noise spectrum of higher order, like q,e oc ~, the phase change can be reduced.

8. CONCLUSIONS We have shown that OFDM system needs a phase recovery loop optimaly designed for best performance. Since in the OFDM case most often the symbols are long enough to invalidate the traditional "analog" loop design assumptions, we have derived the proper loop design methodology. If the optimal loop does not meet the required margin constraints and uncertainties, we propose the use of the QFT synthesis procedure which we extend for this case. The solution is given for arbitrary phase noise and additive noise spectrum, margins and uncertainty, and loop delay. In addition, we give simple low order loop design procedure which may be sufficient in many cases. The case where pilot symbols are used is covered and we combine them for best performance. Finally, we show that a large improvement can be gained also in the phase change over the symbol. Optimal loop design for minimizing phase change is the subject of future research.

9. REFERENCES [1] Armada A.G.and Calvo M., 'Phase noise and sub-carrier spacing effects on the performance of an OFDM communication system', IEEE Communications Letters, Jan. 1998, Vol. 2, 11 - 13 [2] Astrom K. J. and B. Wittenmark, Computercontrolled Systems Theory and Design, Prentice-hall, Third Edition, 1997. [3] Borghesani C., Chait Y., and Yaniv 0., Quantitative Feedback Theory Toolbox, The MathWorks Inc., Natick, Mass., 1994. [4] Chait Y. and O. Yaniv, 1993, 'MultiInput/Single-Output Computer-Aided Control 614