01095422

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-30, NO. 10, OCTOBER 1982 2297

Phase-Locked Loops with Limiter Phase Detectors in the Presence of Noise

WERNER ROSENKRANZ

Abstract-" nonlinear mathematical model of phase-locked loops with limiter phase detectors in the presence of noise is presented. The model, which is an extension of the well-known baseband model of loops with sinusoidal phase detectors without limiters, incorporates a modified nonlinear phase detector characteristic, the form of which is changed if the input carrier-to-noise ratio alters, as well as a modified phase noise as an input to the model. Both the modified phase detector characteristic and the spectral density of the modified noise term are calculated specifically for sinusoidal, triangular, and sawtooth types of limiter phase detectors, allowing the application of various methods to determine the loop noise performance. As an example, the phase error variance of a first-order phase-locked loop is calculated, thereby showing a strong dependence on the specific phase detector realization.

I. INTRODUCTION

P HASE-LOCKED loops (PLL) with hard limiters preceding the phase detector (PD), resulting, for example, in a

sinusoidal, triangular, or sawtooth type of nonlinear characteristic, are of great practical interest [ l ] , [2] . The theoretical examination of their noise performance, however, is dif- ficult because of the hard limiting of the noisy input signal and the following nonlinear operation forming the specific phase detector characteristic of Fig. l(b), (c), (d). Therefore, the very well documented nonlinear theory, equivalent models, and results (e.g., [ l ] , [2]), developed for the sinusoidal phase detector (multiplier) without limiter, as illustrated in Fig. l(a), are not immediately applicable.

Although there are a few papers [3] -161 dealing with some aspects of limiter phase detectors in an open loop condition, very little seems to be known about the noise performance of the closed PLL. Therefore, it is worth looking at the problem more closely, particularly as both the limitation and the specific nonlinear characteristic may strongly influence the noise performance, as we will see later on in an example. The pur- pose of the present paper is to develop a more general equivalent model for phase-locked loops comprising phase detectors with limiters and, to some extent, arbitrary periodic characteristics in the presence of noise. The equivalent PLL model is an extension of the well-known mathematical model [ I ] , 123 mainly for two reasons.

1) The new general model, having the same structure, allows the application of the well-known linear or nonlinear methods for calculating the noise performance.

Manuscript received June 5, 1981; revised December 22, 1981. This

The author is with Lehrstuhl fur Nachrichtentechnik, Universitat work has been supported by the Deutsche Forschungsgemeinscha.

Erlangen-Niirnberg, D-8520 Erlangen, West Germany.

(a) SINUSOIDAL PO WITHOUT LIMITER (TYPE I)

IN $ BP T u o l vco A I

-+- (b) SINUSOIDAL LIMITER PO (TYPE I I )

1

id TRIANGULAR LIMITER PO (TYPE 111)

(d) SAWTOOTH LIMITER PO (TYPE IV)

Fig. 1. Realizations of (a) sinusoidal phase detector without limiter and (b), (c ) , (d) limiter phase detectors.

2) The model clarifies the reasons for nonlinear effects, thereby allowing the limits of the very useful linearization techniques for weak disturbances to be estimated.

11. GENERAL PLL MODEL FOR NOISY INPUT SIGNALS

The usual mathematical baseband model of the PLL (sinusoidal PD without limiter, i.e., an analog multiplier) with addi- tive narrow-band Gaussian input noise n(t) of bandwidth B 1 , variance an2, and zero mean represents the phase detector by means of a nonlinear element sin cp excited by the phase error process cp(t) as defined in Fig. 2. Here sin cp is the low frequency term of the PD output signal resulting from the product of the signal u1 (t) [total input signal: u1 (t) + n(t)]

u l ( t ) = iil sin [o,t + 41(t)] (1)

and the output signal of the voltage-controlled oscillator (VCO)

uz( t ) = i i 2 cos [ q t + 42(t)]. (2)

0090-6778/82/1000-~2297$00.75 0 1982 IEEE

2298 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-30, NO. 10, OCTOBER 1982

*,I I I

‘ -- PD L P vco

V = @,I-*,: p h a s e e r r o r

KO: p h a s e d e t e c t o r g a i n

KO: VCO galn

F(s) : loop f i l ter t ransfer funct ion

PLL input s ignal: u, ( t )+n(t )=

i ,sin(w,t+$,)+n(t)=

A(t)sin(w,t+O,+Q,)

V C O output s ignal : u,(t)= ~,cos(w,t+~,)

Fig. 2. General equivalent PLL model and definitions.

The input noise n(t) is introduced into the model as modified baseband phase noise n’(t) resulting from the product

1 n’(t) = - n(t) - u2(t)

KD

with KD = 1/2 l i l i i2 as phase detector constant. It has been shown [ l ] , [2] that, on the reasonable assumption that the closed loop noise bandwidth BL (one-sided) is much smaller than the input noise bandwidth B , , the modified phase noise n’(t) is independent of’cp or G2 and has variance anr2 = an2/iil ’, bandwidth B1‘ = B 1 , and a power spectral density Sn’(O), which is assumed to be constant in the input band B 1 , i.e., for frequencies -rrBl < o < rrB1.

As a result, the mathematical baseband model is made up from the baseband output signal uo(t) of the phase detector which is described by the sum of a nonlinearity sin cp, and a modified phase noise term n‘(t) with known and simple statistics

uo(t, cp) = [ul(t) + 4t>l * u2(0

= KD [sin q(t) + n’(t)] . (5)

In the case of a limiter phase detector the output signal uo is primarily given as a nonlinear function g(.) (e.g., sinusoidal, triangular, or sawtooth) of the phase difference between the input phase and the phase G2 of the VCO-signal. From

ul(t) + n(t) = A ( t ) sin [olt + @ l ( t ) + #n(t)l (6)

(see Fig. 2 ) with random envelope A and random phase Gn the total input phase is G 1 + &.Therefore,

where KO = const.

In (7) uo is a random process depending on the two random processes cp and Gn via the nonlinear PD characteristic. Pro- ceeding on the same assumption B1 S BL that leads to ( S ) , a phase error cp results which is much smaller and varies much more slowly than the input phase noise G n . To establish an equivalent model where, as in ( S ) , the PD output signal is described as the sum of a noise term and a signal term, we shall first consider cp fixed. Then, uo may be written as a mean ‘depending on the low frequency phase error plus a broad-band zero-mean noise term associated with the input noise n(t)

u O ( t , cp) = KDLEk(q + 4 n ) I + n’(t, cp)]

= KD[g’(P) + I Z ’ ( ~ , (8)

The low frequency component of uo may then be evaluated from the expectation

g’(cp) = Ek(cp + 4 n ) I PI. (9)

g’(cp) is a modified nonlinear PD characteristic which enters the equivalent model in Fig. 2 and is in general not equal to the actual PD characteristic g(-).

The rapidly varying noise term n’(t, cp) is generally also a function of cp. Comparison of (7) to (8) yields

n’(t, CP) = dc~ + G n ) -g‘(V>. (1 0)

The variance O , ‘ ~ ( ~ J > of this noise term has been evaluated for specific g(.) and fixed cp in [4] and [8], with the result that onr2 is not altered much from its maximum value at L,O = 0 even if cp is as large as, say, r / 4 . Moreover, the smaller the input carrier-to-noise ratio is (the case when large values of cp are expected at all), the smaller is the deviation of anr2(cp) from its value at q = 0. Therefore, we neglect the state dependence of the modified phase noise and (IO) becomes simplified in all cases where cp(t) has zero-mean (no static phase error)

n ‘ ( 0 = g(@n>. (1 1)

With the PD output signal given by (8), the equivalent baseband model of the closed loop is depicted in Fig. 2. It contains phase detector, loop filter (LP), and the voltage input- phase output relationship of the voltage-controlled oscillator. Using the key relationships (9) and (1 1) it is now possible to determine quantitatively the modified characteristic g’(cp), and the statistics, i.e., variance a,” and bandwidth B , ’ of n‘ , to yield its spectral density No’ in the interesting range around zero frequency. This is carried out in the following section for three specific characteristics of limiter phase detectors, i.e., of the sinusoidal, triangular, and sawtooth type.

111. THE MODIFIED PHASE DETECTOR CHARACTERISTIC

From [3] -[SI it is known that the characteristic of a limiter phase detector deteriorates from its noise-free sinusoidal, triangular,,or sawtooth form in the presence of noise. Here we show that by applying (9) and expanding the periodic characteristic g(.) in a Fourier series, a relatively clear analytical ex- pression for g’(p) may be found.

R O S E N K R A N Z : P H A S E - L O C K E D LOOPS W I T H L I M I T E R P H A S E DETECTORS 2299

A. Limiter PD with Sinusoidal Characteristic On account of the sinusoidal characteristic s(.) we find

from (9), after a trigonometric manipulation and considering that the input phase noise has zero mean,

B. Limiter PD with Trianguhr Characteristic Expanding the periodic triangular characteristic g(cp + &)

in a Fourier series and averaging, with cp remaining fixed, yields ( k = 2 v - 1 , u E N )

C. Limiter PD with Sawtooth Characteristic

Applying the same procedure as before we obtain (v EN)

1 g’(cp) = 2 * 2 (-1)”‘ - E {cos v&} - sin vcp. (14)

v = 1 , 2 , 3 V

In order to determine g’(cp) readily from(12)-(14) it is necessary to evaluate the expectation

where p(&) is the probability density function (pdf) of the random phase @n of the noisy input signal according to (6). p($,) may be represented [7] in integral form as marginal pdf of the envelope A (0 < A S w) and phase (-n < d n < 4

1 rm

f 1 1

Equation (15) with (16) has been solved elsewhere [3]. The result is known as signal suppression factor pv [8] and may be expressed in terms of modified Bessel functions I of order (v f 1)/2 P I

Here

is the input carrier-to-noise ratio. Now from (1 2)-(14) with the factors p,, in (1 7) the modi-

fied phase detector characteristic which enters the equivalent model may be expressed as in Table I . Numerical evaluation yields the graphs of gr(cp) in Fig. 3 and of gof, the slope of g’(q) at the origin cp = 0 in Fig. 4 as a function of CNR. Both the slope go’ and the maximum values ofg’(cp) are degraded if CNR decreases.

IV. THE MODIFIED PHASE NOISE

The modified phase noise n’(t) which characterizes the noise term at the phase detector output has variance anr2 and spectral bandwidth B1 in the baseband. As will be shown sub- sequently, B1’ is only slightly different from B1 and B1 is much larger than B L , the loop noise bandwidth. As a result, the phase noise nf( t ) is regarded as an approximately white process with constant (two-sided) spectral density

Therefore, Nor will be determined by calculating the power anf2 and the spectral bandwidth B1 from the definition given in (1 1).

A . Limiter PD with Sinusoidal Characteristic Starting with (1 1) the variance becomes

r+ n

with p(&) as in (16). Evaluation [6] of (20) yields (see Fig. 5)

The asymptotic values for very low and very high CNR are, respectively,

on’2 = L , 2

CNR < 1 (22)

The noise bandwidth B I r of the modified phase noise n’(t) is equal to the bandwidth of the noise term at the output of the bandpass-limiter preceding the multiplier [see Fig. l(b)] which has been determined in [ 8 ] . Referring to [8], the bandpass- limiter output noise, being the spectral component of the first zone [8] at the limiter output, has the bandwidth


TABLE I CHARACTERISTIC VALUES O F THE GENERAL EQUIVALENT PLL MODEL WITH DIFFERENTTYPES OF PHASE DETECTORS

CNR =

- 10 I 1.2

1.2 , I

1.0 T y p e I

0.6 -

I 0.0 1 0.1 1 10 -

CNR loo

Fig. 4. Slope go' of the modified phase detector characteristics for different types of phase detectors.

0.0 1 0 1 1 10 - CNR loo

Fig. 5. Variance un'2 of the modified phase noise for different types of phase detectors.

Equation (24) is an approximation in the sense that, although the asymptotic values

B, '=BI , C N R S 1 ( 2 5 )

and (for a rectangular shape of the input noise spectral density)

( c) Fig. 3. Modified phase detector characteristics for different values of

CNR and different limiter phase detectors. (a) Sinusoidal. (b) Tri- angular. (c) Sawtooth.

were calculated analytically, the exponential relationship was postulated on the basis of numerical evaluations. Experimental measurements of the noise bandwidth at the output of a lim-

ROSENKRANZ: PHASE-LOCKED LOOPS WITH LIMITER PHASE DETECTORS 2301

1.4 1 I

t 0.8

0.0 1 0.1 1 10 - 100 C N R

Fig. 6 . Bandwidth B1’ of the modified phase noise for different types of phase detectors.

iter PD, carried out by the author, were also in good agreement with (24) (see Fig. 6).

B. Limiter PD with Triangular Characteristic In order to calculate the variance of the modified phase

noise we first simplify the integration due to the fact that both the pdf p(@, ) and the square of the triangular characteristic g(*) are even functions:

rn

Substitution of the variable in the second term of the sum yields

Equation (28) may be evaluated with the pdf (16) as indicated in the Appendix. We find an infinite sum of expressions of the same form as in (17). Therefore, we write in terms of the signal suppression factors pz, 1 = 2v, v E N .

Equation (29) is plotted in Fig. 5 as a function of CNR. By considering the properties of the signal suppression factors the asymptotic values of unr2 become

The bandwidth B1 of the modified phase noise for a triang-

ular phase dgector may also be approximated for our pur- poses by(24)-(26) as in the sinusoidal case. Apart from the fact that an analytical evaluation does not seem possible, this approximation may be justified from the results of careful measurements of the noise bandwidth at the PD output (see Fig. 6). It is confirmed that B,’ does not alter much if CNR changes. This again justifies the approximation,

C. Limiter PD with Sawtooth Characteristic

In this case the variance is from ( 1 1)

which may be evaluated, as is shown in the Appendix, in terms of p u (v EN) as defined in (1 7) :

m 1

3 v-1.2 (33)

The asymptotic values become (see Fig. 5)

Again, the bandwidth B1 cannot be calculated analytically for arbitrary values of CNR. However, analytical methods may be used to determine the asymptotic values. The high CNR limit is as before B1 ’ = B , , which is easy to show. The maximum value of B1’ is achieved in the limit CNR -+ 0 and has been evaluated in [IO] using some properties of the narrow-band Gaussian input noise. For a rectangular spectral density of the input noise, the asymptotic value becomes

As in the other cases the bandwidth of the noise term is not affected much by the nonlinear phase detector operation. Therefore, it seems to be a good policy to approximate B1 ‘ by an analogous exponential relationship as before.

Measured results are in good agreement with this equation as is shown in Fig. 6.

The derived analytical expressions of the characteristic values of the equivalent PLL model in Fig. 2 are summarized in Table I and compared to the simple multiplier PD case. The diagrams of these values in Figs. 3-6 may be useful in applying the equivalent model as, for example, in calculating the phase error variance, which is shown in the next section.


V. EXAMPLE: PHASE ERROR VARIANCE OF THE FIRST-ORDER LOOP

The phase error variance of a first-order PLL (F(s) = 1) is calculated in order to demonstrate an application of the equivalent model in a simple but important example. In the case of a sinusoidal phase detector without limiter (i.e., multiplier), this problem has been solved using Fokker-Planck techniques [ I ] , [2] to account for the nonlinear behavior in the low CNR case, whereas, in most practical applications, useful approximate results are achieved by linearizing the PLL. Both techniques may be applied directly to the limiter phase detector case because of the common structure of the equivalent model.

Overall Linearization: Linearizing the loop with multiplier means neglecting the nonlinearity of the phase detector characteristic. The PLL (F(s) = 1) may then be regarded, in respect to its ability to restore the noisy input signal, as a linear filter with bandwidth BL = KO ‘ K D / ~ . In this case (sinusoidal PD without limiter and linearization sin cp: = cp) the variance up2 bf the phase error becomes independent of the PD characteristic

where

(39)

and l/a is a common abbreviation for the phase error variance as calculated from the approximation (38). For the multiplier phase detector (38) is a good approximation as long as u92 < 0.2, which means that CNR may be rather small as long as BL is substantially smaller than B , .

However, in the case of limiter phase detectors, the phase error variance is only equal to (38) (i.e., up2 = l/a) and, thus, independent of the specific nonlinear phase detector, for com- paratively high CNR, that is, if the asymptotic values (see Sec- tions I11 and IV)

g‘(lP) = cp >‘d cp (40)

are achieved. This is roughly true if CNR > 10 as is seen in Figs. 4-6. Therefore, the result up2 = l / a , although widely applicable for multiplier phase detectors, is far less useful in the limiter phase detector case. Here linearization of g‘(cp) yields comparable results.

Linearization of Modified Characteristic g’(cp): Linearizing the modified phase detector characteristic g’(cp) around the origin cp = 0 results in a linear equivalent model withg’(cp) re-

placed by g,’.cp. The closed-loop transfer function G(s) is

(43)

Thus, we find the phase error variance from linear systems analysis

with u as in (39) and gor, ont2 , and B1 ‘ from Table I or Figs. 4-6. With CNR = a-v as in (38) it is possible to depict graph- ically the variance as a function of l/a with u as a parameter. This was done in Fig. 7 for a typical value u = K o K D / ( ~ B ~ ) = 1/50. Now, even though the equivalent model is linearized with respect to its nonlinear element g‘(cp), the phase error variance depends on the specific characteristic of the limiter phase detector. From Fig. 7 it is evident that the variance may improve dramatically from that of the no limiter case. Consid- ering the PLL model, this may be explained by the degrada- tion of go’ with increasing noise, thereby decreasing the ef- fective loop noise bandwidth in an adaptive sense. Moreover, (44) is a good approximation as long as the phase error process does not substantially exceed the linear portion of g’(cp).

Fokker-Planck Techniques: If we want to take the nonlinear nature of the phase detector into full account we have to refer to the Fokker-Planck differential equation [2], [7]. Omitting the details, the Fokker-Planck equation, taking the nonlinear equivalent model of Fig. 2 as a basis, yields the pdf of the phase error which must be integrated numerically to achieve the variance. The results obtained by this technique are also depicted in Fig. 7, in order to compare them with the above approximations and with measurements. A description of the experimental setup is given in the Appendix. As can be seen, the experimental points are in good agreement with the nonlinear theoretical results.

The above example demonstrates one of the possible applications, and verifies the accuracy, of the derived PLL model. We recognize that as long as the phase error variance is small (i.e., up2 2 0.2) the model with the linearized modified phase detector characteristic i s applicable, whereas the result u92 = l/a of the overall linearization is restricted to rather high carrier-to-noise ratios (CNR > 10) if limiter phase detectors are applied.

VI. CONCLUSIONS

A nonlinear equivalent phase-locked loop model has been developed (Fig. 2) which describes loops with generalized phase detectors, including limiter phase detectors, in the presence of noise. The model is based on a representation of

ROSENKRANZ: PHASE-LOCKED LOOPS WITH LIMITER PHASE DETECTORS 2303

1.0

0.8

0.6

0.4

0.2

0

/ /

/'

/

/ /'

- F O K K E R - P L A N C K

LINEAR

0.2 0.4 0.6 0.8 10 (a)

- lla ./

F O K K E R - PLANCK LINEAR

0.4 - Y / _ _ _

,/'

0.6 - lla ./ - FOKKER- P L A N C K

/

0.4 - /' --- LINEAR

/ '

/ '

/.

0 0.2 0.4 0.6 0.8 1.0 ( C)

--- LINEAR E

( 4

0 0 0.2 0.4 0.6 0.8 1.0

Fig. 7. Phase error variance of first-order PLL calculated with different approximations, and measured points. u = B L / B ~ , = K O K D / ( ~ B I ) = 1/50. (a) Sinusoidal PD without limiter. (b) Sinusoidal, (c) triangular, (d) sawtooth limiter PD.

the phase detector by a modified phase detector characteristics g'(p) and a modified phase noise n'(t). All necessary data of the model are summarized in Table I or in the diagrams of Figs. 3-6, specifically for sinusoidal, triangular, and sawtooth characteristics.

The phase error variance of a first-order loop is calculated to provide an example of an application of the model. The results are in good agreement with experimental outcomes and show that a specific phase detector realization may strongly influence the noise performance.

Apart from the example demonstrated in this paper, other noise characteristics of phase-locked loops with limiter phase detectors may be obtained from the nonlinear model. Refer- ring to methods which were developed for the multiplier phase dete&r, higher order loops or the cycle-slipping phenomenon, for&!mple, have been studied [ 101 . Also, the acquisition behavior and the performance of the loop as an FM-receiver have been investigated in [ 101 with the aid of the described model.

APPENDIX

Variance of the Modified Phase Noise, Triangular, and Sawtooth Limiter PD

According to (28) and (32), integrals of the form

F= I_: 6' 4n2P(4n) d4n = 2 4n2P(4n> d4rI (-41)

are to be solved. With the pdf ~(4,) as in (16) and with the Anger-Jacobi formula [9]

m

e+-acosGn = ~ ~ ( a ) + 2 (*l>'Iu(a> COS ~4~ u= 1

(AI) takes on the following series representation:

Now the identities [9]

xe-ax21u(bx) dx

and

may be used to yield

The second term contains signal suppression factors as defined in (17). Therefore, for the sawtooth characteristic, the result in (33) is obtained directly.


WHITE 6 , ; NOISE 1 2 , S k H z m GEN.

r-----------i PLL,6 ,=KoK, /4 -256 H z

I I

I I L _ _ _ _ _ _ - _ _ - _ J

VOLTMETER RMS-

Fig. 8. Experimental setup for phase error measurement.

For a triangular phase detector the result (29) may be calculated from (28) in a similar way, bearing in mind the new limits of integration and the substitution in p(@,).

Experimental Setup The outline of the experimental setup for the measurement

of uv2 is shown in Fig. 8. Any one of the four phase detectors in Fig. l(a)-(d) was connected with a VCO to form a first- order PLL. Each PD includes an RC low-pass filter (PD-LP in Fig. 1) with a cutoff frequency sufficiently larger than the PLL bandwidth but small enough to suppress any significant high frequency disturbances. A voltage proportional to the phase error (mod 2n) was obtained by means of a highly linear phase detector (sawtooth). During the measurements all parameters were held constant; only the noise power was changed to yield different values of l / a which were computed from the measured CNR at the PLL input.

ACKNOWLEDGMENT

The author wishes to thank Prof. M. Brunk of the Lehrstuhl fur Nachr ich ten technik , Univers i ty of Erlangen-Nurnberg, who has encouraged this work through numerous discussions and valuable suggestions. He is also grateful to the Deutsche Forschungsgemeinschaft (DFG) for their support.

REFERENCES A. J . Viterbi. Principles of Coherent Communication. New York: McGraw-Hill, 1966. W. C. Lindsey, S-vnchronizution Systems in Communication and Control. Englewood Cliffs, NJ: Prentice-Hall, 1972. A. H. Pouzet, "Characteristics of phase detectors in presence of noise," in Proc. 8rh Int . Telem. Conf., Los Angeles, CA, 1972, pp. 818-828. F. H. Raab. "Square-wave correlation phase detector with VLF atmospheric noise," IEEE Trans. Aerosp. Electron. S w t . . vol.

B. N . Biswar et a l . . "Phase detector response to noisy and noisy fading signals," IEEE Trans. Aerosp. Electron. Syst.. vol. AES- 16. pp. 150-157, 1980. S . A. Butman and J . R. Lesh, "The effects of bandpass limiters on n-phase tracking systems," IEEE Trans. Commun., vol. COM-25, pp. 569-576. 1977. D. Middleton, An Introduction to Srutistical Communication Theory. New York: McGraw-Hill, 1960. J . C. Springett and M. K . Simon, "An analysis of the phase coherent-incoherent output of the bandpass limiter,'' IEEE Trans. Commun. Technol.. vol. COM-19. pp. 42-49. 1971. I . S . Gradshteyn and I . M. Ryzhik, Table oj'lntegruls. Series and Products. New York: Academic, 1965. W. Rosenkranz, "Ein allgemeines Ersatzmodell zur nichtlinearen Berechnung des Storverhaltens von Phasenregelkreisen ( A general equivalent model for the nonlinear calculation of the noise performance of phase locked loops)." Ausgewiihlte Arbeiten uber Nachrichrensysteme no. 44, issued by Prof. Dr.-lng. H. W. Schiissler, Erlangen, West Germany, 1980.

AES-15, pp. 726-732, 1979.

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