Digital communication receivers P2 - P2

Comparing the equivalent model with the synchronizer structure shown in Figure 2-18, we see that the equivalent model replaces the timing error detector output signal by the sum KDge + N

Y(b) Timing Error x(t) ) ,+)

Detector

4 r(t; 8 )

u(t)

Figure 2-18 Error-Tracking Synchronizer Structure

The timing error detector performs a suitable time-invariant nonlinear operation

on the input signal and the local reference, so that its output signal z(t) gives an indication of the instantaneous timing error e = & - g The timing error detector output signal enters a linear time-invariant loop filter Its frequency response in the Laplace domain is denoted by F(s) ; the frequency response in the frequency domain is F(w) The loop filter output signal u(t) determines the instantaneous frequency of the VCO according to (2-31) such that the timing error e is reduced

As the input signal y(t; E) has zero mean, it does not contain a deterministic periodic component, so that a conventional PLL cannot be used to estimate E However, there is still some periodicity embedded in the input signal, because

of its cyclostationarity with period T This cyclostationarity is exploited by the synchronizer in order to make an estimate of E

There exists a large variety of timing error detector circuits; some examples are considered in Section 2.2.3 In spite of the many different types of timing error detector circuits encountered in practice, it will be shown that all error- tracking synchronizers can be represented by the same equivalent model, shown

in Figure 2- 19, so that they can be analyzed in a unified way Comparing the equivalent model with the synchronizer structure shown in Figure 2-18, we see that the equivalent model replaces the timing error detector output signal by the sum KDg(e) + N(t), where g(e) and N(t) are called the timing error detector characteristic and the loop noise, respectively The useful component KDg(e)

of the timing error detector output signal is periodic in e with period 1; also, it will be shown in Section 2.3.3 that g(0) = 0, and the timing error detector gain I(0 is usually chosen such that g’(O), the timing error detector slope at the origin,

is normalized to 1 The loop noise N(t) represents the statistical fluctuations

of the timing error detector output signal These fluctuations are caused by the additive noise n(t) and the random nature of the channel symbols {a,} in the input signal y(t; E) The loop noise is a zero-mean wide-sense stationary process whose power spectral density S(w ; e) is periodic in the timing error e with period

1 The equivalent model and the actual synchronizer structure have the same loop filter The remaining part of the equivalent model corresponds to (231), with AF representing the frequency detuning of the VCO:

(2-34)

1 N(t)

Figure 2-19 Equivalent Model of Error-Tracking Synchronizer

KO /S is the Laplace domain frequency response of an integrator with gain h’o g The equivalent model is a powerful means for studying error-tracking syn- chronizers Indeed, the results obtained from investigating the equivalent model are valid for the whole class of error-tracking synchronizers In order to relate

a specific synchronizer to the equivalent model, one must be able to derive both the useful component Keg(e) of the timing error detector output and the loop noise spectrum S(w; e) from the description of the synchronizer In the following, these quantities will be obtained for the general synchronizer structure shown in Figure 2- 18

Actually, Ilog and S(w ; e) will be derived under “open-loop” conditions This involves opening the feedback loop (say, at the input of the loop filter) and applying a periodic reference signal ~(t; e) with period T and constant value

of Ệ From the resulting timing error detector output signal under open-loop conditions, the useful component Keg(e) and the loop noise spectrum S(w; e) will be determined These quantities are assumed to remain the same when the loop is closed This approximation is valid when the loop bandwidth (which will

be defined in Section 2.3.4) is small with respect to the channel symbol ratẹ

2.3.2 Open-Loop Statistics of the Timing Error Detector

Output Signal

In order to determine statistical properties of the timing error detector output signal under open-loop conditions, we consider an input signal y(t; 6) given by (2-29) with a constant value of E, and a periodic reference signal ~(t; g) with period T and a constant value of 6 The timing error detector output signal is

denoted by 2 (t ; E, 2) in order to indicate explicitly the dependence on the constant parameters E and 2 The case where both E and i are fluctuating in time will be briefly considered also

Taking into account the cyclostationarity of y(t; E) with period T, the periodic- ity of r(t; 2) with period T, and the time-invariant (but not necessarily memoryless) nature of the timing error detector circuit, it follows from Section 1 1.2 that the timing error detector output signal under open-loop conditions is also cyclostation- ary with period T In all cases of practical interest, the synchronizer bandwidth (to be defined in Section 2.3.4) is a small fraction (about one percent or even much less) of the channel symbol rate l/T, so that the synchronizer operation

in closed loop is determined entirely by the lowpass content of the timing error detector output signal Taking into account that the statistics of the timing error detector output signal in closed loop are essentially the same as in open loop, we can replace z(t; e, e) in open loop by its lowpass content ~~p(t; E, Z), without af- fecting the operation of the synchronizer when the loop will be closed The signal ZLP(~;E, e) is defined as

+oO

ZLp (t ; e, t> =

J hLp(t - u) i?$; e, i) du -00

(2-35)

where hop is the impulse response of an ideal lowpass filter with frequency response HLP(w), given by

1 I4 < 4T

0 I4 > r/T

It has been shown in Section 1.1.2 that passing a cyclostationary process through

a lowpass filter with a bandwidth (in hertz) not exceeding 1/(2T) yields a wide-sense stationary process Hence, the lowpass content ZLP(~; &,Z) of the timing error detector output signal is wide-sense stationary This implies that the statistical expectation E[zLP(~; E, e)] and the statistical autocorrelation function E[zLP(~; E, e) ZLP(~ + u; E, e)] do not depend on t As ET ahd ET are time delays, it is easily verified that the joint statistical properties of y(t; e) and r(t; 8) are the same as for y(t - ST; E - 2) and ~(t - tT; 0) Hence, XL& E, 2) and

XLP@ - tT; e - 6,O) have identical statistical properties

The lowpass content ~,p(t; E, t) can be uniquely decomposed into a useful component E[Q,P(~; E, e)] and a zero-mean disturbance N(t; E, i) called the loop noise:

ZLP(t; e, e) = E[ZLP(t; e, EI)] + N(t; e, e) (2-37)

As z&t; E, 2) and XL& - EIT; E - g, 0) have the same statistical properties and are both wide-sense stationary, one obtains

E[z~~(~;E,~)] = E[zLP(~ - C!-';E - O)]

= E[zLp(O; E - i, O)]

= ICD g(E - i)

(2-38)

which indicates that the useful component at the timing error detector output depends on the delays ET and EIT only through the timing error e = E - 2 Similarly, it follows that

E[N(t; E, EI) N(t + u; E, i)] = E[N(t - EIT; E - f, 0) N(t + u - iT; E - i, 0)]

= E[N(O; & - i, 0) N(u; E - i, O)]

= &(u;& - e> (2-39)

so that also the autocorrelation function of the loop noise depends only on the timing error As the reference waveform r(t; e) is periodic in t with period T, it satisfies r(t;e) = r(t;b + 1) so that z~p(t;&,C) = z~p(t;~,i + 1) Hence, the useful component Keg(e) at the timing error detector output and the autocorre- lation function RN (u; e) of the loop noise are both periodic in the timing error e with period 1 The loop noise spectrum SN (w ; e) is the Fourier transform of the autocorrelation function RN (u; e)

The useful component KDg( e) and the loop noise spectrum SN (w ; e) can easily be derived from the statistical expectation and the autocorrelation function

of the timing error detector output signal s(t; 5, e), because z~p(t; E, e) results from a linear filtering operation on z(t; e, i) Taking into account (l-39) and (l-40), one immediately obtains

&v(u; E - El) = [(&(f, t + u; E, e)),lLp (2-41) where ( .) t denotes time-averaging with respect to the variable t , [ .lLp indicates that the quantity between brackets is passed through the ideal lowpass filter, whose frequency response H&w) is given by (2-36), and KZ (t , t + u; &, E”) denotes the autocovariance function of z(t; &, i):

I<&, t + u; E, q = E[z(t; &, t> z(t + u; E, e)] - E[z(t; E, i)] E[z(t + u; E, i)]

(2-42) Equivalently, I<, (t , t + u; &, e) is the autocorrelation function of the cyclostation- ary disturbance z(t; &, 2) - E[z(t; E, g)] at the timing error detector output The important conclusion is that, as far as the operation of the synchronizer

is concerned, the timing error detector output under open-loop conditions can be represented as the sum of:

A useful component Kog( e), which is a periodic function of the timing error

e and

The zero-mean loop noise N(t), whose spectrum S(w ; e) is a periodic function

of the timing error e [for notational convenience, N(t; &, 2) will be denoted

by N(t) from now on]

In the case where both & and i are fluctuating in time, the above representation

of the timing error detector output signal under open-loop conditions is still valid, provided that & and i are so slowly varying that they are essentially constant over

the correlation time of the loop noise (which is in the order of the channel symbol interval T) and over the possibly nonzero memory time of the timing error detector circuit (which is, at most, also in the order of the channel symbol interval T)

2.3.3 The Equivalent Model and Its Implications

Assuming that the loop bandwidth of the synchronizer (which will be defined

in Section 2.3.4) is much smaller than the channel symbol rate l/T, the statistical properties of the timing error detector output signal under open-loop conditions remain valid when the loop is closed This yields the equivalent model shown in Figure 2-19 Note that this equivalent model is the same as for a conventional PLL operating on a sinusoid disturbed by additive noise (see Chapter 3 of Volume 1); hence, an error tracking clock synchronizer and a conventional PLL have very similar properties

In the absence of loop noise N(t) and for zero frequency detuning AF, the stable equilibrium points of the equivalent model are determined by the positive going zero crossings of g(e), assuming a positive open-loop gain Because of the periodicity of g(e), there exists an infinite number of stable equilibrium points By introducing a suitable delay in the definition of the local reference r(t; Z), the stable equilibrium points can be made coincident with e = - 2, - 1 , 0, 1,2, Usually, the constant KD is determined such that the slope g’(0) at the stable equilibrium point is normalized to 1 The negative zero crossings of g(e) correspond to the unstable equilibrium points of the synchronizer

In the absence of loop noise and for nonzero frequency detuning, the stable and unstable equilibrium points correspond to the positive and negative zero crossings, respectively, of g(e) - y, where y = AF/(K&CD.F(O)) is the normalized frequency detuning Hence, frequency detuning gives rise to a steady-state error, unless the DC gain F(0) of the loop filter becomes infinite This can be achieved

by using a perfect integrator in the loop filter

We now show that, in the presence of loop noise, the stable equilibrium points coincide no longer with those for zero loop noise, because the loop noise spectrum SN (w ; e) depends on the timing error e This means that the loop noise contributes to the steady-state error; this phenomenon is called noise-induced drift Let us assume that e = e, is a stable equilibrium point in the presence of loop noise, and that the timing error e( t ) takes on the value e, at t = 0 For t > 0, the timing error exhibits random fluctuations due to the loop noise N(t) On the average, the rate of change Ae/At = (e(At) - e(O))/At over a small time increment must be zero, because e(0) = e, is a stable equilibrium point Indeed, a nonzero average rate of change would imply that some deterministic force pushes e(t) away from its initial position e,, which contradicts our assumption that e, is

a stable equilibrium point Hence, e, is such that

which expresses that, at a stable equilibrium point, the input signal and the

reference signal have the same average frequency The left-hand side of the above equation is the definition of the intensity coefJicient K1 (e), evaluated at the stable equilibrium point (see Volume 1, Section 9.3.1) For a first-order loop [i.e., F(s) = 11, the timing error satisfies the system equation

de z= -Ko Km(e) + AF - KoN(f) (2-44) When the synchronizer bandwidth becomes much smaller than the bandwidth of the loop noise, e(t) converges to the solution of the stochastic differential equation

de = do &g(e) + AF + 1/4K(? d&v(o; e)

de 1 dt + KO SN(O; e) dW G-45) where W(t) is a Wiener process with unit variance parameter, SN(O; e) is the loop noise power spectral density at w = 0, and the term involving the derivative of SN(O; e) is the It6 correction term (see Volume 1, Section 9.3.2) For the above stochastic differential equation, the intensity coefficient Ki (e) is given by

k(e) = -Ko Keg(e) -t AF + 1/4K,2 d&(0; e) de (2-46) The stable equilibrium points coincide with the negative zero crossings of 1C1 (e) When the loop noise spectrum SN(W; e) does not depend on e, the It6 correction term is identically zero, and the stable equilibrium points are the same as for zero loop noise When the loop noise spectrum does depend on the timing error, the stable equilibrium points in the presence of noise are displaced in the direction

of the increasing loop noise spectral density This situation is depicted in Figure 2-20, where el and e, denote the stable equilibrium points in the absence and

dS(a = 0; e)

de

e

Figure 2-20 Illustration of Noise-Induced Drift

presence of loop noise, respectively Also shown is a sketch of the probability density function of the timing error e in the presence of loop noise As there

is more noise in the region e > ei than in the region e < el , the timing error spends more time in the region e > el This explains the strong asymmetry of the probability density function, which is decidedly non-Gaussian In a similar way, it can be shown that a timing-error-dependent loop noise spectrum SN(W; e) affects also the position of the unstable equilibrium points, which coincide with the positive zero crossings of the intensity coefficient I<1 (e)

Most of the time, the timing error e(t) fluctuates in the close vicinity of a stable equilibrium point These small fluctuations, caused by the loop noise, are called jitter This mode of operation can be described by means of the linearized equivalent model, to be considered in Section 2.3.4 Occasionally, the random fluctuations of the error e(t) are so large that the error moves into the domain

of attraction of a neighboring equilibrium point: a cycle slip occurs Cycle slips are nonlinear phenomena and, hence, cannot be investigated by means of a linear model An in-depth treatment of cycle slips is provided in Volume 1, Chapters

6 and 11

Let us consider the case where the initial timing error e(0) is very close to an unstable equilibrium point e, , located between the stable equilibrium points e, and

e, + 1 For the sake of simplicity, let us assume that the synchronizer is a first-order loop, so that the timing error satisfies the stochastic differential equation (2-45) Note that e, corresponds to a positive zero crossing of the intensity coefficient Ii1 (e), given by (2-46) This is illustrated in Figure 2-21, When e(0) is slightly smaller than e,, the intensity coefficient Ki (e) is negative, so that the drift term

Ki (e) dt in (2-45) has the tendency to drive the timing error toward the stable equilibrium point e, However, in the vicinity of the unstable equilibrium point

e, , the intensity coefficient I<1 (e) is very small This can cause the timing error e(t) to spend a prolonged time in the vicinity of the unstable equilibrium point, before it finally reaches the neighborhood of the stable equilibrium point e, , This phenomenon is called hung-up The occurrence of hang-ups gives rise to long acquisition times Acquisition is investigated in Volume 1, Chapters 4 and 5

K,(e)

-

Figure 2-21 Intensity Coefficient Kl( e)

It has been shown in Volume 1, Section 3.2, that for some sequential logic timing error detectors, which make use only of the zero crossings of the input signal and the reference signal, the timing error detector characteristic has a sawtooth shape At the unstable equilibrium points, the sawtooth characteristic jumps from its largest postive value to the opposite value, so that the restoring force near an unstable equilibrium point cannot be considered small Hence, one might wonder whether hang-ups can occur in this case The answer is that hang-ups are possible when additive noise is present at the input of the timing error detector Indeed, Figure 3.2-5 in Volume 1 shows that the timing error detector characteristic has

a sawtooth shape only when additive noise is absent In the presence of additive noise, the slope of the characteristic at the unstable equilibrium point is finite (and decreases with increasing noise level); hence, the restoring force near the unstable equilibrium point is very small, so that hang-ups can occur

2.3.4 The Linearized Equivalent Model

Under normal operating conditions, the timing error e(t), for most of the time, exhibits small fluctuations about a stable equilibrium point e, Suitable measures of performance for this mode of operation are the steady-state error e, and the variance of the timing error It is most convenient to analyze these small fluctuations by linearizing the equivalent model about the stable equilibrium point and applying standard linear filter theory

The nonlinear nature of the equivalent model is caused by:

The nonlinear timing error detector characteristic g(e)

The dependence of the loop noise spectrum SN(W; e) on the timing error e Linearization of the equivalent model involves the following approximations:

s(e) = s(eS) + (e - e,) g’(h) (2-47)

where g’(e$) is the timing error detector slope at the stable equilibrium point for a small steady-state error e, and g’(e,) is well approximated by g’(0) = 1 This yields the linearized equivalent model, shown in Figure 2-22 The steady-state

Figure 2-22 Linearized Equivalent Model

error e, to be used in the linearized equivalent model is the steady-state error for zero loop noise, i.e., a positive zero crossing of g(e) - AF/(K~KD F(0)) This is motivated by the fact that, for a low loop noise level and/or a small synchronizer bandwidth, the Ito correction term has only a minor influence on the position of the stable equilibrium point (see Figure 2-20)

The linearized equivalent model can be transformed into the model shown in Figure 2-23, where H(s), given by

KAx7’(eJ P(s)

which is called the closed-loop frequency response (in the Laplace domain) of the synchronizer Note that H(s) is the frequency response of a lowpass filter with unit gain at DC; consequently, 1 - H(s) is the frequency response of a highpass filter

It is evident from Figure 2-23 that the timing error e(t) consists of three terms: The steady-state error e,

The highpass content of &(t) This reflects that the synchronizer can track with a negligible error only a slowly fluctuating delay ET

l The lowpass content of the disturbance N(t)/Kog’(e,)

An important parameter is the one-sided loop bandwidth BL (in hertz) of the synchronizer, which is a measure of the bandwidth of the closed-loop frequency response:

In many cases of practical interest, e(t) is fluctuating very slowly with respect

to the channel symbol rate l/T Therefore, it is possible to use a small loop

Figure 2-23 Contributions to the Timing Error e(t)

bandwidth for reducing the effect of loop noise on the timing error e(t), while still having a negligible contribution to the timing error from the fluctuations of e(t) Typical values of BLT are in the order of one percent, and, in some cases, even much smaller

2.3.5 The Linearized Timing Error Variance

Assuming that I is essentially constant over many channel symbol intervals,

so that its contribution to the timing error is negligible, and taking into account the wide-sense stationarity of the loop noise N(t), application of the linearized equivalent model yields a wide-sense stationary timing error e(t) Its statistical expectation and variance do not depend on time, and are given by

+oO

1

vaNt)l = [KDg,(eb)12 J lw412 sw3> ijy dw

oo

(2-52)

Because of the small loop bandwidth BL, we expect that the timing error variance

is determined mainly by the behavior of the loop noise spectrum SN(W; e,) in the vicinity of w = 0 When the variation of the loop noise spectrum within the loop bandwidth is small, i.e.,

(2-53) the loop noise spectrum is well approximated by its value at w = 0, as far as the timing error variance eq (2-52) is concerned Taking into account the definition

eq (2-50) of the loop bandwidth, this approximation yields

var[e(t)] 2 (SBLT) ~K(o~~){l~

D e3

(2-54)

However, for many well-designed synchronizers the loop noise spectrum 5’~ (w ; e, ) reaches its minimum value at w = 0; this minimum value can be so small (and in some cases even be equal to zero) that the variations of the loop noise spectrum within the loop bandwidth cannot be ignored, and the approximation eq (2-54)

is no longer accurate

When the variation of SN(W; e,) within the loop bandwidth cannot be ne- glected, we use the Parseval relation to obtain

+oO

Iw4l” SN(W3) - = 27r J hz(f) RN(~) dt (2-55)

where RN(t; e,) is the autocorrelation function of the loop noise [i.e., the inverse Fourier transform of SN(W; e,)], and hs(t) is the inverse Fourier transform of ]H(w)j2 Denoting the inverse Fourier transform of H(w) by h(t), it follows that hz(t) is the autocorrelation function of the pulse h(t) As h(t) is causal [i.e.,