IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-28, NO.
`
`8, AUGUST 1980
`
`1107
`
`Carrier and Bit Synchronization in Data Communication-
`A Tutorial Review
`
`L. E. FRANKS, FELLOW. IEEE
`
`the problems of carrier phase
`paper examines
`Absrmcr-This
`estimation and symbol timing estimation for carrier-type synchronous
`digital data signals, with tutorial objectives foremost. Carrier phase
`recovery for suppressed-carrier versions of double
`sideband (DSB),
`(VSB), and quadrature amplitude
`vestigial sideband
`modulation
`(QAM) signal formats is considered first. Then the problem of symbol
`a baseband pulse-amplitude modulation (PAM)
`timing recovery for
`signal is examined. Timing recovery circuits
`based on elementary
`statistical properties are discussed as well as timing recovery based on
`maximum-likelihood estimation theory. A relatively simple approach
`to evaluation of timing recovery circuit performance in terms of rms
`jitter of the timing parameters is presented.
`
`I
`
`I. INTRODUCTION
`N digital data communication there is a hierarchy of syn-
`chronization problems to be considered. First, assuming
`that a carrier-type system is involved, there is the problem of
`carrier synchronization which concerns the generation of a
`reference carrier with a phase closely matching that of the data
`signal. This reference carrier is used at the data receiver to per-
`form a coherent demodulation operation, creating a baseband
`data signal. Next comes the problem of synchronizing a receiver
`clock with the baseband data-symbol sequence. This is com-
`monly called bit synchronization, even when the symbol alpha-
`bet happens not to be binary.
`Depending on the type
`of system under consideration,
`problems of word-, frame-, and packet-synchronization will be
`encountered further down the hierarchy. A feature that distin-
`guishes the latter problems from those of carrier and bit syn-
`chronization is that they are usually solved by means of special
`design of the message format, involving the repetitive insertion
`of bits or words into the data sequence solely for synchroniza-
`tion purposes. On the other hand, it
`is desirable that carrier
`and bit synchronization be effected without multiplexing spe-
`cial timing signals onto the data signal, which would use up a
`portion of the available channel capacity. Only timing recov-
`ery problems of this
`type are discussed in this paper. This
`excludes those systems wherein the transmitted signal contains
`an unmodulated component of sinusoidal carrier (such as with
`“on-off’ keying). When an unmodulated component or pilot
`is present, the standard approach to carrier synchronization is
`to use a phase-locked loop (PLL) which locks onto the carrier
`component, and has a narrow enough loop bandwidth so as
`not to be excessively perturbed by the sideband components
`of the signal. There is a vast literature on the performance and
`
`design of the PLL and there are several textbooks dealing with
`synchronous communication systems which treat the PLL in
`great detail [l] -[SI. Although we consider only suppressed-
`carrier signal formats here, the PLL material is still relevant
`since these devices are often used as component parts of the
`overall phase recovery system.
`For modulation formats which exhibit a high bandwidth
`efficiency, i.e., which have a large “bits per cycle” figure of
`merit, we find the accuracy requirements on carrier and bit
`synchronization increasingly severe. Unfortunately, it is also
`in these high-efficiency systems that we find it most difficult
`to extract accurate carrier phase and symbol timing informa-
`tion by means of simple operations performed on the received
`signal. The pressure to develop higher efficiency data transmis-
`sion has led to a dramatically increased interest in timing recov-
`ery problems and, in particular, in the ultimate performance
`that can be achieved with optimal recovery schemes.
`We begin our review of carrier synchronization problems
`with a brief discussion of the major types of modulation for-
`mat. In each case (DSB, VSB, or QAM), we assume coherent
`demodulation whereby the received signal is multiplied by a
`locally generated reference carrier and the product is passed
`through a low-pass filter. We can get some idea of the phase
`accuracy, or degree of coherency, requirements for the various
`modulation formats
`by examining the expressions for the
`coherent detector output, assuming a noise-free input. Let us
`assume that the message signal, say, a(t), is incorporated by
`the modulation scheme into the complex envelope @(t) of the
`carrier signal.’
`v(t) = Re [Nt> exp (io) exp 0’2rf0t)l
`and the reference carrier r(t) is characterized by a constant
`complex envelope
`r(t) = Re [exp 0;) exp (j2rfot)] .
`
`(2)
`
`(1)
`
`From (A-8), the output of the coherent detector is
`z1 (f) = 3 Re [p(t) exp (je - ji)] .
`For the case of DSB modulation, we have @(t) = a(t) 4-
`je, sf z l ( t ) i s simply proportional to a(t). The phase error
`8 - 8 in the reference carrier has only a second-order effect
`
`(3)
`
`Manuscript received June 28, 1979;revised March 26, 1980.
`of Electrical and Computer
`The author
`is with the Department
`Engineering, University of Massachusetts, Amherst, MA 01003.
`0090-6778/80/0800-1107 $00.75 0 1980 IEEE
`
`for definitions and basic relations concerning
`See the Appendix
`complex envelope representation of signals.
`
`Dish
`Exhibit 1018, Page 1
`
`

`
`1108
`
`IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-28, NO. 8, AUGUST 1980
`
`(4)
`
`on detector performance. The only loss is that phase error
`to cos’ (6 - b), insignal-
`causes a reduction, proportional
`to-noise ratio at the detector output when additive noise is
`present on the received signal.
`more severe distortion. In this case P(t) = a(t) + jZ(t), where
`For VSB modulation, however, phase error produces a
`Z(t) is related to a(t) by a time-invariant fdtering operation
`which causes a cancellation of a major portion of one of the
`sidebands. In the limiting case of complete cancellation of a
`sideband (SSB), we have Z(t) = ;(t), the Hilbert transform of
`a(t) [ 6 ] . The coherent detector output (3) for the VSB signal
`is
`z1 (t) = 3 a(t> COS (6 - 4) - 3 y t ) sin (6 - 6)
`(4) introduces an interference called
`and the second term in
`quadrature distortion when # 6. As Z(t) has roughly the
`same power level as a(t), a relatively small phase error must be
`maintained for low distortion,
`e.g., about 0.032 radian error
`for a 30 dB signal-to-distortion ratio.
`In the QAM case, two superimposed DSB signals at the
`same carrier frequency are employed by making P(t) = a(t) +
`jb(t), where a(t) and b(t) are two separate, possibly independ-
`ent, message signals. A dual coherent detector, using a refer-
`n/2 phase-shifted version, separates
`the
`ence carrier and its
`received signal into its. in-phase (Z) and quadrature (Q) com-
`ponents. Again considering only the noise-free case, these com-
`ponents are
`cr(t) = 3 a(t) cos (6 - 6) - 3 b(t) sin (6 - 6)
`cQ(t) = 3 b(t) cos (6 - 4) + 4 a(t) sin (6 - 6).
`From (5) it is clear that t$ # 6 introduces a crosstalk interfer-
`ence into the I and Q channels. As a(t) and b(t) can be expec-
`ted to be at similar power levels, the phase accuracy require-
`ments for QAM are high compared
`to straight DSB modula-
`tion.
`From the previous discussion we see that the price for the
`approximate doubling of bandwidth efficiency in
`VSB or
`a greatly increased sensitivity
`QAM, relative to DSB, is
`to
`phase error. The problem is compounded by the fact that car-
`rier phase recovery is much more difficult for VSB and QAM,
`compared to DSB.
`
`( 5 )
`
`11. CARRIER PHASE RECOVERY
`
`recovery circuits for the
`Before examining specific carrier
`suppressed-carrier format, it is helpful to ask, “What proper-
`ties must the carrier signal y(t) possess in order that operations
`on y(t) will produce a good estimate of the phase parameter
`e?” A general answer to this question lies in the cyclostation-
`ary nature of the y(t) process.’ A cyclostationary process has
`statistical moments which are periodic
`in time, rather than
`constant as in the case of stationary processes [2], [ 6 ] , [71.
`TO a large extent, synchronization capability can be character-
`
`ized by the lowest-order moments of the process, such as the
`mean and autocorrelation. The y(t) process is said to be cycle-
`stationary in the wide sense if Eb(t)J and kyv(t + 7 , t ) =
`E b ( t + T)Y(t)] are both periodic functions of
`t. A process
`modeled by (1) is typically cyclostationary with a period
`of
`l/fo or 1/2f0. The statistical moments of this process depend
`upon the value of the phase parameter 6 and it is not surpris-
`ing that efficient phase estimation procedures are similar to
`moment estimation procedures. It
`is important to note here
`that we are regarding 6 as an unknown but nonrandom param-
`eter. If instead we regarded 6 as a random parameter uni-
`formly distributed over a 2n interval, then the y(t) process
`would typically be stationary, not cyclostationary.
`A general property of cyclostationary
`processes is that
`there may be a correlation between components in different
`in contrast to the situation for stationary
`frequency bands,
`processes [8] . For carrier-type signals, the significance lies in
`the correlation between message components centered around
`the carrier frequency (+fo) and the image components around
`(-fo). This correlation is characterized by the cross-correlation
`function. kpp*(7) = E[P(t + 7)P(t)] for a y(t) process as in (1)
`when p(t) is a stationary p r o ~ e s s . ~
`Considering first the DSB case with P(t) = a(t) + j 6 , and
`using (A-10) we have
`krr(t + 7 , t ) = 3 Re [ k , , ( ~ ) exp O’277f0dl
`+ 3 Re [kaa(7) exp (j477fot + j2nh7 +j26)]
`(6)
`where the second term in (6) exhibits the periodicity in t that
`makes y(t) a cyclostationary process.
`We are assuming that y(t) contains no periodic components.
`Consider what happens, however, when y(t) is passed through
`a square-law device. We see immediately from (6) that the out-
`put of the squarer has a periodic mean value, since
`
`E[Y2(t)l = k,,(t, t>
`= 3 kau(0) + kaa(0) Re [exp (j26+j4nfor)J. (7)
`If the squarer output ‘is passed through a bandpass fdter with
`transfer function H(f) as shown in Fig. 1, and if H(f) has a uni-
`ty-gain passband in the vicinity of f = 2f0, then the mean
`value of the filter output is a sinusoid with frequency 2f0,
`phase 26, and amplitude + E [ a 2 ( t ) ] . In thissense, the squarer
`component from the y(t) signal.
`has produced a periodic
`It is often stated that the effect of the squarer is to produce
`a discrete component (a line at 2f0) in the spectrum of its out-
`put signal. This statement lacks precision and can lead to seri-
`ous misinterpretations because y’(t) is not a stationary process,
`so the usual spectral density concept has no meaning.
`A
`stationary process can be dpived fromy2(t) by phase random-
`izing [ 6 ] , but then the relevance to carrier phase recovery is
`lost because the discrete component has a completely indeter-
`minate phase.
`
`2 In [ 21, these processes are called periodic nonstutionary.
`
`3 Despite its appearance, this is not an autocorrelation function, due
`to the definition of autocorrelation for complex processes; see (A-1 1).
`
`Dish
`Exhibit 1018, Page 2
`
`

`
`FRANKS: CARRIER AND BIT SYNCHRONIZATION
`
`BpF(2fo) Timing wove
`
`H(f)
`
`Fig. 1. Timing recovery circuit.
`
`The output of the bandpass filter in Fig. 1 can be used direc-
`tly to generate a reference carrier. Assuming that Hcf) com-
`pletely suppresses the low-frequency terms [see (A-8)] the fil-
`ter output is the reference waveform
`
`w(t) = 4 Re { [ w B p2 ] (t) exp (i2e) exp 0'47rfo t))
`(8)
`where the convolution product [w B p2] represents the filter-
`ing action of H(J) in terms of its
`low-pass equivalent
`in
`(A-5). For the DSB case, p2(t) = a2(t) is real and o(t) is real4
`if Zlcf) has a symmetric response about 2f0. Then the phase of
`the reference waveform is 20 and the amplitude of the refer-
`ence waveform fluctuates slowly [depending on the bandwidth
`of H(J)]. The reference carrier can be obtained by passing w(t)
`through an infinite-gain clipper which removes the amplitude
`fluctuations. The square wave from the clipper can drive a fre-
`quency divider circuit which halves the frequency and phase.
`Alternatively, the bandpass filter output can be tracked by a
`PLL and the PLL oscillator output passed through the fre-
`quency-divider circuit.
`called the
`There is another tracking loop arrangement,
`(VCO)
`Costas loop, where the voltage-controlled oscillator
`operates directly at f o . We digress momentarily to describe the
`Costas loop and to point out that it is equivalent to the squarer
`followed by a PLL [ l ] -[3]. The equivalence is established by
`noting that the inputs to the loop filters in the two configura-
`tions shown in Fig. 2 are identical. In the PLL quiescent lock
`condition, the VCO output is in quadrature with the input sig-
`nal so we introduce a n/2 phase shift into the VCO in the con-
`figurations of Fig. 2.. Then using (A-8) to get the output of the
`multiplier/low-pass filter combinations, we see that the input
`to the loop filter is
`
`(9)
`
`V(t) = + Re [A2p2(t) exp 0'26 - j28 -jr/2)]
`in both configurations if the amplitude of the VCO output is
`taken as f A'
`in the squarer/PLL configuration, and taken as
`A in the Costas loop.
`Going back to (8), we see that phase recovery is perfect if
`[w 0 p2] is real. Assuming w(t) real, a phase error will result
`only if a quadrature component [relative to p2 (t)] appears at
`the output of the
`squarer. This points out the error, from a
`different viewpoint, of using the phase randomized spectrum
`of the squarer output to analyze the phase recovery perform-
`ance because the spectrum approach obliterates the distinction
`between I and Q components. For the DSB case, a quadrature
`component will appear at the squarer output only if there is a
`quadrature component of interference added to the input sig-
`nal y(t). We can demonstrate this effect
`by considering the
`
`4 A real w(t) corresponds to the case where the cross-coupling paths
`between input and output [and Q components in Fig. 10 are absent. If
`the bandpass function H(n does not exhibit the symmetrical amplitude
`response and antisymmetrical phase response about 2fo for a real w(t),
`then there simply is a fwed phase offset introduced by the
`bandpass
`filter.
`
`Squarer
`
`PLL
`
`1109
`
`I
`
`(a)
`
`(b)
`Fig. 2. Carrier phase tracking loops. (a) SquarerlPLL (b) Costas loop.
`input signal to be z(t) = y(t) + n(t) where n(t) is white noise
`with a double-sided spectral density of No W/Hz. We can rep-
`resent n(t) by the complex envelope, [uI(t) + juQ(t)] exp 0'6)
`where, from (A-1 5) the I and Q noise components relative to a
`phase 6 are uncorrelated and have a spectral density of u V o .
`The resulting phase of the reference waveform (8) is
`
`(1 la)
`
`We can approximate the phase error @I = 0 - 0 (also called
`phase jitter because 8 is a quantity that fluctuates with time)
`by neglecting the noise X noise term in the numerator and
`both signal X noise and noise X noise terms in the denomina-
`tor in (10). Furthermore, we replace w C3 p2 by its expected
`value (averaging over the message process) and use the tan-'
`x % x approximation. With all these simplifications, which are
`valid at sufficiently high signal-to-noise ratio and with suffici-
`ently narrow-band H O , it is easy to .derive an expression for
`the variance of the phase jitter.
`var @I = (2NoB)S- '
`(5)-' ('w )
`
`4
`
`=
`
`where
`
`00
`
`m
`
`B G ~ w l n l t ) 1 2 d f =
`
`I H ( n I 2 d f
`
`is the noise bandwidth of the bandpass filter, recalling that we
`have set n(0) = 1. The message signal power is S = E[a2 (t)]
`and for the second version of the jitter formula (1 1 b) we have
`assumed a signal bandwidth of W Hz and have defined a noise
`power over this band of N = 2N0 W. This allows the satisfying
`physical interpretation of jitter variance being inversely pro-
`portional to signal-to-noise ratio and directly proportional
`to
`the bandwidth ratio of the phase recovery circuit and the mes-
`sage signal. For the smaller signal-to-noise ratios, the accuracy
`and convenience of the expression can be maintained by incor-
`porating a correction factor known as the squaring loss [3] .
`
`Dish
`Exhibit 1018, Page 3
`
`

`
`1110
`
`IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-28, NO. 8, AUGUST 1980
`
`When the signal itself carries a significant quadrature com-
`ponent, as in the case of the VSB signal, there will be a quad-
`rature component at the
`squarer output that interferes with
`the phase recovery operation even at high signal-to-noise ratios.
`h t us suppose that the VSB signal is obtained by filtering a
`DSB signal with a bandpass filter with a real transfer function
`of fo. The
`(no phase shift) and with a cutoff in the vicinity
`resulting quadrature component for the VSB signal is Z(t) =
`[PQ @ a] (t) and p ~ ( t )
`is derived from the low-pass equivalent
`transfer function for
`the bandpass filter in accordance with
`(A-7). The real transfer function condition makes pQ(t) an
`odd function of time, which also makes the cross-correlation
`function for a(t) and Z(t) an odd function.
`is that, for P(t) = a(t) + lZ(t), the autocorre-
`The result
`lation for the VSB signal is
`kyy(t + T, t ) = a Re [{k,,(~) + kzz(r) +j2ksiiT(~)}
`exp (j27rfo7)] + 4 Re [{k,,(~) - ~;;;T(T)}
`- exp (j47rfot + j271foT + j20)] .
`
`(1 2)
`
`Comparing (12) with (6), we see that the second, cyclostation-
`ary, term is much smaller for the VSB case than the DSB case
`since the autocorrelation functions for a(t) and Z(t) differ only
`to the extent that some of the low-frequency components in
`Z(t) are missing because of the VSB rolloff characteristic.
`Although the jitter performance will be poorer,
`the phase
`recovery circuit in Fig. 1 can still be used since the mean value
`of the reference waveform is a sinusoid exhibiting the desired
`phase, but with an amplitude which is proportional to the dif-
`ference in power levels in a(t) and Z(t).
`E[ w(t)] = 3 [k,,(O) - k;;(O)] Re [exp (j477f0t +i20)1.
`(1 3)
`
`However, it is not possible to get a very simple formula for the
`variance of phase jitter, as in (1 l), because the power spectral
`density of the quadrature component of P2 (t), which is pro-
`portional to a(t) Z(t), vanishes at f = 0, unlike in the additive
`noise case. An accurate variance expression must
`take into
`account. the particular shape of the 520 filtering function as
`well as the shape of the VSB rolloff characteristic.
`Our examination of phase recovery for DS3 (with additive
`noise) and VSB modulation formats has indicated
`that rms
`phase jitter can be made
`as small as desired by making the
`width of 520 sufficiently small. The corresponding parameter
`in the case of the tracking loop configuration is called the loop
`bandwidth [3]. These results, however, are for
`steady-state
`phase jitter since the signals at the
`receiver input were
`presumed to extend into the remote past. The difficulty with a
`very narrow phase recovery bandwidth is that excessive time is
`taken to get to the steady-state condition when a new signal’
`process begins. This time interval is referred to as the acquisi-
`tion time of the recovery circuit and in switched communica-
`tion networks or polling systems it is usually very important to
`keep this interval
`small, even at the expense of
`the larger
`steady-state phase jitter. One way to accommodate the conflict-
`ing objectives in designing a carrier recovery circuit is to spe-
`
`then adjust
`cify a minimum phase-recovery bandwidth and
`other parameters of
`the system to minimize the steady-state
`phase jitter.
`Another problem with a very narrow-band bandpass fdter is
`in the inherent mistuning sensitivity, where mistuning is a result
`of inaccuracies in filter element values or a result of small inac-
`the carrier frequency. This problem is
`curacies or drift in
`avoided with tracking loop configurations since they lock onto
`the carrier frequency. One the other hand, tracking loops have
`some problems also, one of the more serious being the “hang-
`up” problem
`[9] whereby the nonlinear nature of
`the loop
`can produce some greatly prolonged acquisition times.
`Although we have modeled the phase recovery problem in
`terms of a constant unknown carrier phase, it may be impor-
`tant in some situations to consider the presence of fairly rapid
`fluctuations in carrier
`phase (independent of the message
`process). Such fluctuations are often called phase noise and if
`the spectral density of these fluctuations has a greater band-
`width than that of the phase recovery circuits, there is a phase
`error due to the inability to track the carrier phase. Phase error
`of this type, even in steady state, becomes larger as the band-
`width of the recovery circuits decreases.
`Another practical consideration is a 7r-radian phase ambigu-
`ity in the phase recovery circuits we have been discussing. The
`result is a polarity ambiguity
`in the coherently demodulated
`signal. In many cases this polarity ambiguity
`is unimportant,
`but otherwise some a priori knowledge about the message sig-
`nal will have to be used to resolve the ambiguity.
`For a QAM signal with p(t) = a(t) + jb(t), where a(t) and
`b(t) are independent zero-mean stationary processes, we get
`
`kyy(f + 7, t) = +.Re [{~uu(T) + kbb(7)I exp 0’271fo7)l
`+ 3 Re [{k,,(7) - kbb(7)l
`- exp (j47rfot + j271fO7 +j20)]
`
`(14)
`
`and the situation is very similar to the VSB case (12). In this
`case where a(t) and b(t) are uncorrelated, the mean reference
`waveform has the correct phase, but the amplitude vanishes
`if the power levels in the I and Q channels are the same.
`E[w(t)l = 4 [k,,(O) - kbb(0)l Re [exp U4nfot +j20)l.
`(1 5)
`Hence, unless the QAM format is intentionally unbalanced, the
`squaring approach in Fig. 1 does not work. We briefly examine
`what happens when the squarer is replaced by a fourth-power
`device in the recovery schemes we have
`been considering.
`From (I), we can obtain
`
`Y 4 ( t ) = Re [@(t) exp ( j h f o t +j40)]
`+ 3 Re [ I P(t) 12p2(t) exp (j4.rrfot +j2e)]
`+ 5 I P(O 14.
`Now if we use a bandpass filter tuned to 4f0 which passes only
`
`(16)
`
`Dish
`Exhibit 1018, Page 4
`
`

`
`FRANKS: CARRIER AND BIT SYNCHRONIZATION
`
`1111
`
`in (16), then the mean reference waveform at
`the first term
`the filter output is
`
`E[ w(t)] = + Re [ (3 - 3(2)’} exp (j8n& t + j 4 Q ]
`
`(1 7)
`
`still assuming independent a(t) andb(t) and a balanced QAM
`format, i.e., kaa(0) = kbb(0) = a 2 . Hence, a mean reference
`waveform exists even in the balanced Q W case if a fourth-
`power device is used.’
`One very popular QAM format is quadriphase-shift keying
`(QPSK) where the standard carrier recovery technique is to use
`a fourth-power device followed by a PLL or to use an equiva-
`lent “double” Costas loop configuration [3]. The QPSK for-
`mat, with independent data symbols, can be regarded as two
`independent binary phase-shift-keyed (BPSK) signals in phase
`quadrature. In a nonbandlimited situation each BPSK signal
`can be regarded as DSB-AM where the message waveform has
`a rectangular shape characterized by a(t) = f 1. In this case,
`the complex envelope of the QPSK signal is characterized by
`or P(t) = exp (j(n/4) + j(n/2)k) with k =
`P(t) = (f 1 t j)/*
`0, 1, 2, or 3. The result is that p4 (t) = -1 and the 4f0 com-
`ponent in
`(16) is a pure sinusoid with no fluctuations in
`either phase or amplitude. For PSK systems with a larger alpha-
`bet of phase positions, the result of (17) cannot generally be
`used as the I and Q components are no longer independent.
`Analysis of the larger alphabet cases shows that higher-order
`nonlinearities are required for successful phase recovery [3],
`[lo] . For any balanced QAM format, such as QPSK, the phase
`recovery circuits discussed here give a n/2-radian phase ambig-
`uity. This problem is often handled by use of a differential
`PSK scheme, whereby the information is transmitted as a
`sequence of phase changes rather than absolute values of phase.
`
`111. PAM TIMING RECOVERY
`The receiver synchronization problem
`in baseband PAM
`transmission is to find the correct sampling instants for extract-
`ing a sequence of numerical values from the received signal.
`For a synchronous pulse sequence with a pulse rate of 1/T, the
`sampler operates synchronously at the same rate and the prob-
`lem is to determine the correct sampling phase within a T-
`second interval. The model for the baseband PAM signal is
`
`m
`
`x(t) =
`k=--
`
`a&(t -kT-
`
`T)
`
`(18)
`
`the message sequence and g(t) is the signaling
`where{ak} is
`pulse. We want to make an accurate determination of T , from
`on x(t). We assume that g(t) is so
`operations performed
`defined that the best sampling instants are at t = k T + T;
`k = 0, t 1, k 2, ... . The objective is to recover a close replica
`x(kT + +)}, assuming a normalization of g(0) = 1. In the
`of the message sequence {ak} in terms of the sequence {hk =
`noise-free case, the difference between ak and dk is due to
`intersymbol interference which can be minimized by proper
`shaping of the data pulse g(t). With perfect timing (? = T), the
`
`intersymbol interference is
`
`and this term can be made to vanish for pulses satisfying the
`Nyquist criterion, i.e., g(nT) = 0 for n # 0. For bandlimited
`Nyquist pulses, the intersymbol interference will not be zero
`when i # T, and if the bandwidth is not significantly greater
`than the Nyquist bandwidth (1/2T) the intersymbol inter-
`ference can be quite
`severe even for small values of timing
`error. The problem
`is especially
`acute for multilevel (non-
`binary) data sequences where timing accuracy of only a few
`percent of the symbol period is often required.
`Symbol timing recovery is remarkably similar in most
`respects to carrier phase recovery and we find that similar
`signal processing will yield
`suitable estimates of the param-
`eter T. In the discussion to follow, we assume that {ak} is a
`zero-mean stationary sequence with independent elements.
`The resulting PAM signal (18) is a zero-mean cyclostationary
`process, although there are no periodic components present
`[61. The square of the PAM signal does, however, possess a
`periodic mean value.
`
`Using the Poisson Sum Formula [ 6 ] , we can express (20) in
`the more convenient form of a Fourier series whose coefficients
`are given by the Fourier transform of g2(t).
`
`where
`
`For high bandwidth efficiency, we are often concerned with
`data pulses whose bandwidth is at most equal to twice the
`I > 1/T and there
`Nyquist bandwidth. Then I GCf) I = 0 for If
`are only three nonzero terms (l= 0, f 1) in (21).
`This result suggests the use of a timing recovery circuit of
`the same form as shown in Fig. 1, where now the bandpass fii-
`ter is tuned to the symbol rate, 1/T. Alternate zero crossings
`of w(t), a timing wave analogous to the reference waveform in
`Section 11, are used as indications of the correct sampling in-
`stants. Letting H(l/T) = 1, the mean timing wave is a sinusoid
`with a phase of -2nr/T, for a real GQ.
`
`exp
`
`j - - j -
`
`2 3 1
`.
`
`(22)
`
`a2 [
`
`E[w(t)] = - R e A ,
`T
`
`( 2;t
`
`-
`-
`5 Unless o(t) and b(r) are Gaussian processes, for then a4 = 3(u*)2.
`
`
`
`We see that the zero crossings of the mean timing wave are at a
`fined time offset (T/4) relative to the desired sampling instants.
`
`Dish
`Exhibit 1018, Page 5
`
`

`
`Sompler
`
`Prefilter
`
`Fig. 3. Baseband PAM receiver with timing recovery.
`
`of A l .
`very small values
`off characteristic can result in
`Although the class IV partial response format exhibits a
`rel-
`[ 1 2 ] , it is likely that
`atively high tolerance
`to timing error
`some other recovery scheme may have to be used. Some of
`the proposed schemes [13] , [14] closely resemble the data-
`aided approach discussed in Section IV.
`the actual zero
`Calculation of the statistical properties of
`crossings of the timing wave is difficult: A useful approxima-
`tion can be obtained by locating the zero
`crossings by linear
`extrapolation using the mean slope at the mean zero crossing.
`When this approach
`is used, the expression for timing
`jitter
`variance becomes [ 1 1 1
`
`logic in the
`This timing offset can be handled by counting
`clock circuitry, or by designing H ( f ) to incorporate a n/2
`phase shift at f = 1 IT.
`zero crossings of w(t) fluctuate about the
`The actual
`desired sampling instants because the timing wave depends on
`the actual
`realization of the entire data
`sequence. Different
`zero crossings result for different data sequences and for this
`reason the fluctuation
`in zero crossings is sometimes called
`pattern-dependent jitter to distinguish it from jitter produced
`by additive noise on the PAM signal. To evaluate the statistical
`nature of the pattern-dependent jitter,
`we need to calculate
`the variance of the timing wave. This is a fairly complicated
`expressioil in terms of H y ) and Cy) but it can be evaluated
`numerically to study the effects of a variety of parameters
`(bandwidth, mistuning, rolloff shape, etc.) relating
`to data
`pulse shape and the bandpass filter transfer function [l 11 . For
`a relatively narrow-band real H y ) and real C(f) bandlimited as
`mentioned previously, the variance expression has the form
`
`4n
`var w(tj = Co + Cl COS - (t - 7)
`(23)
`T
`where Co 2 C1 > 0 are constants depending on Cy) and Hy).
`The cyclostationarity of the timing wave is apparent from this
`expression. As the bandwidth of H y ) approaches zero,
`the
`value of C1 approaches Co so that the variance has a great
`fluctuation over one symbol period. Note
`that the minimum
`variance occurs just at the instant of the mean zero crossings,
`In order to reduce this pattern-dependent jitter, there is
`hence the fluctuations
`in zero crossings are much
`less than
`fortunately an attractive alternative to making the bandwidth
`would be expected from a consideration of
`the average vari-
`of H y ) very small, which increases acquisition time
`in the
`ance of the timing wave over a symbol period. This again points
`same manner as for carrier phase recovery circuits, or to mak-
`out the error in disregarding the cyclostationary nature of the
`symmetry
`ing the bandwidth of Cy) very large. There are
`timing wav? process as, for example, 'in using the power spec-
`conditions that can be imposed upon Hcf) and Cy) that make
`tral density of the squarer output
`to analyze the jitter phe-
`C1 = Co in (24), resulting in nonfluctuating zero crossings.
`nomenon.
`These conditions are simply that Cy) be a bandpass charac-
`The mean timing wave (22) can be regarded as a kind of
`teristic symmetric about 1/2T, with a.bandwidth not exceed-
`discriminator characteristic
`or S-curve for measuring
`the
`ing 1/2T, and H y ) be symmetric about 1/T. The symmetry in
`parameter 7. For the bandlimited case we are discussing here,
`by prefiltering the PAM signal
`GCf) can be accomplished
`this 5'-curve is just a sinusoid, with a zero crossing at the true
`before it enters the squarer [ 111 , [ 1.51 . Since the timing recov-
`value of the parameter. Discrimination is enhanced by increas-
`ery path is distinct from the data signal path, the prefdtering
`ing the siope at the zero crossing. As this slope is proportional
`can be performed wi