Digital communication receivers P8

If the frequency offset is restricted to small values, roughly IL?ZTI < 0.15, timing have superior tracking performance compared to the algorithms operating with 8.6 frequency estimation

Chapter 8 Frequency Estimation

8.1 Introduction / Classification of Frequency Control Systems

In this chapter we are concerned with frequency estimation We could have studied this problem earlier in Chapter 5 by including an additional parameter Q

in the set 8 = (0, E, a} The main reason why we choose to include a separate chapter is that for a sizeable frequency offset St we must first compensate this

implies that frequency offset estimation algorithms must work independently of the values of the other parameters The operation of the algorithms is nondata aided and nonclock aided The only exception occurs for small frequency offset

Section 8.2 we derive estimators which work independently of the other parameters

likelihood function with respect to the parameter R (Section 8.3) The algorithms of the first two sections operate on samples {rf (ICT,)} which are sufficient statistics

If the frequency offset is restricted to small values, roughly IL?ZTI < 0.15, timing

have superior tracking performance compared to the algorithms operating with

8.6 frequency estimation for MSK signals is studied In summary, the rate-l/T8 algorithms can be regarded as coarse acquisition algorithms reducing the frequency

with improved accuracy can be employed in a second stage running at symbol rate l/T

8.1.1 Channel Model and Likelihood Function

We refer to the linear channel model of Figure 3-l and Table 3-1 The equivalent baseband model is shown in Figure 3-3 The input signal to the channel

is given by zl(t)e jeT(‘) In the presence of a frequency offset Q we model the

445

Digital Communication Receivers: Synchronization, Channel Estimation, and Signal Processing

Heinrich Meyr, Marc Moeneclaey, Stefan A Fechtel Copyright  1998 John Wiley & Sons, Inc Print ISBN 0-471-50275-8 Online ISBN 0-471-20057-3

(0 constant phase offset) The input signal to the channel is then given by

We require that the frequency response of the channel C(f) and that of the prefilter F(f) are flat within the frequency range

when B is the (one-sided) bandwidth of the signal u(t), and f&,,, is the maximum frequency uncertainty (see Figure 8-l) Only under this condition the signal sf (t , Q) can be written as

is violated, the frequency estimator algorithms presented here produce a biased estimate This issue will be discussed when we analyze the performance of the frequency estimators

Assuming a symmetrical prefilter IF(w) 1’ about 1/2T, and Nyquist pulses, the normalized likelihood function is given by eq (4-157) Using the signal definition

’ wm ,_r- -. - r , ,J ww(~~

passband of the channel- ’

Figure 8-l Frequency Translation and Passband of

the Channel G(w) and Prefilter F(w)

8.1 Introduction / Classification of Frequency Control Systems 441

I > hF(kT,) ====i’ polator Inter- = - ~ L - I

Timing independent Timing dependent

Figure 8-2 Classification of ML Frequency Estimation

Algorithms: Typical Signal Flow

(8-5) the matched filter output in the presence of a frequency uncertainty is given by

To obtain z~(&, n> the signal is first multiplied by e-jnT81c This operation must

be performed for each realization Ra of the trial parameter 0 Subsequently each signal is filtered by the matched filter The sampling rate (see Figure 8-l) must obey the inequality

The only exception to the signal flow diagram shown in Figure 8-2 occurs if the frequency offset is small, lOT( < 1 In this case it may be advantageous to

(compare Figure 8-3)

Decimator

I FF Timing dependent

Phase

II e @Tn

Frequency Translator

Figure 8-3 Classification of ML Frequency Estimation

Algorithms: Alternative Signal Flow for IRTl < 1

Then, as we will see in Section 8.4, it is convenient to introduce a mathemat- ically equivalent formulation of (8-6) Equation (8-6) can be written in the form

00 zn(izr Q) = e-jnnT C rf(kT,) gMF(nT + 67 - kT,)e-jhl(kT8-nT) (8-8)

k=-co The sum describes a convolution of the received signal rf (Ha) with a filter with impulse response

and frequency response

(8- 10) The approximation of gn (kT, ) by gMF( IcT,) as it is suggested in Figure 8-3 is discussed later on in Section 8.4

Besides the above distinction between timing-directed (DE) and non-timing- directed (ND&), the algorithms employed in the frequency synchronization unit can be further classified as discussed earlier (NDA, DA, DD)

Remark: Before we are going into further detail of the frequency synchronization schemes, it should be emphasized that in practice (8-7) and the relationship

any structure (Figures 8-2 and 8-3) can cope with The value Bfl stands for the (one-sided) frequency range of the analog prefilter, where the signal passes undistorted If the key parameters (prefilter shape and sampling rate) are fixed, the pull-in range - determined by the maximal manageable frequency offset - can only be increased by an additional control loop for the analog oscillator in front

of the analog prefilter

0 There are open-loop (FF) and closed-loop (FB) structures

8.2 Frequency Estimator Operating Independently

8.2 Frequency Estimator Operating Independently of Timing Information 449

unbiased estimator of R which requires no knowledge of E In other words, we can first estimate the frequency offset St, compensate for 52, and then estimate the remaining synchronization parameters

To qualitatively understand why this separation is possible we recall the main result of the section on timing parameters estimation via spectral estimation, (see Section 5.4) In that section, we expanded the timing wave ]z( Kf + ET, 0) I2 into a Fourier series Due to the band-limitation of ]z(lT + ET, !G!) I2 only three coefficients (c-1, cc, cl} of the Fourier series have a nonzero mean Therefore,

Fourier series contribute to the random disturbance only

2 I%(lT$ &T, fql” = CO + 2 Re[clej2”“] + c cn ejannr (8-12)

random disturbance

As will be shown, the expected value of CO depends on Q but is independent of

E Furthermore, E[cc] is shown to be maximum if Q assumes the true value 520 Hence, the value fi which maximizes the coefficient co(Q) is an unbiased estimate:

Remark: Despite the fact that E[lci)] is also a function of Q, we may seek the maximum of the likelihood function (8-11) by maximizing c,-,(R) This, of course,

is possible only because (8-13) alone provides an unbiased estimate of a

(9

We maintain that under the following conditions:

The sampling rate fulfills l/777 > 2(1 + a)/T (twice the rate required for the data path)

(ii) The ratio T/T3 = Ma is an integer

(iii) i.i.d data {an},

the following sum

(8-14)

defines an unbiased estimate It is remarkable that no conditions on the pulse g(t) are required in order to get an unbiased estimate But be aware that our derivation

Much of the discussion that follows is similar to that on timing parameters

estimation via spectral estimation (Section 5-4) The reader is therefore urged to

reconsult Chapter 5 in case the following discussion is found to be too concise

The matched filter output z( IT,, s2) in the presence of a frequency offset C!

is given by

co

%(ZT,, i-2) = c r#T,) e-jnTak gMF(zTs - ICT,) (8-15)

k=-00 Replacing in the previous equation the received signal samples rf (/CT,) by

we can write for Z( IT,, 0)

8.2 Frequency Estimator Operating Independently of Timing Information 451

interval is negligible The expected value then is a periodic function

h,(t, AC!) = 2 (h(t - nT - eoT, A!2)12 (8-21)

n= 00 which can be represented by a Fourier series

1 E-e -j %iaoT

-Xl

From the definition of h(t, AQ) the spectrum of h(t, ACI) is found to be H(w, An) = G(w) G*(w - AQ) G(w) is band - limited to B [Hz]

(8-24) B=

(a excess bandwidth) Since squaring the signal h(t, Ai2) doubles the bandwidth, the spectrum of Ih(t, As1)12 is limited to twice this value

Blhl a = $(l+a) (8-25)

From this follows that only the coefficients d-1, do, dl are nonzero in the Fourier series Hence, the expected value of E [ Iz(iTs, AC!) 12] can be written in the form

where the coefficients do, dl are defined by (8-23), and M, = T/T8 is an integer

We have now everything ready to prove that the estimate of 6 of (8- 12) is indeed unbiased First, let us take expected value of the sum (8-14)

LM,-1

I=-LM,

(8-27)

Next, replace E [ ]z(zT~, AO)~~]

Ail = S&J - R = 0 Hence

LM,-1

I=-LM,

is an unbiased estimate (Figure 8-4)

Regarding the manageable frequency uncertainty Rc we refer to Section 8.1.1 and the relation (8-7) and obtain

8.2 Frequency Estimator Operating Independently of Timing Information 453

G! The timing parameter & can subsequently be found as

fi But keep in mind that our derivation requires that the transmission pulse g(t)

be known at the receiver

8.2.1 Frequency Estimation via Spectrum Analysis

Consider the estimation rule of (8-14):

(8-36) different realization of the estimator We write z(lT, , s2) in the form

where gn( ICT,) is the impulse response of

A block diagram corresponding to (8-35) and (8-36) is shown in Figure 8-5 The samples znj (/CT,) are obtained as output of a freque,ncy translated matched filter, G* (ej (w-aj )LTB) , where Qj is the jth trial value The number of parallel branches

in the matched filter bank is determined by the required resolution AR = v

8.2 Frequency Estimator Operating Independently of Timing Information 455

Since

(8-38)

the multiplication of the output zaj (aT, ) by a complex exponent e-jnzTa can be avoided The squared absolute value lznj (ICT,) 1 2 is averaged and the maximum

of all branches is determined

The above estimator structure can be interpreted as a device that analyzes the power spectral density of a segment t E { -TE/~, TE/~} of the received signal

TE/~

Then the Parseval theorem (approximately) applies

TE/~ -

J lznj (t) I2 dt

of G(w - aj),

Since Figure 8-5 is merely a different implementation of the algorithm de- scribed by Figure 8-4 the maximum manageable frequency uncertainty 00 is again given by

rf (kTs)

- -

complex coefficients

real decision variables

Figure 8-6 Block Diagram of the DF’I’ Analyzer, Xj [compare (8-40) and (8-44)]

frequency resolution

To obtain a reliable estimator the result of several DFT (Discrete Fourier Trans- form) spectra must be averaged The resulting structure is shown in Figure 8-6 The operation required to generate a frequency estimate is illustrated in Figure

8-6

(i) N, spectra are generated by the DFT using NaV nonoverlapping sequences { rf (ICT,)} of length NFFT

(ii) These N,, spectra are accumulated in the next operation unit

(iii) In the spectrum analyzer an estimate of Ro is generated via

fij = arg rnax

3 m=-(&FT/2-1) n’=zT’2 I&& Af)l’jG(m Af - (j Af))l”

A performance measure for a maximum seeking algorithm is the probability that

an estimation error exceeds a given threshold

Table 8-l shows some simulation results Parameters are NFFT, iv,, , the ratio

denote how often the estimation error As2 exceeds a given threshold as a function

of the SNR The results obtained for SNR=200 dB indicate that the above algorithm suffers from self-noise phenomena, too The self-noise influence can be lowered

by increasing the number of the F’FT spectra to be averaged This, in turn, means that the estimation length and the acquisition length, respectively, are prolonged

8.2 Frequency Estimator Operating Independently of Timing InfomMion 457

total number of trials is 10,000 for each entry

8.2.2 Frequency Estimation via Phase Increment Estimation

Up to now we were concerned with maximum-seeking frequency estimators The basic difference of the present estimator compared to the maximum-seeking algorithms is that we directly estimate the parameter 0 as the argument of a rotating phasor exp(jRi) We discuss a non-data-aided, non-timing-directed feedforward (NDA-NDE-FF) approach

We start from the basic formulation of the likelihood function introduced in Chapter 4:

an unbiased estimate of G?, as will be proved below

Now let us conjecture the following approximation:

Sj(M-3) M PTa Sj((k - 1)7-y,) (8-48)

Obviously, the larger the ratio T/T’ is, the better this approximation becomes

Also, for high SNR we have

In a practical realization the summation is truncated to LF symbols

A block diagram of the estimator is shown in Figure 8-7

In the sequel we analyze the performance of the algorithm (8-53) We start

by proving that the algorithm has (for all practical purposes) a negligible bias Rather than averaging fiT we take the expected value of the complex phasor

c

M&F ) w(-) -=G

Figure 8-7 Block Diagram of the Direct Estimator Structure

8.2 Frequency Estimator Operating Independently of Timing Information 459

shown to be equal 0eT, then it follows that the algorithm is unbiased Writing

‘J (ICT,) as the sum of useful signal plus noise we obtain after some straightforward algebraic steps

For a properly designed prefilter F(w) the noise samples are uncorrelated,

so R,,(T,) = 0 It remains to evaluate the first term Interchanging the order of summation we obtain

g(kT, - nT)g*((k - 1)X - nT) (8-55)

n=- N - - k=-hcf,LF

In all reasonable applications the number of transmitted symbols is larger than the estimation interval, i.e., N >> L F Therefore, the correlation-type algorithm suffers from self-noise caused by truncation of the estimation interval This effect was previously discussed in Section 5.2.2 If the number D of symbols which essentially contribute to the self-noise is much smaller than LF (i.e., D << LF),

these contributions to the self-noise effects are small The summation over ICT, then equals approximately

co Cg(kTd-nT)g*((lc-l)T,-nT) M $ / 9(t) 9* (t - T.9 > &

= h,(q) This expression is clearly independent of the other estimation parameters { 0, E‘, a} Provided h,(t) is real, which is the case for a real pulse g(t), the algo- rithm (8-53) yields a (nearly) unbiased estimate for a sufficiently large estimation interval Simulation results confirmed the assertion If g(t) is known, the condition

on g(t) can be relaxed [compare with the Remark following eq (8-34)] because the bias caused by a particular filter shape g(t) is known a priori and therefore can be compensated

Our next step is to compute the variance of the estimate:

It proves advantageous to decompose the variance into

which reflects the fact that the frequency jitter is caused by (signal x signal),

BPSK and QPSK modulation we get the expressions listed below

E theory rolloff =O.S

lxxl simulation rolloff =0.5

Es/No WI

Figure 8-8 Estimation Variance of hT/27r for BPSK Transmission,

8.2 Frequency Estimator Operating Independently of Timing Information

3 (noise x noise) for QPSK and BPSK:

Figure 8-9 Estimation Variance of fiT/27r for QPSK

Transmission,T/T, = 4, Estimation Length LF;

dotted lines: SX S, SxN, NxN contribution for LF = 800 and a = 0.9

unity The good agreement between analytically obtained curves and the simulation proves the approximations to be valid

Further simulations not reported here confirm that the estimate properties are independent of the parameter set (0,) EO}, as was predicted from our theoretical

schemes such as 8-PSK and M-QAM

is directly proportional to the bandwidth of the analog prefilter; second, because

large if the variance of the estimate has to be smaller than a specified value An appropriate solution for such a problem is illustrated in Figure 8-10

The basic idea is that the frequency offset to be estimated is lowered step by step In the first stage of Figure 8-10 a coarse estimate is generated Although this coarse estimate may still substantially deviate from the true value, the quality

of this first estimate should guarantee that no information is lost if the frequency- adjusted samples of (ICT,) e-JSlkT* are fed to a digital lowpass filter Note that the decimation rate T,, /Y& and the bandwidth of the digital lowpass filter have to be properly selected according to the conditions on sufficient statistics In the second stage the same frequency estimator can operate in a more benign environment than

in the first stage with respect to the sampling rate and the noise power

The last comment concerns the behavior of the estimator if, instead of statis- tically independent data, periodic data pattern are transmitted Simulation results show that the estimation properties remain intact in the case of an unmodulated

cases

Bibliography

[l] F M Gardner, “Frequency Detectors for Digital Demodulators Via Maximum

463

8.3 Frequency Error Feedback Systems Operating

Independently of Tlming Information

To obtain a frequency error signal we follow the familiar pattern We differentiate the log-likelihood function of Section 8.2 [eq 8-131 with respect to s2 to obtain (neglecting an irrelevant factor of 2):

The derivative of the matched filter (MF) output equals

The time-weighting of the received signal can be avoided by applying the weighting

to the matched filter We define a time-invariant frequency matched filter (FMF) with impulse response:

gFMF(kTd) = gMF(kT,)jkT, (8-68) The output of this filter with of (ICT,) as input is

= &%(kT,, s-2) + jkT, %(kT,, n)

(8-69) The last equality follows from the definition of d/aQ( z(lcT,, Q)) and of z( IcT,, a) Replacing

in (8-66), we obtain the error signal

z(~cT,) = Re{z(kT,, Q)[z~MF(~TJ~ Q) + jkT,z*(kT,, n)l)

= Re{z(kT,,Q) ~~~~(6T3,n)+j~~It(leT,,n)l~}

= Re{z(kT,, 0) &I&T,) n)}

(8-71)

8.3 Frequency Error Feedback Systems Operating Independently 465

Figure 8-11 Frequency Response of the Digital Frequency Matched Filter GFMF (ejwTs) for a Root-Raised Cosine Pulse G(w), LY = 0.5

readily appreciated when we consider the frequency response of the filter:

(8-72)

= &GMF (ejwT8)

response of the signal matched filter A dual situation was found for the timing matched filter which is the time derivative of the signal matched filter in the time domain and weighted by (jw) in the frequency domain

Example

Figure 8-l 1 for a root-raised cosine signal pulse

The operation of the frequency matched filter is best explained if we write

= Iz(kT,, fi) + zFMF(&, n)l” - I+%, fi) - zFMF(&, fi)j”

(8-73)

- (q(kT,)e -jnTEk @ [gMF(k%) - gFMF(kT,)112

The first term in the difference equals the signal power at the output of the filter

second term equals the output power of a filter gMF( kT,) - gFMF( kT, ) when the same signal is applied

(RRCF stands for root-raised cosine filter, and FMF for

the corresponding frequency matched filter)

For the sake of a concise explanation we consider a root-raised cosine pulse The two filters in (8-73) differ in the rolloff frequency range only, see Figures 8-11 and 8-12 The power difference remains unchanged if we replace the two filters by filters with a frequency response which is nonzero only in these regions (compare Figure 8-13) Furthermore, the two filters may be simplified appropriately The simplified filter HP ( ejwTa) passes those frequencies which lie in the positive rolloff region, while the filter HN (ejwTa) passes the negative frequencies

Let us see now what happens when a signal rf (t) shifted by 00 is applied to these filters Due to the symmetry of the filter frequency response and the signal spectrum, the output of the two filters is identical for sic = 0

8.3 Frequency Error Feedback Systems Operating Independently 467

For all other values of IQ, 1 < 2B an error signal is generated The maximum

with rolloff factor a,

Figure 8-13 Frequency Response of the Power Equivalent

Filters HN (ejwTn) and Hp (ejwTn)

The frequency error signal equals the difference of the shaded areas

An S-curve expression is obtained by taking the expected value of the error signal with respect to the random data and the noise,

For the case that all filters are tailored to a root-raised cosine transmission pulse

it can be shown from (8-76) and (8-73) that the error signal has a piecewise defined structure The location of the piece boundaries, and even the number of pieces depends upon the excess bandwidth factor a which is obvious from the above interpretation of the frequency error detector in the frequency domain The formulas in Table 8-2 are valid for a 5 0.5

The expressions in Table 8-2 and the simulation results of Figure 8-14 present the error signal for the case that the transmission pulse energy is normalized to unity and all normalization of the MF and FMF was performed in such a way that the fully synchronized output of the MF holds z(n) = a, + N(n), with a, the transmitted symbol and N(n) the filtered noise with variance var[N(n)] = No/E, From the foregoing discussion it is clear that the filters H~(ej**.) and

HP (ej**#) defined by the signal and frequency matched filter can be approximated

by a set of simple filters which perform a differential power measurement The resulting algorithms are known as dualJilter and mirror imagecfilter in the literature [ l]-[3] They were developed ad hoc Again, it is interesting that these algorithms can be derived systematically from the maximum-likelihood principle by making suitable approximations Conditions on the filter HN (ej**n) and Hp (ej**e) to produce an unbiased estimate, and, preferably, a pattern-jitter-free error signal will

be discussed later on

Symmetric about Zero, Afl = St0 - fi = 27rAf

8.3 Frequency Error Feedback Systems Operating Independently 469

S-Cutve of NDA NED Frequency Control Loop

Figure 8-14 S Curve of the NDA NDe Frequency Control Loop for

a = 0.5 and E [lci2] = 1; Data of Table 8-2

8.3.1 Tracking Performance Analysis

In this section we analyze the tracking performance of the algorithm In a first step, we discuss the self-noise phenomena Self-noise phenomena are the reasons why estimates are disturbed although thermal noise is absent As a result, this leads to an irreducible degradation of the estimate properties Such an impact can only be mitigated if the algorithm itself is modified

The algorithm of Figure 8-15 serves as an example of how to analyze self- noise effects of a tracking loop The first step was carried out in the previous section by the determination of the S curve We next determine the variance of the estimate We follow the methodology outlined in Section 6.3 where we have shown that the variance can be approximated by

(8-77)

where Kn is the slope of the S curve at the origin, 2l3~T is the equivalent two- sided loop bandwidth, and SZ (ejzrjT) stands for the power spectral density of the frequency error output The approximation in (8-77) is valid for a small loop bandwidth and a nearly flat power spectral density about f = 0

In Figure 8-17 the total variance ai is plotted versus E,/iVo Also indicated are the analytical results of the self-noise variance aiXs (The details of its calculation are delegated to Section 83.2.) We observe that the algorithm suffers from strong self-noise disturbances in the moderate and high SNR region It should therefore only be used to acquire initial frequency lock in the absence of a timing information In the tracking mode, timing-directed algorithms as discussed

in Sections 8.4 and 8.5 are preferable since they can be designed to be free of self-noise (pattern jitter)

Finally we concisely touch the question of acquisition time As for any feedback loop, the time to acquire lock is a statistical parameter depending on the frequency uncertainty region and the SNR

To get a rough guess about the time to acquire, the acquisition length L, can be assessed from a linearized model of the tracking loop where the frequency

8.3 Frequency Error Feedback Systems Operating Independently 471

Self -Noise in NDA Frequency Estimation

Table 8-3 Acquisition Length L,,

detector characteristic in Figure 8-14 is stepwise linearized In Table 8-3 we have summarized the acquisition lengths, measured in required symbol intervals, for different loop bandwidths The calculations and the simulation, respectively, were performed for an acquisition process in the noiseless case where the initial frequency offset was set to &T/27r = 1, and we defined the acquisition process to

be successfully terminated if the difference between the estimated frequency offset and the true frequency offset AfT = (l/fL?r)Jfi~T-fIT( was less than 0.01 These results should give only a rough guess about the order of magnitude of the length

of the acquisition process The exact analysis of the loop acquisition time requires more sophisticated mathematical tools as mentioned in Chapter 4 of Volume 1 Only if Ifl,,,JT < 1 and the time to acquire is uncritical should feedback structures be employed For the case of a burst transmission, open-loop structures are to be used Here for a given estimation interval the probability of exceeding a threshold can be computed The time to acquire lock is then constant and identical

to the estimation interval

8.3.2 Appendix: Calculation of the Self-Noise Term

It can easily be verified that the output of the frequency error detector X( lT, )

is a cyclostationary process, because its statistics are invariant to a shift of the time origin by multiples of the symbol interval T Therefore, the power spectrum density at the origin is

To get a manageable expression, we replace in the above expression T by LT,

where L is an integer, and we approximate the above integral by

R,,(k) 2 &

[CL- 1wwf~

8.3 Frequency Error Feedback Systems Operating Independently 473

where we have suppressed in the second line the apostrophe (k’ -f k) for the sake

For the frequency error detector signal

we demonstrate exemplarily how the expressions can be numerically calculated

We start with

Pl =2 c D[E c c c

LMJ l,(LM*) k,a ~I,N ma,N ms,N mr,N

E[aI,m al,,] # 0 only for m = n, and E[aQ,m a*,,] # 0 analogously, After interchanging the order of summation we arrive at

l,(LM,)

(8-86)

X c h(kT, - mlT)h$MF(kTs - mzT) k,m

Defining the function