Digital communication receivers P9

Chapter 9 Timing Adjustment by Interpolation In this chapter we focus on digital interpolation and interpolator control.. 9.1 Digital Interpolation The task of the interpolator is to c

Chapter 9 Timing Adjustment

by Interpolation

In this chapter we focus on digital interpolation and interpolator control In Section 9.1 we discuss approximations to the ideal interpolator We first consider FIR filters which approximate the ideal interpolator in the mean square sense

A particularly appealing solution for high rate applications will be obtained if the dependency of each filter tap coefficient on the fractional delay is approximated by

a polynomial in the fractional delay It is shown that with low-order polynomials excellent approximations are possible

In Section 9.2 we focus on how to determine the basepoint m, and fractional delay Pi, considering either a timing error feedback system or a timing estimator 9.1 Digital Interpolation

The task of the interpolator is to compute intermediate values between signal samples z(lcT,) The ideal linear interpolator has a frequency response (Section 4.2.2):

9 with

HI@, pT’) = { ~exdjw~T.) lw/24 < 1/2T,

elsewhere and is shown in Figure 9-l

P-2)

Figure 9-1 Frequency Response of the Ideal Interpolator

505

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

506 Timing Adjustment by Interpolation

Figure 9-2 Impulse Response of the Ideal Interpolator

The corresponding impulse response (Figure 9-2) is the sampled si( z) function

h~(nT,, j iT,) = si

[ +T,+pT,) 1 (n= ‘) -1,OJ , ) (9-3)

s Conceptually, the filter can be thought of as an FIR filter with an infinite number of taps

b&J) = hr(nT,, I-G)

= si [+T,+ST,)] (n= -1,OJ ,.‘.) (9-4)

s

9.1 Digital Interpolation 507

Figure 9-3 FIR Filter Structure of the Ideal Interpolator

The taps are a function of p For a practical receiver the interpolator must be approximated by a finite-order FIR filter

In Figures 9-3 and 9-4, the FIR filter structures of the ideal and the fourth-order interpolating filter, respectively, are shown

The filter performs a linear combination of the (11 + 12 + 1) signal samples X( nT8) taken around the basepoint ?-r&k :

Y(mk~ + Pz) = c x[(mk - n)Z] h,&)

9.1.1 MMSE FIR Interpolator

The filter coefficients h,(p) must be chosen according to a criterion of optimality A suitable choice is to minimize the quadratic error between the

Figure 9-4 Fourth-Order Interpolating Filter

508 Timing Adjustment by Interpolation

impulse response of the ideal interpolator and its approximation

2xB

e2(14 =

Jl ejwTBp _ 5 hn(p)e-jWTan 2cZ~ + min (9-7)

-27rB 7a3- I1

where B is the one-sided signal bandwidth The optimization is performed within the passband of the signal x(t) No attempt is made to constrain the frequency response outside B

The result of the optimization for p = 0 and ~1 = 1 are immediately evident For 1-1 = 0 the integrand becomes zero in the entire interval for

L(O) = {; else n=O

by Oetken [ 11

Example: MiWE FIR Interpolator with 8 Taps

Table 9-1 Coefficients of the MMSE interpolator with B = 1/4T, and N = 4

9.1 Digital Interpolation 509 The values for ~1 > 0.5 are obtained from those for p < 0.5 from

510 Timing Adjustment by Interpolation

Figure 9-6 Combined Plot of Coefficients h,(p), Where

II&) = h(t/T,) at t = (n-t p)T,

The values &(p) equal the function h(t/T,) taken at t = (7~ + p)T, The normalized delay error

(9- 12) and the normalized amplitude error

eA(w, p) = 1 - l&t (ejwT8, P) 1 (9- 13) are plotted versus o and p in Figure 9-7

We observe level plateaus for the frequency range of interest Since no attempt was made to constrain the out-of-band response, a steep error increase can be observed for w > 27rB (B = 1/4T,)

The accuracy of the interpolation is a function of the interpolator filter as well

as the signal spectrum Let us consider the stochastic signal z(nT,) with power spectrum S, ( ejwTn) The mean square error between the output of the ideal and approximate interpolator, given z(nT,) as input, is

9.1 Digital Interpolation 511

Figure 9-7 (a) Normalized Delay Error; (b) Normalized Amplitude Error where a$ is the variance of the input signal The function has been numerically evaluated to obtain the fractional delay p max corresponding to the worst-case

~$-nax * In Figure 9-8 we have plotted the number of filter taps 2N for various values of o:,,,~ as a function of the useful bandwidth B

From Figure 9-8 it is possible to select the order of the FIR interpolating filter to meet a specified error requirement This seems an appropriate performance measure for communication applications Since the values of p are equiprobable the variance a: averaged over all possible values of p is also of interest:

512 Timing Adjustment by Interpolation

9.1 Digital Interpolation 513

The 2N coefficients of the FIR filters must be precomputed and stored in memory for a number of L possible values ~1 = l/L As a consequence, the recovered clock suffers from a maximum discretization error of L/2

Assume the coefficients hn(pl) are represented by a word length of W bits For each interpolation these 2N words must be transferred to the filter structure The complexity of the transfer thus can easily become the limiting factor in a VLSI realization An alternative in high-speed applications is to use a bank of parallel filters where each filter corresponds to a different quantized value of p

Implementation complexity depends upon the structure parameters, viz order

of filter 2 N, time discretization L, and word length W

9.1.2 Polynomial FIR Interpolator

We approximate each coefficient hn(p) by a (possibly different) polynomial

p We impose the following constraints on (g-19),

Since we restrict the function hn (p) to be of polynomial type, the quadratic error of the polynomial interpolator will be larger than for the MMSE interpolator discussed previously, although it can be made arbitrarily small by increasing the degree of the polynomial Since the polynomial interpolator performs worse, why then consider

it at all? The main reason is that the polynomial interpolator can be implemented very efficiently in hardware as will be seen shortly

For simplicity (though mathematically not necessary) we assume that all polynomials have the degree of M(n) = A4, Inserting for h, (CL) the polynomial

514 Timing Adjustment by Interpolation

expression of (9-17) the FIR transfer function reads

For every degree m of the polynomial the expression in the

describes a time-invariant FIR filter which is independent of p:

up This structure was devised by Farrow [2]

The Farrow structure is attractive for a high-speed realization The (M + 1) FIR filters with constant coefficients can be implemented very efficiently as VLSI circuits Only one value for p must be distributed to the M multipliers which can

be pipelined (see Figure 9-10) The basic difference of the polynomial interpolator with respect to the MMSE interpolator is that the coefficients are computed in real time rather than taken from a table

Example: Linear Interpolator

The simplest polynomial interpolator is obtained for M = 1 and N = 1 The four coefficients are readily obtained from the constraints

(9-24) The interpolator (Figure 9-11) performs a linear interpolation between two samples,

(9-25) The coefficients Cm(n) for a set of parameter values of practical interest are tabulated Section 9.1.4

516 Timing Adjustment by Interpolation

x[(k+l &I

v z-1

C,(4)= 1

t c, (0) = -1 2” C&-l) = 0

Co(O) = 1

b Y(‘$ 1

Figure 9-11 Linear Interpolator The number of taps of the polynomial interpolator versus normalized band- width is plotted in Figure 9- 12 for various values of a:

A number of interesting conclusions can be drawn from this figure For a given signal bandwidth BT, there is a trade-off between signal degradation and signal processing complexity, which is roughly estimated here by the number of taps 2N and polynomial degree M It can be seen that for a signal bandwidth

BT, = 0.25 already 2N = 2 taps (independent of the polynomial order) will suffice

to produce less than -20 dB signal degradation For BT, = 0.45 and -20 dB signal degradation the minimal number of taps is 2N = 6 with a polynomial order

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

bandwidth BTs

Figure 9-12 Number of Taps versus Normalized Bandwidth for Constant Fz

9.1 Digital Interpolation 517

M > 2 In the plotted domain (2N 5 10) almost no difference between results for the third- and fourth-order polynomial has been observed The independence of the linear interpolator of the number of taps 2N is a consequence of the constraints (9-24)

In Figure 9- 13 the variance ??:/a: is plotted versus the excess bandwidth cx for

a random signal with raised cosine spectrum For most cases a linear interpolation between two samples will be sufficient Doubling the number of taps is more effective than increasing the order of the polynomial to reduce the variance

-15 _

/ 2N=2,4,6,8 -2o= -

-80 -

40

Figure 9-13 SNR Degradation versus Rolloff Factor

a for (a) T/Ts = 2 and (b) T/Z” = 4

518 Timing Adjustment by Interpolation

9.1.3 Classical Lagrange Interpolation

In numerical mathematics the task of interpolation for our purposes can be stated as follows Given a function z(t) defined for t-(~-l), , to, , TV, find

a polynomial P(t) of degree (2N - 1) which assumes the given values CC(&), q&a> = &d (n = -(N - l), * , -l,O, 1, * *, N) (9-26) (see Figure 9-14) In general, the points t, do not need to be equidistant, nor does the number of points have to be an even number 2N As shown in any textbook

on numerical mathematics the solution to this problem is unique There exists a Lagrange polynomial of degree 2 N - 1:

p(t) = 5 &a [(t - L(N-1)) ’ * * (t - tn-1)(t - t& * * * (t - tJJ)] 11: (tn)

n=-(N-l)

(9-27) with

x n=

( t, -t -(N-l)) (t.-t.~)(t.-tn+~) (t~-tN) (9-28) Using the definition of (9-27) it is easily verified that for every tra, we have P(tn) = x(tn) as required The polynomial P(t) is a linear combination of the values z(tn),

N n=-(N-l)

with the so-called Lagrange coefficients

Figure 9-15 Farrow Structure of Lagrange Interpolation

Since the FIR filter also computes a linear combination of sample values z(kT,),

it is necessary to point out the differences between the two approaches

As we are only interested in the interpolated values in the central interval O<t<l,wesett= p in the definition of the Lagrange coefficients qn (t) Every qn(p) is a polynomial in p which can be written as

520 Timing Adjustment by Interpolation

There is a single polynomial valid for the entire range t-(~-r) < t < TV For the polynomial interpolator, to each tap h, (p) there is associated a different polynomial valid just for this tap

The coefficients of the Lagrange interpolator are completely specified given z(tn), while the polynomial interpolator coefficients are the result of an optimization Thus, even for quadratic polynomial interpolator structures they have nothing in common with the Lagrange array coefficients d,(n)

For more details on Lagrange interpolation see the work of Schafer and Rabiner [3] Example: Cubic Lugrange Interpolator

522 Timing Adjustment by Interpolation

-0.01807 0.09361 -0.17945 0.10350 0.10350 -0.17945 0.09361 -0.01807

9.2 Interpolator Control

This section focuses on the control of the interpolator The task of the interpolator control is to determine the basepoint m, and the corresponding fractional delay prs based on the output of a timing estimator or of a timing error estimator The control algorithm will be different for the two cases

9.2.1 Interpolator Control for Timing Error Feedback Systems

Error detectors produce an error signal at symbol rate l/T using fractionally spaced samples rCT’ = W/MI (MI integer) Most error detectors work with two

or four samples per symbol Since the sampling rate l/T8 is not an exact multiple

of the symbol rate, the samples { ICT’) have to be mapped onto the time scale { kT, ) of the receiver Mathematically, this is done by expressing the time instant kTI + EIT~ by multiples of TJ plus fractional rest,

kT + EITI = &NT [kZ + EITI] Z + plcT,

524 Timing Adjustment by Interpolation

Position of sample

Transmitter time reference

Receiver time reference

Figure 9-17 Definition of (~1, 7”) and Their Relation to the

Receiver Time Axis { ICT, )

Mapping of ET1 onto mkT, is exemplarily shown for one sample and MI=2

as an FIR filter with time-variant coefficients A new sample is read into the filter at constant rate l/Ts, while a new output is computed only at the basepoints mkT, (Figure g-19)

9.2 Interpolator Control 525 kTs

(from timing error detector path) Coefficients h,(p) ; i = -(N-l), N

Figure 9-19 Interpolator and Decimator Implemented as Time-Variant FIR Filter

Based on the fractionally spaced samples the error detector produces an error signal at symbol rate Therefore, a second decimation is performed in the error detector This second decimation process is slaved to the first one It selects exactly every Mlth basepoint ??‘&k (h = n&iI), to output one error signal

The error detector signal is further processed in a loop filter The output of the loop filter e(m,M,) is used to adjust the control word of the timing processor

w(md4,) = w(m(,-Ip4,) + JL ++-I)MJ

In the absence of any disturbance the control word assumes its correct value:

Basepoint and fractional delay are recursively computed in the timing processor

as follows we express IcTr + E~TI as function of (mk , p]E), see eq (9-34) The next sample at (Ic + ~)TJ + EITI is then given by