IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-X, NO. 8, AUGUST i984 935

AFC Tracking Algorithms FRANCIS D. NATAL1

Abstract-The automatic frequency control (AFC) loop is beginning to play an important role in digital data links. There is very little published literature which deals with AFC loop implementations and performance in noise (in contrast to the large body of literature dealing with phase-locked

This paper presents a brief discussion of eight different AFC loop imple- mentations, including the block diagram, discriminator characteristic, and loop tracking error in the presence of gaussian noise. Tracking perform- ance of the various loops is compared in the presence of noise for CW, DPSK, and MFSK signaling.

loops).

D 1. ~NTRODUCTION

IGITAL data transmission over satellite links has been dominated by coherent biphase-shift-keyed (BPSK)

and quadrature-phase-shift-keyed (QPSK) systems. The "heart" of the coherent demodulator is some form of phase-locked loop (PLL) and, as a result, a wealth of published informafion is available on this subject.

The coherent PSK systems have excellent performance in the presence of Gaussian noise; however, they are not very tolerant of other link disturbances such as high Doppler rates, specular multipath, and ionospheric anomalies. These effects have bekome more important in recent years, particularly with the proliferation of airborne users. As a consequence,, there is motivation for moving toward differentially coherent detection of PSK signals (DPSK) and multiple frequency- shift-keyed (MFSK) systems. MFSK signaling is also used frequently in frequency-hopped (FH) systems. These systems generally require automatic frequency tracking (AFC) loops which are more robust in the presence of link disturbances than the PLL. AFC loops are also being used in conjunction with PLL's to perform initial frequency acquisition of the received signal. Furthermore, burst mode QPSK modems al- most universally employ an AFC loop of some form (of ten operating on a CW o r BPSK preample so that the AFC loop does not operate during QPSK data transmission).

In contrast to the situation with PLL's, there has been relatively little published on the performance of AFC loops. Section I1 of this paper gives a brief discussion of eight dif- ferent AFC loop configurations (seven of which are digital implementations) which have been analyzed at ST1 over the past few years. The noise performance results are presented without derivations due to space limitations. Several, but not all,, of the results have been compared with both simulation and experimental data for verification. Data are presented where available and show good agreement with analytical results.

Section 111 contains a summary of results and compares the performance of different configurations, where appropriate, for CW, DPSK, and MFSK signaling.

Paper approved by the Editor for Satellite and Space Communication of the IEEE Communications Society for publication after presentation at the Interns- t i O d Conference on Communications, Boston, MA, June 1983. Manuscript received June 27, 1983; revised November 22, 1983

94086. The author is with Stanford Telecommunications, Inc., Sunnyvale, CA

11. AFC CONFIGURATIONS The AFC loop configurations discussed below employ

three basic frequency discriminator techniques. They may be categorized as

differentiate and multiply discriminator; cross product discriminator discrete Fourier transform discriminator.

These'discriminator techniques are suitable for CW, and in some cases DPSK and FSK. signals. Several variations on these basic configurations which are particularly well suited to DPSK and MFSK signaling are also presented. A total of eight configurations are discussed.'

The received signal is assumed to be of the form

s ( t ) z= A COS [mot + @(t)] + n ( t ) (2-1 1 where n ( t ) is white Gaussian noise with two-sided spectral density

' S n ( f ) = N o / 2 . (2-2)

The noise process n ( t ) may be represented as

n(t) = n l ( t ) cos w0t + n 2 ( t j sin wet. (2-3)

The processes / z l ( t ) and 122 ( t ) are independent, Gaussian processes with spectral densities

s,*,(O = sn,(t> = N o . (2-4)

The AFC loops to be discussed have the basic configuration of Fig. 1, The VCO output is assumed to be of the form

v,(t) = 2 cos 0 1 t (2-5)

which results in baseband outputs of

I(t) = A cos [ A u t -t @(t)]

+ n l ( t ) cos A u t + n 2 ( t ) sin A u t

and

Q(t) = A sin [ A u t -I- @(t)]

+ n l ( t ) sin A u t + n 2 ( t ) cos A u t

where

Am = - ~ 1 .

A. Dif ferent iator AFC (DIFAFC) The differentiator AFC (DIFAFC) configuration is shown

in Fig. 2 . The discriminator is that described by Park [ 3 ]

author. Credit is given when possible. ' The originator of each ofthe algorithms is, in some cases, not known by the

0090-6778/84/0800-0935$01 .OO O 1984 IEEE

. . > 1 '

936 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-32, NO. 8 , AUGUST 1984

4 I

I F iNPUT

I- DISCRIMINATOR

vco t LOOP F I L T E R

Fig. 1. AFC loop configuration.

@++ F I L T E R LOOP

Fig 2. DIFAFC configuration.

as the "ideal detector with inverse limiting." The DIFAFC is particularly amenable to analog implementation in contrast to the other configurations to be discussed, which are best implemented digitally. Results pertaining to other analog AFC loop implementations are given in [ 1 1 and [ 2 ] .

The discriminator output, which is the loop error control voltage, can be shown to be

This characteristic is plotted in Fig. 3 for two different low- pass filters. By analogy with phase-locked loops, a reasonable estimate of threshold is given by

(2.1-2)

where ,AF(peak) is the frequency error corresponding to the peak of the discriminator S-curve.

The DIFAFC is useful for tracking CW, DPSK, or the average frequency of an FSK signal. The input bandwidth, B L p l must be wide enough to pass the input signal spectrum, without serious distortion, for proper operation.

The noise performance of the DIFAFC has been shown to be [ 5 ]

B ~ ~ = LOW PASS LILTER NOISE BANDWIDTH(S1NGLE-SIDED)

Y Fig. 3. Discriminator characteristic.

B. Cross Product AFC(CPAFCJ An AFC configuration which employs a "cross product

discriminator" is shown in Fig. 4 . The CPAFC is similar t o the DIFAFC with the.differentiators replaced by delays. This is very convenient for a digital implementation as shown. Integrate-and-dump (I /D) filters are used as presampling filters t o limit the input noise. The filter noise bandwidth is

The discriminator output is represented by

(2.2-1)

(2.2-2)

which is the cross product of the vectors ps(k) and p s ( k - I ) , where

vs(k) = Zk + jQk (2.2-3)

is the baseband representation of the input signal. The discriminator characteristic for a sinusoidal input is

D,(w) = A ~ sinc2 ( - A:T1) sin A U T , (2.2-4)

where

sinc (x) = __ . sin x

.Y

This characteristic is shown in Fig. 5 . Loop threshold is estimated to occur when [see (2.1-2)]

U A ~ T , = 0.07. (2.2-5)

The performance of the CPAFC in the presence of Gaussian noise can be shown to be [21

for

fo r BL < BI. (2.1-3) u - BT = single-sided loo^ noise bandwidth

NATALI: AFC TRACKING ALGORITHMS 93 7

I .o t"

( .

oo NCO FILTER C-

t k - t k - l = T 1

Fig. 4. Cross product AFC (CPAFC).

. I . O L

Fig. 5. CPAFC discriminator curve.

The CPAFC is useful for tricking a CW carrier (or the average value of an FSK signal). A DPSK signal of the form

S( t ) = A d ( t ) cos (woT + 4) (2.2-7)

where d i t ) = ? I is the differentially encoded data, generates a loop error voltage

I I tk

I t Tn-I

Fig. 6 . Data bit with period To = NT,.

The discriminator function remains the same as for the CPAFC and threshold is estimated to occur when

OAfT, = 0.07. (2.2-10)

The loop performance in Gaussian noise can be shown to be [61

0.05 1 (BL/Bz)Bz2 O a f 2 =

SNRz [ 1 - y] 1 +- ___ 1) 2 S N R ]

(2.2-1 1)

(2.2-8) for

thus causing the error voltage to have the wrong sign when dk ' # d k - 1 , Several variations of the CPAFC suitable for use with DPSK'signals are given below.

1 ) Cvoss ,Product AFC with Sample Excising fov DPSK (SEAFC): The CPAFC qan be made to operate with DPSK signals by excising samples which occur around data transi- tions. Let us consider the case

N T , = Td (2.2-9)

where Td is the data period as shown in Fig. 6. In this case, if we excise those values of v,(n) [see (2.2-8)l which are derived around a data transition, then d, = d,*-1 , and the loop error voltage has the proper sign. Thus, the CPAFC can be made to work with DPSK data by excising the sample formed at the data transition and using the N - 1 remaining samples as shown in Fig. 7 .

N N > 2 and N T 1 =- = Td

BZ

Note that the above operation implies that bit sync is available. In the absence of bit sync, as during signal acquisition. the

loop can be operated without sample excising, as discussed above, as long as the number of samples per bit, N , is relatively large (N > 8). The large N will ensure that performance will not be degraded more than about 1.5 dB from that given by (2.2-6). N should be decreased to N = 2 and sample excis- ing begun as soon as bit sync is achieved in order to take advantage of a reduced input bandwidth (E1 = N J T d ) .

A comparison of performance predicted by (2.2-1 1) and digital simulation results is shown in Fig. 8 . Excellent agree- ment is obtained.

2) Angle Doubling A F C ( A D A F C ) .for DPSK Signals:

93 8 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-32, NO. 8, AUGUST 1984

I I I

t k - t k - l - '1 .

Fig. 7. Cross product AFC with sample excising for DPSK.

J

B I - 100 Hz

AGC INPUT BANDWIDTH, BI(AGC) = lKHz

EL - 1 H z (C/No - 00)

N = 2

3 - . A DIGITAL SIMULATION RESULTS - ANALYTICAL RESULTS

2 -

1 -

1 10 100 1000

SNRI

Fig. 8. Comparison of computed and digit simulation results for an SEAFC loop with DPSK signaling. The effect of AGC on the loop bandwidth is included. (Data provided by G. Spcci, STI.)

!

Another modification of the CPAFC for operation with DPSK This technique can be readily extended to a configuration is shown in. Fig. 9. Basically, the instantaneous phase of the suitable for QPSK signals by introducing a second angle carrier, relative to the receiver reference, is doubled, thus doubling operation, su'ch that angle quadrupling is achieved. removing the DPSK data. To show this, consider (2-6) and Alternatively, an analog frequency quadrupler followed by (2-7 j for the noise-free case. the CPAFC will accomplish the same thing.

The ADAFC can be made independent of data synchroniza- I(fn) = A d , COS [O(t,>] (2.2-12) tion by using a time invariant (as opposed to integrate-and-

Q(t,) =Ad, sin-[O(t,>l (2.2-13) The discriminator bandwidth may be reduced by summing dump) input filter and selecting an appropriate sample rate.

the angle doubled output samples as shown in Fig. '9. where d, = +1 is the differentially encoded data. Then, refer- The discriminator characteristic can be shown to be ring t o Fig. 8, M

X = I 2 - (2 2 = A 2 d n n n 2 COS [2e(tn>I (2.2-14) D,(Af ) = A 4 sinc4 ( n A f T , ) x m - sin (m411AfT1)

m = l Y , = 21nQ, = A 2 d , 2 sin [2O(f,)] (2.2-15) + (M - m ) sin [(m + M ) 4IIAjT, ] . (2.2-16)

where This result is plotted in Fig. 10 with M as a parameter. As ill

d,? = 1. increases, the tracking range decreases approximately as 1/M. Note that for higher values of M, the discriminator character-

It should be noted that analog frequency doubling followed istic passes through u, = 0 with positive slope and sufficient by . the CPAFC is mathematically equivalent to the ADAFC amplitude such that false lock could occur. Loop threshold (when bit sync is available). is estimated to occur when AfT, = 0.04 for M = 1 .

NATALI: AFC TRACKING ALGORITHMS

r - - - - - - r - - - - - - - 7

V,(t) B I T SYNC

t. I I I I

X - 1 I " k I I I

B I T S I N C I I I . I L _ _ - _ _ _ - J L _ - _ _ _ _ _ J ANGLE DOUBLING O I S C R I M I N A T O R

Fig. 9. Angledoubling AFC (ADAFC).

-3 - 2 'I . S".ETTJf

I Fig. 10. ADAFC discriminator characteristic for some values of M.

The loop performance in Gaussian noise is [ 7 ]

0.05 1 (BL/BI)BIZ

SNRI ( 1 - 7) O a f 2 ( M = 1 , N > I ) =

1 2.5 2 0.5 - +- N SNRI SNRI' S N R I ~ (2.2-17)

+-+-I 0.05 1 (BL/BI)B12

SNR, ( 1 - 7) 0af2(M > 1, N = 1) =

+ M 2 OS S N R I ~ 1

where

BL - < l BI

Threshold is estimated to occur when 0.04

M O A f T 1 =-

939

k )

3) Cross-Dot Product AFC (CDPAFC) for DPSK Signals: A third variation of the CPAFC configuration, which is suit- able for use with DPSK signals, is shown in Fig. 11. In this scheme, both the cross and dot products of adjacent signal samples are computed and multiplied together to remove the data modulation from the loop error signal.

The (CDPAFC) is mathematically equivalent to the ADAFC with M = 1 . The discriminator characteristic and noise per- formance are the same as shown for the ADAFC (M = 1 ) . Note that , as in the case of the ADAFC, the CDPAFC can be made independent of data synchronization by using a time invariant input filter and selecting an appropriate sample rate.

4 ) Cross-Product AFC w i t h Decision Feedback (CPAFCDF) for DPSK Sigmling: The CPAFCDF is a variation of the cross- dot product AFC and is of considerable interest for DPSK signals. A block diagram of this AFC configuration is shown in Fig. 12 . The cross and dot products of the input samples are formed in the same manner as for the CDPAFC; however, data decisions are performed on the dot product, and the estimated data are used to resolve the sign ambiguity of the loop error voltage.

The discriminator characteristic is given by DE(A.f) = A,2 sinc2 (nAfT1) sin ( ~ I I A ~ T , ) (2.2-19)

sgn [cos ( 2 n A f T l ) ]

which is shown in Fig. 13. Threshold is es t imated to occur when

o A f T 1 = 0.07. (2.2-20)

Comparing (2.2-20) and (2.2-18), we see that this configura-

940 IEEE TRANSACTIONS ON COMMUNICATIONS. VOL. COM-32, N O . a , AUGUST I 984

V J k ) =

Fig. 11. Cross-dot product AFC (CDPAFC) .

VJk)

Fig. 12. Cross product AFC with decision feedback (CPAFCDF).

- 1 . d

Fig. 13. AFCDF discriminator curve.

NATALI: AFC TRACKING ALGORITHMS 94 1

t ion, is considerably more tolerant to frequency fluctuations than the ADAFC or CDPAFC configurations before losing lock.

The loop performance in Gaussian noise is approximated by [41

0.05 1 (BL/BI)B12 oaf =

(1 - 2 P,)2 SNRI (1 - 2 N B L / B I )

1 + 20 P,(BL/BI)( 1 - 1.6 P,)

( 1 -- 2 P,)2 (2.2-21)

where

1 1 P, = - exp (-SNRI) for oaf Q -

2 T d

A comparison of computed and measured performance is shown in Fig. 14. These data were taken with a PN/DPSK receiver with code tracking loop and AGC loop operating. A second-order CPAFCDF was employed since the loop was required to track a high Doppler rate in the intended applica- tion. The computed curve includes 0.8 dB correlation loss as indicated by data error rate measurements. The computed curve also takes into account the change in loop bandwidth due to AGC signal suppression, at low SNRI. This is given, for a second-order loop, by

1 +- 1 ,

4i-0 where

BLo = loop bandwidth at SNRl 3 0 dB

.6

. 3

- .4

- . 5 -

-

- - - -

.2 - -

0.1 I I I I I I I I I I I I I -3 -2 -1 0 1 2 3 . 4 5 6 7 8 9 10 11

SNRI(dB)

Fig. 14. Comparison of computed and measured CPAFCDF performance for a PNlDPSK system. Computed curve assumes 0.8 dB of correlation loss as evidenced by data error rate measurements. Change in bandwidth due to AGC signal suppression due to low SNR, is also included. (Data provided by M. Paff, STI.)

to = loop’damping factor at SNRI 3 0 dB I and

we have

(2.3-3)

SNR, = Signal-to-noise ratio in AGC input IF bandwidth. 1 M - 1

The computed and measured results show good agreement, M m=O with the computed result being slightly optimistic near thresh- old,

C. Discrete Fourier Transform AFC (DFTAFC) ak = R K +jMk (2.3-5)

The configuration of the DFTAFC is shown in Fig. 15. and Im and Q, are samples of ~ ( t ) and Q ( f ) given by (2-6) The DFT can be viewed as a spectrum analysis algorithm and (2-7). which provides the equivalent of a “fi l ter bank.” The DFT The ‘‘p0wer7) of the kth output a t t ime T~ is computes the coefficients.

Mk = - x [Q, cos (7) - I , sin ( y )] (2.4-4)

where

P k = ( a k I 2 = R k 2 + M k 2

(2.3-1) The discr iminator output error es t imate is

1 M - 1 (2.3-6) ak = - X(mTs)e-ikm-OTs12

M m = 0

where T,y = sample period

T o = M T , is the analysis interval

2 n

TO WO =- ‘

If we let where

* X(rnT,) = I , -t jQ, (2.3-2) A f = frequency error.

(2.3-7)

942 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-32, NO. 8 , AUGUST 1984

L P F j R k + A I D

S ( t ) DISCRETE P = k

Rk2 t Mk2 FOURIER TRANSFORM

L L P F d + A I D Q l l

f B I T SYNC

1

NCO f- LOOP M

O0 - F I L T E R I

Fig. 15. DFT AFC configuration.

The discriminator characterist ic is plotted in Fig. 16. Thresh- old is estimated to occur when

U A ~ T O = 0.17. (2.3-8)

The loop noise performance is given by [ 8 I ,

where

I B I = - .

TO Note that when the DFT analysis interval is synchronized with the bit timing interval, and the DFT interval is equal to or less than ‘a bit interval, then the DFTAFC can be used with DPSK and FSK signaling (tracks “average”. frequency in the FSK case). Degraded operation can be achieved before bit syn- chronization is available by employing an analysis interval that is considerably less than the data bit interval (To T d ) .

1 ) Discrete Fourier Transform AFC with Decision Feed- back (DFTAFCDF) for MFSK Signaling: One of the principal

’ applications of the DFTAFC configuration is for MFSK signaling. The MFSK signal consists of M tones with spacing 1/T, (Hz) and frequencies

fo f (2m - 1)/2To (2.3-10)

. for

m = 1 , 2, .... M / 2 .

The DFT may be viewed as a filter bank of (sin x/x)* filters. .The frequency response of the kth filter is

Fig. 16. Discriminator curve of “DFT” AFC algorithm.

where

2rI wo =-

TO To = DFT analysis interval

k = 0, f 1, f 2 , ....

Thus, the f i l ters corresponding to k = f (2m - 1) [designated Pk(Dj1 are matched to the M data tones. The filter responses are shown in Fig. 17 for M = 8 . The filter noise bandwidth is ] / T O . The filters corresponding to

k = 0, +2m (2.3- 12)

fo r

(2,3-11) and designated Pk (AFC), are spaced in between the FSK tone frequencies as shown by the dotted lines of Fig. 17 .

In the case of the DFTAFCDF, the kth received tone is detected by the DFT f i l ter P,,. and the discriminator output

NATALI: AFC TRACKING ALGORITHMS 943

-4 -3 -2 -1 2 3 4

A f T

Fig. 17. Frequency response of DFT coefficients.

is formed by

D(w) = Pk+ (AFC) - f‘k- 1 (AFC) (2.3-13)

as shown in Fig. 18 [9]. That is, once the decision is made as to what data tone was sent, the discriminator error number is formed from the adjacent DFT filter bins. This constitutes decision feedback. The DFT analysis interval is assumed to be synchronized with the data tone period.

This scheme results in considerably improved performance over configurations which track “average frequency” since it can employ a narrower discriminator bandwidth and the result does not depend on the data tone distribution.

The discriminator characteristic has the same form as that given by (2.3-7), centered on the transmitted tone. The threshold i s estimated to occur when uafTo = 0.17.

The MFSK detector must, of course, have an initial fre- quency uncertainty of less than one-half the tone spacing for proper operation. In applications where the initial uncertainty is greater than this amount, a frequency acquisition phase employing the DFTAFC discussed above may be employed.

The loop noise performance in Gaussian noise is given by [ 81

0.31 BL

( P , / N o ) ~ o ~ ( l - ~ e ) ~ ( 1 - ~ N B L T , ) a A j 2 =

(2.3-14)

where

* exp (P ,TO/~NO)

1

’ To B I = - .

The computed performance is compared with measured per- formance in Fig. 19. The experimental data were taken with an FH/8-FSK receiver employing a second-order DFTAFCDF with AGC. The computed result includes the effect of loop bandwidth change due to AGC signal suppressions at low SNRI. This is given for a second-order loop by (2.2-22). ’

NCO LOOP

FILTER

10 r 1 - a - 7 -

6 -

50 - 40 -

30 -

20 -

uA f ( H Z )

10 -

8 -

6 -

4 -

2 -

1 - 16

Fig. 18. DFTAFCDF configuration.

I 1

0 CfLCULATED MEASURED DPTA - I I

C/No (dB)

Fig. 19. Second-order AFC performance in noise. Measured data include effects of hop-timing loop jitter, quantization noise, and AGC. (Data provided by B. Campeau, STI.)

The computed and measured results show excellent agree- ment. There is some divergence at high C/No where the A/D quantization noise becomes evident in the measured data.

111. PERFORMANCE COMPARISONS The choice of a particular AFC cpnfiguration is based on

0 signal modulation compatibility 0 hardware considerations 0 performance.

The compatibility of the various AFC algorithms with CW, DPSK, and MFSK signaling formats is summarized in Table I.

The loop performance equations are summarized in Table 11. Detailed comparisons are given below for the different signal formats.

944

AFC 0NFlGURATl

UIF4FC

CPAFC

UFTAFC

IUFIAFCUF

SEAFC

AUAFC ( N = l

AOAFC (m=1

and CDPAFC

CPAFCDF

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-32, NO. 8 , AUGUST 1984 . .

TABLE I

CONFIGURATION AFC I cw

CPAFC

SEAFC

CDPAFC

ADAFC

CPAFCDF

DFTAFC

DFTAFCD

MODULATION FORMAT

I

* THESE CONFIGURATIONS TRACK AVERAGE FREQUENCY.

THE LOOP BANDWIDTH MUST BE CONSIDERABLY LESS

THAN THE SIGNALING RATE FOR GOOD TRACKING.

OF THE DATA TONE OCCURENCES.

PERFORMANCE ALSO DEPENDS ON T H E S T A T I S T I C S

TABLE I1 AFC NOISE PERFORMANCE SUMMARY

B 'Terms of t h e fnrm 1- & a r e general'ly s m a l l and are neglected here

I

SNRl<<OdB

NBL/BI<<l

0.1126 (BL/BI) B I 2

SNRI4

0.026 (BL /B l ) B 1 2

(1-2 P e ) ' SNRI2

20 Pe (BL/BI) C1-1.6 P e ) 1

,051 ( B L / b I ) d l Z

M 5NK.

OISCKI I I INAT0K TRACKING

RANGE (dz)

? .38 B I

i .21 B I

t .51 8,

+ .51 B I

k .21 RI

t .21 s1

NATALI: AFC TRACKING ALGORITHMS

A . C W Signals In some application, a CW carrier or residual carrier com-

ponent is transmitted for synchronization or Doppler tracking purposes. The DIFAFC, CPAFC, and DFTAFC are potential choices for this application. In order to get a meaningful performance comparison, we must first select parameters such that the loops have equivalent tracking ranges, i.e., from Table 11.

0.38 BI (DIFAFC) = 0.2 1 BI (CPAFC)

945

0.5 1

To (DFTAFC) - -

which leads t o 2 .43

BI (CPAFC) = - TO

1.34 BI (DIFAFC) = -

TO

(3.1-1)

(3.1-2)

(3.1-3)

where To is the DFT analysis interval. Comparing performance for SNRI < 1 with the above values for B I , we find

U A ~ ' (DIFAFC) = 0.40 BL

(PSI& 1' T , for SNRI < 1 (3.

0 .3 7 BL O a f 2 (CPAFC) = -- f o r SNR, < 1,

(~s /No)2 'To

NBL/BI< 1 (3.

-4)

-5

NBL/BI < 1. (3.1-6)

The conclusion is that these configurations provide essentially identical performance (within the accuracy of the approximate performance solutions) at very low SNRI. Normalized per- formance is plotted in Fig. 20 versus SNRI. The DIFAFC exhibits better performance for the higher' values of SNRI than the other considerations. The DFTAFC and CPAFC (N = 1 ) have essentially identical performance for the chosen parameter set.

These results indicate that tracking performance is very similar for all the loops as they approach threshold. The analog DIFAFC gives improved performance for SNRI > 0 . The performance of the CPAFC can be improved at SNRI > 0 dB by increasing N ; however, this improvement is 'limited by the restriction that NBL T < 1 .

B. DPSK Signals Several of the AFC configurations discussed above are

suitable for use with DPSK modulated signals. Referring to Table I the DIFAFC, SEAFC, CDPAFC, ADAFC, CPAFCDF, and DFTAFC are recommended candidates.

The DIFAFC is the only configuration which does not require bit sync for full performance. The CDPAFC and ADAFC can, be made independent of bit sync by subsituting a low-pass filter (LPF) for the input integrate-and-dump (I&D) filter. However, the LPF must be at least twice as wide as the I&D filter to prevent degradation due to intersymbol interference, and poorer loop performance will result.

The AFC loop generally has an acquisit ion mode to accom- modate the lack of, bit sync and large initial frequency un- certainties. This mode is characterized by a wider input band-

1.0

0.8

0.6

0.4

0.2 Y u g.5 fT B a 0.1

:: .Of

0 .O€

w

w z - 4 ar

Y N

i Q

w

6 .04 z

.92

.01

I I I I I I

BLT-O.Ol

I I I I I I -4 - 2 0 2 4 6

SNRI ( d B )

Fig. 20. Loop tracking performance comparison for CW signals

width BI and possibly loop bandwidth BL than used during tracking. Lack of bit sync during acquisit ion can be'accom- modated, for those configurations'that use it , by taking several samples per bit such that the percentage of samples affected by the data transitions is small. This is consistent with the wider input bandwidth and pull-in range often required to handle initial frequency uncertainties since BI = N / T d (Sec- t ion 11-B-1). A wide 'BL is often employed for rapid loop setting. The acquisition mode is not 'discussed here in detail; however, performance can be inferred from results already presented.

The AFC loop generally operates with the minimum input bandwidth during tracking, in order to achieve the best per- formance. The tracking 'mode parameters for the various loops are summarized in Table 111. A performance co.mpari- son is shown in Fig. 21. ThaCPAFCDF appears to achieve the best noise performance of any configuration for DPSK signal- ing (assuming perfect bit sync). The CDPAFC and ADAFC also provide excellent noise performance but threshold almost 3 dB higher (this situation may change if there is significant bit timing jitter, since it will probably have a greater effect on the CPAFCDF) .

The DFTAFC and DIFAFC are not as efficient in noise as the cross product detector loops; howtrver, they exhibit as threshold about 2 dB lower than the CPAFCDF.

The SEAFC exhibits a threshold approximately equivalent t o t h a t o f the CPAFCDF but has significantly poorer noise performance.

C. MFSK Signals The DFTAFCDF (Section 11-C-1) was derived specifically

for use with MFSK signals and provides very efficient per- formance in noise as given by (2.3-14).

An "average frequency" tracking loop such as the DIFAFC requires that the discriminator curve span the entire signal

946 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-32, NO. 8, AUGUST 1984

AFC

i O N F I tiURA1 ION

D l F A F C

S E A F C ( N = 2 )

CDPAFC d AORFC ( M = 1 )

CPAFC OF

OFTAFC

*Es /No = __ and Ne R C

TABLE I11 LOOP PERFORMANCE SUMMARY FOR DPSK SIGNAL TRACKING

RAllGE. AF/Ru T R A C K I N G

t 0.75

r 0.12

t 0.12

! 0.2

t 0.51

THRESHOLO

O A f " U

0.25

0. I 4

0.04

0.07

0. I7

symbol I ' a te

LOOP ERROR 2 2

" A f / R U

0.051 ( E L / R o ) 2 . 5 E 5'"O r1-mL'8;J [' ' ' (ES/No) + (ES/No) 0.5 ' I

where P = _I_ exp ( - [$ /No) e 2

ENERGY-TO-NOISE DENSITY RATIO,ES/N,(dB)

Fig. 21. Loop tracking performance comparison for DPSK signal modula- tion. (parameters selected for best performance after bit sync has been ac- quired.)

NATALI: AFC TRACKING ALGORITHMS 947

0.9 1.0

0.8 0.7

0.6

0.5

0 .4

PI 0 PI PI w

0 . 3

0.2 z w 3 G- U T E c+Af 0

CL 0 0

0 .09 0.1

5 .07 z .06

.05

.04

y .08 4

0

.03

.02

.O’

1

I -6 -4 -2 0 2 4 6 8

P s To/N, ( d B )

. .

Fig. 22. Performance comparison for DFTAFCDF and DIFAFC for an 8- FSK signal in noise.

bandwidth. For example, a n 8-FSK signal with tone spacing l / T o uses tone frequencies of f o * o.5T0, f o * 1 .STo , f o * l .STo, fo * 3.5T0 about the carrier frequency fo. Let us set the discriminator peaks of the DIFAFC to coincide with the tonesfo k 3.5To. Then from Table 11,

Bz (DIFAFC)=- = 9.21 To. 3.5To

0.38 (3.3-1)

A comparison of the performance of the DFTAFCDF and the DIFAFC is shown in Fig. 22 for an 8-FSK system. The ad- vantage of the DFTAFCDF is clearly shown for this signal format.

ACKNOWLEDGMENT This paper presents previously unpublished results ob-

tained by the author and his associates at Stanford Telecom- munications, Inc. The DIFAFC noise performance, (2.1-3), represents a joint effort by Q. Hua and the author (1978) . The result for the SEAFC, (2.2-11), was obtained by A. J . VanDierendonck (1 98 1 ). The remaining noise performance results were obtained by the author.

Preliminary work on the ADAFC and CDPAFC algorithms was performed by B . Currivan and S. Chu. The DFTAFCDF algorithm was proposed by M. Miller and J. Ohlson. The CPAFCDF algorithm was first suggested, to the best of t he author’s knowledge, by the Mission Research Corporation.

REFERENCES D. Richman, “Color-carrier reference phase synchronization accuracJ in NTSC color television,” Proc. IRE, vol. 42, Jan. 1954. L. H. Enloe, “Decreasing the threshold in FM by frequency feedback.” Proc. IRE, vol. 50, Jan. 1962. J. H. Park, “AM FM detector for low SIN,” IEEE Trans. Comrnun. Technol., vol. COM-18, Apr. 1970. F. D. Natali, “Noise performance of a cross-product AFC with decision feedback for DPSK signals.” in Conf. Rec. Naf . Telecommun. Conf., Nov. 1983. Q : Hau and F. Natali, unpublished, 1978. A. J. Van Dierendonck, unpublished, 1981. F. Natali, unpublished, 1981. -,unpublished, 1982. M. Miller and J. Ohlson, private communications. 1982.

* Francis D. Natali received the B.S. degree in electrical engineering from Rensselaer Polytechnic Institute, Troy, NY, in 1960, and the M.S. and Ph.D. degrees in electrical engineering from the State University of New York at Buffalo in 1964 and 1967, respectively.

He joined Stanford Telecommunication, Inc., Sunnyvale, CA. shortly after it was formed in 1973. He has been a technical contributor in the areas of spread-spectrum signaling, demodulators, and rang- ing receivers, including applications to the Global

Positioning System. Previous to joining STI, he was employed by Philco-Ford in the Communications Sciences Department, and Cornell Aeronautical Labo- ratories in the Systems Research Department.