Abstract:
An apparatus for obtaining carrier synchronization acquisition in a digital burst mode communication system is provided. An accurate estimate of the carrier phase of the unmodulated preamble, θ, is obtained by determining which of 256 intervals X o   2  +X e   2  fall into, and by evaluating which of 256 intervals Y o   2  +Y e   2  fall into. A quantized value is assigned to the generated output code for both X and Y inputs. The square root and arc tangent are evaluated to determine the value θ.

Description:
FIELD OF THE INVENTION 
     The present invention is directed to a method and apparatus for carrier synchronization acquisition in a digital burst communication system. Specifically, a simple programmable read-only memory (PROM) implementation for acquisition of carrier synchronization is provided given samples taken from a binary alternating preamble at the rate of two samples/symbol. 
     BACKGROUND OF THE INVENTION 
     In a digital burst mode communication system, a binary alternating preamble normally precedes the data. At a receiver, an oscillator of the same frequency as the transmitted waveform is used, but the phase difference between the received sinusoidal waveform and the oscillator is unknown and needs to be estimated for coherent demodulation. This is known as the carrier acquisition problem. 
     In a digital demodulator implementation, samples are taken from the in-phase and quadrature components x(t) and y(t), respectively, as shown in FIG. 1. A recovered or estimated carrier is passed through a phase shifter 10 which provides an inphase carrier to mixer 12 and a quadrature carrier to mixer 14. This results in a data stream I=x(t) for the in-phase channel and a data stream Q =y(t) for the quadrature channel, and these are sampled at reference numerals 16 and 18. 
     The samples are usually taken at a rate equal to or larger than two samples/symbol. Normally, two samples/symbol makes for an efficient implementation, and this is the case considered below. Samples taken on the X channel during the preamble are sequentially numbered as X 1 , X 2 , X 3 , X 4 , X 5 , . . . 
     Due to the alternating nature of the binary preamble, and moreover, since the samples are taken at a rate of two samples/symbol, it follows that X 1  =-X 3  =X 5  . . . in the absence of noise. Therefore, in order to decrease the effect of noise, a quantity X odd  is formed by combining the odd-numbered samples on the X channel in the following manner: 
     
         X.sub.odd =X.sub.1 -X.sub.3 +X.sub.5... 
    
     This has the effect of averaging out the value of the odd-numbered samples. The same procedure is repeated to obtain X even , Y odd , and Y even . 
     Given four values (X e , X 0 , Y e , Y 0 ) of 8 bits (1 byte) each, it is desired to find a simple PROM implementation to evaluate ##EQU1## such that the maximum error in evaluating θ (because of the finite precision resulting from the finite PROM size) is as small as possible. 
     This problem arises when implementing a digital demodulator for digital burst mode communication. As noted above, an alternating preamble usually precedes the data. At the receiver, an oscillator of the same frequency as the transmitted waveform is used, but the phase difference, θ, between the received sinusoidal waveform and the oscillator is unknown, and an estimate of it is desired. As shown in FIG. 1, samples are available from the in-phase and quadrature components x(t) and y(t), respectively, sampled at the rate of two samples/symbol (half a sinusoidal period represents one symbol). By denoting the samples on the X channel as X 1 , X 2 , X 3 , X 4  ..., and the samples on the Y channel as Y 1 , Y 2 , Y 3 , ..., the quantities X 0 , X e , Y 0 , and Y e  are formed as follows: 
     
         X.sub.0 =X.sub.1 -X.sub.3 +X.sub.5 -X.sub.7... 
    
     
         X.sub.e =X.sub.2 -X.sub.4 +X.sub.6 -X.sub.8... 
    
     
         Y.sub.0 =Y.sub.1 -Y.sub.3 +Y.sub.5 -Y.sub.7...             (2) 
    
     
         Y.sub.e =Y.sub.2 -Y.sub.4 +Y.sub.6 -Y.sub.8... 
    
     where 
     
         X.sub.1 =cos θ sin α+noise 
    
     
         X.sub.2 =-cos θ cos α+noise 
    
     
         X.sub.3 =-cos θ sin α+noise                    (3) 
    
     
         X.sub.4 =cos θ cos α+noise 
    
     repeats ever four samples and 
     
         Y.sub.1 =sin θ sin α+noise 
    
     
         Y.sub.2 =-sin θ cos α+noise 
    
     
         Y.sub.3 =-sin θ sin α+noise                    (4) 
    
     
         Y.sub.4 =sin θ cos α+noise 
    
     repeats ever four samples, and where α is the phase displacement between the sinusoidal signal and the sampling clock. 
     The purpose of computing X 0 , X e , Y 0 , and Y e  as above before estimating θ is to decrease the noise variance by averaging out over several symbols before performing squaring operations. Clearly, 
     
         Y.sub.o.sup.2 +Y.sub.e.sup.2 ≈sin.sup.2 θ 
    
     
         and 
    
     
         X.sub.o.sup.2 +X.sub.e.sup.2 ≈ cos.sup.2 θ 
    
     from which it follows that equation (1) is an estimate of θ as stated above. 
     Note that the value of θ estimated above will be in the first quadrant (i.e., between 0° and 90°). Therefore, there is a four-fold ambiguity in the value of θ that needs to be resolved. This can be taken care of in the detection of a unique word that follows the preamble. It is also possible to reduce the four-fold ambiguity to a two-fold ambiguity (which must then be resolved by the unique word) by examining the, sign of X 0  X e  +Y 0  Y e . 
     This method of averaging several symbols before computing an estimate of the carrier phase is well known and widely used. The present invention is directed to finding a simple PROM implementation to obtain an accurate estimate of the carrier phase given four quantities X odd , X even , Y odd , and Y even . 
     A method of estimating the carrier phase known in the art is shown in FIG. 2. Basically, the method of the prior art consists of squaring and adding operations performed on X odd , X even , Y odd , and Y even . 
     First, (X e ) 2  is provided at the output of squaring circuit 20, with the most significant byte being loaded into adder 22 and the least significant byte loaded into adder 24. (-Xo) 2  is then provided at the output of squaring circuit 20, and is added to (X e ) 2  in adders 22 and 24. The log of (X e ) 2  +(X o ) 2  is then calculated in log circuit 26. The log of (Y e ) 2  +(Y o ) 2  is similarly provided by circuits 28-34. The quantity (Y 0   2  +Y e   2  /(X 0   2  +X e   2 ) is then obtained by subtracting the output of log circuit 34 from the output of log circuit 26 in subtracters 36 and 38. Circuit 40 then obtains the square root by dividing by 2, and calculates θ by taking the arctan of the result. 
     The additions are performed with full precision, i.e., 2 bytes, obtained at the adder&#39;s outputs because of the desire to obtain an accurate estimate. Next, a division operation is performed. However, in the method of the prior art, the division cannot be accomplished in a PROM since the numerator and the denominator are each 2 bytes long. Therefore, division is accomplished by computing logarithms, subtracting the results, and then taking exponentials, all implemented in PROMs. Finally, an inverse tangent operation is performed to obtain the desired carrier phase estimate. The disadvantage of the method of the prior is that several PROMs and latches (not shown) are required in order to estimate the desired angle. 
     SUMMARY OF THE INVENTION 
     An object of the present invention is to overcome the problems inherent in the prior art by providing a method and apparatus which can accurately estimate the carrier phase of the unmodulated preamble in a digital burst mode communications system with a minimum amount of hardware. 
     With the present invention, it is possible to obtain an accurate estimate of the carrier phase of the unmodulated preamble by using only two PROMS and a latch. Mathematically, this corresponds to evaluating the inverse tangent of a square root. The quantity under the square root is a ratio of sums of squares. The numerator is Y 2   odd  +Y 2   even  and the denominator is X 2   odd  +X 2   even . 
     The first of the two PROMS is referred to as the coded quantized sum of squares (CQSS) PROM performing the following tasks. Given two inputs of one byte (8 bits) each, the sum of squares of these two inputs is obtained, resulting in a twobyte-long quantity. The output of the CQSS PROM must be confined to one byte, however, because of the input requirements of the second PROM. Therefore, the two-byte-long sum of squares must be reduced to one byte with a minimum loss of accuracy. This is achieved by using a quantization scheme based on logarithmic quantization. 
     When the quantity to be quantized is small, a further improvement over logarithmic quantization is possible. This is due to the highly non-uniform distribution of sums of squares of small integers. The number of quantization intervals is chosen as 256 in the preferred embodiment. 
     The output of the CQSS PROM is a code number indicating which of the quantization intervals the sum of squares falls into. Thus, the sum of squares representing the numerator is accurately quantized and the result is stored in a latch using only one byte. This is then repeated for the denominator. The quantities (one byte each) are then applied to the second PROM, where the code number is translated into its actual value: division, square root, and inverse tangent operations take place, the results being precomputed and stored in the PROM. The carrier estimate then appears at the output. 
    
    
     DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a block diagram showing portions of a conventional digital demodulator; 
     FIG. 2 is a block diagram showing a prior art system for estimating carrier phase array; 
     FIGS. 3A and 3B illustrate a preferred embodiment to the present invention; 
     FIG. 4 illustrates the relationship between X 1 , X 2 , Y, θ, and ε; and 
     FIG. 5 illustrates resulting quantization errors of X and Y. 
    
    
     DESCRIPTION OF THE INVENTION 
     FIGS. 3A and 3B illustrate the preferred embodiment of the present invention. As shown in FIGS. 3A and 3B, an accurate estimate of the carrier phase of an unmodulated preamble is obtained using only 2 PROMS and a latch. 
     As background information, a simple mathematical derivation follows. Given X 1  &gt;0, it is desired to determine the largest value of X 2 , for which an angle ε of FIG. 4 is less than or equal a fixed amount ε max  for any given Y. It can be mathematically shown that the answer is X 2  =aX 1  where ##EQU2## and that the value of Y resulting in ε=ε max  is 
     Also, as noted above, a similar procedure gives identical results if the X&#39;s and Y&#39;s above are interchanged. Accordingly, the following conclusion can be derived from the results. 
     If it is desired to find the value of 
     
         φ=tan.sup.-1 (Y/X), 
    
     and X is quantized into intervals of the form (X sin , aX min ), (aX min , a 2  X min ) . . . and similarly for Y, and the quantization value is taken as the geometric mean of the interval, then the quantization of X and Y will cause an error of no greater than 2ε max  in the value of φ, as illustrated in FIG. 5. If on the other hand, it is desired to find the value of ##EQU3## and X and Y are quantized as above, then the error resulting from the quantization will be no greater than ε max . This is true because quantizing X in steps of a is equivalent to quantizing √ in steps of √ . For a small ε max , it is easy to show that if a corresponds to ε max , then the √ corresponds approximately to 1/2 ε max . Relating the above to the present invention, values along the X axis are of the form X 0   2  +X e   2  where X 0  and X e  are each 8 bits, i.e., they are integers between -127 and +127. 
     Assigning a quantization value of 0 for the case X 0   2  +X e   2  =0, intervals can be formed beginning with X 0   2  +X e   2  =1, the intervals being of the form: (1, a), (a, a 2 ), (a 2 , a 3 ) . . ., where the value of a is chosen such that A 255  =127 2  +127 2 , i.e., such that 255 intervals are assigned to all non-zero values of X 0   2  +X e   2 . Accordingly, this provides the results that a =1.04155, i.e., √a=1.02056. 
     From equations (1) and (2) above, the corresponding maximum error in evaluating θ would be about 1°. 
     Further improvements are possible by noticing that for small values of X 0   2  +X e   2 , the interval (X n , aX n ) could be empty. Indeed, since X 0   2  +X e   2  cannot take values between 1 and 2, it follows that the 16 intervals ##EQU4## will all be empty. 
     Therefore, a list of the values of X o   2  +X e   2  in the order of increasing magnitude is shown in Table 1. It has been determined by experimentation that there were 42 values of the sum of squares under 100. Since a 42  =5.5≦100, it is possible to assign each of these 42 values a quantized value equal to itself (i.e., perfect quantization), while at the same time providing better quantization for the values ≧ 100. This follows because only 42 intervals would have been assigned so far, as opposed to 113 (a 113  ≈ 100) intervals that would have been assigned had deviation from the logarithmic quantization rule not occurred. 
     For values X 0   2  +X e   2  &lt;100, the quantization intervals collapse to single points, the regions between those points being values that are impossible for X 0   2  +X e   2  to obtain. 
     Proceeding with the larger values of X 0   2  +X e   2 , because the sum of squares of integers follows a regular pattern, an advantage can be obtained by distorting the shape of the intervals from the logarithmic rule, thus finding better quantization values. This is performed for values of X o   2  +X e   2  up to 442 as shown in Table 1. For larger values, the benefits of deviating from the logarithmic quantization rules become increasing small and not worth pursuing. Thus, the values from 445 (the sum of squares following 442) up to 32,258 were divided into 160 intervals (since 96 out of the 256 intervals had already been assigned). By setting a 160  32 32, 258/445, it is determined that a =1.027133. An examination of Table 1 reveals that for all 256 intervals, the ratio between the upper end of the interval to the lower end is ≦ a. 
     The logarithmic rule of quantization assigns a geometric mean of the interval as the quantization value. However, since √a ≈ 1+(a/2), the arithmetic mean is used instead for simplicity. The complete listing is shown in Table 1. 
     With value of a=1.027133, or equivalent √a=1.0134757, the maximum error in computing θ resulting from quantization may be found from equations (1) and (2), and in this case, comes out to be 0.39°. 
     The PROM implementation that carries out the method of the present invention is shown in FIGS. 3A and 3B. Two PROMs are used in the implementation. As shown in FIG. 3A, the inputs to the first PROM 100 are X 0  and X e  (and later Y 0  and Y e , as shown in FIG. 3B), and the output is an 8-bit code indicating which of the 256 intervals X 0   2  +X e   2  falls into, and this is stored in latch 102. The output is more accurate than having the quantized value itself at the output, since an 8-bit representation of the quantized values will introduce substantial round-off errors for small values. The second PROM 104 assigns the quantized value to the output code of the first PROM for both the X and Y inputs, evaluates the square root and the arc tangent, and a value for θ is provided at the output. 
     The value of θ at the output is limited to 8 bit accuracy. Therefore, if the 90° interval is quantized for θ into 256 values, a quantization interval of (90/256)=0.35 is obtained. This value is smaller than the 0.39° value obtained above. Accordingly, this means that the maximum error in evaluating θ will be 0.7°. If the quantization interval is increased to 0.40°, this provides a guarantee that the maximum error in evaluating θ will be less than or equal to 0.40°. 
     Thus, the present invention provides an accurate estimate of the carrier phase of the unmodulated preamble using only 2 PROMs and a latch. 
     
                       TABLE 1______________________________________VALUE OF SUM     ASSIGNEDOF SQUARES       CODE______________________________________ 0                0 1                1 2                2 4                3 5                4 8                5 9                610                713                816                917                1018                1120                1225                1326                1429                1532                1634                1736                1837                1941                2145                2249                2350                2452                2553                2658                2761                2864                2965                3068                3172                3273                3374                3480                3581                3682                3785                3889                3990                4097                4198                42100-101           43104               44106               45109               46113               47116-117           48121-122           49125               50128               51130               52136-137           53144-146           54148-149           55153               56157               57160               58162-164           59169-170           60173               61178               62180-181           63185               64193-194           65196-197           66200-202           67205-208           68212               69218-221           70225-229           71232-234           72241-245           73250               74256-257           75260-261           76265-269           77272-274           78277-281           79288-290           80292-293           81296-298           82305-306           83313-317           84320-328           85333-340           86346-349           87353-356           88360-365           89369-377           90386-389           91392-397           92400-410            93416-425           94433-442           95445-457           96458-469           97470-482           98483-495           99496-508          100509-522          101521-536          102537-551          103552-566          104567-581          105582-597          106598-613          107614-630          108631-647          109648-664          110665-682          111683-701          112702-720          113721-740          114741-760          115761-780          116781-801          117802-823          118824-846          119847-869          120870-892          121893-916          122917-941          123942-967          124963-993          125 994-1020        1261021-1048        1271049-1076        1281077-1105        1291106-1135        1301136-1166        1311167-1198        1321199-1230        1331231-1264        1341265-1298        1351299-1333        1361334-1369        1371370-1407        1381408-1445        1391446-1484        1401485-1524        1411525-1566        1421567-1608        1431609-1652        1441653-1697        1451698-1743        1461744-1790        1471791-1838        1481839-1888        1491889-1940        1501941-1992        1511993-2046        1522047-2102        1532103-2159        1542160-2218        1552219-2278        1562279-2340        1572341-2403        1582404-2468        1592469-2535        1602536-2604        1612605-2675        1622676-2747        1632748-2822        1642823-2898        1652899-2977        1662978-3058        1673059-3141        1683142-3226        1693227-3314        1703315-3404        1713405-3496        1723497-3591        1733592-3688        1743689-3788        1753789-3891        1763892-3997        1773998-4105        1784106-4217        1794218-4331        1804332-4448        1814449-4569        1824570-4693        1834694-4821        1844822-4951        1854952-5086        1865087-5224        1875225-5365        1885366-5511        1895512-5661        1905662-5814        1915815-5972        1925973-6134        1936135-6300        1946301-6471        1956472-6647        1966648-6827        1976828-7013        1987014-7203        1997204-7398        2007399-7599        2017600-7805        2027806-8017        2038018-8235        2048236-8458        2058459-8688        2068689-8923        2078924-9166        2089167-9414        2099415-9670        2109671-9932        211 9933-10202      21210203-10478      21310479-10763      21410764-11055      21511056-11355      21611356-11663      21711664-11979      21811980-12304      21912305-12638      22012639-12981      22112982-13333      22213334-13695      22313696-14067      22414068-14448      22514449-14840      22614841-15243      22715244-15657      22815658-16082      22916083-16518      23016519-16966      23116967-17426      23217427-17899      23317900-18385      23418386-18884      23518885-19396      23619397-19923      23719924-20463      23820464-21018      23921019-21589      24021590-22174      24122175-22776      24222777-23394      24323395-24029      24424030-24681      24524682-25351      24625352-26038      24726039-26745      24826746-27471      24927472-28216      25028217-28982      25128983-29768      25229769-30576      25330577-31405      25431406-32258      255______________________________________