A QAM CONSTELLATION WHICH IS ROBUST IN THE PRESENCE OF PHASE NOISE; ENCODER AND DECODER FOR THIS CONSTELLATION

Abstract: A circularly symmetric QAM constellation reduces phase noise. Points of the constellation are situated on concentric circles. The constellation may be encoded and decoded either coherently or noncoherently. A differential encoder for the constellation does not operate by straight subtraction. A differential decoder for the constellation can use an estimated metric to recover the original signal.

Description: A QAM constellation which is robust in the presence of phase noise; encoder and decoder for this constellation BACKGROUND OF THE INVENTION A. Field of the Invention The invention relates to the a QAM transmission system, a transmitter, a receiver and a QAM signal B. Related Art Higher order Quadrature Amplitude Modulation (QAM) schemes (like 64 QAM) have traditionally been associated with coherent detection. Such schemes are used in environments that require high spectral efficiency and good performance in the presence of Additive White Gaussian Noise (AWGN). Coherent detection suffers in environments which have degradations other than AWGN, such as phase noise. Phase noise in particular results in a high error floor. Phase noise typically results from tuners and can be reduced only by extremely stringent requirements on oscillators. Such stringent requirements are incompatible with affordability in the area of consumer electronics. Non-coherent detection is usually used in such environments to reduce cost. However, non-coherent detection, which requires differential encoding and decoding, is usually associated with Phase Shift Keying (PSK), such as disclosed in D. Divsalar et al., "Multiple-symbol differential detection of MPSK", IEEE Trans. Comm. , vol. 38, no. 3, pp. 300-308, March 1990. In PSK, information is present solely in the phase of the transmitted signal, unlike QAM where both the envelope and the phase carry information. Accordingly, PSK performs worse than QAM in the presence of phase noise. On the other hand QAM has not been well suited to differential encoding and decoding. A proposal has been made for a QAM signal which can be differentially encoded and decoded in D. Makra is et al., "Trellis coded noncoherent QAM: a new bandwidth and power efficient scheme", 39th IEEE Vehicular Tech. Conf. , San Francisco, PP. 95-100, May 1989. This scheme uses the traditional rectangular QAM constellation and differentially encodes only the phase between neighboring symbols. This type of differential encoding will however make the transmitted constellation different from the constellation used for encoding the information bits. In this respect it differs from differentially encoded PSK in which the differentially encoded symbol constellation is the same as the uncoded constellation. Having the same constellation for the encoded and uncoded signal is useful where adaptive equalizers are necessary in a receiver. The error signal for the adaptation algorithm is obtained by comparing the equalizer output with the transmitted constellation. The transmitted constellation of Makrakis et al. is much denser than the original QAM constellation which degrades equalizer performance. Moreover, such a scheme does not lend itself to differentially coherent demodulation. Another constellation which could perform well in the presence of phase noise is disclosed in W. T. Webb, "QAM: The Modulation Scheme for Future Mobile Radio Communications", Electronics and Communcication Engineering Journal, Aug. 1992, pp. 167-176. This 32 point constellation could perform well in the presence of phase noise, because it is circularly symmetric; however, it is not energy efficient. The Webb constellation, if extended to 64 points, has an energy efficiency of 28.353. The energy efficiency is defmed as the ratio between the average power of the points of the constellation and the minimum squared disance between a pair of points of the constellation. 2. SUMMARY OF THE INVENTION Accordingly it is an object of the invention to provide a transmission system according to the preamble which is robust in the presence of phase noise, allows coherent and non-coherent reception, differential encoding and decoding, has the same constellation before and after differential encoding and is energy efficient. It is a further object of the invention to provide transmitters and receivers for the new constellation. 3. BRIEF DESCRIPTION OF THE DRAWING The invention will now be described by way of non-limitative example with reference to the following drawings. Fig. 1 shows a constellation according to the invention. Fig. 2 shows simulated performance of the constellation in comparison with rectangular QAM in the presence of white Gaussian phase noise. Fig. 3 shows theoretical performance of the constellation in the presence of white phase noise with a Tikhonov distribution. Fig. 4 shows an encoder according to the invention. Fig. 5 shows a decoder accordmg to the invention. 4. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS The constellation of Fig. 1 includes the following 64 points, expressed in polar coordinates, with angles in radians <d l t Q),(d 3 ,q), (d 5 ,0), (d 7 ,0), (d 2 ,τ/8), (d 4 ,χ/8), (dβ τ/8), (dg,x/8), (d l f τ/4), (d 3 ,x/4), (d 5 ,x/4), (d 7 ,x/4), (d 2 ,3x/8), (d 4 ,3τ/8), (d 6 ,3x/8), (d 8 ,3τ/8), (d j .x/2), (d 3 ,x/2), (d 5 ,x/2), (d 7 ,x/2), (d 2 ,5x/8), (d 4 ,5x/8), (d 6 ,5τ/8), (d 8 ,5τ/8), (d, ,3x/4), (d 3 ,3x/4), (d 5 ,3x/4), (d 7 ,3τ/4), (d 2 ,7τ/8), (d 4 ,7x/8), (d 6 ,7x/8), (d g , 7τ/8), (d^x), (d 3 ,x), (d 5 ,x), (d 7 ,τ), (d 2 ,9τ/8), (d 4 ,9x/8), (d 6 ,9x/8), (d 8 ,9x/8), (d^x/4), (d 3 ,5x/4), (d 5 ,5x/4), (d 7 ,5x/4), (d 2 ,l lx/8), (d 4 ,l lx/8), (d 6 ,l lx/8), (dg.l lτ/8), (d„3x/2), (d 3 ,3x/2), (d 5 ,3x/2), (d 7 ,3x/2), (d 3 ,13x/8), (d 5 ,13x/8), (d 7 , 13x/8), (d 2 , 7x/4), (d 4 ,7x/4), (d 6 , 7τ/4), (d 8 ,7x/4), (d l t 5x/8), (d 3 , 15x/8), (d 5 , 15x/8), (d 7 , 15x/8) where d 2 = 2.07 , d 3 = 2 - 52 dmin d 4 = 3.07 d^ d 5 = 3.52 d min d 6 = 4.07 d d 7 = 4.52 ( d 8 = 5-07 d^ d ττύn is the minimum distance between constellation points. The constellation has 8 concentric circles, each having 8 points. The points on adjacent circles are offset from each other by 22.5° or _π radians. The values d j ,...,d 8 are radii of 8 the concentric circles. This constellation results in a minimum distance between constellation points of d jnj ,.. The constellation minimizes an energy value which is determined according to the following equation In (1) d 2 (X 0) is the squared distance of the point x-, of the constellation to the origin A quantity which is widely used to compare constellations is the energy efficieny. This is the ratio between the average power -f of the points of the constellation and the minimum squared distance between points of the constellation. The smaller this ratio, the better the energy efficiency and performance of the constellation in AWGN and coherent detection. For the constellation described above, <£7d 2 min = 12.1051 , which is 0.62dB worse than the rectangular 64 QAM constellation for which ^fd 2 πήa = 10.5 . The constellation of the invention is, however, 3.696 dB better than the Webb constellation cited above. The small difference in performance between the constellation of the invention and a rectangular constellation, in the presence of AWGN, is offset by the superior performance of the invention in phase noise environments. Fig. 2 shows simulated performance of the proposed and rectangular 64 QAM constellations in white Gaussian phase noise. From this it can be seen that the constellation of Fig. 1 is roughly comparable to rectangular 64 QAM in the presence of AWGN but significantly better in the presence of 2° rms phase noise. The rectangular 64 QAM saturates at a bit error rate (BER) of 10 "5 irrespective of signal to noise ratio (SNR), with phase noise. The circular constellation, though 2dB worse at a BER of 10 "6 than with AWGN, does not saturate until the BER drops to about 10 "9 . In this respect, the reader is referred to Fig. 3 where the performance of the constellation of the invention is shown in the presence of phase noise with a Tikhonov distribution. As mentioned above, it is desirable in differential encoding that the encoded constellation be the same as the uncoded constellation. In order to achieve this goal with the constellation of Fig. 1 a new differential encoding rule is employed. Let a k , k= 1 , 2, ... be the sequence of data symbols taken from the constellation shown in Fig. 1, Let x k be the sequence of differentially encoded symbols. Let C = {dj, d 3 , d 5 , d 7 } be the set of radii corresponding to every alternate circle in the constellation. Then, the sequence x k is related to the sequence a k as follows: a ι = x ι where The above mapping ensures that the transmitted sequence x k has symbols from the same constellation as the data sequence a . The amplitude of each transmitted symbol is the same as the corresponding data symbol, i.e. |x | = |a k | , but the phase is equal to the phase difference between the previous transmitted symbol X j .^ and the present data symbol a k plus the bias term β j -. j . An encoder which encodes according to the invention is shown in Fig. 4. In this encoder, symbol a k conjugated in box 401 to yield complex conjugate * which is a k fed to multiplier 402. Encoded symbol x is available at an output of multiplier 402. The output of multiplier 402 is also fed back to delay 403. The output of delay 403 is variously supplied, directly to multiplier 404, to element 405, and to element 406. Element 405 provides an output which is one over the magnitude of the input of element 405. Element 406 provides an output which is the phase angle e ^ θ * -ι of the input element 406. Since there are only 8 possible magnitudes and 16 possible phase angles in this signal constellation, elements 405 and 406 can readily be implemented as ROM lookup tables. The output of multiplier 404 is therefore given by j k ι x^ . The symbol at the receiver after demodulation and matched filtering is given by: y k - x k e» ♦ n k (4) where φ is the unknown phase of the receiver oscillator and n k is AWGN. The phase φ is assumed to be uniformly distributed between (0,2x). If the phase remains constant over N symbols, it can be shown that the optimum noncoherent detector should choose the sequence that maximizes the following metric In the case where N=3, the above equation can be rewritten explicitly in terms of the data sequence as follows: Thus maximizing τ;(k) with respect to a k , , and a k . 2 jointly will give an estimate of a k and a k . j but will only give an estimate of la^l . Such a maximization operation will involve 64x64x8 comparisons for every two data symbols decoded. This number of comparisons allows the various points of the signal constellation to be tried in place of a k , a k . j , and a k _ 2 until a maximum is found. However, this number is still inordinately large, and hence a suboptimal decoding procedure must be used. An example of such a procedure follows: 1. Estimate a 3 and a 2 by maximizing »7(3). This will involve 64x64x8 comparisons and the phase of a j will not be recoverable. These estimates will be called and a 2 respectively. 2. for k>3 generate an estimate a k for a k according to the following expressions: and The second step, which makes decisions for the present symbol based on the decisions made for past symbols, only involves 64 comparisons per data symbol, which is considerably less than the number in the first step. The above sub-optimal procedure is easily extended for N greater than 3. Simulation results show that for N=4, the loss as compared to coherent detection is only about 1 dB. A decoder operating according to these principles is shown in Fig. 5. Box 500 is shown which produces η ' ( ) . Identical boxes produce - • ( ) , - • ( ) , ••• η ' ( (k) • Box 550 outputs S k by choosing symbol i for which , i.e. which maximizes the metric - ' ( ) . In box 500, a received symbol y k is input at 501. Delay element 502 produces delayed input signal y k _γ. Delay element 503 produces delayed input signal y k . 2 A feedback loop via delay element 504 provides the previous estimated symbol SJ-.J. Delay element 505 provides delayed estimated symbol __ k _ 2 . Elements 506 and 507 generate from S k . j and S k _ 2 , respectively. Elements 506 and 507 can be look up tables operating according to equation (8) above. Element 508 takes one over the absolute value of its input and therefore outputs i . Multiplier 509, fed by elements 506, 507, and SJ-.J , ¥i outputs - • . Multiplier 510, fed by elements 508, 509, and y k , outputs Multiplier 515 is fed by ai and the output of element 510. Multipliers 515 and 516 are fed by a j because this is the box for estimating the value rj j (k). In general, the box which estimates ^ ' (k) will be fed by a-, at the elements which correspond to multipliers 515 and 516. The output of multiplier 515 is therefore Multiplier 511, fed by element 507, y k _ 1 % and S k . l 5 outputs __.( _ • Element 512 takes the absolute value of a^. Multiplier 513, fed by element 512 and y k _ 2 outputs y k _ 2 |a k _ 2 | . Adder 514, fed by element 513 and element 511, oωpms - i,-.*- " ♦ y.-.K Adder 516 is fed by elements 514 and 515 and therefore outputs a λ-ι-5 :-ι e * - .-2 a _.-2 Box 517 takes the absolute value of the output of box 516. Adder 516, fed by l 2 , la^ l 2 , and |a k.2 | 2 , yields |a j | 2 + |a k.2 | 2 + |a k _ ! | 2 . Multiplier 518 multiplies the output of element 516 by 0.5. Thus the output of adder is the value of equation (7) for i= 1. Element 550 then chooses the maximum of the values of equation (7) and guesses, for the received symbol, that a, giving the maximum value.

Claims: CLAIMS 1. Digital transmission system comprising a transmitter for differentially modulating a carrier according to a QAM constellation, said transmitter being arranged for transmitting said modulated carrier via a transmission medium to a receiver, characterised in that the QAM constellation is invariant to differential encoding, and in that the ratio between the average power of the points of the QAM constellation and the squared minimum distance between points of the QAM constellation is smaller than 28.353 2. Transmission system according to claim 1 , characterised in that the ratio between the average power corresponding to the points of the QAM constellation and the squared minimum distance between the points of the QAM constellation is smaller than or equal to 12.1051 3. Transmission system according to claim 1 or 2, characterised in that the QAM constellation comprises substantially the following points, expressed in polar coordinates: (d„0), (d 3 ,0), (d 5 ,0), (d 7 ,0), (d 2 ,χ/8), (d 4 ,x/8), (d 6 x/8), (d 8 ,x/8), (d l t x/4), (d 3 ,x/4), (d 5 ,x/4), (d 7 ,x/4), (d 2 ,3x/8), (d 4 ,3x/8), (d 6 ,3x/8), (d 8 ,3x/8), (d j .x/2), (d 3 ,x/2), (d 5 ,x/2), (d 7 ,x/2), (d 2 ,5x/8), (d 4 ,5x/8), (d 6 ,5x/8), (d 8 ,5x/8), (d, ,3x/4), (d 3 ,3x/4), (d 5 ,3x/4), (d 7 ,3x/4), (d 2 ,7x/8), (d 4 ,7x/8), (d 6 ,7x/8), (d 8 , 7x/8), (d- x), (d 3 ,x), (d 5 ,x), (d 7 ,x), (d 2 ,9x/8), (d 4 ,9x/8), (d 6 ,9x/8), (d 8 ,9x/8), (d, ,5x/4), (d 3 ,5x/4), (d 5 ,5x/4), (d 7 ,5x/4), (d 2 , l lx/8), (d 4 , l lx/8), (d 6 ,l lx/8), (d 8 , l lx/8), (d, ,3x/2), (d 3 ,3x/2), (d 5 ,3x/2), (d 7 ,3x/2), (d j , 13x/8), (d 3 , 13x/8), (d 5 ,13x/8), (d 7 , 13x/8), (d 2 , 7x/4), (d 4 ,7x/4), (d 6 , 7x/4), (d 8 ,7x/4), (d j , 5x/8), (d 3 , 15x/8), (d 5 , 15x/8), (d 7 ,15x/8) where d^ l.SO d^ d 2 = 2.07 d^ d 3 = 2.52 d^ d 4 = 3.07 d^ d 5 = 3.52 d^ d 6 - 4.07 ( d 7 = 4.52 d^ d jj ^ jj is the minimum distance between constellation points. 4. Transmitter for differentially modulating a carrier according to a QAM constellation, characterised in that the QAM constellation is invariant to differential encoding, and in that the ratio between the average power of the points of the QAM constellation and the squared minimum distance between points of the QAM constellation is smaller than 28.353 5. Transmitter according to claim 4, characterised in that the ratio between the average power of the points of the QAM constellation and the squared minimum distance between points of the QAM constellation is smaller than or equal to 12.1051 6. Transmitter according to claim 4 or 5, characterised in that the QAM constellation comprises substantially the following points, expressed in polar coordinates: (d j ,0), (d 3 ,0), (d 5 ,0), (d 7 ,0), (d 2 ,x/8), (d 4 ,x/8), (d 6 x/8), (d 8 ,x/8), (d, ,x/4), (d 3 ,x/4), (d 5 ,x/4), (d 7 ,x/4), (d 2 ,3x/8), (d 4 ,3x/8), (d 6 ,3x/8), (d 8 ,3x/8), (d, ,x/2), (d 3 ,x/2), (d 5 ,x/2), (d 7 ,x/2), (d 2 ,5x/8), (d 4 ,5x/8), (d 6 ,5x/8), (d 8 ,5x/8), (d, ,3x/4), (d 3 ,3x/4), (d 5 ,3x/4), (d 7 ,3x/4), (d 2 ,7x/8), (d 4 ,7x/8), (d 6 ,7x/8), (d 8 , 7x/8), (d,,τ), (d 3 ,x), (d 5 ,x), (d 7 ,x), (d 2 ,9x/8), (d 4 ,9x/8), (d 6 ,9x/8), (d 8 ,9x/8), (dι ,5x/4), (d 3 ,5x/4), (d 5 ,5x/4), (d 7 ,5x/4), (d 2 , l lx/8), (d 4 , l lx/8), (d 6 , l lx/8), (d 8 ,l lx/8), (d„3x/2), (d 3 ,3x/2), (d 5 ,3x/2), (d 7 ,3x/2), (d l f 13τ/8), (d 3 ,13x/8), (d 5 ,13x/8), (d 7 , 13x/8), (d 2 , 7x/4), (d 4 ,7x/4), (d 6 , 7x/4), (d 8 ,7x/4), (d j , 5x/8), (d 3 ,15x/8), (d 5 ,15x/8), (d 7 ,15x/8) where d-i - l^O d,^ d 2 = 2.07 d^ d 3 = 2 -52 dπ ά a d 4 = 3.07 d^ d 5 = 3.52 dm ! ,. d 6 = 4.07 d^ d 7 = 4.52 d^ d 8 = 5.07 djuin d j u j u is the minimum distance between constellation points. 7. Receiver for receiving a signal comprising a carrier modulated according to a QAM constellation, characterised in that the QAM constellation is invariant to differential encoding, and in that the ratio between the average power of the points of the QAM constellation and the squared minimum distance between points of the QAM constellation is smaller than 28.353 8. Receiver according to claim 7, characterised in that the ratio between the average power corresponding to the points of the QAM constellation and the squared minimum distance between the points of the QAM constellation is smaller than or equal to 12.1051 9. Receiver to claim 7 or 8, characterised in that the QAM constellation comprises substantially the following points, expressed in polar coordinates: (d,,0), (d 3 ,0), (d 5 ,0), (d 7 ,0), (d 2 ,x/8), (d 4 ,x/8), (d 6 x/8), (d 8 ,x/8), (d l 5 x/4), (d 3 ,x/4), (d 5 ,x/4), (d 7 ,x/4), (d 2 ,3x/8), (d 4 ,3x/8), (d 6 ,3x/8), (d 8 ,3x/8), (d^x/2), (d 3 ,x/2), (d 5 ,x/2), (d 7 ,x/2), (d 2 ,5x/8), (d 4 ,5x/8), (d 6 ,5x/8), (d 8 ,5x/8), (d, ,3x/4), (d 3 ,3x/4), (d 5 ,3x/4), (d 7 ,3x/4), (d 2 ,7x/8), (d 4 ,7x/8), (d 6 ,7x/8), (d 8 , 7x/8), (d j ,x), (d 3 ,x), (d 5 ,x), (d 7 ,x), (d 2 ,9x/8), (d 4 ,9x/8), (d 6 ,9x/8), (d 8 ,9x/8), (d,,5x/4), (d 3 ,5x/4), (d 5 ,5x/4), (d 7 ,5x/4), (d 2 ,l lx/8), (d 4 ,llx/8), (d 6 ,l lx/8), (d 8 ,l lx/8), (d ! ,3x/2), (d 3 ,3x/2), (d 5 ,3x/2), (d 7 ,3x/2), (d ! , 13x/8), (d 3 , 13x/8), (d 5 , 13x/8), (d 7 , 13x/8), (d 2 , 7x/4), (d 4 ,7x/4), (d 6 , 7x/4), (d 8 ,7x/4), (d j , 5x/8), (d 3 ,15x/8), (d 5 ,15x/8), (d 7 ,15x/8) where ^ = 1.30 (1^ d 2 = 2.07 d^ d 3 = 2.52 d^ d 4 = 3.07 d^ d 5 = 3.52 djn f c. d 6 = 4.07 d^ d 7 = 4.52 d^ d 8 = 5.07 d^ d-nj- j is the minimum distance between constellation points. 10. Signal comprising a carrier modulated according to a QAM constellation, characterised in that the QAM constellation is invariant to differential encoding, and in that the ratio between the average power of the points of the QAM constellation and the squared minimum distance between points of the QAM constellation is smaller than 28.353 11. Signal according to claim 10, characterised in that the ratio between the average power corresponding to the points of the QAM constellation and the squared minimum distance between the points of the QAM constellation is smaller than or equal to 12.1051 12. Signal according to claim 10 or 11, characterised in that the QAM constellation comprises substantially the following points, expressed in polar coordinates: (d, ,0), (d 3 ,0), (d 5 ,0), (d 7 ,0), (d 2 ,x/8), (d 4 ,x/8), (d 6 χ/8), (d 8 ,x/8), (d,,x/4), (d 3 ,x/4), (d 5 ,x/4), (d 7 ,x/4), (d 2 ,3x/8), (d 4 ,3x/8), (d 6 ,3x/8), (d 8 ,3x/8), (d, ,x/2), (d 3 ,x/2), (d 5 ,x/2), (d 7 ,x/2), (d 2 ,5x/8), (d 4 ,5x/8), (d 6 ,5x/8), (d 8 ,5x/8), (d lt 3τ/4), (d 3 ,3x/4), (d 5 ,3x/4), (d 7 ,3x/4), (d 2 ,7x/8), (d 4 ,7x/8), (d 6 ,7x/8), (d 8 , 7x/8), (d j ,x), (d 3 ,x), (d 5 ,x), (d 7 ,x), (d 2 ,9x/8), (d 4 ,9x/8), (d 6 ,9x/8), (d 8 ,9x/8), (d lt 5x/4), (d 3 ,5x/4), (d 5 ,5x/4), (d 7 ,5x/4), (d 2 ,llx/8), (d 4 ,llx/8), (d 6 ,llx/8), (d 8 ,llx/8), (d-.,3x/2), (d 3 ,3x/2), (d 5 ,3x/2), (d 7 ,3x/2), (d ! ,13x/8), (d 3 ,13x/8), (d 5 ,13x/8), (d 7 , 13x/8), (d 2 , 7x/4), (d 4 ,7x/4), (d 6 , 7x/4), (d 8 ,7x/4), (d-,, 5x/8), (d 3 ,15x/8), (d 5 ,15x/8), (d 7 ,15x/8) where d 2 = 2.07 d d 3 = 2.52 dmi j . d 4 = 3.07 d^ d 5 = 3-52 d^ d 6 = 4.07 d^ d 7 = 4-52 d^ d 8 = 5.07 ( du jj n is the minimum distance between constellation points.

Inventors: GHOSH MONISHA, GHOSH, MONISHA
Applicants: PHILIPS ELECTRONICS NV, PHILIPS NORDEN AB, PHILIPS ELECTRONICS N.V.
Application year: 1995
Main classification: H04L27
IPC codes: H04L27