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Probability of Error vs Integration Length
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TECHNIQUE FOR AUTOMATICALLY DETECTING THE CONSTELLATION SIZE OF A QUADRATURE AMPLITUDE MODULATED
(QAM) SIGNAL 5
BACKGROUND OF THE DISCLOSURE
1. Field of the Invention
The invention relates to communications receivers, and particularly, to apparatus for use in such a receiver for automatically detecting a constellation size of a received quadrature amplitude modulated (QAM) signal.
2. Description of the Prior Art
Various communications systems utilize quadrature amplitude modulation (QAM) for transmission of relatively high data rate information within a limited transmission bandwidth. Typically, QAM communications systems use a fixed symbol constellation for all transmis- 2Q sions, e.g., sixteen positions or points within a constellation. Conventional QAM receivers are capable of receiving transmissions of only a single symbol constellation size.
However, recently, sophisticated QAM communica- 25 tions systems are capable of transmitting variable symbol rate transmissions using two or more symbol constellations. For example, such a QAM system could vary its symbol rate between a 32-ary constellation and a 16-ary constellation depending upon the presence of 30 atmospheric noise. Specifically, during periods of low atmospheric noise, such a QAM system can use 32-ary transmissions. When atmospheric noise, as measured at the transmitter, has increased above a pre-established noise threshold, the constellation size is then decreased 35 to a 16-ary constellation. During periods of extremely low noise, the constellation size could be increased to transmissions of 32-ary, 64-ary or even 128-ary constellations. Alternatively, a QAM system may be required to transmit one constellation size over-the-air and a 40 second constellation size over a cable broadcast system. As such, a "cable ready" receiver must be able to receive both constellation sizes.
One such variable constellation size communication system is being considered by the Federal Communica- 45 tions Commission (FCC) as a standard transmission format for high definition television (HDTV). The particular standard would permit HDTV broadcasters to use either 16-ary or 32-ary QAM symbol constellations for broadcasting HDTV signals. As such, a given 50 HDTV receiver may receive a 16-ary transmission when viewing one particular channel and a 32-ary transmission when viewing a second channel. Additionally, a given HDTV broadcaster may change its transmitted symbol constellation from 16- to 32-ary, or vice 55 versa, whenever noise conditions permit the higher (lower) rate of transmission to a majority of the broadcast audience. Thus, an HDTV receiver must be capable of automatically determining whether a received broadcast is a 16- or 32-ary transmission. Such a deter- 60 mination must be accomplished whenever a user changes channels or the broadcaster changes transmission rates. Additionally, the constellation size determination must be accomplished relatively quickly such that a user will not notice the change in the constella- 65 tion size of the broadcast, i.e., notice a loss of signal reception while the receiver adjusts to a new constellation size.
Furthermore, HDTV cable broadcasts may utilize 64-ary transmissions. As such, a "cable ready" HDTV receiver must be capable of receiving 16, 32 and 64-ary transmissions.
Typically, a receiver of variable constellation size transmissions contains demodulator circuits capable of demodulating each expected size of symbol constellation. In particular, a receiver capable of receiving both 16- and 32-ary symbol constellations would contain both a 16-ary demodulator and a 32-ary demodulator. In this manner, both symbol constellations are demodulated simultaneously even though only one constellation is transmitted. Thus, one demodulator produces no signal, while the other demodulator demodulates the received symbols and generates information therefrom. Though such a receiver accomplishes reception of either symbol constellation, such redundant demodulators are complex and costly to manufacture.
Using an alternative technique to determine constellation size, an HDTV receiver contains a single demodulator having two modes of operation, i.e., one for each expected symbol constellation size. As such, the demodulator attempts to demodulate one of the constellation sizes, e.g., a 16-ary symbol constellation, using one of its two modes of operation. Meanwhile, circuitry within the receiver monitors an error rate from a Reed-Solomon decoder within the demodulator. If the error rate exceeds a pre-defined threshold, the receiver assumes that the demodulator is set to demodulate the incorrect symbol constellation size. In response, the receiver switches the demodulator to a second mode of operation, e.g., a 32-ary symbol constellation mode, capable of demodulating a second symbol constellation size, e.g., 32-ary. Simply stated, the error rate at the output of the Reed-Solomon decoder indicates which constellation is presently being demodulated. However, for a Reed-Solomon decoder to operate properly, a carrier recovery circuit must be locked onto a carrier for the transmission. Otherwise, the Reed-Solomon decoder produces a high error rate for both modes of operation. As such, this form of receiver first requires that a carrier lock be achieved before determining the constellation size. Detrimentally, this two-step process is relatively slow to determine the constellation size of the received broadcast. Consequently, a large amount of information can be lost while the receiver is achieving carrier lock and then determining the constellation size before beginning to demodulate the broadcast.
Therefore, a need exists in the art for apparatus, particularly though not exclusively for use in a HDTV QAM receiver, for automatically detecting the constellation size of a QAM transmission without requiring the QAM demodulator to, a priori, acquire carrier lock. Additionally, to minimize any noticeable signal impact to a viewer, this detection should be performed relatively quickly.
SUMMARY OF THE INVENTION
My invention advantageously overcomes the disadvantages heretofore associated with receivers that require a priori carrier lock before determining constellation size of a received QAM signal. Specifically, through my invention the constellation size of a QAM signal, received by a QAM receiver, is directly determined without advantageously requiring carrier lock or redundant demodulators.
In general, my invention determines the constellation size, e.g., 4, 16, 32-ary, of a particular received QAM
signal by analyzing the probability density function (pdf) of that signal. To properly analyze this signal, its magnitude is first squared and then normalized to a preset value, e.g., 2.0. As such, the signal power in any of the various constellations is normalized to a fixed 5 level. In addition, squaring the magnitude of the QAM signal removes the requirement to achieve carrier lock prior to attempting to determine the constellation size. Then, through generating a histogram of the squared and normalized QAM signal, the number of levels of 10 modulation contained in the QAM signal is determined. Each constellation size has a unique number of modulation levels and thus a unique histogram. For example, a 4-ary constellation contains one modulation level, 16-ary contains three modulation levels and 32-ary con- 15 tains five modulation levels. Each modulation level produces a histogram. Thus, by generating a histogram, my invention determines the constellation size of a QAM signal without requiring a priori carrier lock before accomplishing such a determination. 20
Specifically, in accordance with my inventive teachings, a number of histogram bins is established for each different QAM constellation in a QAM signal that is expected to be received. Each bin has a pre-defined width and is located at an expected peak in a pdf of a 25 corresponding one of the QAM signals. The peaks are located at various normalized power levels of the expected QAM signal; the bin widths are defined by a range of power levels around each peak location. The number of bins and their specific locations depend upon 30 the number of different constellation sizes and the size of those constellations that are expected to be received. In operation, a counter associated with each bin counts (accumulates) the number of symbols that fall therein. The symbols are counted over a fixed time period. At 35 the end of the period, my technique compares, to one another, the number of symbols counted by each counter. The result of the comparison indicates the number of modulation levels for the QAM signal being received which, in turn, indicates the constellation size 40 of that QAM signal.
BRIEF DESCRIPTION OF THE DRAWINGS
The teachings of the present invention can be readily understood by considering the following detailed description in conjunction with the accompanying drawings, in which:
FIG. 1 graphically shows 16- and 32-ary quadrature amplitude modulation (QAM) signal constellations having equal power levels;
FIG. 2 depicts a gTaph of the probability density functions (pdfs) of magnitude squared 16- and 32-ary QAM signals;
FIG. 3 depicts a graph of the probability of error versus bin width for both 16- and 32-ary QAM signals;
FIG. 4 depicts a graph of the probability of error versus integration length for either 16- or 32-ary QAM signals; and
FIG. 5 depicts a block diagram of my inventive modulation level detector circuit 500.
To facilitate understanding, identical reference numerals have been used, where possible, to designate identical elements that are common to the Figures.
After considering the following description, those skilled in the art will clearly realize that the teachings of my invention can be readily utilized in various commu
nications receivers that receive quadrature amplitude modulated (QAM) signals of various constellation sizes. In particular, my invention is used to determine the constellation size of a QAM signal without requiring carrier lock prior to making such a determination.
In general, my inventive apparatus first generates a square of the magnitude of a received QAM signal, i.e., a magnitude squared QAM signal. The magnitude squared QAM signal is then accumulated over a predefined time period to produce a probability density function (pdf) for the QAM signal. Each QAM signal constellation size has a unique pdf, e.g., a 32-ary QAM signal produces a pdf having five peaks (maximums), a 16-ary QAM signal produces a pdf having three peaks. Each peak corresponds to a concentric ring of symbol amplitudes, i.e., a modulation level, within the constellation. The resulting pdf of the received QAM signal is monitored, with the constellation size being determined therefrom. The constellation size determination is then passed to a demodulator which in turn, takes whatever action is necessary to demodulate a QAM signal having that particular constellation size.
The following detailed discussion of my invention illustratively focuses upon using my invention to determine whether a QAM signal has a 16- or 32-ary constellation. However, those skilled in the art will realize that the invention can be used to determine whether a QAM signal is any one of a plurality of M-ary constellation sizes.
FIG. 1 depicts graph 100 of a 16-ary QAM signal constellation (each symbol is shown as an X) superimposed over a 32-ary QAM signal constellation (each symbol is shown as an O), where each constellation contains equal power. In other words, the constellations have been equalized to have equivalent power content. For each constellation, each symbol within that constellation falls upon one of a number of pre-defined concentric rings that represent symbol magnitude. Specifically, the 16-ary constellation has each of its symbols fall on rings 102, 104 and 106 (all shown as dashed lines). In contrast, the 32-ary constellation has the magnitude of its symbols aligned with rings 108,110,112,114 and 116 (all shown as solid lines). Each ring represents a modulation level within the QAM signal.
FIG. 2 depicts graph 200 of probability density functions (pdfs) for magnitude squared 16- and 32-ary QAM signal constellations when the constellations, as shown in FIG. 1, are scaled to contain equal power. The pdf of the 16-ary constellation is shown in FIG. 2 using solid line 202 and the pdf of the 32-ary constellation is shown using dashed line 204. The pdfs are produced by first squaring the magnitude of a QAM signal having a 30 dB signal to noise ratio (SNR) and then accumulating the magnitude squared signal over a pre-determined period of time. Consequently, a peak (maximum) appears in graph 200 wherever a ring of symbols appeared in FIG. 1. The locations of the peaks are unique for each constellation size. Thus, by analyzing the pdfs to determine the locations of peaks therein, the constellation size of a QAM signal can be uniquely determined.
In general, analyzing the QAM signal constellation in accordance with my inventive technique requires establishing a plurality of histogram bins. The specific number of bins depends on particular design parameters that are discussed in detail below. Each bin has a width defined by a range of normalized power that is centered at a normalized power level represented by each peak in graph 200. A counter associated with each of the histo
If a large number of symbols is histogrammed into bins surrounding each of the pdf peaks, the expected number of symbols to be counted in each bin would be the total number of symbols (N) expected during a given time period multiplied by the expected pdf amplitude of the peak associated with a particular bin. However, in the presence of noise, the histogram is degraded. Additive Guassian noise alters the pdf of the QAM signal by spreading and shifting the peaks. For relatively high signal-to-noise ratios (SNR), the pdf peaks will stand out above the noise, but at low SNR, the peaks smear together and become indistinguishable from one another. Typically, the higher magnitude peaks, e.g., peaks 4 and 5 in 32-ary QAM, are most severely affected by additive noise and become indistinguishable before the lower magnitude peaks, e.g., peaks 1, 2 and 3 in 32-ary QAM, become indistinguishable. These lower magnitude peaks all occur below a normalized power of 1.0. To optimize performance when selecting between 16- or 32-ary constellations, the peaks above the normalized power of 1.0 are not examined by my inventive technique.
Additionally, for the illustrative use of my technique 45 to differentiate 16- from 32-ary QAM, counting the number of symbols that occur in each bin would require eight bins and a rather significant amount of circuitry to compare the counts of each counter associated with each bin to determine the constellation size. The circuitry required to differentiate between 16- and 32-ary QAM signals can be significantly reduced by using only two bins. Specifically, a bin (bin 2) is located at a normalized power level of 0.5, e.g., peak 2 of the 32-ary QAM signal pdf, and a bin (bin 3) is located at a normalized power level of 1.0, e.g., approximately at peak 3 of the 32-ary QAM signal pdf and at peak 2 of the 16-ary QAM signal pdf. As such, over a given time period, a 32-ary QAM signal constellation generates a higher count in bin 2 than in bin 3, i.e., the maximum value of 6X1 the pdf at peak 2 is larger than at peak 3. In contrast, the 16-ary QAM constellation generates a higher count in bin 3 than in bin 2. Consequently, the count in these two bins over a pre-established period uniquely defines the QAM signal constellation present. The error rate for 65 such a determination method depends upon the width of the bins and the length of the period over which the count is taken. To optimize the bin widths and accumu
lation time, a statistical analysis of the interaction of these parameters is necessary.
FIG. 3 depicts graph 300 of probability of error in detecting a symbol versus bin width for both 16- and 32-ary QAM signals. To derive graph 300, the bins can be thought of as random variables each having a binomial distribution. The probability of a hit (symbol) falling within a given bin is computed by integrating the magnitude squared symbol plus noise pdf (FIG. 2) over the power range (width) of the bin. The number of symbols integrated is the number of trials in the binomial distribution. From the number of hits in each bin, a random variable X is determined. Specifically, X is a difference between a number (N2) of hits in bin 2 and a number (N3) of hits in bin 3. Consequently, the overall probability of error for my technique is represented by equation (1), as follows:
X is a difference between a number (N2) of hits in
bin 2 and a number (N3) of hits in bin 3.
To evaluate the probability density of X, certain assumptions and approximations must be made about N2 and N3. Since the number of symbols examined is assumed to be very large, e.g., 10,000 or more, and the probability of a hit in a given bin is relatively small, then N2 and N3 can each be approximated by a Poisson distribution. Also, for numerical simplicity, it is assumed that N2 and N3 are independent variables, which, with respect to QAM modulated signals, is not strictly true. However, when the number of symbols is relatively large and the SNR is relatively high, the accumulation in each bin has little effect upon the accumulation in another bin. Under these assumptions and approximations, the pdf of X is a discrete convolution of two Poisson distributions representing N2 and N3 and, thus, can be numerically evaluated.
The defining parameter for each of the Poisson distributions is an expected value of the number of hits in each bin. The value of this parameter is the probability of one hit in a bin (p) times the number of symbols examined (N). Thus, the probability of error for detecting a symbol correctly is determined by the bin width (which determines p) and the number of symbols accumulated (integrated). Each of these parameters can be graphed separately against the probability of error.
In particular, FIG. 3 depicts graph 300 of the probability of error for both a 16- and a 32-ary QAM signals versus bin width for a fixed number of accumulated symbols. Curve 302 was formed using a 16-ary QAM signal having a 10 dB SNR and curve 304 was formed using a 32-ary QAM signal having a 13 dB SNR. The probabilities were calculated using a fixed accumulation (N) of 10,000 symbols. The graph shows the optimal bin width for minimizing the probability of error for both 16- and 32-ary QAM is between 0.25 and 0.3 of the normalized power.
FIG. 4 depicts graph 400 of the probability of error for either a 16- or 32-ary QAM signal versus the integration time measured in number of symbols (N) for a fixed bin width. As would be expected, this graph shows that the probability of error decreases linearly as the integration time is lengthened. However, in practice, it desirable to minimize the integration time, i.e., minimize the signal acquisition time of the receiver in which my