OBJECTIVE: To introduce a novel, standardised approach to evaluating AI prediction models in balancing effectiveness, efficiency and utility, using a sepsis prediction model case study. MATERIALS AND METHODS: Retrospective patient data from electronic medical records of 7 public hospitals was used to retrain and evaluate a machine learning sepsis prediction model. Four conventional metrics-area under the receiver operating curve (AUROC), sensitivity, positive predictive value, and specificity-were compared with a novel graphical display integrating metrics of predictive accuracy (effectiveness), alert burden (efficiency) and lead time of alerts relative to clinical events (utility) for different alert thresholds. RESULTS: The dataset comprised 977,506 inpatient admissions. The novel methodology produced a plot of four vertically aligned graphs that enables decision-makers to identify an alert threshold that optimally balances effectiveness, efficiency and utility (EEU) at the level of an entire admission, and which differs from that derived using conventional metrics. DISCUSSION: Conventional evaluation metrics do not consider alert timing relative to clinical events and are often applied to different evaluation datasets (sample and admission level), introducing bias and confusion. In contrast, the EEU methodology (i) generates admission level evaluations at different alert thresholds; (ii) measures alert timing relative to clinical events; and (iii) provides a visual display that enables identification of the alert threshold that optimally balances EEU factors. CONCLUSION: Evaluations of prediction models for adverse events in hospitalised patients should incorporate the EEU approach in assessing model suitability and selecting alert thresholds.

This paper applies channel sounding measurements to enable physical-layer security coding. The channel measurements were acquired in an indoor environment and used to assess the secrecy capacity as a function of physical location. A variety of Reed-Muller wiretap codes were applied to the channel measurements to determine the most effective code for the environment. The results suggest that deploying physical-layer security coding is a three-point design process, where channel sounding data guides 1) the physical placement of the antennas, 2) the power settings of the transmitter, and 3) the selection of wiretap coding.

In this paper we provide a real-world test of physical-layer security using channel sounding techniques in an indoor wireless network. We consider the possibility of exploiting issues that arise in practical receivers to manufacture a discrete memoryless wiretap channel model from the Gaussian case, and show how the secrecy capacity of the Gaussian wiretap channel model changes as these issues are considered. Results indicate that the secrecy capacity is a function of the manufactured channel model, physical antenna location, and power settings at the transmitter. These parameters can be optimized to maximize the potential for secrecy in the network.

Inverse, or negative binomial, sampling is often used when the observation of interest occurs extremely infrequently. As this is the case in bit error rate (BER) simulations, especially in high signal-to-noise ratio cases, negative binomial sampling can be advantageously employed in a computationally economic fashion to compare bit error rates between different systems. When the results of two negative binomial sampling tests are compared, point estimates and interval estimates quantify the performance relationship between the results of the tests. This paper derives a new, optimal, logarithmically symmetric confidence interval estimator for the ratio of BER estimates derived from two negative binomial tests. In addition, a three-sided hypothesis test with a single significance level is derived to quantify the confidence of the relationship between the two systems. Low-BER approximations for the confidence interval and decision thresholds are derived based on the F-distribution. The approximation is shown to work with BERs as high as 10-2. An example inspired by bit interleaved coded modulation shows how the technique can be used to reduce simulation time by an order of magnitude and facilitate straightforward interpretation and comparison between different systems. Negative binomial sampling is recommended for comparison experiments where BER is the key metric.

Data-aided (DA) and non-data-aided (NDA) maximum likelihood (ML) estimators for the SNR of CPM in the presence of phase and frequency offset are derived and analyzed. Cramér-Rao bounds for both are obtained and compared to the simulated performance of these estimators for a full response example, CPSFK and a partial response example, GMSK. Analysis and simulations show that the performance of the DA ML estimator suffers from an unremovable bias caused by uncompensated frequency offset due to frequency offset estimation errors. As a consequence, the estimator error variance of the DA ML estimator is not able to achieve its lower bound. In contrast, the estimator error variance of the NDA ML estimator is capable of achieving its lower bound (although the lower bound for NDA ML estimator is higher than the lower bound for the DA ML estimator). This is because the NDA ML estimator is not burdened with the requirement of relying on estimates of nuisance parameters. The NDA estimator only achieves its lower bound when the observation length or true SNR are sufficiently large.

We compute the pilot sequences that minimize the Cramér-Rao bound(CRB) for carrier phase offset, carrier frequency offset, and symbol timing offset estimation. If we require the pilot sequence to be drawn from a discrete set of constellation points, then the alternating sequence is the CRB-minimizing sequence. If we relax the constellation constraint and apply a total energy constraint, we obtain three different sequences that minimize the CRB for each of the synchronization parameters. We show that the maximum-likelihood (ML) estimator achieves the CRB for any of the sequences presented. We also explore the impact of using the pilot symbols in a preamble with unknown data symbols immediately following. The interference caused by the unknown data symbols reduces the performance of the ML estimator beyond an SNR threshold.

This paper examines the binomial and negative binomial tests for estimating the bit error rate using Monte Carlo simulations. Whereas the estimator error variances and simulation times do not recommend one test over the other, confidence interval analysis and test design criteria recommend the negative binomial test for estimating a small bit error rate based on a limited number of error events. The Clopper-Pearson interval length is much more consistent for the negative binomial test in the small bit error rate case. The logarithmically centered confidence interval is introduced and analyzed. For a given confidence interval length and confidence probability, the negative binomial test requires fewer error events than the binomial test. The asymptotic analyses of the Clopper-Pearson and logarithmically-centered confidence intervals yield useful design criteria for the negative binomial test in a way that is not possible for the binomial test.

The maximum likelihood estimator for a constant channel gain based on offset QPSK (OQPSK) training symbols is derived and analyzed. The estimator is unbiased and only asymptotically efficient. The estimator error variance is a function of the variance of an inphase/quadrature (I/Q) interference term that creates a floor for the estimator error variance. The I/Q interference term is not present in estimators based on QPSK and constitutes the primary difference between the estimators for OQPSK and QPSK. Examples using the non-return to zero (NRZ) and square-root raised cosine (SRRC) pulse shapes demonstrate that for a fixed training sequence length, the performance floor for the estimator error variance is lower (better) with full response pulse shapes (NRZ) than with partial response pulse shapes (SRRC). The criterion for training sequences that minimize the undesired performance floor is also set forth.

Simulation, point estimation, and interval estimation of the bit error rate are rigorously examined. Exact confidence intervals based on the F-distribution approximation for the binomial and negative binomial distributions are derived. Simulations demonstrate 1) that the analysis gives accurate confidence intervals for bit error rate estimators and 2) that the performance of the binomial and negative binomial estimators give similar performance. The negative binomial estimator has the advantage that an a priori estimate or guess of the true bit error probability is not required.

Two low-complexity data-aided frequency estimators, suitable for use in burst-mode communications, are described and analyzed. The estimators are based on pilot symbols organized into disjoint blocks embedded in the burst. The first estimator is a generalization of the phase increment frequency estimation technique and the second estimator is a generalization of the autocorrelation frequency estimation technique. The generalizations are needed to account for the spacings between the pilot blocks. It is shown that the frequency estimators exhibit good accuracy while maintaining useful operating ranges.

This paper presents a novel IP core reuse strategy which reduces design time from days to hours for communication circuits such as digital radio receivers. This design productivity is obtained by leveraging a highly parameterized library of communication specific cores. These cores are described in IP-XACT XML with vendor extensions describing the timing behavior of their communication interfaces. A synthesis tool, called Ogre, was created that generates the communication interfaces between cores described in IP-XACT and synthesizes full designs from structural synchronous dataflow specifications. Design productivity improvements are demonstrated with several radio receiver designs.

We compute the training sequences that minimize the Cramer-Rao Bound (CRB) corresponding to the joint estimator for carrier phase, carrier frequency, and symbol timing offset. It is shown that if the symbols in a training sequence are drawn from the set of complex numbers, the resulting CRB is lower than if the symbols are drawn from a discrete set of constellation points.

Reconfigurable radios implemented on FPGAs operating in high-radiation environments are subject to single-event- upsets (SEUs). The traditional mitigation method of applying triple modular redundancy (TMR) to the entire design does not have to be used in this application. This is because the majority of the SEUs impact the overall performance (measured by bit error rate) in the same way additive noise does. The results of this paper show which sections must be protected from SEUs and provide a guide for the bit error rate performance versus FPGA area tradeoff as a function of SEU mitigation.

In this letter we examine the application of space-time block codes (STBC) to offset QPSK (OQPSK). We present the maximum-likelihood (ML) decoder for STBC-OQPSK as well as a simplified decoder. We show through analysis and simulation that the simplified decoder matches the performance of the ML decoder.

A practical data-aided frequency estimator, suitable for use in burst-mode communications, is described and analyzed. The estimator is based on pilot symbols organized into disjoint blocks embedded in the burst, and it is a generalization of the autocorrelation frequency estimation technique. The generalizations are needed to account for the spacings between the pilot blocks. It is shown that the frequency estimator exhibits good accuracy while maintaining useful operating ranges.

This paper describes the use of orthogonal space- time block codes to overcome the performance and complexity difficulties associated with the use of an offset modulation with space-time trellis codes in multiple-input multiple-output communication systems. The orthogonal space-time block code is applied to offset QPSK modulated waveforms in the same way it would be applied to symbols. The procedure allows the receiver to orthogonalize the link. The main benefits of this orthogonalization are 1) the removal of the noise correlation at the input to the space-time trellis decoder and 2) the elimination of I/Q interference resulting from multi-antenna transmission. These benefits are achieved at the expense of data rate. The data rate reduction is a function of the rate of the orthogonal space-time block code.

A detector architecture capable of detecting both shaped offset quadrature phase shift keying (SOQPSK-TG) and Feher's quadrature phase shift keying (FQPSK-JR) is developed and analyzed. Both modulations are embodied as fully interoperable modulations in the Interrange Instrumentation Group (IRIG) standard IRIG-106. It is shown that the common detector achieves near optimal bit error rate performance without knowledge of which modulation is used by the transmitter. The detection techniques are based on a common trellis-coded modulation representation and a common continuous phase modulation (CPM) representation for these two modulations. In addition the common pulse amplitude modulation (PAM) decomposition of the common CPM representation is developed. The common PAM-based detector offers the best performance- complexity trade-off among the detectors considered.

This letter considers the pulse amplitude modulation (PAM) representation of continuous phase modulation (CPM) with a ternary data alphabet. This technique is applied to the problem of constructing reduced-complexity detectors with near-optimum performance. The usefulness of this approach is demonstrated using the ternary CPM variant known as shaped-offset quadrature phase-shift keying (SOQPSK).

The Alamouti space-time code (STC) is attractive because of the low complexity of its associated detection algorithm. However, when this code is applied to offset modulations such as offset QPSK (OQPSK), the offset complicates the detection process. The Maximum Likelihood (ML) detector for Alamouti encoded OQPSK is much more complex than the ML detector for QPSK. We present two alternate detection algorithms: one which duplicates the simplicity of the non- offset QPSK Alamouti detector with a cost in bit error rate performance and one which matches the performance of the ML detector with less complexity than the ML detector.

A hybrid automatic repeat-request (ARQ) code combining scheme employing different multidimensional space- time trellis codes (MSTTCs) over a multiple-input, multiple- output (MIMO) channel is described. The retransmission codes are designed using sub-optimal partition chains of the MSTTC super-constellation using a relatively simple search. The MSTTCs designed using the sub-optimal partition chain are, by themselves, not optimal codes. But when combined with the coded packet used for previous transmission(s), they provide better error control than using the same code for all transmissions. Theoretical performance analysis and simulation results are provided.

We develop a reduced-complexity approach for the detection of coded shaped-offset quadrature phase-shift keying (SOQPSK), a highly bandwidth-efficient and popular constant-envelope modulation. The complexity savings result from viewing the signal as a continuous-phase modulation (CPM). We give a simple and convenient closed-form expression for a recursive binary-to-ternary precoder for SOQPSK. The recursive nature of this formulation is necessary in serially concatenated systems where SOQPSK serves as the inner code. We show that the proposed detectors are optimal in the full-response case, and are near-optimal in the partial-response case due to some additional complexity reducing approximations. In all cases, the proposed detectors achieve large coding gains for serially concatenated coded SOQPSK. These gains are similar to those reported recently by Li and Simon, which were obtained using a more complicated cross-correlated trellis-coded quadrature modulation (XTCQM) interpretation.

We consider multiple-bit differential detection (MBDD) of differentially encoded shaped-offset quadrature phase-shift keying (SOQPSK), a highly bandwidth-efficient and popular constant-envelope modulation. We propose two MBDD schemes that are based on a recent continuous phase modulation interpretation of SOQPSK. We show that the performance of these MBDD schemes approaches that of coherent detection (CD) as the multiple-bit observation N window increases. The first scheme uses a detection window that spans the full-bit observation window (F-MBDD), and is shown to require very large values of N to approach the performance of CD. This presents a practical problem since the complexity of MBDD grows exponentially with N. The second scheme is an improved version (I-MBDD) with a detection window that is shortened to N-2 bit intervals. Although the complexity of I-MBDD also increases exponentially with N, it represents a significant improvement since only modest values of N are needed for high performance. These performance characteristics are identified via a detailed performance analysis, which provides asymptotic formulas for the bit error probability that are confirmed with computer simulations. The analysis is also used to find the symmetric frequency pulse shapes with the best and worst error performance. Finally, we develop a simplified and practical decision feedback differential detection algorithm that achieves near-optimal performance with complexity that grows only linearly with N.

We consider multiple-bit differential detection (MBDD) of differentially encoded shaped-offset quadrature phase-shift keying (SOQPSK), a highly bandwidth-efficient and popular constant-envelope modulation. We propose two MBDD schemes that are based on a recent continuous phase modulation (CPM) interpretation of SOQPSK. We show that the performance of these MBDD schemes approaches that of coherent detection as the multiple-bit observation window N increases. The first scheme uses a detection window that spans the full N-bit observation window (F-MBDD) and is shown to require very large values of N to approach the performance of coherent detection. This presents a practical problem since the complexity of MBDD grows exponentially with N. The second scheme is an improved version (I-MBDD) with a detection window that is shortened to N - 2 bit intervals and requires more modest values of N for high performance. These performance characteristics are identified via a detailed performance analysis, which provides asymptotic formulas for the probability of bit error that are confirmed with computer simulations.

This paper shows that the SW and GBN retransmission protocols must be generalized when used in a multichannel communications system. The generalization takes the form of packet-to-channel assignment rules. A general condition governing the packet-to-channel assignment rule is derived and important special cases are identified. Simulation results were used to demonstrate that 1) packet-to-channel assignment impacts channel utilization when the channels are different and 2) the optimal assignment rule produces a channel utilization that is better than the channel utilization that results from doing something else or nothing at all

This paper shows that the SW and GBN retransmission protocols must be generalized when used in a multichannel communications system. The generalization takes the form of packet-to-channel assignment rules. A general condition governing the packet-to-channel assignment rule is derived and important special cases are identified. Simulation results were used to demonstrate that 1) packet-to-channel assignment impacts channel utilization when the channels are different and 2) the optimal assignment rule produces a channel utilization that is better than the channel utilization that results from doing something else or nothing at all.

Symbol-by-symbol detection of SOQPSK and FQPSK using detectors designed for offset QPSK represents a simple common detector architecture for these two interoperable waveforms. Unfortunately, this detection method results in a 2 dB loss in bit error rate performance. This paper describes detection methods for recovering this loss without the need for knowing which modulation is used by the transmitter. An equivalent cross-correlated trellis-coded quadrature modulation (XTCQM) representation for SOQPSK is developed which forms the basis of a common TCM detector. An equivalent CPM representation for FQPSK is developed which forms the basis for a common CPM detector. The common XTCQM detector performs slightly better than the common CPM detector, but achieves this gain at the expense of higher complexity

In this paper we develop a reduced-complexity detection scheme for shaped-offset quadrature phase-shift keying (SOQPSK), a highly bandwidth-efficient constant-envelope modulation. The detector is based on the well-known pulse amplitude modulation (PAM) representation of continuous phase modulation (CPM). Since SOQPSK is a ternary CPM, we show how the binary-based PAM technique is extended to accommodate the ternary case. We demonstrate that a detector based on the first two PAM components requires a simple trellis of only 4 states. We show that near-optimum performance is achieved using this reduced-complexity detector. The potential complexity reduction can be quite large, since one version of SOQPSK requires a trellis of 512 states.

We develop a reduced-complexity approach to the detection of military-standard (MIL-STD) shaped-offset quadrature phase-shift keying (SOQPSK), a highly bandwidth-efficient constant-envelope waveform. The complexity savings result from viewing the signal as a continuous phase modulation (CPM). The proposed detector has a matched filter bank which is one-fourth as complex as previously reported designs. We establish that this reduced-complexity detector achieves optimal performance for uncoded MIL-STD SOQPSK. We also show a simple and convenient closed-form expression for a recursive precoder for SOQPSK, which is necessary in serially concatenated systems. Finally, we show that the proposed reduced-complexity detector achieves the large coding gains that have recently been reported for serially concatenated coded MIL-STD SOQPSK.

It is well understood that the pulse amplitude modulation (PAM) representation of continuous phase modulation (CPM) can lead to reduced-complexity detectors with near optimum performance. It has recently been shown that the PAM representation also extends to CPM schemes with multiple modulation indexes (multi-h CPM). In this paper, we present a detector for multi-h CPM which is based on the PAM representation. We also give an exact expression for the pairwise error probability for the entire class of PAM-based CPM detectors (single- and multi-h, optimal, and reduced-complexity) over the additive white Gaussian noise (AWGN) channel and show that this bound is tighter than the previously published bound for approximate PAM-based detectors. In arriving at this expression, we show that PAM-based detectors for CPM are a special case of the broad class of mismatched CPM detectors. We also show that the metrics for PAM-based detectors accumulate distance in a different manner than metrics for other CPM detectors. These distance properties are especially useful in applications with greatly reduced trellis sizes. We give thorough examples of the analysis for different single- and multi-h signaling schemes. We also apply the new bound in comparing the performance of PAM-based detectors with other reduced-complexity detectors for CPM.

It is known that any multilevel continuous phase-modulated (CPM) signal with a single modulation index can be exactly represented by a sum of pulse-amplitude modulated (PAM) waveforms. In this paper, we show how multi-h CPM signals can also be represented in this manner. The decomposition is presented in general terms as a function of the alphabet size, modulation indexes, and phase pulse of the CPM scheme. The number of pulses required to exactly construct the signal is shown to increase over that previously given for single-h schemes; this increase is in proportion to the number of modulation indexes. We propose an approximation which significantly reduces the number of signal pulses and which minimizes the mean-squared error for an arbitrary set of modulation indexes. We show that this approximation can have two objectives: 1) to reduce the number of pulses in the same manner as has been proposed for single-h schemes; and/or 2) to reduce the number of multi-h pulses; we also show the conditions where this latter objective is most practical. We compare this minimum mean-squared error approximation with another method which was recently proposed for CPM. We also give numerical results on detection performance which demonstrate the practicality of the proposed approximation.

This paper describes data-aided signal level and noise variance estimators for GMSK using a linear receiver based on the Laurent AMP representation. The estimators are based on the maximum likelihood principle. The variances of these estimators are shown to satisfy the Cramer-Rao bound and hence are efficient estimators. Simulation results are included to confirm the performance of the estimators.

We present an optimal maximum likelihood sequence estimating (MLSE) receiver which is based on the pulse amplitude modulation (PAM) representation of multilevel multi-h continuous phase modulation (CPM). We also give four different approaches to constructing suboptimal receivers which use a significantly reduced number of signal terms. We apply these techniques to two multi-h schemes in current use and evaluate receiver performance with computer simulations. In one instance we reduce the number of trellis states from 512 to 32, and the number of matched filters from 96 to 3, with a degradation of 0.7 dB. The simulations also show that the complexity reductions preserve the performance advantage multi-h CPM schemes have over single-index schemes.

A system that combines MTCM modified for type-I hybrid-ARQ error control with Spatio-Temporal Vector Coding (STVC) for use over a slowly varying MIMO channel is presented. An idealistic retransmission protocol is defined that maximizes the channel utilization is described and analyzed. Numerical examples, based on a set of simple 8-state trellis codes providing a granularity of 0.5 bit per 2-dimensional symbol, demonstrate that this simple type-I hybrid-ARQ system can reduce the code gap by the same amount as an FEC system based on a set of 64-state trellis codes. This shows that type-I hybrid-ARQ STVC can close the code gap (the SNR gap between actual performance and channel capacity) without an increase in the code complexity or a decrease in performance. We also demonstrate the limitations of this approach: as the bit error rate increases and the probability of retransmission increases, the channel utilization drops. As a consequence, the code gap does not decrease or even increases.

This paper examines the bias in data-aided maximum likelihood carrier phase estimation for GMSK based on a linear detector structure using only the first Laurent pulse. The bias in the ML phase estimate increases with decreasing BT/sub b/ and decreasing training sequence length. Three methods for bias compensation are described and their performance examined. Simulation results show that the performance of the bias compensation techniques can achieve the Cramer-Rao bound.

Digital signal processing techniques are applied to maximum likelihood carrier phase synchronization for QPSK and QAM in an all-digital sampled data receiver. To achieve the flexibility required by modern software defined radios (SDRs), this task must either be performed in a DSP processor (reconfigurable software) or in an FPGA (reconfigurable hardware). This paper describes the design process for an FPGA-based design and summarizes the FPGA resources required for QPSK carrier phase synchronization.

This paper describes the use of a polyphase filterbank to perform the interpolations required for symbol timing synchronization in a sampled-data receiver. The polyphase filterbank possesses advantages over architectures based on separate matched and interpolation filters. Interpolations are realized by filterbank index selection and a separate interpolating filter following the matched filter is not required. Maximum likelihood timing synchronization techniques can be easily incorporated into the polyphase filter bank in a natural way. An M-stage polyphase filterbank with input data sampled at approximately N samples/symbol can be used in a loop that operates at MN samples/symbol, N samples/symbol, or 1 sample/symbol. When operating at 1 sample/symbol, auxiliary control must also be included to adjust the clocking of data into the filter bank to account for small differences in the sample clock and N times the data clock. Examples are presented to illustrate loop performance and control.

The selection of interleaver depth for the land-mobile satellite channel is investigated by analyzing the performance of convolutionally encoded BPSK and DPSK over a frequency-nonselective correlated Ricean fading channel. By extending previous analyses of partially interleaved convolutional codes over such channels, a new figure of merit-which is a function of the channel correlation function-is identified. Simulation results for the first-order Butterworth, Jakes (1974), and Divsalar (1991) fading spectra are presented which demonstrate the applicability of the figure of merit as an aid in interleaver design. The figure of merit also tracks the nonmonotonic, or quasi-oscillatory, behavior of the decoded bit error rate with increasing interleaver depths in those situations where this occurs.

We show that a differentially encoded maximal-length shift register sequence (or M-sequence) is also an M-sequence or the logical complement of an M-sequence. The differentially encoded sequence is a cyclically shifted version of the original M-sequence or its complement.

Software defined radios (SDR) are highly configurable hardware platforms that provide the technology for realizing the rapidly expanding third (and future) generation digital wireless communication infrastructure. Many sophisticated signal processing tasks are performed in a SDR, including advanced compression algorithms, power control, channel estimation, equalization, forward error control and protocol management. While there is a plethora of silicon alternatives available for implementing the various functions in a SDR, field programmable gate arrays (FPGAs) are an attractive option for many of these tasks for reasons of performance, power consumption and configurability. Amongst the more complex tasks performed in a high data rate wireless system is synchronization. This paper is about carrier and timing synchronization in SDRs using FPGA based signal processors. We describe and examine a QPSK Costas loop for performing coherent demodulation, and report on the implications of an FPGA mechanization. Symbol timing recovery is addressed using a differential matched filter control system. A tutorial style approach is adopted to describe the operation of the timing recovery loop and considerations for FPGA implementation are outlined.

Two soft-decision incomplete decoding rules based on geometric considerations are proposed and analyzed for binary antipodal signaling on an additive white Gaussian channel. It is seen that for high signal to noise ratios, low decoder error probabilities are attainable requiring approximately 2 dB less power than equivalent hard-decision decoding schemes without a significant loss in performance due to rejections. The incomplete decoding rules allow the implementation of a new reduced complexity decoding algorithm based on a processing loop constructed around an encoder. Initial results indicate the performance of this new decoder compares favorably with that of the ideal decoder.>

Generalized minimum distance (GMD) decoding is modified to allow simultaneous error detection and error correction. The soft-decision information retained by the generalized minimum distance concept is used as a source of reliability information which determines when a retransmission of the received packet should be requested. The performance of hybrid-ARQ GMD decoding is shown to be comparable to the ideal reliability metrics proposed by Loeliger (1990) and superior to the performance of hard-decision decoders. Performance expressions are derived for low to moderate signal-to-noise ratios and bounds are derived for high signal-to-noise ratios which are loose by 0.5 to 1.0 dB.>