samples of a continuous stochastic process, such as Gaussian noise. A structure for producing the shown in Fig. 78.8. In this case the BSR has been replaced by an LCA [Knuth, 1981]. The LCA is very similar to BSR in that it requires a seed value, is clocked once for each symbol generated, and will generate a periodic sequence. One can generate a white noise process with an arbitrary first-order probability density function (pdf) by passing the output of the LCa through an appropriately designed nonlinear, memoryless mapping Simple and well-documented algorithms exist for the uniform to Gaussian mapping. If one wishes to generate a nonwhite process, the output can be passed through the appropriate filter. Generation of a wide-sense stationary Gaussian stochastic process with a specified power spectral density is a well-understood and -documented problem. It is also straightforward to generate a white sequence with an arbitrary first-order pdf or to generate a specified power spectral density if one does not attempt to control the pdf. However, the problem of generating a noise source with an arbitrary pdf and an arbitrary power spectral density is a significant allenge [Sondhi, 1983 78.10 Transmitter, Channel, and Receiver Modeling Most elements of transmitters, channels, and receivers are implemented using standard DSP techniques. Effects that are difficult to characterize using mathematical analysis can often be included in the simulation with little additional effort. Common examples include gain and phase imbalance in quadrature circuits, nonlinear amplifiers, oscillator instabilities, and antenna platform motion. One can typically use LPE waveforms and devices to avoid translating the modulator output to the carrier frequency. Signal levels in physical systems often vary by many orders of magnitude, with the output of the transmitters being extremely high energy signals and the input to receivers at very low energies. To reduce execution time and avoid working with extremely large and small signal level simulations, one often omits the effects of linear amplifiers and attenuators and uses normalized signals. Since the performance of most systems is a function of the signal-to-noise ratio, and not of absolute signal level, normalization will have no effect on the measured performance. One must be areful to document the normalizing constants so that the original signal levels can be reconstructed if needed. Even some rather complex functions, such as error detecting and correcting codes, can be handled in this manner. If one knows the uncoded error rate for a system, the coded error rate can often be closely approximated by applying a mathematical mapping. As will be pointed out below, the amount of processor time required to produce a meaningful error rate estimate is often inversely proportional to the error rate. While an uncoded error rate may be easy to measure, the coded error rate is usually so small that it would be impractical to execute simulation to measure this quantity directly. The performance of a coded communication system is most often determined by first executing a simulation to establish the channel symbol error rate. An analytical mapping can then be used to determine the decoded BER from the channel symbol error rate Once the signal has passed though the channel, the original message is recovered by a receiver. This can A receiver encounters a number of clearly identifiable problems that one may wish to address independently For example, receivers must initially synchronize themselves to the incoming signal. This may involve detecting that an input signal is present, acquiring an estimate of the carrier amplitude, frequency, phase, symbol synchronization, frame synchronization, and, in the case of spread spectrum systems, code synchronization Once acquisition is complete, the receiver enters a steady-state mode of operation, where concerns such as symbol error rate, mean time to loss of lock, and reaction to fading and interference are of primary importance. To characterize the system, the user may wish to decouple the analysis of these parameters to investigate relationships that may exist. For example, one may run a number of acquisition scenarios and gather statistics concerning the probability of acquisition within a specified time interval or the mean time to acquisition. To isolate the problems face synchronization from the inherent limitation of the channel, one may wish to use perfect synchronization information to determine the minimum possible BER. Then the symbol or carrier synchronization can be held at fixed errors to determine sensitivity to these parameters and to investigate worst-case performance. Noise processes can be used to vary these parameters to investigate more typical performance. The designer may also e 2000 by CRC Press LLC© 2000 by CRC Press LLC samples of a continuous stochastic process, such as Gaussian noise. A structure for producing these samples is shown in Fig. 78.8. In this case the BSR has been replaced by an LCA [Knuth, 1981]. The LCA is very similar to BSR in that it requires a seed value, is clocked once for each symbol generated, and will generate a periodic sequence. One can generate a white noise process with an arbitrary first-order probability density function (pdf) by passing the output of the LCA through an appropriately designed nonlinear, memoryless mapping. Simple and well-documented algorithms exist for the uniform to Gaussian mapping. If one wishes to generate a nonwhite process, the output can be passed through the appropriate filter. Generation of a wide-sense stationary Gaussian stochastic process with a specified power spectral density is a well-understood and -documented problem. It is also straightforward to generate a white sequence with an arbitrary first-order pdf or to generate a specified power spectral density if one does not attempt to control the pdf. However, the problem of generating a noise source with an arbitrary pdf and an arbitrary power spectral density is a significant challenge [Sondhi, 1983]. 78.10 Transmitter, Channel, and Receiver Modeling Most elements of transmitters, channels, and receivers are implemented using standard DSP techniques. Effects that are difficult to characterize using mathematical analysis can often be included in the simulation with little additional effort. Common examples include gain and phase imbalance in quadrature circuits, nonlinear amplifiers, oscillator instabilities, and antenna platform motion. One can typically use LPE waveforms and devices to avoid translating the modulator output to the carrier frequency. Signal levels in physical systems often vary by many orders of magnitude, with the output of the transmitters being extremely high energy signals and the input to receivers at very low energies. To reduce execution time and avoid working with extremely large and small signal level simulations, one often omits the effects of linear amplifiers and attenuators and uses normalized signals. Since the performance of most systems is a function of the signal-to-noise ratio, and not of absolute signal level, normalization will have no effect on the measured performance. One must be careful to document the normalizing constants so that the original signal levels can be reconstructed if needed. Even some rather complex functions, such as error detecting and correcting codes, can be handled in this manner. If one knows the uncoded error rate for a system, the coded error rate can often be closely approximated by applying a mathematical mapping. As will be pointed out below, the amount of processor time required to produce a meaningful error rate estimate is often inversely proportional to the error rate. While an uncoded error rate may be easy to measure, the coded error rate is usually so small that it would be impractical to execute a simulation to measure this quantity directly. The performance of a coded communication system is most often determined by first executing a simulation to establish the channel symbol error rate. An analytical mapping can then be used to determine the decoded BER from the channel symbol error rate. Once the signal has passed though the channel, the original message is recovered by a receiver. This can typically be realized by a sequence of digital filters, feedback loops, and appropriately selected nonlinear devices. A receiver encounters a number of clearly identifiable problems that one may wish to address independently. For example, receivers must initially synchronize themselves to the incoming signal. This may involve detecting that an input signal is present, acquiring an estimate of the carrier amplitude, frequency, phase, symbol synchronization, frame synchronization, and, in the case of spread spectrum systems, code synchronization. Once acquisition is complete, the receiver enters a steady-state mode of operation, where concerns such as symbol error rate, mean time to loss of lock, and reaction to fading and interference are of primary importance. To characterize the system, the user may wish to decouple the analysis of these parameters to investigate relationships that may exist. For example, one may run a number of acquisition scenarios and gather statistics concerning the probability of acquisition within a specified time interval or the mean time to acquisition. To isolate the problems faced in synchronization from the inherent limitation of the channel, one may wish to use perfect synchronization information to determine the minimum possible BER. Then the symbol or carrier synchronization can be held at fixed errors to determine sensitivity to these parameters and to investigate worst-case performance. Noise processes can be used to vary these parameters to investigate more typical performance. The designer may also 8574/ch078/frame Page 1757 Wednesday, May 6, 1998 11:08 AM