s of Complexity in Software Integration EAN-13 in Software s of Complexity

Examples of Complexity using software todeploy ean/ucc-13 for web,windows application Microsoft Windows Official Website The complexity of c Software EAN13 ommunication systems varies widely. We now consider three communications systems of increasing complexity. We will see that for the rst system, simulation is not necessary.

For the second system, simulation, while not necessary, may be useful. For the third system, simulation is necessary in order to conduct detailed performance studies. Even the most complicated of the three systems considered here is still simple by today s standard.

. i i i i i TranterBook Software EAN-13 2003/11/18 16:12 page 3 #21. Section 1.1. Examples of Complexity Figure 1.1 Analytically tractable communications system. The Analytically Tractable System A very simple commu nications system is shown in Figure 1.1. This system should remind us of the basic communications system studied in a rst course on communications theory.

The data source generates a sequence of symbols, dk . The symbols are assumed to be discrete. The source symbols are assumed to be elements from a nite symbol library.

For a binary communication system, the source alphabet consists of two symbols, which are usually denoted {0, 1}. In addition, the source is assumed to be memoryless, which means that the k th symbol generated by the source is independent from all other symbols generated by the source. A data source satisfying these properties is referred to as a discrete memoryless source (DMS).

The role of the modulator is to map the source symbols onto waveforms, with a di erent waveform representing each of the source symbols. For a binary system, we have two possible waveforms generated by the modulator. This set of waveforms may be denoted {s1 (t), s2 (t)}.

The transmitter, in this case, is simply assumed to amplify the modulator output so that the signals generated by the modulator are radiated with the desired energy per bit. The next part of the system is the channel. In general, the channel is the most di cult part of the system to model accurately.

Here, however, we will assume that the channel simply adds noise to the transmitted signal. This noise is assumed to have a power spectral density (PSD) that is constant for all frequency. Noise satisfying this constant PSD property is referred to as white noise.

The noise amplitude is also assumed to have a Gaussian probability density function. Channels in which the noise is additive, white, and Gaussian are referred to as AWGN channels. The function of the receiver is to observe the signal at the receiver input and from this observation form an estimate, denoted dk , of the original data signal,.

i i i i i TranterBook 2003/11/18 16:12 page 4 #22. The Role of Simulation 1 . dk . The receiver i Software EAN13 llustrated in Figure 1.1 is referred to as an optimum receiver because the estimate of the data symbol is made so that the probability of error, PE , is minimized.

We know from basic digital communication theory that the optimum receiver for the system described in the preceding paragraphs (binary signaling in an AWGN environment) consists of a matched lter (or, equivalently, a correlation receiver), which observes the signal over a symbol period. The output of the matched lter is sampled at the end of a symbol period to generate a statistic, Vk , which is a random variable because of the addition of noise to the transmitted signal in the channel. The statistic, Vk , is compared to a threshold, T .

If Vk > T the decision, dk , is made in favor in one of the data symbols. If Vk < T the decision is made in favor of the other data signal. We refer to this system as an analytically tractable because, with a knowledge of basic communication theory, analysis of the system is carried out with ease.

For example, the probability of error is found to be PE = Q k Es N0 (1.1). where Es represents the average energy, calculated over a symbol period, associated with the set of waveforms {s1 (t), s2 (t)}, and N0 represents the single-sided power spectral density of the additive channel noise. The parameter, k, is determined by the correlation of the waveforms {s1 (t), s2 (t)}. As an example, for FSK (frequencyshift keying) transmission, the waveforms {s1 (t), s2 (t)} are sinusoids having di erent frequencies and equal power.

Assuming that the frequencies are chosen correctly, the signals are uncorrelated and k = 1. For the PSK case (phase-shift keying), the signals used for data transmission are assumed to be sinusoids having the same frequencies and equal power but di erent initial phases. If the phase di erence is radians, so that s2 (t) = s1 (t), the signals are anticorrelated and k = 2.

The performance of the system illustrated in Figure 1.1 is easily determined using traditional analysis techniques, and we are therefore able to classify the system as analytically tractable. Why is this system analytically tractable The rst and most obvious reason deals with the AWGN channel and the fact that the receiver is linear.

Since the noise is Gaussian and the matched lter is a linear system, the decision statistic, Vk , is a Gaussian random variable. We are therefore able to calculate the bit error rate (BER) analytically as a function of the parameters of the receiver lter and determine the values of those parameters that result in a minimum BER. There are a number of other factors leading to the fact that the system shown in Figure 1.

1 is analytically tractable. These relate to the simplicity of the system model, which results from a number of assumptions. The data source was assumed memoryless, which may or may not be true in practice.

In addition, perfect symbol synchronization was assumed, so that we have exact knowledge of the beginning and ending times of the data symbols. This assumption allows the decision statistic, Vk , to be correctly extracted. Would simulation ever play a role in an analytically tractable system The answer is yes, since the system shown in Figure 1.

1 may well be the basic building.
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