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These relationships are given for reference only and will not be used further in this book. An important parameter for carrier systems is the ratio of the average carrier power to the noise power density, usually denoted by [C/N0]. The [Eb/N0] and [C/N0] ratios can be related as follows. The average carrier power at the receiver is PR W. The energy per symbol is therefore PR/Rsym J, with Rsym in symbols per second. Since each symbol contains m bits, the energy per bit is PR/mRsym J. But mRsym Rb, and therefore, the energy per bit, Eb, is Eb PR Rb (10.22)

As before, let N0 represent the noise power density. Then Eb/N0 PR/RbN0. But PR/N0 is the carrier-to-noise density ratio, usually denoted by C/N0, and therefore, Eb N0 C/N0 Rb (10.23)

Rearranging this and putting it in decibel notation gives c C d N0 c Eb N0 d [Rb] (10.24)

If the rms in a perfectly competitive industry are making short-run profits, more rms will enter the industry in the long run. This increases market supply of the commodity and reduces the market price until all profits are competed away and all rms just break even. The exact opposite occurs if we start with rms with short-run losses. As a result, all rms in a perfectly competitive industry with long-run equilibrium produce where P = lowest LAC and resources are utilized in the most ef cient way.

It should be noted that whereas [Eb/N0] has units of decibels, [C/N0] has units of dBHz, as explained in App. G.

the required [Eb /N0] at the ground station receiver is 9.5 dB. Calculate the required [C/N0].

The transmission rate in decibels is [Rb]

10 log(61

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The equations giving the probability of bit error are derived on the basis that the filtering provides maximum signal-to-noise ratio. In practice, there are a number of reasons why the optimal filtering may not be achieved. The raised-cosine response is a theoretical model that can only be approximated in practice. Also, for economic reasons, it is desirable to use production filters manufactured to the same specifications for the transmit and receive filter functions, and this may result in some deviation from the desired theoretical response. The usual approach in practice is that one knows the BER that is acceptable for a given application. The corresponding ratio of bit energy to noise density can then be found from Eq. (10.18) or from a graph such as that shown in Fig. 10.17. Once the theoretical value of Eb/N0 is found, an implementation margin, amounting to a few decibels at most, is added to allow for imperfections in the filtering. This is illustrated in the following example.

1. In a perfectly competitive industry, each rm can affect the commodity price. 2. The marginal revenue of a rm in perfect competition is equal to the commodity price. 3. The perfectly competitive rm maximizes pro ts at the quantity where its MR curve intersects the rising portion of its MC curve. 4. A rm breaks even when price equals its average variable cost. 5. All rms in perfect competition break even in the long run. Answers: 1. False; 2. True; 3. True; 4. False; 5. True

5 rate of no more than 10 , the implementation margin being 2 dB. Calculate the required Eb /N0 ratio in decibels.

The graph of Fig. 10.17 shows that Eb /N0 is around 9 dB for a BER of 10 5. By plotting this region to an expanded scale, a more accurate value of Eb/N0 can be obtained. This is shown in Fig. 10.18. from which [Eb/N0] is seen to be about 9.65 dB. This is without an implementation margin. The required value, including an implementation margin, is 9.65 2 11.65 dB.

To summarize, BER is a specified requirement, which enables Eb/N0 to be determined by using Eq. (10.18) or Fig. 10.17. The rate Rb also will be specified, and hence the [C/N0] ratio can be found by using Eq. (10.24).

1 10 4

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