## S

where Dr is the diameter of the receive antenna.

The antenna gain may also be defined as the ratio of its effective aperture area, A,., to the effective area of a hypothetical isotropic antenna, A2/4ti, where A is the wavelength of the transmitted signal. For the receive antenna

where C is the received power and 0./4nS)2 is defined as the space loss, Ls.

In digital communications, the received energy per bit, Eb, is equal to the received power times the bit duration, or

where C is in W, R is the data rate in bps, and Eb is in W-s or J.

The noise power at the receiver input usually has a uniform noise spectral density, Na, in the frequency band containing the signal. The total received noise power, N, is then N„B, where B is the receiver noise bandwidth. (B is determined by the data rate and the choice of modulation and coding, as discussed later in this chapter.) N0 and N

are related to the system noise temperature, Ts, by:

where N0 is in W/Hz, Nis in W, k is Boltzmann's constant = 1.380 x 10"23 J/K, Ts is in K, and B is in Hz. By combining Eqs. (13-9) and (13-10) with Eq. (13-8), we obtain our original link equation, Eq. (13-4).

### 13.3.2 Link Design Equations

The link equation is a product of successive terms and, therefore, can be conveniently expressed in terms of decibels oidB.A number expressed in dB is just 10 logl0 of the number. Thus, a factor of 1,000 is 30 dB and a factor of 0.5 is -3 db. If the number has units, they are attached to the dB notation. For example, 100 W is 20 dBW. Antenna gain is the ratio of radiated intensity in a specific direction to that of an isotropic antenna Tadiating uniformly in all directions and, therefore, is a pure number which should, in principle, be expressed in dB. However, we use dBi (dB relative to isotropic) as the units for antenna gain to be consistent with standard practice in the industry.*

Eq. (13-4) can be rewritten in decibels as

E^Na= P + k + G, + LpT + 4 + L,, + Gr + 228.6 - 10 log7J - 10 logff (13-13)

where Eb/No, Lt, Gt, La, Lpn and Gr are in dB, P is in dBW, Ts is in K, R is in bps, and 10 log k = -228.60 dBW7(Hz-K). This can also be written as

Eb/N0 = EIRP + Lpr + Ls + La + Gr+ 228.6 - 10 logTs - 10 log« (13-14)

* Editor's Note: Of course, modem computers are fully capable of multiplying real numbers instead of adding logarithms. Like much of astronautics, this peculiar nomenclature remains intact primarily to ensure the fill] employment of communications systems engineers.

where the EIRP is in dBW and sensitivity of the receiving station, Gr/Ts = Gr -10 log Tp is expressed in dB/K. Eq. (13-14) is preferred when specifying the transmitter EIRP and receiver Gr/Ts separately .Note ¿at in these equations, Gr and Ts must be specified at the same point, usually the junction between the receive antenna terminal and the Low Noise Amplifier.

We can find the carrier-to-noise-density-ratio, C/Na by multiplying E//N0 by the data rate, R [see Eq. (13-9)]. Carrier power in W is the energy per bit in J times the number of bits per second.

From Eq. (13-12), the carrier-to-noise-ratio, C/N is C/NaB or, in dB, is C/Na -10 log B. Combining this with Eq. (13-15b):

C/N = EIRP + Ls + La + Gr + 228.6 - 10 logTs -10 log B (13-16)

where B is the noise bandwidth of the receiver in Hz. C/N also equals Eb/N„+ 10 log(RM).

The Received Isotropic Power, or RIP, is the power received if the receive antenna gain is OdB. If we substitute Gr= 1 (0 dB) into Eq. (13-8), then C = RIP. Combining this expression with Eq. (13-4), and converting to dB yields:

where RIP is in dBW. A good way to specify the receiving system performance is to specify the bit error rate (the probability a data bit is incorrectly received) required for a given RIP. The designer then has the freedom to trade off his demodulator design (which determines the Eb/N0 required to meet the specified bit error rate) against the antenna gain and noise temperature (G/Ts) to meet the RIP specification at minimum cost.

We can similarly convert Eq. (13-7) to dB. Using the relationship/ = c/A, where c is the velocity of light in free space =» 3 x 108 m/s, we obtain the following equation for the antenna gain, G, in dB:

G = 20 log 7t + 20 log D + 20 log/ + 10 log 7) - 20 log c (13-18a)

G = -159.59 + 20 log D + 20 log/+ 10 log ij (13-18b)

For a circular antenna beam the half-power beam width, Q, is the angle across which the gain is within 3 dB (50%) of the peak gain. We may estimate Qfrom the following empirical relationship:

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