# On Distributed Communications Series

## VI. Mini-Cost Microwave

#### III. Required Signal/Noise Ratio

The analysis upon which is based the choice of signal-to-noise (S/N) ratio is somewhat different than that for conventional analog modulation transmission because of the particular network properties in which these links are used. As a consequence, a lower S/N ratio than normally used can be tolerated. A secondary factor contributing to the reduction of link S/N ratio and fading margin is the use of short repeater spacing distances, thereby reducing the severity of fading. Even more important is the system's capability for using low-reliability links since short-time outages are of much less concern, unlike conventional analog microwave transmission systems.

The spans, each covering a distance of up to 200 miles, are composed of mounted microwave repeater stations, at average intervals of 20 miles. The worst-case error rate occurs when each of the links forming the span fades independently.[1] Because the error rate is low and low outage times are anticipated, and will represent only a small percentage of the total up-time, the expedient of summing the individual link error rates and outage times will be employed to describe the performance of a span. This simplification will, at worst, merely introduce a trivial pessimistic bias. A further simplification is to multiply the down-time of a single link by the number of links in tandem. Finally, it shall be assumed that there are ten links in one span.

As opposed to the case of *spans* which are made up of tandem links, the worst *network* error case occurs when all the spans connected to a single Switching Node fade in a *dependent* manner. Good statistics on the degree of such dependence are lacking; however, the need for weighty analysis can be avoided by using the assumption that the first link of every span connected to a Switching Node is shorter than normal and is of such path clearance that fading will rarely ever exceed the threshold margin, even under worst-case propagation conditions. This assumption appears moderately reasonable inasmuch as high-clearance paths are not subject to the same intensity of fading as are almost-grazing paths.

### Magnitude of Errors Caused by Fading and Noise

In the microwave link, two separate phenomena corrupt the transmitted signal. The first is the multiplicative type modification of signal strength caused by fading. The second is additive Gaussian noise.

Reiger,[2] for example, has shown that extremely low error rates, on the order of l0^{6}, are easily achieved with Gaussian noise S/N ratios on the order of 12-15 db. Even using simple modulation and detection means, it will thus be seen that the greatest cause of lost bits will be due to fading.

Goldman and Sommer[3] describe the determination of error rates in a cascaded chain of binary repeater links corrupted by both Rayleigh fading and Gaussian noise. Because of the nature of the data format being transmitted in the mini-cost microwave system, an alternative approach to that used by Goldman and Sommer has been chosen. The rationale is simple: Fading errors are highly clustered and exhibit the property that if a bit is received in error due to fading, adjacent bits have a high probability of also being in error. (Since "Message Blocks" of 1024 bits are being transmitted,[4] it would be highly deceptive to regard each erroneous bit as being statistically independent of its neighbor.) Thus, instead of computing the *average* error rate on the channel, we shall take advantage of that property of the Switching Nodes (see ODC-VII) which avoids using those links exhibiting high error rates. All spans and links shall be characterized as being "up" (operative) a certain portion of the time and "down" part of the time. It is only during the up-time that the link error rate computation becomes meaningful.

At this point in the investigation, a search for statistics on the durations of fades was made in order to provide something against which to compare Message Block times. Unfortunately, the data found was meager and gave only the *median duration* of the fade. While it is not precisely the information we were seeking, a brief perusal of Fig. 1 reveals that fades of shorter duration than the 2/3-millisecond Message Block time may be expected to occur only on deep fades greater than 50 db below normal.

### Models for Fading

Since the fading component is the more important effect, it is well to review the statistical distribution used to describe fading. Much of the American literature relies upon the Rayleigh distribution, as given by Bullington.[5]

Fagot and Magne[6] review the derivation of the Rayleigh distribution in a microwave link and show that it is really a worst-case upper-bound value. Empirical measurements are also shown which indicate that other distributions are more appropriate, particularly over the shorter paths over land. Figure 2 compares alternative estimates of microwave link fading and plots the depth of fade in decibels versus the fractional time the fade is in excess of the abscissa. Three separate curves are shown: a) The Rayleigh curve; b) the Durkee[7] curve; and c) the curve used by the French P.T.T. [8] The Durkee curve appears most representative of the conditions which can be anticipated in the proposed system. Using either the Durkee or the French P.T.T. curve, it would appear that if the system had a 25-db fading margin allowance, each link would be inoperative only about 0.0007 fractional part of the time. In other words, a 25-db fading margin reserve would be exceeded only at comparatively rare intervals. If, alternatively, the Rayleigh fading distribution is chosen, the link would be down 0.003 fractional part of the time.

If Rayleigh fading is assumed, then a span composed of ten tandem links is operative 97 per cent[9] of the time; with the Durkee or the French B.T.T. curves, the figure is 99.3 per cent.[10].

To this point, it has been assumed that when the 15-db S/N ratio threshold is exceeded, an error is guaranteed. Actually, the condition is not quite so absolute. Figure 3, taken from Goldman and Sonirner,[11] shows, for non-coherent frequency shift-keying, that even when a fade dips below the 25-db fading margin allocation, the error probability does not increase markedly from the 15-db Gaussian noise margin value of 10^{-7} until an additional 5-db or so greater loss is experienced. Thus, it can be stated that the composite error rate for ten tandemly connected repeaters forming a span is equal to or less than ten times theindividual link error rate (which is less than about 10^{-7}) giving a 10^{-6} bit-error rate for the span. Under the worst possible distribution of errors (where the errors are spaced 1000 or more bits apart for a Message Block of 1000 bits), only one in every 1000 Message Blocks will be corrupted by error during the 99.3 per cent up-time period, assuming, under worst-case conditions, a total composite signal-to-noise and fading-margin ratio of 40 db. This low total S/N is much less than that commonly associated with micro-wave systems designed for analog modulation.[12] Thus, it can be seen that the restriction to digital modulation can greatly reduce the required radiated power and antenna size of the microwave equipment. No standby or duplicate "protection channel" capability is assumed in the proposed system. It will be seen that the diversity needed to cope with extreme fading conditions is better obtained through the spatial distribution of the spans forming the network--provided the network is designed to utilize this spatial redundancy.

### Rain Attenuation

Atmospheric and rain attenuation at microwave frequencies is described in detail by Harvey,[13] with attenuation values given. However, from the discussion by Fagot and Magne, it appears that rain and deep propagation fades are negatively correlated; that is, a turbulent atmosphere caused by rain will produce stable propagation conditions and very little fading. Because of the 25-db fading margin, we may expect to retain much of this gain margin as a reserve available to cope with the modest attenuation due to most rain. For the relatively short links being considered, it is expected that outages due to rain will be of relatively short duration and limited in extent. Only extremely heavy downpours are expected to give cause for concern. Hathaway and Evans[14] describe rain attenuation at 11 Gc.[15]

Spatial diversity, which already exists in the distributed network, will be extremely helpful with respect to minimizing attenuation. It is to be also noted that in an all-digital transmission system, the effect of rain over a widespread area is less damaging than for analog transmission, because of the complete regeneration of the digital signal at each repeater point. Moderate rainstorms may cover vast areas, producing high cumulative losses, but very heavy downpours are usually rather localized.

Unlike the analog case, we should not be troubled by tandem deterioration. It is only extremely heavy downpours that can cause trouble. At 11 Gcs, an excess path loss of 1 db/mi would not occur except in the case where the rainfall rate exceeds 0.8 inches per hour. As such extremely heavy downpours are localized over small areas, it is unlikely that difficulty from rain in a network spatially organized as that described in ODC-I would be significant.

### Summary

An overall 40-db signal-to-noise ratio, split 25 db reserved for fading margin and 15 db for additive white noise, appears adequate for the construction of 200-mi spans, operative 99.3 percent of the time, and with fewer than one in every 1000 Message Blocks transmitted incorrectly. As this figure is only about one-tenth that assumed as the error rate in the calculations of ODC-VII, in which A user-to-user error rate of less than l0~8 was established, it would appear that a very-low-power microwave system may be designed for this application.

[1] If the links fade simultaneously, the total amount of time the links will be "out" will be less than the sum of the individual link outage times. Thus, the independent fading case is the worst one for tandemly connected links.

[2] Reiger, S., "Error Probabilities of Binary Data Transmission Systems in the Presence of Random Noise," *IRE National Convention Record*, Part 8, Information Theory, 1953, pp. 72-79.

[3] Goldman, H. D., and R. C. Soinmer, "An Analysis of Cascaded Binary Communication Links," *IRE Trans. Comm. Syst.*, Vol. CS-b, September 1962, pp. 291-299.

[5] Bullington, K., "Radio Propagation Fundamentals," *Bell System Tech. J.*, Vol. 36, May 1957, pp. 593-626.

[6] Fagot, J., and P. Magne, *Frequency Modulation Theory, Application to Microwave Links*, Pergamon Press, London, 1961.

[7] The Durkee curve is also taken from Fagot and Magne, ibid.

[8] Libois, L. J., and M. Thue, *Study CNET*, No. 304T, Private publication; and, L. J. Libois, *Study CNET,* No. 316T, Private publication; and, W. Klein, and L. J. Libois, *Onde Elect.*, December 1953.

[10] (1-(0.0007)(10)) - 99.3%.

[12] For example, Miller describes the design of a T-J Bell microwave system operating at 11 Gcs where 43 db is set aside for fading margin alone, and another 32 db for the FM audio signal; Miller, Arthur G., "An ll,000-Mc Radio System Across Chesapeake Bay," *IRE Trans. Com. Syst.*, Vol. CS-9, No.4, December 1961, pp. 455-459.

[13] Harvey, A. F., *Microwave Engineering*, Academic Press, New York, 1963, pp. 1074-1078.

[14] Hathaway, S. D., and H. W. Evans, "Radio Attenuation at 11 kmc and Some Implications Affecting Relay System Engineering, * Bell System Tech. *__J.__, Vol. 38, No. 1, January 1959, pp. 73-97.

[15] It is to be noted that rain affects both the primary and the "protection" channels used in traditional practice.