In the minimum signal-to-noise ratio it is assumed that the echo signal received from a particular target did not vary with time. In practice, however, the echo signal from a target in motion is almost never constant. Variations in the echo signal may be caused by meteorological conditions, the lobe structure of the antenna pattern, equipment instabilities, or variations in the target cross section. The cross sections of complex targets (the usual type of radar target) are quite sensitive to aspect. Therefore, as the target aspect changes relative to the radar, variations in the echo signal will result.
- One methodof accounting for a fluctuating cross section in the radar equation is to select a lower bound, that is, a value of cross section that is exceeded some specified (large) fraction of time. The fraction of time that the actual cross section exceeds the selected value would be close to unity (0.95 or 0.99 being typical). For all practical purposes the value selected is a minimum and the target will always present a cross section greater than that selected. This procedure results in a conservative prediction of radar range and has the advantage of simplicity. The minimum cross section of typical aircraft or missile targets generally occurs at or near the head-on aspect.
- However, to properly account for target cross-section fluctuations, the probability- density function and the correlation properties with time must be known for the particular target and type of trajectory.
- The probability-density function gives the probability of finding any particular value of target cross section between the values of σ and σ dσ, while the autocorrelation function describes the degree of correlation of the cross section with time or number of pulses.
- A more economical method to assess the effects of a fluctuating cross section is to postulate a reasonable model for the fluctuations and to analyze it mathematically. Swerling has calculated the detection probabilities for four different fluctuation models of cross section. In two of the four cases, it is assumed that the fluctuations are completely correlated during a particular scan but are completely uncorrelated from scan to scan. In the other two cases, the fluctuations are assumed to be more rapid and uncorrelated pulse to pulse. The four fluctuation models are as follows:
Case 1: The echo pulses received from a target on any one scan are of constant amplitude throughout the entire scan but are independent (uncorrelated) from scan to scan. This assumption ignores the effect of the antenna beam shape on the echo amplitude. An echo fluctuation of this type will be referred to as scan-to-scan fluctuation. The probability-density function for the cross section σ is given by the density function
Where, σav is the average cross-section over all target fluctuations.
Case 2: The probability-density function for the target cross section is also given by Eq. above but the fluctuations is more rapid than in case 1 and are taken to be independent from pulse to pulse instead of scan to scan.
Case 3:In this case the fluctuation is assumed to be independent from scan to scan as in case 1 but the probability-density function is given by
Case 4: The fluctuation is pulse to pulse according to Eq. above. The probability-density function assumed in cases 1 and 2 applies to a complex target consisting of many independent scatterers of approximately equal echoing areas. Although, in theory, the number of independent scatterers must be essentially infinite, in practice the number may be as few as four or five. The probability-density function assumed in cases 3 arid 4 is more indicative of targets that can be represented as one large reflector together with other small reflectors. In all the above cases, the value of cross section to be substituted in the radar equation is the average cross section σav. The signal-to-noise ratio needed to achieve a specified probability of detection without exceeding a specified false-alarm probability can be calculated for each model of target behavior. For purposes of comparison, the non-fluctuating cross section will be called case 5.