# 10.6 Target Detection

Target detection by radar is governed by two factors: (1) the signal energy received by the radar receiver due to reflection of part of the transmitted energy by the target and (2) the noise energy generated by the receiver. Figure 10-13 depicts the output response of a radar receiver as a function of time and shows the signals due to two targets displayed against the noise contributed by external sources as well as by the devices making up the receiver. The random variations exhibited by the noise may at times make it difficult to distinguish the signal reflected by the target from a noise spike. In Fig. 10-13, the mean noise-power level at the receiver output is denoted by P n o=G rec P n i, where G rec is the receiver gain and P n i is the noise level referred to the receiver’s input terminals. The power levels P r 1 and P r 2 represent the echoes of the two targets observed by the radar. Because of the random nature of noise, it is necessary to set a threshold level, P r min, for detection. For threshold level 1 indicated in Fig. 10-13, the radar will produce the presence of both targets, but it will also detect a false alarm. The chance of this occurring is called the false-alarm probability. On the other hand, if the threshold level is raised to level 2 to avoid the false alarm, the radar will not detect the presence of the first target. A radar’s ability to detect the presence of a target is characterized by a detection probability. The setting of the threshold signal level relative to the mean noise level is thus made on the basis of a compromise that weighs both probabilities.

# Figure 10-13 The output of a radar receiver as a function of time.

Figure 10 13 Full Alternative Text

To keep the noise level at a minimum, the receiver is designed so that its bandwidth B is barely wide enough to pass most of the energy contained in the received pulse. Such a design, called a matched filter, requires that B be equal to the reciprocal of the pulse length τ (i.e., B=1/τ). Hence, for a matched-filter receiver, Eq. (10.10) becomes

P n i=K T sys B=K T sys τ. (10.18)

The signal power received by the radar, P r, is related to the transmitted power level, P t, through the radar equation. We will first derive the radar equation for the general case of a bistatic radar configuration in which the transmitter and receiver are not necessarily at the same location, and then we will specialize the results to the monostatic radar case where the transmitter and receiver are colocated. In Fig. 10-14, the target is at range R t from the transmitter and at range R r from the receiver. The power density illuminating the target is given by

S t=P t 4 π R t 2 G t (W/m 2), (10.19)

where (P t/4 π R t 2) represents the power density that would have been radiated by an isotropic radiator, and G t is the gain of the transmitting antenna in the direction of the target. The target is characterized by a radar cross section (RCS) σ t (m 2), defined so that the power intercepted and then reradiated by the target is

P rer=S t σ t=P t G t σ t 4 π R t 2 (W). (10.20)

# Figure 10-14 Bistatic radar system viewing a target with radar cross section (RCS) σ t.

This reradiated power spreads out over a spherical surface, resulting in a power density S r incident upon the receiving radar antenna. Hence,

S r=P rer 4 π R r 2=P t G t σ t(4 π R t R r)2 (W/m 2). (10.21)

With an effective area A r and radiation efficiency ξ r, the receiving radar antenna intercepts and delivers (to the receiver) power P r given by

P r=ξ r A r S r=P t G t ξ r A r σ t(4 π R t R r)2=P t G t G r λ 2 σ t(4 π)3 R t 2 R r 2, (10.22)

where we have used Eqs. (9.29) and (9.64) to relate the effective area of the receiving antenna, A r, to its gain G r. For a monostatic antenna that uses the same antenna for the transmit and receive functions, G t=G r=G and R t=R r=R. Hence,

Unlike the one-way communication system for which the dependence on R is as 1/R 2, the range dependence given by the radar equation goes as 1/R 4, which is the product of two one-way propagation processes.

The detection process may be based on the echo from a single pulse or on the addition (integration) of echoes from several pulses. We will consider only the single-pulse case here. A target is said to be detectable if its echo signal power P r exceeds P r min, which is the threshold detection level indicated in Fig. 10-13. The maximum detectable range R m a x is the range beyond which the target cannot be detected, corresponding to the range at which P r=P r min in Eq. (10.23). Thus,

R m a x=P t G 2 λ 2 σ t(4 π)3 P r min 1/4. (10.24)

The signal-to-noise ratio is equal to the ratio of the received signal power P r to the mean input noise power P n i given by Eq. (10.18):

S n=P r P n i=P r τ K T sys, (10.25)

and the minimum signal-to-noise ratio S min corresponds to when P r=P r min:

S min=P r min τ K T sys. (10.26)

Use of Eq. (10.26) in Eq. (10.24) gives

The product P t τ is equal to the energy of the transmitted pulse. Hence, according to Eq. (10.27), it is the energy of the transmitted pulse rather than the transmitter power level alone that determines the maximum detectable range. A high-power narrow pulse and an equal-energy, low-power long pulse will yield the same radar performance as far as maximum detectable range is concerned. However, the range-resolution capability of the long pulse is much poorer than that of the short pulse [see Eq. (10.16)].

The maximum detectable range R m a x also can be increased by improving the signal-to-noise ratio. This can be accomplished by integrating the echoes from multiple pulses in order to increase the total amount of energy received from the target. The number of pulses available for integration over a specified integration time is proportional to the PRF. Hence, from the standpoint of maximizing target detection, it is advantageous to use as high a PRF as allowed by other considerations.