5-3] DETECTION PROBABILITY FOR A PULSE RADAR 143 



antenna. The antenna pattern is approximated by a constant gain over the 

 antenna beamwidth 9, and zero gain outside of this region. The received 

 signal on a single scan will consist of n pulses. In the absence of target size 

 fluctuations, these pulses will all be of the same size. The number of pulses 

 is given by the product of the repetition frequency /r and the beamwidth, 

 divided by the scan velocity co^: 



Number of pulses in a scan = n = frQ/cjis- (3-11) 



The received signal is assumed to be a pulsed sinusoid. The signal power 

 during a pulse is denoted by S, and the internal noise power referred to the 

 same point in the system is denoted by A^. In the case of a fluctuating 

 target an average signal power S will be defined. 



The essential parts of the receiver for this analysis consist of a predetec- 

 tion amplifier, a square law detector, a pulse integrator, and a decision 

 threshold. 



The predetection amplifier is normally the intermediate frequency (IF) 

 amplifier, and it is assumed to be matched to the envelope of the pulse 

 shape. That is, the bandwidth of this amplifier is approximately equal to 

 the reciprocal of the pulse length. Noise with a uniform power density is 

 assumed to be introduced into the system at the input to this amplifier. 

 The power spectrum of the noise at the amplifier output will thus be equal 

 to the power transfer function of the amplifier. The peak signal-to-noise 

 ratio at the output of the predetection amplifier is S jN, as was indicated 

 above. 



A square-law detector is assumed to generate a video voltage equal to the 

 square of the envelope of the predetection signal plus noise. In this case, 

 the development in Paragraph 5-7 is applicable and can be used to establish 

 the amplitude distribution and power-density spectrum of the video signal 

 plus noise. The assumption of a square-law detector rather than a linear 

 detector is primarily for mathematical convenience. It does not represent 

 a serious restriction because the basic results are only slightly dependent 

 on the detector law. 



The pulse integrator combines the n pulses received during a scan over the 

 target. In Paragraph 5-10 it will be shown that the linear operation which 

 gives the greatest signal-to-noise ratio for a signal consisting of n pulses 

 corresponds to the addition of these pulses to form a sum signal. Accord- 

 ingly, in order to provide the greatest possible signal-to-noise ratio at the 

 decision threshold — and thus the greatest reliability of detection — these 

 n pulses are assumed to be added together by a pulse integrator.^ 



^In many practical sj^stems, integration is provided by the memory of the human operators 

 or by retention of the signal on the face of the cathode ray display tube. In such cases, the 

 integration is not a perfect summing process, and degradation in the S/N ratio is experienced. 

 This degradation is discussed later in this chapter and also in Chapter 12, on radar displays. 



