274 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 



Y(o:) = transfer function of receiver-filter 

 n(i) = noise input to receiver-filter 



D = power density of noise input to receiver-filter 

 no(f) = noise output of receiver-filter 



0-2 = noise power in output of receiver-filter 



z^ = peak signal-to-nolse power ratio in output of receiver-filter 



/o = observation time 



The target echo is represented by a signal input to the receiver-filter 

 denoted by s{t) with a spectrum S((a). The signal output of the filter and 

 its spectrum are denoted by So(t) and «S'o(aj). The transfer function of the 

 filter is represented by F(co), and the output signal spectrum is equal to the 

 product of this transfer function and the input signal spectrum: 



So{c^) = FM .S'(co). (5-109) 



The output waveform will, of course, be simply the inverse Fourier trans- 

 form of So{oi) : 



i/. 



Output signal = r„(/) = -^ / Yico) S{oo) ^^"Wco. (5-110) 

 It J -ex, 



We choose to make our observation of the output at the time to- It is 

 supposed that /o is selected so that the whole of the input signal is available 

 to the filter. The signal power in the output of the filter at the observation 

 time will be Sg^Uo), while the noise power in the filter output is denoted by 

 0-2. The input noise is assumed to be Gaussian with a uniform or "white" 

 spectrum with power density D. The output noise power will thus be 



1 f" 

 Output noise power =^ 0-2 = / D|y(a))|Vw. (5-111) 



The output signal-to-noise ratio at the time /^ is denoted by 2-: 



Output signal-to-noise ratio = 2- = So-(/o)/(r~ = 



kl. 



y(co).V(a;)^^'-Vco 



■/ D|y(co)|Va; 



(5-112) 



The minimum value of this ratio can be determined by means of 

 Schwarz's inequality. This can be derived in the following fashion. Suppose 

 that the functions /(;c) and g{x) and the parameter n are real. Then the 



