244 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 



h{t)e-^'''dt = 1. (5-18) 



/. 



Thus the spectrum of an impulse is constant or uniform. Such a spectrum 

 is often referred to as "white noise" in view of the fact that al! frequencies 

 are equally represented. If the spectrum of an impulse (unity) is multiplied 

 by the transfer function of a network, the spectrum of the network output 

 is seen to be simply the transfer function itself. Thus, formally at least, the 

 transfer function of a network is the Fourier transform of the transient 

 response of the network to an impulse function input as was noted in 

 Paragraph 5-2. 



The constant spectrum given by Equation 5-18 does not have a finite 

 integral and so does not properly have an inverse Fourier transform. We 

 can, however, approximate this spectrum by one which is unity for |co| < A 

 and zero for |aj| > A^ where A is very large but finite, and this approxi- 

 mation will have an inverse transform. This inverse transform should have 

 characteristics very similar to the finite impulse pictured in Fig. 5-2 and 

 should approach an impulse function as ^^ — ^ <^ . This turns out to be true 

 and gives us a second representation for impulse functions: 



./ N 1- 1 / ,^, 7 r sin At .. .^. 



bit) = hm ;:- / e'"' do: = lim (5-19) 



A-^o.livJ-A /l^co irt 



This expression occurs often in signal and noise studies. Many important 

 functions cannot be transformed from the time to the frequency domain 

 because the Fourier integral Equation 5-1 does not converge with time. 

 Approximations to this integral, however, can often be derived on the 

 same basis as for the uniform spectrum. When this is done, the resulting 

 expression often contains expressions which can be interpreted as impulse 

 functions in the limit. 



For example, it was just established that the Fourier transform of a 

 constant/(/) = 1 is an impulse, 27r6(co), which denotes concentration of the 

 frequency spectrum at zero frequency. 



Similarly, a sinusoid will have a spectrum which may be derived as 



lim / 



.l-^ccj-.l 



COS (j^it e '"^ dt — lim d 



sin (co + cji).y sin (co — o:\)A 



-|- coi o) — aj]rf 



(5-20) 



= f/7r[5(co + coi) + 5(co — wi)]. 



This expression indicates a spectrum which is concentrated at the positive 

 and negative values at the frequency of the sinusoid. 



In deriving the impulse function representation given in Equation 5-19, 

 a constant over the entire range of w was approximated by a truncated 

 function which approached che constant function in the limit. Other 



