By Gordon Joseph Alexander Bird

Ebook by way of chicken, Gordon Joseph Alexander

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**Example text**

F(t) = H ence n-I L ;= 0 la(t - ik ) 1 cos [ wo t + ¢ ( t - ik ) ] fa ( t) = n-I L ;= 0 a(t - ik ) e i w o t which c orresponds t o C; = 1 . T h u s t h e coeffic i e n t o f X ( T + mk, w ) is a ge ometric p rogression given b y e -i w m k p + e - i w k + . . + e -i W k (n - l - m ) } { -i W k (n - m ) . - e - J wm k e k e -i w _ l _ I} Similarly the coefficien t of X (T - m k , w) is By defining e = �k { e -i W �(n - m ) - I e -J W k _ 1 } and u si ng the identity ( I _ e - 2 i x ) = 2j e - i x sin (x) the above coefficients can be written as follows : X ( T + m k , w ) � e - j(n - I + m w X( T - mk , w) � e j (n - l - m W { { sin [(n - m)8 ] .

24. 25 let n b(t) = I Ci a(t - ik) i= O Hence b(t) = Coa(t)+C\ a(t-k) . . +Cn a(t-nk) b*(t+r)= C�a*( t+r)+cra*(t+r--k) . . +Ga*(t+r- nk) givi ng b(t )b*(t+r) = Co C�a(t)a*(t+r) +C\ C�a(t-k)a*(t+r) . . + + Cn C�a(t - nk)a*(t+r) + Co Cfa(t)a*( t+r- k) + C\ Cfa(t - k)a*(t+r- k) . . + + Cn Cfa(t -n k)a*(t +r-k) + Co Q'a(t)a*(t+ r-n k)+ c\ CJa(t - k)a*(t +r- nk) . . + +Cn C�a(t-nk)a *(t +r- n k) 26 The mathematical treatment of the uncertainty function Since j J 00 �� a(t + x) a * (t + y ) e - j w t dt = i t follows t h a t J -� a(t ) a * (t -x + y ) e - j w ( t - x) dt 00 X I ( T , W) = b (t) b *(t + T) e - j w t dt + n + e - i w k c;, Q' X ( T + n k , w) = Co Cl)X( T , w) + Co crX ( T - k, w) + e - j w k C I Cl' X [ T , w ] .

N + e - j w k Cn C fx [ T + (n - l )k , w ] + Co c,iX( T - nk , w) + e - j w k CI G X [ T - (n - l )k . w ] + + e - j w n k Cn C* n X(T W) Hen ce X I ( T , w) = e - j w n k X(T + nk , w ) { c;, Co' } + . . I , + e - i wk X( T + k , wH cl Co' + C2 ci e - j w k . . + Cn G I e - j w (n - l )k � n C: c;, ci . . cl k w + w k e j Co T, X( w e j + + } H C� + X ( T - k , w H Co C; + C I G e - j w k . . + c;, _ I c: e - j w (n - I ) k } . . + X( T - l I k , w H Co G } which leads to X J T, w ) = n L m =1 + e - j w m k X( T + mk , w ) n L m =0 X( T - m k , w) n -m L ;= 0 n -m L ;= 0 G + m Ge - j w ;k C G+ m e - j w ;k ; so that the new 'n' i s e q u a l t o the n u m b e r o f Re p l a c i n g n b y n p u lses give s n-I n- I-m m w w = X I ( T, ) L e j k X( T + mk , w) L C; + m G e - j w;k m=1 1=0 + 11 - 1 L m=O X( T - mk, w) n - I-m L ;=0 c;c7+m e - j w;k The math ema tical treatmen t of the uncertain ty function 27 2.