Proof of orthogonality of the complex exponentials ”=1 1-e2π(k-r） N 1-e() 1, k-r=mN,m is an integer Problem 0, otherwise 8.54 =∑6[k-r-mW]=p[k-r N points k-小 K-r -N-N+1·-2-101 2 N-1NN+1N+2. the periodic impulse train9  - -    m  k r r k mN p  =− = − =  1 2 ( ) 0 2 - ( ) 2 ( ) 1 1 1 1 N j k n r j k r N j k n r N e N N e e    − = − − − = −  Proof of orthogonality of the complex exponentials  1, - , is 0, k r mN m aninteger otherwise = = k-r 0 1 -N -N+1…… -2 -1 2 …… N-1 N N+1 N+2 …… N points 1 pk -r the periodic impulse train Problem 8.54