COMMUNICATION IN PRESENCE OF NOISE 67 



than F. A shortcoming of this nature must be accepted since it is impos- 

 sible to have a signal possessing both finite duration and finite bandwidth. 

 The first step of the encoding process is to compute the integer A^ given 

 by 



N < FT < N + 1 (2-1) 



We assume that FT is not an integer in order to avoid borderhne cases. 

 Let W s be the average signal power available for transmission and define 

 the standard deviation a of the o- universe introduced in Section 1 by 

 (A^ + 1/2)0-- = W s- To encode the first message, draw 2N -\- 1 numbers 

 A-N, • • •,Ao'\- ■ • A^^^ at random from the a universe. The signal correspond- 

 ing to the first message is then taken to be 



/o(/) - 2-''l4r + Z (Ai'^ COS iTTUt/T + A^-'l sin 2x»//r) (2-2) 



71 = 1 



The remaining A' messages are encoded in the same way, the signal repre- 

 senting the ^th message being 



hit) - 2"^'-^^'"' -H Z Un^ coslirni/T + A^-l sin 2Tni/T). (2-3) 



n=I 



It is apparent that each signal consists of a d-c term plus terms corre- 

 sponding to A' discrete frequencies, the highest being N/T < F, and that 

 the average power (assuming hiO to flow through a unit resistance) in the 

 ^th signal is 



T-' f' Ilit) dl = 2-\Al''f + Z 2-\{A':'f + {A'Jlir-] (2-4) 



''- 7-/2 71=1 



Since the .I's were drawn from a universe of standard deviation cr, the ex- 

 pected value of the right hand side is (2A" + \)a~/2 which is equal to the 

 average signal power W s, as required. 



We pick one of the A' + 1 messages at random and send the correspond- 

 ing signal over a transmission system subject to noise. We choose our nota- 

 tion so that the sent signal is represented by /o(/) as given by {2-2). Let the 

 noise be given by 



.V 



/(/) = 2~"-B^ -f Z {Bn cos 2irnt/T + 5_„ sin 27r»//r) (2-5) 



74 = 1 



where 5_.v, • • • , 7?n, • • • , B^^ are (2A^ + 1) numbers drawn at random from 

 the normally distributed v universe mentioned in the introduction. The 

 standard deviation v of the universe is given by (A^ + l/2)v~ — W x, Wn 

 being the average noise power. We call ./(/) simply "noise" rather than 



