log,MN from )=x()+n()by matched filtering y4=),p1-kT)》=∫)p1-kT)d=x4+n4 (2.15) The sequence is a sequence of i.i.d.Gaussian N-vectors,whose elements have variance .Thus the AWGN waveform channel is converted to a discrete-time vector AWGN channel,which is equivalent to N real discrete-time AWGN channels operating in parallel. Let P(f)=[P().P().(f)],where p(f)is the Fourier transform of p (t) Then the frequency-domain condition for orthonormality is the generalized Nyquist criterion G(-cm ∑p'f+m/T)Pf+m/T) me =w, f≤1/（270 (2.16 wheredenotes conjugate transpose.G=(P().sn.n'N-1)is the NxN matrix,and I is the NxN identity matrix.It can be shown that the total one-sided bandwidth for which some P(f).f is nonzero must be at least N/(27)Hz. Multicarrier modulation is the generic term used for any OPAM scheme in which each of the orthogonal pulses is roughly localized in the frequency domain.It includes as special cases OFDM and DMT.In OFDM,the pulse set p (t))is 1N= P(0= N2eg-1、cn=0. 2.5 Discrete-time and Continuous-time Parameters We now examine the between the parameters of continuous-time and discrete-time AWGN channel models.The parameters involved are,namely,W,SNR,pand Continuous-time band-limited AWGN channel model:M(r)=x()+n() Discrete-time (real or complex)AWGN channel model:y=x+n The nominal bandwidth W of the continuous-time channel is equal to the inverse 1//of the symbol interval of the discrete-time channel when the latter is complex, or to 1/2T when it is real. The nominal spectral efficiency of the continuous-time channel is defined as 2-18 2-18 2 2 log 2log /(2 ) N M M T N T  = =  . An optimum demodulator may recover noisy estimates zk of xk from y(t) = x(t) + n(t) by matched filtering ( ), ( ) ( ) ( ) k k k = − = − = + y t t kT y t t kT dt  y p p x n (2.15) The sequence {nk} is a sequence of i.i.d. Gaussian N-vectors, whose elements have variance 2  n . Thus the AWGN waveform channel is converted to a discrete-time vector AWGN channel, which is equivalent to N real discrete-time AWGN channels operating in parallel. Let 0 1 1 ( ) [ ( ), ( ), , ( )] N f P f P f P f P = − , where ( ) P f n is the Fourier transform of ( ) n p t . Then the frequency-domain condition for orthonormality is the generalized Nyquist criterion 1 ( ) ( / ) m f f m T T  G G = +  1 † ( / ) ( / ) m f m T f m T T  = + + P P = IN, | f |  1/(2T) (2.16) where “ † ” denotes conjugate transpose,   * ' ( ) ( ) ( ),0 , ' 1 n n G f P f P f n n N =   − is the NN matrix, and IN is the NN identity matrix. It can be shown that the total one-sided bandwidth for which some ( ) P f n , f 0 is nonzero must be at least N/(2T) Hz. Multicarrier modulation is the generic term used for any OPAM scheme in which each of the orthogonal pulses is roughly localized in the frequency domain. It includes as special cases OFDM and DMT. In OFDM, the pulse set { ( )} n p t is 1 2 / 0 1 ( ) ( / ) N j nk N n k p t e g t kT N N  − = = −  , for n = 0, ., N-1 2.5 Discrete-time and Continuous-time Parameters We now examine the relationship between the parameters of continuous-time and discrete-time AWGN channel models. The parameters involved are, namely, W, SNR,  and C. Continuous-time band-limited AWGN channel model: y(t) = x(t) + n(t) Discrete-time (real or complex) AWGN channel model: y = x + n ◼ The nominal bandwidth W of the continuous-time channel is equal to the inverse 1/T of the symbol interval T of the discrete-time channel when the latter is complex, or to 1/2T when it is real. ◼ The nominal spectral efficiency of the continuous-time channel is defined as