transmission over an AWGN channel and down-modulation,the received signal is 0=x(0+0 where n()is independent ofx(r)and can be represented as n)=∑n:p-kT)+广) Here is a sequence of i.i.d.zero-mean complex Gaussian variables,each with variance No,and n(r)is a Gaussian process with<n(),p(=0for all k. The output sequence of a QAM demodulator is given by y=(0),p1-kT)）=x+n which represents an equivalent discrete-time AWGN channel model obtained with an orthonormal QAM system 2.4.3 Nyquist Criterion and Minimum Bandwidth 由信号检测理论可知，在AWGN下，接收滤波器采用匹配滤波器，即q)=p'(-), 能铭使得系练在采洋时别的输出信噪出达到品大，闲出匹配滤波婴（成等效地，相关 常被用作最佳接收机的前端 ,这是从抗噪声性能的角度来考虑的。从抗SI(无 失真传输)的角度来讲，我们需要选择发送滤波器p)、接收滤波器g)使得接收波形 y)=∫x(r)q(t-r)dr-∑xgt-kT) 在kT时的样值ykT)=x4 for eachk其中g)=p)*q()=∫广p(r)qt-r)dr 考虑上述两个因索，我们要求g)=p)*p'(-)满足gO)=1 and g(kT)=0for each nonzero integer k.Waveforms with this property are said to be idea Nyquist. We now consider the design of the modulation pulse p()such that the sym ol interference.(That is,find the condition under whichp( ■Nyquist Pulses In 1928,Nyquist recognized that the sinc pulse in the Sampling theorem was significant in two ways: the shifts of sinc pulse are orthogonal: the sinc pulse has the all sampling points other than at time 0 Nyquist realized that,to avoid Isl,the crucial property was to find a pulse whose shifis are orthogonal We now find the conditions under which a real waveform p(r)has the property: [1,fork=0 p(t-KT)p'(1)di= 2.8) 0,fork≠0 Note that the Fourier transform Fp(-kT)]=eP(f),we have 212-11 transmission over an AWGN channel and down-modulation, the received signal is y(t) = x(t) + n(t) where n(t) is independent of x(t) and can be represented as ( ) ( ) ( ) k k n t n p t kT n t ⊥ = − +  Here {nk} is a sequence of i.i.d. zero-mean complex Gaussian variables, each with variance N0, and n t( ) ⊥ is a Gaussian process with < n t( ) ⊥ , p(t-kT)> = 0 for all k. The output sequence of a QAM demodulator is given by ( ), ( ) k k k y y t p t kT x n = − = + which represents an equivalent discrete-time AWGN channel model obtained with an orthonormal QAM system. 2.4.3 Nyquist Criterion and Minimum Bandwidth 由信号检测理论可知，在 AWGN 下，接收滤波器采用匹配滤波器，即 * q t p t ( ) ( ) = − ， 能够使得系统在采样时刻的输出信噪比达到最大，因此匹配滤波器（或等效地，相关 器）通常被用作最佳接收机的前端。这是从抗噪声性能的角度来考虑的。从抗 ISI（无 失真传输）的角度来讲，我们需要选择发送滤波器 p(t)、接收滤波器 q(t)使得接收波形 ( ) ( ) ( ) ( ) k k y t x q t d x g t kT     −  = − = −   在 t=kT 时的样值 ( ) k y kT x  = for each k, 其中 g t p t q t p q t d ( ) ( )* ( ) ( ) ( )     − = = −  . 考虑上述两个因素，我们要求 * g t p t p t ( ) ( ) ( ) =  − 满足 g(0) 1 = and g kT ( ) 0 = for each nonzero integer k. Waveforms with this property are said to be idea Nyquist. We now consider the design of the modulation pulse p(t) such that the PAM/QAM system has no intersymbol interference. (That is, find the condition under which {p(t-kT)} forms a orthonormal set.) ◼ Nyquist Pulses In 1928, Nyquist recognized that the sinc pulse in the Sampling theorem was significant in two ways: - the shifts of sinc pulse are orthogonal; - the sinc pulse has the value 0 at all sampling points other than at time 0. Nyquist realized that, to avoid ISI, the crucial property was to find a pulse whose shifts are orthogonal. We now find the conditions under which a real waveform p(t) has the property: * 1, for 0 ( ) ( ) 0, for 0 k p t kT p t dt k  −  = − =     (2.8) Note that the Fourier transform 2 [ ( )] ( ) j kTf p t kT e P f −  − = , we have