0122229现代数字通信与编码理论 September 1,2010 XDU Fall 2010 Lecture Notes Chapter 1 Preliminaries In this chapter we will briefly review some basic concepts and principles,which will be used as the basis of discussions later. A.Phase splitter and analytic signal Ifx(t)is a real-valued signal,then its Fourier transform X()satisfies the symmetry property X(-f)=X'(f) where is the complex conjugate of ) ■A phase] also kno wn as Hilbert filter)is a complex filter with impulse response()and transfer function.where f≥0 10,f<0 h-(0 XA() x(t) 40 Figure 1.0A Hilbert filter If the real-valued input to a phase splitter isx(),then the output is x,0=方0+0.or X,(f)=2x(f)H.(f) where()is the Hilbert transform of x(t).We introduce the factor 2 so that x(t)andx(r)have the same energy (or power).Notice thatx(r)is a complex-valued signal. Asignal with only nonnegative frequency components is called an analytic signal. x(t)can be recovered from x(t)by x()=x] 1010 0122229 现代数字通信与编码理论 September 1, 2010 XDU, Fall 2010 Lecture Notes Chapter 1 Preliminaries In this chapter we will briefly review some basic concepts and principles, which will be used as the basis of discussions later. A. Phase splitter and analytic signal „ If x(t) is a real-valued signal, then its Fourier transform X(f) satisfies the symmetry property * X ( ) () − = f Xf where X* (f) is the complex conjugate of X(f). The symmetry property says that knowing X(f) for f ≥ 0 is sufficient to entirely describe X(f) and thus to describe x(t). „ A phase splitter (also known as Hilbert filter) is a complex filter with impulse response h+(t) and transfer function H+(f), where ⎩ ⎨ ⎧ < ≥ + = 0, 0 1, 0 ( ) f f H f h+(t) x(t) xA(t) x(t) xA(t) 1 πt j xˆ( )t Figure 1.0 A Hilbert filter „ If the real-valued input to a phase splitter is x(t), then the output is 1 ( ) [ ( ) ( )] ˆ 2 A x t x t jx t = + , or () 2 () () X A f XfH f = + where xˆ( )t is the Hilbert transform of x(t). We introduce the factor 2 so that x(t) and xA(t) have the same energy (or power). Notice that xA(t) is a complex-valued signal. „ A signal with only nonnegative frequency components is called an analytic signal. „ x(t) can be recovered from xA(t) by x() 2 () t xt = R[ A ]