DSP离散时间信号处理第4章.pptVIP

Chapter 4 Sample of Continuous-Time Signals Periodic Sampling A continuous-time signal xa(t) should be converted to a discrete form xa(nT) or x(n) before it is processed digitally. This process is called as sampling. T is the sampling interval. After the discrete-time signal is processed, the discrete-time output y(n) or ya(nT) is converted back to the continuous-time domain, i.e., ya(t) . The sampling theorem establishes the relationship between the continuous-time signal and the discrete-time signal, which gives the conditions under which a continuous-time signal can be recovered from its samples. Basic principles The Fourier transform of a continuous-time signal f (t) is given by and the corresponding inverse transform is defined as So, the spectrum（频谱）of xa(t) is Basic principles If x(t)=a(t)b(t), then where X(jΩ), A(jΩ), and B(jΩ) are the Fourier transforms of x(t), a(t), and b(t) respectively. If x(t) is periodic with period T, then we can express it by its Fourier series（傅立叶级数）defined by Sampling theorem（抽样定理） A continuous-time signal xa(t) is sampled by a sampling function（抽样函数）p(t), which is defined as and the corresponding discrete-time signal is xi(t). This process can be expressed as Sampling theorem Sampling theorem Since Now we must first determine the Fourier transform of p(t), P(jΩ) Sampling theorem So, the p(t) can be expressed by a Fourier series as As the Fourier transform of is , then the Fourier transform of p(t) is Sampling theorem Substituting this expression for P(jΩ) in the former equation, we have that Since the convolution of a function F(jΩ) with a shifted impulse δ(Ω-Ω0) is the shifted function F(jΩ-jΩ0) ,then Sampling theorem Sampling theorem Explanation of the Xi( jΩ) The spectrum（频谱）of xi( t), Xi( jΩ), is composed of infinite shifted copies of the spectrum of xa(t), Xa( jΩ), with the shifts in frequency being multiples of the sampling frequency, Ωs= 2π/T .