[经济学]Lecture 5 Fourier Series_ frequency domain representation of LTI system_Sampling and reconstruction.ppt

,The Discrete-Time Fourier Representation of Periodic Sequence The frequency domain representation of LTI system Sampling and reconstruction of analog signals,1 The Discrete-Time Fourier Representation of Periodic Sequence,Periodic Sequence ：？,Absolutely numerable sequences：,1) Fourier series of Periodic Sequence (周期序列的离散傅里叶级数,（证明参考中文教材）,where,ak is also a periodic Sequence,Let,: periodic Sequence with fundamental period N ， called DFS(Discrete Fourier Series),(*) 式表明将周期序列分解成N次谐波,第k个谐波频率为k=(2/N)k, k=0, 1, 2 N-1， 幅度为 。 基波分量的频率是2/N， 幅度是 。 一个周期序列可以用其DFS表示它的频谱分布规律。,(*),(*),例1 设x(n)=R4(n)， 将x(n)以N=8为周期，进行周期延拓，得到如图1(a)所示的周期序列 ，周期为8， 求 的DFS。,解：,(1) ej0n,integer,2） The Discrete-Time Fourier Representation of Periodic Sequence,The DTFT of ej0n,(2),verify：,k=0, 1, 2 N-1,where,The Discrete-Time Fourier Representation of Periodic Sequence,例 2 求例1中周期序列的DTFT。 解： 由例1,代入,图 2 例2图,对比图1， 对于同一个周期信号， 其DFS和DTFT分别取模的形状是一样的， 不同的是DTFT用单位冲激函数表示(用带箭头的竖线表示)。 周期序列的频谱分布用其DFS或者DTFT表示都可以， 但画图时应注意单位冲激函数的画法。,The frequency domain representation of LTI system,The Fourier transform representation is the most useful signal representation for LTI systems. It is due to the following result: Response to a complex exponential Response to sinusoidal sequences Response to arbitrary sequences,Response to a complex exponential,Frequency response: the DTFT of an impulse response is called the frequency response (or transfer function) of an LTI system and is denoted by,Response to a complex exponential,The output sequence is the input exponential sequence modified by the response of the system at frequency 0,In general, the frequency response H is a complex function of . The magnitude |H| is called the magnitude (gain) response function（幅度响应）, and the the angle is called the phase response function （相位响应）.,extend,Response to sinusoidal sequences,Steady-state response,Response to arbitrary sequences,Condition: Absolutely summable sequence LTI system,Frequency response function from difference equations,When an LTI system is represented by the difference equation, then to evaluate its frequency response , we would need the impulse response h(n). We know that when , then y(n) must be,Sampling and reconstruction of analog signals,Analog signals can be converted into discrete signals using sampling and quantization operations: analogy-to-digital conversion, or ADC Digital signals can be converted into analog signals using a reconstruction operation: digital-to-analogy conversion, or DAC Using Fourier analysis, we can describe the sampling operation from the frequency-domain view-point, analyze its effects and then address the reconstruction operation. We will also assume that the number of quantization levels is sufficiently large that the effect of quantization on discrete signals is negligible.,Sampling(采样),Continuous-time Fourier transform and inverse CTFT,Absolutely integrable Omega is an analogy frequency in radians/sec,Sampling,Sample xa(t) at sampling interval Ts sec apart to obtain the discrete-time signal x(n),1. relationship between analog signals and sampling signals in frequency-domain. 2. The above relation is known as the aliasing formula,The analog and digital frequencies,Fs: the sampling frequency, sam/sec,Amplitude scaled factor: 1/T; Frequency-scaled factor: =T (=02) Frequency-translated factor: 2k/T;,Xa(j)以周期s=2/T进行周期延拓,=T,aliasing formula,Ban-limited signal（带限信号）,A signal is band-limited if there exists a finite radians frequency c such that Xa(j ) is zero for | | c. The frequency Fc= (c /2)pi is called the signal bandwidth in Hz,the frequency response of sampling signals(采样信号的频谱),fs2 fc，,fs2 fc,Suppose signal band is limited to c,Reconstruction of the original continuous-time signal from its sampling version (采样恢复 ),Sampling Principle,A band-limited signal xa(t) with bandwidth Fc can be reconstructed from its sample values x(n)=xa(nTs) if the sampling frequency Fs=1/Ts is greater than twice the bandwidth Fc of xa(t) , Fs 2 Fc. Otherwise aliasing would result in x(n). The sampling rate of 2Fc for an analog band-limited signal is called the Nyquist rate.,Reconstruction,1. Lowpass filter band-limited to the -Fs/2,Fs/2 band 2. The ideal interpolation is not practically feasible because the entire system is noncausal and hence not realizable.,Interpolating formula,Practical D/A converters,Zero-order-hold (ZOH) interpolation: In this interpolation a given sample value is held for the sample interval until the next sample is received. It can be obtained by filtering the impulse train through an interpolating filter of the form,Zero-order-hold (ZOH) interpolation,The resulting signal is a piecewise-constant (staircase) waveform which requires an appropriately designed analog post-filter for accurate waveform recon