《信号与系统教学资料》9-DT Fourier Transforms.pptVIP

* * * * * * * * * * * Calculating Periodic Convolutions Suppose we integrate from - to : Example: Calculating Periodic Convolutions Duality in Fourier Analysis Fourier Transform is highly symmetric CTFT: Both time and frequency are continuous and in general aperiodic Suppose f(?) and g(?) are two functions related by Example of CTFT duality Square pulse in either time or frequency domain DTFS Suppose f(?) and g(?) are two functions related by Duality in DTFS Then Let m=n and r=-k Let r=n and m=k Duality between CTFS and DTFT CTFS-DTFT Duality Suppose f(?) is a CT and g(?) a DT sequence related by then * * * * * * * * * * * * * * * * * * * * * * * * * Signals and Systems Spring 2014 Lecture #15 22 April, 2014 Chen Zubin (陈祖斌) Room 304, Geology Palace, 0431 czb@jlu.edu.cn The Discrete-Time Fourier Transform The Discrete-Time Fourier Transform CT Fourier Transform pair The Discrete-Time Fourier Transform ? Discrete frequency points become denser in ω as T increases T=4T1 T=8T1 T=16T1 The Discrete-Time Fourier Transform x[n] –aperiodic and (for simplicity) of finite duration N is large enough so that x[n]=0 if for and periodic with period N for any n as DTFT Derivation (Continued) DTFT Derivation (Continued) Convergence Issues Synthesis Equation: None, since integrating over a finite interval Analysis Equation: Need conditions analogous to CTFT, e.g. Finite energy Absolutely summable Examples Parallel with the CT examples in Lecture #8 1 2 shifted unit sample More Examples 3 Exponentially decaying function Infinite sum formula Still More 4) DT Rectangular pulse (Drawn for N1= 2) Still More 5) Properties of the DT Fourier Transform