信号与系统英文版课件：Chap5 The Discrete-Time Fourier Transform

1、The Discrete-Time Fourier TransformChapter 5 Lecture 1Signals and Systems Spring 2015Response of LTI Sys to Complex Exphnz0 : a complex constantH(z0): a constant determined by z0 and hnResponse of LTI Sys to Complex ExpIf fn can be represented as linear combinations of complex exponentials, thenh(t)

2、hnzk : a complex constantH(zk): a constant determined by zk and hn?Fourier Series Representation of Periodic SignalsThe complex exponential is periodic with period N=2/0. is called the kth (k integer) harmonically related complex exponential of , which is also periodic with period N. must be periodi

3、c with period N=2/0, Fourier Series Representation of Periodic SignalsFurther notice thatFor a DT periodic signal fn with period N=2/0, how to find ak?h(t)?Consider a DT periodic signal fn with period N=2/0 that can be represented as k and m both integersConsider a DT periodic signal fn with period

4、N=2/0 that can be represented as Fourier series coefficientFourier series representationFourier Series Representation of Periodic SignalsFourier Series Coefficients is periodic with period N, i.e. Fourier coeffs of f*n isIf fn is real, then or If fn is real and even, then its Fourier coeffs is real

5、and even, i.e. If fn is real and odd, then its Fourier coeffs is imaginary and odd, i.e. The Discrete-Time Fourier TransformChapter 5 Lecture 2Signals and Systems Spring Example 1 Given a DT periodic signal as shown below. Find the Fourier series coefficient of it.Solution: Period of fn is N. Fourie

6、r coefficientswhere .Let , thenFourier Series and LTI SystemsKey steps for computing the response of a LTI system to a periodic signal: Compute Fourier series coefs of the signal Compute the eigenvalue of the LTI systemh(t)Example Given an LTI system with impulse response . Determine the response of

7、 this system to the periodic signal Solution: is periodic with period ,Fourier Transform for DT signals Problem: How to express aperiodic signal as a summation of complex exponentials?Fourier series coefficient is (Assume N is even) As Define can be expressed as Define can be expressed asFourier transformInverse Fourier transformExamples for DT Fourie