[工学]Chapter3 - Discrete Time Fourier Transform.ppt

Chapter 3 Discrete-Time Fourier Transform Discrete-Time Fourier Transform The Definition The Theorems Frequency Response of an LTI DTS Phase and Group Delays Continuous-Time Fourier Transform We already discussed this topic in Chapter 1. You can read this part(§3.1) in the textbook p117-122. Continuous-Time Fourier Transform Pair: Discrete-Time Fourier Transform Definition - The discrete-time Fourier transform (DTFT) X(ej?) of a sequence x[n] is given by Discrete-Time Fourier Transform Example - The DTFT of the unit sample sequence d[n] is given by Discrete-Time Fourier Transform Its DTFT is given by Inverse Discrete-Time Fourier Transform Is the DTFT of x[n] periodic? Inverse Discrete-Time Fourier Transform Can be seen as the Fourier series expansion of the periodic function X(ejw) Inverse Discrete-Time Fourier Transform Inverse discrete-time Fourier transform: Inverse Discrete-Time Fourier Transform The order of integration and summation can be interchanged if the summation inside the brackets converges uniformly, i.e. X(ej?) exists Then Inverse Discrete-Time Fourier Transform Now Basic Properties In general, X(ej?) is a complex function of the real variable w and can be written in rectangular form as Basic Properties It follows that Basic Properties X(ej?) can alternately be expressed in the polar form as X(ej?) = | X(ej?) |ej?(?) where ?(?) = arg{X(ej?) } Basic Properties | X(ej?) | is called the magnitude function ?(?) is called the phase function Both quantities are again real functions of w In many applications, the DTFT is called the Fourier spectrum Likewise, | X(ej?) | and ?(?) are called the magnitude spectrum and phase spectrum Basic Properties The magnitude and phase of the DTFT X(ej?) = 1/(1 – 0.5e-j?) are shown below Basic Properties The relation between the rectangular and polar forms of X(ej?) is given by Basic Properties Note: X(ej?) = | X(ej?) |ej?(?+2?k) = | X(ej?) |ej?(