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## Chapter 5 The Discrete-Time Fourier Transform

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**In this chapter, we will learn---**• The Discrete-Time Fourier Transform (DTFT) • The Fourier Transform for Periodic Signals • Properties of the DTFT**Notation:**• CFS: ( the continuous Fourier series,FS) • DFS: ( the discrete Fourier series) • CTFT: ( the continuous time Fourier Transform,FT) • DTFT: ( the discrete time Fourier Transform)**5.1 Representation of Aperiodic Signals: DTFT**5.1.1 Development of DTFT (from DFS to DTFT) (1) Fourier series (periodic signal)**x[n]**ak Fourier series (periodic signal)**x[n]**0 0 (2) Fourier transform (aperiodic signal)**Since over**When So DTFT**Where is the spacing of the samples in the frequency**domain. The coefficients akare proportional to X(ejw) samples of**The discrete-time Fourier transform pair**Synthesis equation Analysis equation • That means, x[n] is composed of ejwn at different frequencies w with “amplitude” X(ejw)dw/2p ; X(ejw)is referred to the spectrum of x[n].**Note:**1. Relation between of DFS and DTFT Discrete <--> periodic Periodic <--> discrete Discrete <--> periodic Aperiodic <--> continuous**2. Comparing DTFT to FS**Duality Continuous-time、Discrete-frequency Discrete-time、Continuous-frequency Eq.(5.9) can be regarded to FS in frequency-domain; Eq.(5.8) is just that x[n] is the FS coefficients of X(ejw).**3. Relation between FT and DTFT**Let and so**Example Poisson formula**If then**Note:**That implies It is a Fourier Series in Frequency-Domain.**5.1.2 Examples of DTFT**Example 5.1**We can conclude:**real and even real and even**Example 5.3**Similar to a sinc, but periodic with a period 2π.**Note:**If the periodization of x[n] which period N is made, We can get DFS coefficients: Just same as (3.104) in P218**1**w 0 n 0 Example 5.4**Example 5.5**Ideal DT LPF A discrete sinc in time domain**or**5.1.3 Convergence of DTFT • Convergence conditions: –– Absolutely summable — Finite energy**W=p/4**W=3p/8 W=3p/4 W=p/2 W=p W=7p/8**There are not any behaviors like the Gibbs phenomenon in**evaluation the discrete-time Fourier transform synthesis equation. (see page 368 fig5.7)**5.2 DTFT for periodic signals**Recall CT result: What about DT: a) We expect an impulse (of area 2π) at ω=ωo b) But X(ejω) must be periodic with period 2π, In fact**w**1 w 0 0 n Ifw0=0**Now consider a periodic sequence**See page 370 Fig5.9**Example**1. 2.**5.3 Properties of DTFT**5.3.1 Periodicity 5.3.2 Linearity 5.3.3 Time Shifting and Frequency Shifting**5.3.4 Conjugation and Conjugate Symmetry**If x[n] is real 5.3.5 Differencing and Accumulation**if n is a multiple of k**if n is not a multiple of k. 5.3.6 Time Reversal 5.3.7 Time Expansion**5.3.8 Differentiation in Frequency**5.3.7 Parseval’s Relation**5.4 The Convolution Property**P386 Ex5.14**5.5 The Multiplication Property**Periodic convolution Homework: P400-5.2* 5.10*