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*