文档介绍：Outline of chapter 3
The Discrete-Time Fourier Transform
序列的(离散时间)傅立叶变换
(Time Domain – Discrete)
(Frequency Domain – Continuous)
Fourier Transform
Definition:
Let t=n*T, Ώ= ω/T, T=sampling period (sec)
n-integer, ω-Digital Frequency(rad)
DTFT and IDTFT
离散时间傅立叶正反变换
(时域离散,频域连续)
DFT and IDFT
离散傅立叶正反变换
(时域,频域均离散)
See Chapter 5
Calculation of DTFT
Two Important Properties:
Periodicity calculate in ω=[-π,π] or [0,2π]
Symmetry calculate in ω=[0, π]
Ex030100 Analytical solution of infinite seq.
Ex030200 Numerical solution of finite seq.
Ex030300 Numerical solution of infinite seq.
Ex030400 Finite seq. on discrete frequencies
Ex030500 periodicity of DTFT
Ex030600 Symmetry of DTFT for real seq.
DTFT Calculation as Matrix-vector multiplication
Other Properties of DTFT
Linearity
Shifting (in time and in frequency)
Conjugation symmetry
Convolution (in time and in frequency)
Parseval’s Theorem
Examples to prove:
Ex030700 linearity of two random seq.
Ex030800 time shift property
Ex030900 frequency shift property
Ex031000 conjugate property
Ex031100 folding property
Ex031200 symmetry property
Even-odd verse sym-antisym
pose time sequence into even/odd (conj. symmetry/anti-.. plex seq.)
Corresponds to
pose its FT into Real/Imaginary parts
For real sequence: FT
序列及其变换的奇偶与虚实
时域序列
频域DTFT变换
实偶序列
共扼对称(复)序列
实部
实奇序列
共扼反对称(复)序列
虚部
实序列
共扼对称:
实部偶,虚部奇
LTI Systems in Frequency Domain
Response plex exponential
Frequency Response
Response to sinusoidal seq.
Response to arbitrary seq.
Frequency Response from difference Eq.
Determined by numerator coefficients b
and denominater coefficients a.