Chapter 5-Finite-length Discrete TransformA

M-1
x[n] y[n],
0 n N 1
4.2 Sampling the DTFT
y[n]
m
x(n mN )
x ( n)
n M-1

0 1 2
x '(n)
0 1 2
N
n
the length of x(n) M > N
4.2 Sampling the DTFT
Finite length
X (s)
DTFT:
jt
x ( n) X ( k )
DFT:
DFT ( )
Fourier Transform:
X ( ) x( t )e


dt
X ( )
n


x(n)e j n
j n
X (k ) x(n)e
k 0
n 0
L 1
2 j nk N
,0 k N 1;
N反映DFT点数 L反映序列长度

L和N可以相等，也可以不等；若L<N, 可以对 输入序列补零，使补零后序列长度等于N
1. Definition
1. Definition
1. Definition
2. Matrix Relations
Chapter 5
Finite-Length Discrete Transforms
Review of Transforms
Analog Signal
x(t ) X ()
L () F () DTFT ( ) Z ()
Discrete-time Signal
x(n) X ( ) X ( z)
0 Re WN
3)对称性
4) W
N 2 N
WN
(k
N ) 2
k WN
3 WN 2 WN
1 WN
W
N 2 N
N 8
1
2n n W W N 5) N 2
1. Definition
1. Definition
1. Definition

长度为L的序列x[n]的N点DFT
X (k ) x(n)e
4.1 DTFT from DFT by interpolation
4.1 DTFT from DFT by interpolation
4.2 Sampling the DTFT
4.2 Sampling the DTFT
4.2 Sampling the DTFT
Y[k]
4.2 Sampling the DTFT
2. Matrix Relations
2. Matrix Relations
2. Matrix Relations
3. DFT Computation Using MATLAB



The functions to compute the DFT and the IDFT are fft and ifft These functions make use of FFT algorithms which are computationally highly efficient compared with the direct computation Figure in the next slide shows the DFT and DTFT of the sequence cos(6πn/16) 0≤n≤15
Part A: Discrete Fourier Transform (DFT)

Definition
Matrix Relations
DFT Computation Using MATLAB
Relation between DTFT and DFT and their
inverses
Chapter 5 Finite-Length Discrete Transforms
x(n+4)
0
1
2
3
4
5
x(n)
0
1
2
3
4
5
x(n-4)
0
1
2
3
4
5
y(n)
4
6
2
3
4.3 Numerical Computation of the DTFT using DFT
4.3 Numerical Computation of the DTFT using DFT
通过末尾补 零增加序列 长度
N 1
j
2 nk N
x( t ) X ( )e


j t
d 2
x(n) X ( )e


d 2
0 k N 1;
2 j nk 1 N 1 x(n) X (k )e N N k 0 0 n N 1.
Laplace Transform：
1. Definition
1. Definition
1. Definition
twiddle factor (旋转因子)
WN e
j 2 N
j Im
( n N ) k N
1)周期性 W
nk N
W
W
n( k N ) N
4 WN
5 WN
6 WN 7 WN
nk ( N n)k n( N k ) ( W ) W W 2)共轭性 N N N
4.2 Sampling the DTFT
y(n)
m
x(n mN )
x ( n)
n

0 1 2
M-1
x '(n)
0 1 2
M-1 N
n
the length of x(n)
M N
the length of x(n)
M N
x ( n)
'
0 1 2
N
n
y ( n)
n
0 1 2
5.2 The Discrete Fourier Transform 5.3 Relation Between the FT and the DFT 5.4 Circular Convolution 5.5 Classification of Finite-Length Sequences 5.6 DFT Symmetry Relations 5.7 Discrete Fourier Transform Theorems 5.8 Fourier-Domain Filtering 5.9 Computation of the DFT of Real Sequences 5.10 Linear Convolution using the DFT
4.3 Numerical Computation of the DTFT using DFT
Example
3n x[n] cos( ), 8
0 n 15
Homework
Problems: 5.2，5.9，5.10, 5.20, 5.25，5.28, 5.43，5.45, 5.55
3. DFT Computation Using MATLAB
4. Relatiheir inverses
4.1 DTFT from DFT by interpolation
4.1 DTFT from DFT by interpolation
Z-Transform：
X (s) x(t )est dt


X ( z)
n
x(n)z

n
Ch.5 Finite-Length Discrete Transforms

Part A: Discrete Fourier Transform (DFT) Part B: Operations on Finite-Length Sequences and Discrete Fourier Transform Properties