DSP-3-07

Chapter 3 1 Introduction
DFT and FFT
Several forms of Fourier transform: (1) Continuous time, continuous frequency -----non-periodic signal Fourier Transform (2) Continuous time, discrete frequency -----Fourier Series (3) Discrete time, continuous frequency -----Fourier transform of sequence Discrete-Time Fourier Transform (4) Discrete time, discrete frequency -----Discrete Fourier Series -----Discrete Fourier Transform
the circular convolution sum is
The condition of not causing an aliasing is:
5 Frequency-domain sampling
If x(n) is the original sequence, xp(n) is the reconstructed sequence, then
3.2 Properties (1) Linearity
(2) Symmetry A real sequence: (3) Circular shift of a sequence (循环移位) If , then
(4) Circular convolution sum （循环卷积）
The N-point circular convolution sum of x1(n) and x2(n) is defined by
And according to the symmetry
We can get the equations as:
k X (k) X(k)=X (k)+ W 1 N 2 k+N X(k+N)=X (k+N)+ W 2 X (k+N) 1 N 2 2 2 2 k X (k) =X (k)− W 1 N 2
Frequency sampling theorem: If X(z) is the z-transform of x(n), then two conditions must be met for L samples of X(z) on the unit circle uniquely determining the X(z). (1) x(n) is a finite length sequence; (2)
~(n) = ~ (n)∗~ (n) y x x2 1 = ∑~ (m ~ (n−m x )x2 ) 1
m=0 N− 1
calculation steps:
If
~ ~ Y(k) = X1(k)⋅ X2(k) ,
then ~(n) = ~ (n)∗~ (n) y x x2 1
If ~ ( n ) = ~ ( n) ⋅ ~ ( n) , y x1 x2
1 x(n) = N
~
~ ∑ X (k ) e
k =0
N −1
j
2π nk N
~ ( n) = 1 x N
DFS pair :
~ ∑ X (k ) e
k =0
N −1
j
2π nk N
2.2 Properties of DFS (1) Linearity
(2) Shift
(3) Periodic convolution summation (周期卷积)
k=01⋅⋅⋅, N− ,, 1 2
k=01⋅⋅⋅, N− ,, 1 2
X1(k)
k X1(k)+wN X2(k)
X2(k)
k wN
−1
k X1 (k ) − wN X 2 (k )
6.3 DIF 6.3.1 Principle of algorithms 6.4 FFT algorithm for a composite number N
Determine the convolutions of the following sequences.
4 Linear convolution evaluated by the circular convolution Assume the length of x1(n) and x2(n) are M and N, the linear convolution sum is
calculation steps:
x Step1: periodically extending → ~1 ( n),
~ ( n) x2 x x Step2: periodical convolution summation →ing the principal values → [ ~ ( n) ∗ ~ ( n)] ⋅ R ( n)
X(k) = D [x(n)] = ∑x(n)Wnk FT N
n=0 N− 1
0 ≤ k ≤N −1
1 N−1 x(n) = ID [X(k)] = ∑X(k)W−nk FT N N n=0
0 ≤ n ≤N −1
The DFT steps of a sequence: sequence:
Step1: periodically extending (周期延拓) Step2: DFS transforming Step3: taking the principal value（取主值）
∞
X(k) = D [x(n)] = ∑x(n)Wnk FT N
n=0
N− 1
0 ≤ k ≤N −1
3.4 Discussion
(1) Spectrum leakage (2) Windows effect (3) Fence effect (栅栏效应) (4) Zero padding
4 Linear convolution evaluated by the circular convolution
1 ~ ~ ~ then Y (k ) = X 1 (k ) * X 2 (k ) N
3 Discrete Fourier transform and the properties 3.1 DFT
Definition: Definition:
DFT is the FT of a finite length sequence:
signal≤1.25kHz, (2) resolution between frequencies≤5Hz. Determine the following parameters: (a) minimum record length, (b) maximum time between samples, and (c) minimum number of points in a record.
1 2 N
Circular convolution
3.3 The relationships between DFT, z-transform and DTFT.
∞
X(z) = Z[x(n)] = ∑x(n)z−n ,
n=−∞
X(e jω ) = F[x(n)] = ∑x(n)e− jωn
n=−∞
Number of samples in a record: N
example: A FFT processor is to be employed in the spectral analysis of a random real signal. Assume that the number of points permitted by the processor must be an integer power of two and that no special data modifications are given: are (1) used. highest The following in specifications frequency
7.2 Convolution using FFT
(1) Linear convolution using FFT
(2) Overlap-add convolution
(3) Overlap-save convolution
1. The sequence x(n) is showed as follow: (1) Draw the picture of x(n)④x(n) (1) Draw the picture of x(n)⑧x(n) 2. Give the principle of DIF algorithm. 3. Draw the 4-point FFT flow chart with DIF algorithm. Assume , determine the result X(k)
6.2 DIT 6.2.1 Principle of algorithms Assume the sequence length N=2L, where L is an integer
then
∵
X(k) = X (k)+ k X (k) W 1 N 2
Where
According to the periodicity
Chapter 3
1. Introduction
DFT and FFT
2. Discrete Fourier Series and the Properties 3. Discrete Fourier transform and the properties 4. Linear convolution evaluated by the circular convolution 5. Frequency-domain sampling 6. FFT 7. Application of FFT