Real motion and sampling
Ideal solution
Higher-order methods
Aliasing and Nyquist frequency
例子：
t u U sin 2 T t 1
2 U sin(t ), T
当ω△t≈O(1)，误差的量级为1，所以有100%的误差。 对于二次有限差分，ω△t≤ 0.8，使得误差小于 10%。 △t≤ T/8
Aliasing and Nyquist frequency
利用泰勒展开
du |t ( a2 a1 a0 a1 a2 )u (t ) dt du (2a2 a1 a1 2a2 )t |t dt t 2 d 2 u (4a2 a1 a1 4a2 ) | 2 t 2 dt t 3 d 3u (8a2 a1 a1 8a2 ) |t 6 dt 3 t 4 d 4 u (16a2 a1 a1 16a2 ) |t 24 dt 4 t 5 d 5 u (32a2 a1 a1 32a2 ) | 5 t 120 dt O ( t 6 )
海洋调查方法之采样
程灵巧 上海海洋大学
Real motion and sampling
Ideal solution
Higher-order methods
Aliasing and Nyquist frequency
现实：连续型 采样：离散型
tn t0 nt n 1, 2,...
曲线上某点的斜率：
du dt
采样所得相邻两点间的斜率：
u (t t ) u (t ) t
Real motion and sampling
Ideal solution
Higher-order methods
Aliasing and Nyquist frequency
理想：采样所获的信息尽量接近现实情况
du u (t t ) u (t ) lim dt t 0 t du u (t t ) u (t ) dt t
Ideal solution
Higher-order methods
Aliasing and Nyquist frequency
Taylor series:
du t 2 d 2u t 3 d 3u 4 u (t t ) u (t ) t O ( t ) 2 3 dt 2 dt 6 dt
U t 2 2 T U t 3 3 T t 4 U T4
du u (t t ) u (t ) t U O dt t T T
t 两边同除以量级U/T后，切断误差为： O T
t T
Real motion and sampling
Ideal solution
Higher-order methods
Aliasing and Nyquist frequency
理想：采样所获的信息尽量接近现实情况
现实：1、观测仪器分解率有限 2、计算机内存有限 3、现场观测时间有限...等。
CTD（Sea-bird 911plus）：24 Hz
有其它办法吗？
Real motion and sampling
Ideal solution
Higher-order methods
Aliasing and Nyquist frequency
Forward
Backward
du t 2 u (t t ) u (t ) t dt 2 du t 2 u (t t ) u (t ) t dt 2
（1）
尽量减小△t，使得：
（2）
du u (t t ) u (t ) t dt
du t 2 d 2u t 3 d 3u 4 u (t t ) u (t ) t O ( t ) 2 3 dt 2 dt 6 dt
尽量小
Taylor series:
Real motion and sampling
Real motion and sampling
Ideal solution
Higher-order methods
Aliasing and Nyquist frequency
a2 a1 a0 a1 a2 0 (2a2 a1 a1 2a2 )t 1
4a2 a1 a1 4a2 a1 a1 16a2 0
du 4 u (t t ) u (t t ) 1 u (t 2t ) u (t 2t ) |t dt 3 2t 3 4 t
Real motion and sampling
Ideal solution
Higher-order methods
d 2u t 3 2 dt 6 d 2u t 3 2 dt 6
d 3u 4 O ( t ) （4） 3 dt d 3u 4 （5） O ( t ) dt 3
（4）式-（5）式，centered difference approximation
du u (t t ) u (t t ) 2 O(t ) dt 2t
Real motion and sampling
Ideal solution
Higher-order methods
Aliasing and Nyquist frequency
一般化
假设
du |t a2u (t 2t ) a1u (t t ) a0u (t ) a1u (t t ) a2u (t 2t ) dt