测试与检测技术基础----19.ppt

文档介绍：Fundamentals of Measurement Technology(5)
Prof. Wang Boxiong
The discrete Fourier transform (DFT):
where
Since x(n) may plex we can write
Fast Fourier transform (FFT)
()
()
()
From Eq. () it is clear that
For each value of k, the putation of X(k) requires 4N real multiplication and (4N-2) real additions.
X(k) must puted for N different values of k, the putation of the discrete Fourier transform of a sequence x(n) requires 4N2 real multiplications and N(4N-2) real additions or, alternatively, plex multiplications and N(N-1) complex additions.
Fast Fourier transform (FFT)
For the putation of the discrete Fourier transform, 4N2 real multiplications and N(4N-2) real additions are required.
The amount putation, and thus putation time, is approximately proportional to N2, so the number of arithmetic operations required pute the DFT by the direct method es very large for large values of N.
Conclusion: computational procedures that reduce the number of multiplications and additions are of considerable interest.
Fast Fourier transform (FFT)
Improving the efficiency of putation of the DFT exploits one or both of the following special properties of the quantities
:
Symmetry
Periodicity
Fast Fourier transform (FFT)
()
()
For example: using the first property, we can group terms in Eq. () as
By this method, the number of multiplications can be reduced by approximately a factor of 2.
The second property, ., the periodicity of plex sequence , can be employed in achieving significantly greater reductions of putation.
Fast Fourier transform (FFT)
In 1965, Cooley and Tukey published an algorithm for putation of the discrete Fourier transform that is applicable when N is posite number; ., N is the product of two or more integers. The algorithms are known as fast Fourier transform, or simply FFT, algorithms.
Fast Fourier transform (FFT)
The fundamental principle: pose putati