# # 离散傅立叶变换（numpy.fft）

## # 标准 FFTs

fftopen in new window(a[, n, axis, norm])计算一维离散傅立叶变换。
ifftopen in new window(a[, n, axis, norm])计算一维离散傅立叶逆变换。
fft2open in new window(a[, s, axes, norm])计算二维离散傅立叶变换
ifft2open in new window(a[, s, axes, norm])计算二维离散傅立叶逆变换。
fftnopen in new window(a[, s, axes, norm])计算N维离散傅立叶变换。
ifftnopen in new window(a[, s, axes, norm])计算N维逆离散傅立叶变换。

## # 实际 FFTs

rfftopen in new window(a[, n, axis, norm])计算实数输入的一维离散傅立叶变换。
irfftopen in new window(a[, n, axis, norm])对于实输入，计算n点DFT的逆。
rfft2open in new window(a[, s, axes, norm])计算实数组的二维FFT。
irfft2open in new window(a[, s, axes, norm])计算实数组的二维逆FFT。
rfftnopen in new window(a[, s, axes, norm])计算实输入的N维离散傅立叶变换。
irfftnopen in new window(a[, s, axes, norm])计算实输入的N维FFT的逆。

## # 厄米特 FFTs

hfftopen in new window(a[, n, axis, norm])计算具有厄米特对称性的信号的FFT，即实谱。
ihfftopen in new window(a[, n, axis, norm])计算具有厄米特对称性的信号的逆FFT。

## # 帮助

fftfreqopen in new window(n[, d])返回离散傅立叶变换采样频率。
rfftfreqopen in new window(n[, d])返回离散傅立叶变换采样频率(用于rfft、irfft)。
fftshiftopen in new window(x[, axes])将零频率分量移至频谱中心。
ifftshiftopen in new window(x[, axes])fftshiftopen in new window的逆。

## # 实施细节

The DFT is in general defined for complex inputs and outputs, and a single-frequency component at linear frequency is represented by a complex exponential , where is the sampling interval.

The values in the result follow so-called “standard” order: If A = fft(a, n), then A[0] contains the zero-frequency term (the sum of the signal), which is always purely real for real inputs. Then A[1:n/2] contains the positive-frequency terms, and A[n/2+1:] contains the negative-frequency terms, in order of decreasingly negative frequency. For an even number of input points, A[n/2] represents both positive and negative Nyquist frequency, and is also purely real for real input. For an odd number of input points, A[(n-1)/2] contains the largest positive frequency, while A[(n+1)/2] contains the largest negative frequency. The routine np.fft.fftfreq(n) returns an array giving the frequencies of corresponding elements in the output. The routine np.fft.fftshift(A) shifts transforms and their frequencies to put the zero-frequency components in the middle, and np.fft.ifftshift(A) undoes that shift.

When the input a is a time-domain signal and A = fft(a), np.abs(A) is its amplitude spectrum and np.abs(A)**2 is its power spectrum. The phase spectrum is obtained by np.angle(A).

The inverse DFT is defined as

It differs from the forward transform by the sign of the exponential argument and the default normalization by 1/n.

## # Normalization

The default normalization has the direct transforms unscaled and the inverse transforms are scaled by 1/n. It is possible to obtain unitary transforms by setting the keyword argument norm to "ortho" (default is None) so that both direct and inverse transforms will be scaled by .

## # Real and Hermitian transforms

When the input is purely real, its transform is Hermitian, i.e., the component at frequency is the complex conjugate of the component at frequency , which means that for real inputs there is no information in the negative frequency components that is not already available from the positive frequency components. The family of rfftopen in new window functions is designed to operate on real inputs, and exploits this symmetry by computing only the positive frequency components, up to and including the Nyquist frequency. Thus, n input points produce n/2+1 complex output points. The inverses of this family assumes the same symmetry of its input, and for an output of n points uses n/2+1 input points.

Correspondingly, when the spectrum is purely real, the signal is Hermitian. The hfftopen in new window family of functions exploits this symmetry by using n/2+1 complex points in the input (time) domain for n real points in the frequency domain.

In higher dimensions, FFTs are used, e.g., for image analysis and filtering. The computational efficiency of the FFT means that it can also be a faster way to compute large convolutions, using the property that a convolution in the time domain is equivalent to a point-by-point multiplication in the frequency domain.

## # 参考文献

[CT]Cooley, James W., and John W. Tukey, 1965, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19: 297-301.

[NR]Press, W., Teukolsky, S., Vetterline, W.T., and Flannery, B.P., 2007, Numerical Recipes: The Art of Scientific Computing, ch. 12-13. Cambridge Univ. Press, Cambridge, UK.