# NDK_IDFT

 int __stdcall NDK_IDFT ( double * amp, double * phase, size_t nSize, double * X, size_t N )

Calculates the inverse discrete fast Fourier transformation, recovering the time series.

Returns
status code of the operation
Return values
 NDK_SUCCESS Operation successful NDK_FAILED Operation unsuccessful. See Macros for full list.
Parameters
 [in] amp is an array of the amplitudes of the fourier transformation components. [in] phase is an array of the phase angle (radian) of the Fourier transformation components . [in] nSize is the number of spectrum components (i.e. size of amp and phase). [out] X is the filtered (recovered) time series output [in] N is the original number of observations used to calculate the fourier transform.
Remarks
1. The input time series may include missing values (NaN) at either end, but they will not be included in the calculations.
2. The input time series must be homogeneous or equally spaced.
3. The first value in the input time series must correspond to the earliest observation.
4. The frequency component order, $$k$$, must be a positive number less than $$N$$, or an error (#VALUE!) is returned.
5. The DFT returns the phase angle in radians; i.e. $$0 \lt \phi \lt 2 \times \pi$$.
6. The discrete Fourier transformation (DFT) is defined as follows: $X_k = \sum_{j=0}^{N-1} x_j e^{-\frac{2\pi i}{N} j k}$ Where:
• $$k$$ is the frequency component
• $$x_0,...,x_{N-1}$$ are the values of the input time series
• $$N$$ is the number of non-missing values in the input time series
7. The Cooley-Tukey radix-2 decimation-in-time fast Fourier transformation (FFT) algorithm divides a DFT of size N into two overlapping DFTs of size $\frac{N}{2}$ at each of its stages using the following formula: $X_{k} = \begin{cases} E_k + \alpha \cdot O_k & \text{ if } k \lt \dfrac{N}{2} \\ E_{\left (k-\frac{N}{2} \right )} - \ \alpha \cdot O_{\left (k-\frac{N}{2} \right )} & \text{ if } k \geq \dfrac{N}{2} \end{cases}$ Where:
• $$E_k$$ is the DFT of the even-indicied values of the input time series, $$x_{2m} \left(x_0, x_2, \ldots, x_{N-2}\right)$$
• $$O_k$$ is the DFT of the odd-indicied values of the input time series, $$x_{2m+1} \left(x_1, x_3, \ldots, x_{N-2}\right)$$
• $$\alpha = e^{ \left (-2 \pi i k /N \right )}$$
• $$N$$ is the number of non-missing values in the time series data
8. The unit frequency of the DFT is $$\frac{2\pi}{N}$$, where $$N$$ is the number of non-missing observations.
Requirements
Examples



 Namespace: NumXLAPI Class: SFSDK Scope: Public Lifetime: Static
 int NDK_IDFT ( double[] amp, double[] phase, UIntPtr nSize, double[] data, UIntPtr nWindowSize )

Calculates the inverse discrete fast Fourier transformation, recovering the time series.

Return Value

a value from NDK_RETCODE enumeration for the status of the call.

 NDK_SUCCESS operation successful Error Error Code
Parameters
 [in] amp is an array of the amplitudes of the fourier transformation components. [in] phase is an array of the phase angle (radian) of the Fourier transformation components . [in] nSize is the number of spectrum components (i.e. size of amp and phase). [out] data is the filtered (recovered) time series output [in] nSize is the original number of observations used to calculate the fourier transform.
Remarks
1. The input time series may include missing values (NaN) at either end, but they will not be included in the calculations.
2. The input time series must be homogeneous or equally spaced.
3. The first value in the input time series must correspond to the earliest observation.
4. The frequency component order, $$k$$, must be a positive number less than $$N$$, or an error (#VALUE!) is returned.
5. The DFT returns the phase angle in radians; i.e. $$0 \lt \phi \lt 2 \times \pi$$.
6. The discrete Fourier transformation (DFT) is defined as follows: $X_k = \sum_{j=0}^{N-1} x_j e^{-\frac{2\pi i}{N} j k}$ Where:
• $$k$$ is the frequency component
• $$x_0,...,x_{N-1}$$ are the values of the input time series
• $$N$$ is the number of non-missing values in the input time series
7. The Cooley-Tukey radix-2 decimation-in-time fast Fourier transformation (FFT) algorithm divides a DFT of size N into two overlapping DFTs of size $\frac{N}{2}$ at each of its stages using the following formula: $X_{k} = \begin{cases} E_k + \alpha \cdot O_k & \text{ if } k \lt \dfrac{N}{2} \\ E_{\left (k-\frac{N}{2} \right )} - \ \alpha \cdot O_{\left (k-\frac{N}{2} \right )} & \text{ if } k \geq \dfrac{N}{2} \end{cases}$ Where:
• $$E_k$$ is the DFT of the even-indicied values of the input time series, $$x_{2m} \left(x_0, x_2, \ldots, x_{N-2}\right)$$
• $$O_k$$ is the DFT of the odd-indicied values of the input time series, $$x_{2m+1} \left(x_1, x_3, \ldots, x_{N-2}\right)$$
• $$\alpha = e^{ \left (-2 \pi i k /N \right )}$$
• $$N$$ is the number of non-missing values in the time series data
8. The unit frequency of the DFT is $$\frac{2\pi}{N}$$, where $$N$$ is the number of non-missing observations.
Exceptions
Exception Type Condition
None N/A
Requirements
Namespace NumXLAPI SFSDK Public Static NumXLAPI.DLL
Examples

References
* Hamilton, J .D.; Time Series Analysis , Princeton University Press (1994), ISBN 0-691-04289-6
* Tsay, Ruey S.; Analysis of Financial Time Series John Wiley & SONS. (2005), ISBN 0-471-690740
* D. S.G. Pollock; Handbook of Time Series Analysis, Signal Processing, and Dynamics; Academic Press; Har/Cdr edition(Nov 17, 1999), ISBN: 125609906
* Box, Jenkins and Reisel; Time Series Analysis: Forecasting and Control; John Wiley & SONS.; 4th edition(Jun 30, 2008), ISBN: 470272848