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Inverse Fast Fourier Transform

MuPAD® notebooks are not recommended. Use MATLAB® live scripts instead.

MATLAB live scripts support most MuPAD functionality, though there are some differences. For more information, see Convert MuPAD Notebooks to MATLAB Live Scripts.


numeric::invfft(L, <mode>, <ReturnType = t>, <Clean>)
numeric::invfft(M, <mode>, <ReturnType = t>, <Clean>)
numeric::invfft(A, <mode>, <ReturnType = t>, <Clean>)


numeric::invfft(data) returns the inverse discrete Fourier transform.

The one-dimensional inverse discrete Fourier transform L = invfft(F) of N data elements Fk stored in the list F = [F1, …, FN] is the list L = [L1, …, LN] given by


invfft transforms the data by a Fast Fourier Transform (FFT) algorithm.

The d-dimensional inverse discrete Fourier transform A = invfft(F) is given by

with j1 = 1, …, n1, …, jd = 1, …, nd.

Data provided by a list, or one-dimensional array, or hfarray are transformed according to the one-dimensional transform. Data provided by matrices are transformed according to the two-dimensional transform. Data provided by multidimensional arrays or hfarrays are transformed according to the multidimensional transform matching the format of the input array.

If the data size N factorizes as N = pq, the inverse discrete Fourier transform can be computed by p different inverse Fourier transforms of subsets of the data, each subset having the data size q. The corresponding 'divide and conquer' algorithm is known as FFT ('Fast Fourier Transform'). The invfft routine employs the FFT algorithm. It is most efficient, when the data size N is an integer power of 2 ('radix 2 FFT'). In this case, the algorithm needs elementary operations.


More generally, FFT is efficient, if the data size is the product of many small factors.

Following Bluestein, the inverse Fourier transform is written as a convolution if the data size N is a prime. The data are zero-padded to a data length that is an integer power of 2. The convolution is then computed via radix 2 FFTs. Thus, the algorithm needs elementary operations even if N is a prime.

Environment Interactions

Without the option Symbolic, the function is sensitive to the environment variable DIGITS, which determines the numerical working precision.


Example 1

Compute one-dimensional transformations using lists. By default, numerical expressions are converted to floating-point values:

L := [1, 2^(1/2), 3*I, PI]: 
F := numeric::fft(L)


numeric::invfft(F, Clean)

To use exact arithmetic, specify the option Symbolic:

F := numeric::fft(L, Symbolic)

numeric::invfft(F, Symbolic)

numeric::invfft accepts symbolic expressions. Internally, the default method HardwareFloats (with DIGITS < 16) fails because of the symbolic parameter x. The following results are computed with the software arithmetic provided by the MuPAD® kernel:

L := [x, 2, 3, x]: 


numeric::fft(L, Symbolic)

numeric::invfft(F, Symbolic)

delete L, F:

Example 2

Compute the following two-dimensional transformation using an array with two indices:

A := array(1..2, 1..4, [[1, 2, 3, 4], [a, b, c, d]]):
F := numeric::fft(A, Symbolic)

numeric::invfft(F, Symbolic)

delete A, F

Example 3

Data of arbitrary length can be transformed:

L := [1, 2 + I, PI/3]:

delete L



A list, or a one-dimensional array(1 .. N, [Symbol::hellip]), or a one-dimensional hfarray(1 .. N, [Symbol::hellip]) of arithmetical expressions.


A matrix of category Cat::Matrix of arithmetical expressions.


A d-dimensional array( 1..n_1,Symbol::hellip,1..n_d, [Symbol::hellip] ) or a d-dimensional hfarray( 1..n_1,Symbol::hellip,1..n_d, [Symbol::hellip] ) of arithmetical expressions.


One of the flags Hard, HardwareFloats, Soft, SoftwareFloats, or Symbolic


Hard, HardwareFloats, Soft, SoftwareFloats

With Hard (or HardwareFloats), computations are done using fast hardware floating-point arithmetic from within a MuPAD session. Hard and HardwareFloats are equivalent. With this option, the input data are converted to hardware floating-point values and processed by compiled C code. The result is reconverted to MuPAD floating-point values and returned to the MuPAD session.

With Soft (or SoftwareFloats) computations are done using software floating-point arithmetic provided by the MuPAD kernel. Soft and SoftwareFloats are equivalent. SoftwareFloats is used by default if the current value of DIGITS is larger than 15 and the input matrix A is not of domain type DOM_HFARRAY.

Compared to the SoftwareFloats used by the MuPAD kernel, the computation with HardwareFloats can be much faster. Note, however, that the precision of hardware arithmetic is limited to about 15 digits. Further, the size of floating-point numbers can not be larger than approximately 10308 and not smaller than approximately 10- 308.

If no HardwareFloats or SoftwareFloats are specified, the following strategy is used. If the current value of DIGITS is smaller than 16 or if the matrix A is a hardware floating-point array of domain type DOM_HFARRAY, then hardware arithmetic is tried. If this is successful, the result is returned.

If the result cannot be computed with hardware floating-point values, software arithmetic by the MuPAD kernel is tried.

If the current value of DIGITS is larger than 15 and the input matrix A is not of domain type DOM_HFARRAY, or if one of the options Soft, SoftwareFloats or Symbolic is specified, MuPAD computes the result with its software arithmetic without trying to use hardware arithmetic first.

There can be several reasons for hardware arithmetic to fail:

  • The current value of DIGITS is larger than 15.

  • The data contains symbolic objects.

  • The data contains numbers larger than 10308 or smaller than 10- 308 that cannot be represented by hardware floating-point values.

If neither HardwareFloats nor SoftwareFloats is specified, the function does not indicate whether hardware floating-point values or software floating-point values are used.

If HardwareFloats are specified but fail due to one of the reasons above, a warning is issued that the (much slower) software floating-point arithmetic of the MuPAD kernel is used.

Note that HardwareFloats can only be used if all input data can be converted to floating-point numbers.

With Soft and SoftwareFloats, symbolic objects are accepted even if they cannot be converted to floating-point numbers. The result consists of arithmetical expressions involving both floating-point numbers as well as symbolic objects. See Example 1.

The trailing digits in floating-point results computed with HardwareFloats and SoftwareFloats can differ.


This option prevents conversion of the input data to floating-point values.

Without this option, the floating-point converter float is applied to all input data. Use this option if no such conversion is desired. Exact arithmetic is used to compute the Fourier transformation.


Option, specified as ReturnType = t

Return the result in a container of domain type t. The following return types t are available: DOM_LIST, or DOM_ARRAY, or DOM_HFARRAY, or matrix, or densematrix.

This option determines the domain type t of the result.

If no return type is specified by this option, the result if of the same type and format as the input data.

If the return type DOM_LIST is specified, the result is always a plain list of floating-point numbers. If the input data are given by a matrix or a multidimensional array, the returned list represents the operands of the multidimensional Fourier data. For example, if an n1×n2 matrix is entered, the return value is a list with n1n2 values representing the entries of a n1×n2 matrix. The first n2 entries of the list represent the first row of the result, the next n2 entries represent the second row, and so on.

With ReturnType = matrix or ReturnType = densematrix, only the results of one- and two-dimensional Fourier transformations can be represented.


Reduce roundoff garbage in the result. All entries of the result with absolute values smaller than 10^(-DIGITS) times the maximal absolute value of all operands of the result are set to 0.0. Further, the routine numeric::complexRound is applied to all entries of the result.


The postprocessing of the result is done on the software level. When using hardware floating-point values, this option can increase the runtime significantly.

This option is ignored when used in conjunction with the option Symbolic.

Return Values

List, array, hfarray, or matrix of the same length and format as the first input parameter L, A, or M, respectively. The type of the return value can be changed with the option ReturnType.

See Also

MuPAD Functions

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