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Inverse FFT (Real) (G Dataflow)

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    Last Modified: January 9, 2017

    Computes the inverse discrete Fourier transform (IDFT) of a sequence. You can use this node when the input sequence is the Fourier transform of a real time-domain signal.

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    FFT{x}

    The complex valued input sequence, which should be conjugated centrosymmetric except for the first element. This node uses only the anterior half of FFT {x}.

    This input can be a 1D or 2D array of complex double-precision, floating-point numbers.

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    shift?

    A Boolean that determines whether the DC component is at the center of the FFT of the input sequence.

    True The DC component is at the center of the FFT{x}.
    False The DC component is not at the center of the FFT{x}.

    How This Input Affects 1D FFT

    The following table illustrates the pattern of the elements of FFT{x} with various length of the FFT, when shift? is False. Y is FFT{x} and n is the length of the FFT:

    n is even (k = n/2) n is odd (k = (n-1)/2)
    Array Element Corresponding Frequency Array Element Corresponding Frequency
    Y0 DC component Y0 DC component
    Y1 Δ f Y1 Δ f
    Y2 2 Δ f Y2 2 Δ f
    Y3 3 Δ f Y3 3 Δ f
    Yk-2 ( k 2 ) Δ f Yk-2 ( k 2 ) Δ f
    Yk-1 ( k 1 ) Δ f Yk-1 ( k 1 ) Δ f
    Yk Nyquist Frequency Yk k Δ f
    Yk+1 ( k 1 ) Δ f Yk+1 k Δ f
    Yk+2 ( k 2 ) Δ f Yk+2 ( k 1 ) Δ f
    Yn-3 3 Δ f Yn-3 3 Δ f
    Yn-2 2 Δ f Yn-2 2 Δ f
    Yn-1 Δ f Yn-1 Δ f

    The following table illustrates the pattern of the elements of FFT{x} with various length of the FFT, when shift? is True. Y is FFT{x} and n is the length of the FFT:

    n is even (k = n/2) n is odd (k = (n-1)/2)
    Array Element Corresponding Frequency Array Element Corresponding Frequency
    Y0 -(Nyquist Frequency) Y0 k Δ f
    Y1 ( k 1 ) Δ f Y1 ( k 1 ) Δ f
    Y2 ( k 2 ) Δ f Y2 ( k 2 ) Δ f
    Y3 ( k 3 ) Δ f Y3 ( k 3 ) Δ f
    Yk-2 2 Δ f Yk-2 2 Δ f
    Yk-1 Δ f Yk-1 Δ f
    Yk DC component Yk DC component
    Yk+1 Δ f Yk+1 Δ f
    Yk+2 2 Δ f Yk+2 2 Δ f
    Yn-3 ( k 3 ) Δ f Yn-3 ( k 2 ) Δ f
    Yn-2 ( k 2 ) Δ f Yn-2 ( k 1 ) Δ f
    Yn-1 ( k 1 ) Δ f Yn-1 k Δ f

    How This Input Affects 2D FFT

    The illustration below shows the effect of shift? on the 2D FFT result:

    2D input signals FFT without shift FFT with shift

    Default: False

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    error in

    Error conditions that occur before this node runs. The node responds to this input according to standard error behavior.

    Default: No error

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    x

    The inverse real FFT of the complex valued input sequence.

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    error out

    Error information. The node produces this output according to standard error behavior.

    Algorithm Definition for 1D Inverse FFT

    For a 1D, N-sample, frequency domain sequence Y, the inverse discrete Fourier transform (IDFT) is defined as:

    X n = 1 N k = 0 N 1 Y k e j 2 π k n / N

    for n = 0, 1, 2, ..., N-1.

    Algorithm Definition for 2D Inverse FFT

    For a 2D, M-by-N frequency domain array Y, the inverse discrete Fourier transform (IDFT) is defined as:

    X ( m , n ) = 1 M N u = 0 M 1 v = 0 N 1 Y ( u , v ) e j 2 π m u / M e j 2 π n v / N

    for m = 0, 1, ..., M-1, n=0, 1, ..., M-1.

    Conjugated Centrosymmetric Property of 1D Inverse Real FFT

    When shift? is False and FFT{x} is the Fourier transform of a 1D real time-domain signal with length N, the posterior half part of FFT{x} can be constructed by the anterior half part. The centrosymmetric relationship between the anterior and posterior half part of FFT{x} can be written as

    f N 1 = f i * , i = 1 , 2 , ... , N 2

    where fi is the element in FFT{x}.

    This node uses only the anterior half part, from f0 to f _ N 2 to perform the inverse real FFT, where means the floor operation.

    Conjugated Centrosymmetric Property of 2D Inverse Real FFT

    When shift? is False and FFT{x} is the Fourier transform of a 2D real time-domain signal with M rows and N columns, the lower half part of FFT{x} can be constructed by the upper half part. The centrosymmetric relationship between the upper and lower half part of FFT{x} can be written as

    { f M i , j = f i , N j * i = 1 , 2 , ... , M 2 , j = 1 , 2 , ... , N 1 f M i , j = f i , j * i = 1 , 2 , ... , M 2 , j = 0

    where fi,j is the element in FFT{x}.

    This node uses only the upper half part, from f0,0 to f _ M 2 , N 1 to perform the inverse real FFT, where means the floor operation.

    Where This Node Can Run:

    Desktop OS: Windows

    FPGA: Not supported


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