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# Discrete Cosine Transform API

## DCT API quick reference

Note: The forward DCTs are Type-II. The inverse of the Type-II DCT is the Type-III DCT, so Type-II and Type-III are supported here.

Table 7 DCT functions - quick reference
Brief | Forward Function | Inverse Function
--- | --- | ---
6-point DCT | `dct6_forward()` | `dct6_inverse()`
8-point DCT | `dct8_forward()` | `dct8_inverse()`
12-point DCT | `dct12_forward()` | `dct12_inverse()`
16-point DCT | `dct16_forward()` | `dct16_inverse()`
24-point DCT | `dct24_forward()` | `dct24_inverse()`
32-point DCT | `dct32_forward()` | `dct32_inverse()`
48-point DCT | `dct48_forward()` | `dct48_inverse()`
64-point DCT | `dct64_forward()` | `dct64_inverse()`
8-by-8 2-dimensional DCT | `dct8x8_forward()` | `dct8x8_inverse()`

group dct_api

  Functions

  void dct6_forward(int32_t y[6], const int32_t x[6])

    6-point 32-bit forward DCT.

    This function performs a 6-point forward type-II DCT on input vector \(\bar x\) , and populates output vector \(\bar y\) with the result. To avoid possible overflow or saturation, output \(\bar y\) is scaled down by a factor of \(2^4\) (see dct6_exp).

    This operation may be safely performed in-place if `x` and `y` point to the same vector.

    `x` and `y` must point to 8-byte-aligned addresses.

    Operation Performed

      \[\begin{split}\begin{aligned} & y_k \leftarrow \frac{1}{2^4} \left( 2\sum_{n=0}^{N-1} x_n \cos\left( k\pi \frac{2n+1}{2N} \right) \right) \\ & \qquad\text{for } k = 0,1,\dots,(N-1) \\ & \qquad\text{with } N = 6 \\ \end{aligned}\end{split}\]
    Parameters:

      - y – [out] Output vector \(\bar y\)

      - x – [in] Input vector \(\bar x\)

    Throws ET_LOAD_STORE:

      Raised if `x` or `y` is not double word-aligned (See Note: Vector Alignment)

  void dct8_forward(int32_t y[8], const int32_t x[8])

    8-point 32-bit forward DCT.

    This function performs a 8-point forward type-II DCT on input vector \(\bar x\) , and populates output vector \(\bar y\) with the result. To avoid possible overflow or saturation, output \(\bar y\) is scaled down by a factor of \(2^4\) (see dct8_exp).

    This operation may be safely performed in-place if `x` and `y` point to the same vector.

    `x` and `y` must point to 8-byte-aligned addresses.

    Operation Performed

      \[\begin{split}\begin{aligned} & y_k \leftarrow \frac{1}{2^4} \left( 2\sum_{n=0}^{N-1} x_n \cos\left( k\pi \frac{2n+1}{2N} \right) \right) \\ & \qquad\text{for } k = 0,1,\dots,(N-1) \\ & \qquad\text{with } N = 8 \\ \end{aligned}\end{split}\]
    Parameters:

      - y – [out] Output vector \(\bar y\)

      - x – [in] Input vector \(\bar x\)

    Throws ET_LOAD_STORE:

      Raised if `x` or `y` is not double word-aligned (See Note: Vector Alignment)

  void dct12_forward(int32_t y[12], const int32_t x[12])

    12-point 32-bit forward DCT.

    This function performs a 12-point forward type-II DCT on input vector \(\bar x\) , and populates output vector \(\bar y\) with the result. To avoid possible overflow or saturation, output \(\bar y\) is scaled down by a factor of \(2^7\) (see dct12_exp).

    This operation may be safely performed in-place if `x` and `y` point to the same vector.

    `x` and `y` must point to 8-byte-aligned addresses.

    Operation Performed

      \[\begin{split}\begin{aligned} & y_k \leftarrow \frac{1}{2^7} \left( 2\sum_{n=0}^{N-1} x_n \cos\left( k\pi \frac{2n+1}{2N} \right) \right) \\ & \qquad\text{for } k = 0,1,\dots,(N-1) \\ & \qquad\text{with } N = 12 \\ \end{aligned}\end{split}\]
    Parameters:

      - y – [out] Output vector \(\bar y\)

      - x – [in] Input vector \(\bar x\)

    Throws ET_LOAD_STORE:

      Raised if `x` or `y` is not double word-aligned (See Note: Vector Alignment)

  void dct16_forward(int32_t y[16], const int32_t x[16])

    16-point 32-bit forward DCT.

    This function performs a 16-point forward type-II DCT on input vector \(\bar x\) , and populates output vector \(\bar y\) with the result. To avoid possible overflow or saturation, output \(\bar y\) is scaled down by a factor of \(2^7\) (see dct16_exp).

    This operation may be safely performed in-place if `x` and `y` point to the same vector.

    `x` and `y` must point to 8-byte-aligned addresses.

    Operation Performed

      \[\begin{split}\begin{aligned} & y_k \leftarrow \frac{1}{2^7} \left( 2\sum_{n=0}^{N-1} x_n \cos\left( k\pi \frac{2n+1}{2N} \right) \right) \\ & \qquad\text{for } k = 0,1,\dots,(N-1) \\ & \qquad\text{with } N = 16 \\ \end{aligned}\end{split}\]
    Parameters:

      - y – [out] Output vector \(\bar y\)

      - x – [in] Input vector \(\bar x\)

    Throws ET_LOAD_STORE:

      Raised if `x` or `y` is not double word-aligned (See Note: Vector Alignment)

  void dct24_forward(int32_t y[24], const int32_t x[24])

    24-point 32-bit forward DCT.

    This function performs a 24-point forward type-II DCT on input vector \(\bar x\) , and populates output vector \(\bar y\) with the result. To avoid possible overflow or saturation, output \(\bar y\) is scaled down by a factor of \(2^{10}\) (see dct24_exp).

    This operation may be safely performed in-place if `x` and `y` point to the same vector.

    `x` and `y` must point to 8-byte-aligned addresses.

    Operation Performed

      \[\begin{split}\begin{aligned} & y_k \leftarrow \frac{1}{2^{10}} \left( 2\sum_{n=0}^{N-1} x_n \cos\left( k\pi \frac{2n+1}{2N} \right) \right) \\ & \qquad\text{for } k = 0,1,\dots,(N-1) \\ & \qquad\text{with } N = 24 \\ \end{aligned}\end{split}\]
    Parameters:

      - y – [out] Output vector \(\bar y\)

      - x – [in] Input vector \(\bar x\)

    Throws ET_LOAD_STORE:

      Raised if `x` or `y` is not double word-aligned (See Note: Vector Alignment)

  void dct32_forward(int32_t y[32], const int32_t x[32])

    32-point 32-bit forward DCT.

    This function performs a 32-point forward type-II DCT on input vector \(\bar x\) , and populates output vector \(\bar y\) with the result. To avoid possible overflow or saturation, output \(\bar y\) is scaled down by a factor of \(2^{10}\) (see dct32_exp).

    This operation may be safely performed in-place if `x` and `y` point to the same vector.

    `x` and `y` must point to 8-byte-aligned addresses.

    Operation Performed

      \[\begin{split}\begin{aligned} & y_k \leftarrow \frac{1}{2^{10}} \left( 2\sum_{n=0}^{N-1} x_n \cos\left( k\pi \frac{2n+1}{2N} \right) \right) \\ & \qquad\text{for } k = 0,1,\dots,(N-1) \\ & \qquad\text{with } N = 32 \\ \end{aligned}\end{split}\]
    Parameters:

      - y – [out] Output vector \(\bar y\)

      - x – [in] Input vector \(\bar x\)

    Throws ET_LOAD_STORE:

      Raised if `x` or `y` is not double word-aligned (See Note: Vector Alignment)

  void dct48_forward(int32_t y[48], const int32_t x[48])

    48-point 32-bit forward DCT.

    This function performs a 48-point forward type-II DCT on input vector \(\bar x\) , and populates output vector \(\bar y\) with the result. To avoid possible overflow or saturation, output \(\bar y\) is scaled down by a factor of \(2^{13}\) (see dct48_exp).

    This operation may be safely performed in-place if `x` and `y` point to the same vector.

    `x` and `y` must point to 8-byte-aligned addresses.

    Operation Performed

      \[\begin{split}\begin{aligned} & y_k \leftarrow \frac{1}{2^{13}} \left( 2\sum_{n=0}^{N-1} x_n \cos\left( k\pi \frac{2n+1}{2N} \right) \right) \\ & \qquad\text{for } k = 0,1,\dots,(N-1) \\ & \qquad\text{with } N = 48 \\ \end{aligned}\end{split}\]
    Parameters:

      - y – [out] Output vector \(\bar y\)

      - x – [in] Input vector \(\bar x\)

    Throws ET_LOAD_STORE:

      Raised if `x` or `y` is not double word-aligned (See Note: Vector Alignment)

  void dct64_forward(int32_t y[64], const int32_t x[64])

    64-point 32-bit forward DCT.

    This function performs a 64-point forward type-II DCT on input vector \(\bar x\) , and populates output vector \(\bar y\) with the result. To avoid possible overflow or saturation, output \(\bar y\) is scaled down by a factor of \(2^{13}\) (see dct64_exp).

    This operation may be safely performed in-place if `x` and `y` point to the same vector.

    `x` and `y` must point to 8-byte-aligned addresses.

    Operation Performed

      \[\begin{split}\begin{aligned} & y_k \leftarrow \frac{1}{2^{13}} \left( 2\sum_{n=0}^{N-1} x_n \cos\left( k\pi \frac{2n+1}{2N} \right) \right) \\ & \qquad\text{for } k = 0,1,\dots,(N-1) \\ & \qquad\text{with } N = 64 \\ \end{aligned}\end{split}\]
    Parameters:

      - y – [out] Output vector \(\bar y\)

      - x – [in] Input vector \(\bar x\)

    Throws ET_LOAD_STORE:

      Raised if `x` or `y` is not double word-aligned (See Note: Vector Alignment)

  void dct6_inverse(int32_t y[6], const int32_t x[6])

    6-point 32-bit inverse DCT.

    This function performs a 6-point inverse DCT (same as type-III DCT) on input vector \(\bar x\) , and populates output vector \(\bar y\) with the result.

    This operation may be safely performed in-place if `x` and `y` point to the same vector.

    `x` and `y` must point to 8-byte-aligned addresses.

    Operation Performed

      \[\begin{split}\begin{aligned} & y_k \leftarrow \frac{1}{N} \left( \frac{x_0}{2} + \ \sum_{n=1}^{N-1} x_n \cos\left( n\pi \frac{2k+1}{2N} \right) \right) \\ & \qquad\text{for } k = 0,1,\dots,(N-1) \\ & \qquad\text{with } N = 6 \\ \end{aligned}\end{split}\]
    Parameters:

      - y – [out] Output vector \(\bar y\)

      - x – [in] Input vector \(\bar x\)

    Throws ET_LOAD_STORE:

      Raised if `x` or `y` is not double word-aligned (See Note: Vector Alignment)

  void dct8_inverse(int32_t y[8], const int32_t x[8])

    8-point 32-bit inverse DCT.

    This function performs a 8-point inverse DCT (same as type-III DCT) on input vector \(\bar x\) , and populates output vector \(\bar y\) with the result.

    This operation may be safely performed in-place if `x` and `y` point to the same vector.

    `x` and `y` must point to 8-byte-aligned addresses.

    Operation Performed

      \[\begin{split}\begin{aligned} & y_k \leftarrow \frac{1}{N} \left( \frac{x_0}{2} + \ \sum_{n=1}^{N-1} x_n \cos\left( n\pi \frac{2k+1}{2N} \right) \right) \\ & \qquad\text{for } k = 0,1,\dots,(N-1) \\ & \qquad\text{with } N = 8 \\ \end{aligned}\end{split}\]
    Parameters:

      - y – [out] Output vector \(\bar y\)

      - x – [in] Input vector \(\bar x\)

    Throws ET_LOAD_STORE:

      Raised if `x` or `y` is not double word-aligned (See Note: Vector Alignment)

  void dct12_inverse(int32_t y[12], const int32_t x[12])

    12-point 32-bit inverse DCT.

    This function performs a 12-point inverse DCT (same as type-III DCT) on input vector \(\bar x\) , and populates output vector \(\bar y\) with the result.

    This operation may be safely performed in-place if `x` and `y` point to the same vector.

    `x` and `y` must point to 8-byte-aligned addresses.

    Operation Performed

      \[\begin{split}\begin{aligned} & y_k \leftarrow \frac{1}{N} \left( \frac{x_0}{2} + \ \sum_{n=1}^{N-1} x_n \cos\left( n\pi \frac{2k+1}{2N} \right) \right) \\ & \qquad\text{for } k = 0,1,\dots,(N-1) \\ & \qquad\text{with } N = 12 \\ \end{aligned}\end{split}\]
    Parameters:

      - y – [out] Output vector \(\bar y\)

      - x – [in] Input vector \(\bar x\)

    Throws ET_LOAD_STORE:

      Raised if `x` or `y` is not double word-aligned (See Note: Vector Alignment)

  void dct16_inverse(int32_t y[16], const int32_t x[16])

    16-point 32-bit inverse DCT.

    This function performs a 16-point inverse DCT (same as type-III DCT) on input vector \(\bar x\) , and populates output vector \(\bar y\) with the result.

    This operation may be safely performed in-place if `x` and `y` point to the same vector.

    `x` and `y` must point to 8-byte-aligned addresses.

    Operation Performed

      \[\begin{split}\begin{aligned} & y_k \leftarrow \frac{1}{N} \left( \frac{x_0}{2} + \ \sum_{n=1}^{N-1} x_n \cos\left( n\pi \frac{2k+1}{2N} \right) \right) \\ & \qquad\text{for } k = 0,1,\dots,(N-1) \\ & \qquad\text{with } N = 16 \\ \end{aligned}\end{split}\]
    Parameters:

      - y – [out] Output vector \(\bar y\)

      - x – [in] Input vector \(\bar x\)

    Throws ET_LOAD_STORE:

      Raised if `x` or `y` is not double word-aligned (See Note: Vector Alignment)

  void dct24_inverse(int32_t y[24], const int32_t x[24])

    24-point 32-bit inverse DCT.

    This function performs a 24-point inverse DCT (same as type-III DCT) on input vector \(\bar x\) , and populates output vector \(\bar y\) with the result.

    This operation may be safely performed in-place if `x` and `y` point to the same vector.

    `x` and `y` must point to 8-byte-aligned addresses.

    Operation Performed

      \[\begin{split}\begin{aligned} & y_k \leftarrow \frac{1}{N} \left( \frac{x_0}{2} + \ \sum_{n=1}^{N-1} x_n \cos\left( n\pi \frac{2k+1}{2N} \right) \right) \\ & \qquad\text{for } k = 0,1,\dots,(N-1) \\ & \qquad\text{with } N = 24 \\ \end{aligned}\end{split}\]
    Parameters:

      - y – [out] Output vector \(\bar y\)

      - x – [in] Input vector \(\bar x\)

    Throws ET_LOAD_STORE:

      Raised if `x` or `y` is not double word-aligned (See Note: Vector Alignment)

  void dct32_inverse(int32_t y[32], const int32_t x[32])

    32-point 32-bit inverse DCT.

    This function performs a 32-point inverse DCT (same as type-III DCT) on input vector \(\bar x\) , and populates output vector \(\bar y\) with the result.

    This operation may be safely performed in-place if `x` and `y` point to the same vector.

    `x` and `y` must point to 8-byte-aligned addresses.

    Operation Performed

      \[\begin{split}\begin{aligned} & y_k \leftarrow \frac{1}{N} \left( \frac{x_0}{2} + \ \sum_{n=1}^{N-1} x_n \cos\left( n\pi \frac{2k+1}{2N} \right) \right) \\ & \qquad\text{for } k = 0,1,\dots,(N-1) \\ & \qquad\text{with } N = 32 \\ \end{aligned}\end{split}\]
    Parameters:

      - y – [out] Output vector \(\bar y\)

      - x – [in] Input vector \(\bar x\)

    Throws ET_LOAD_STORE:

      Raised if `x` or `y` is not double word-aligned (See Note: Vector Alignment)

  void dct48_inverse(int32_t y[48], const int32_t x[48])

    48-point 32-bit inverse DCT.

    This function performs a 48-point inverse DCT (same as type-III DCT) on input vector \(\bar x\) , and populates output vector \(\bar y\) with the result.

    This operation may be safely performed in-place if `x` and `y` point to the same vector.

    `x` and `y` must point to 8-byte-aligned addresses.

    Operation Performed

      \[\begin{split}\begin{aligned} & y_k \leftarrow \frac{1}{N} \left( \frac{x_0}{2} + \ \sum_{n=1}^{N-1} x_n \cos\left( n\pi \frac{2k+1}{2N} \right) \right) \\ & \qquad\text{for } k = 0,1,\dots,(N-1) \\ & \qquad\text{with } N = 48 \\ \end{aligned}\end{split}\]
    Parameters:

      - y – [out] Output vector \(\bar y\)

      - x – [in] Input vector \(\bar x\)

    Throws ET_LOAD_STORE:

      Raised if `x` or `y` is not double word-aligned (See Note: Vector Alignment)

  void dct64_inverse(int32_t y[64], const int32_t x[64])

    64-point 32-bit inverse DCT.

    This function performs a 64-point inverse DCT (same as type-III DCT) on input vector \(\bar x\) , and populates output vector \(\bar y\) with the result.

    This operation may be safely performed in-place if `x` and `y` point to the same vector.

    `x` and `y` must point to 8-byte-aligned addresses.

    Operation Performed

      \[\begin{split}\begin{aligned} & y_k \leftarrow \frac{1}{N} \left( \frac{x_0}{2} + \ \sum_{n=1}^{N-1} x_n \cos\left( n\pi \frac{2k+1}{2N} \right) \right) \\ & \qquad\text{for } k = 0,1,\dots,(N-1) \\ & \qquad\text{with } N = 64 \\ \end{aligned}\end{split}\]
    Parameters:

      - y – [out] Output vector \(\bar y\)

      - x – [in] Input vector \(\bar x\)

    Throws ET_LOAD_STORE:

      Raised if `x` or `y` is not double word-aligned (See Note: Vector Alignment)

  headroom_t dct8x8_forward(int8_t y[8][8], const int8_t x[8][8], const right_shift_t sat)

    8-by-8 2D 8-bit forward DCT.

    This function performs a 2-dimensional 8-by-8 type-II DCT on 8-bit input tensor \(\bar x\) (with elements \(x_{rc}\) ). Output tensor \(\bar y\) (with elements \(y_{rc}\) ) is populated with the result.

    This 2D DCT is performed by first applying a 1D 8-point DCT across each row of \(\bar x\) , and then applying a 1D 8-point DCT to each column of that intermediate tensor.

    The output is scaled by a factor of \(2^{-\mathtt{sat}-8}\) . With \(\mathtt{sat}=0\) this scaling is just enough to avoid any possible saturation. If saturation is considered acceptable, or known a priori to not be possible, negative values for \(\mathtt{sat}\) can be used to increase precision on the output.

    This operation may be safely performed in-place if `x` and `y` point to the same vector.

    `x` and `y` must point to 8-byte-aligned addresses.

    Operation Performed

      \[\begin{split}\begin{aligned} & y_{rc} \leftarrow \frac{4 \sum_{m=0}^{N-1} \sum_{n=0}^{N-1} \left( \ x_{mn} \cos\left( c\pi\frac{2n+1}{2N} \right) \cos\left(r\pi\frac{2m+1}{2N} \right)\right)}{2^{\mathtt{sat}+8}}\\ & \\ & \qquad\text{for } r,c \in \{0,1,\dots,(N-1)\} \\ & \qquad\text{with } N = 8 \\ \end{aligned}\end{split}\]
    Parameters:

      - y – [out] Output vector \(\bar y\)

      - x – [in] Input vector \(\bar x\)

      - sat – [in] Additional output scaling exponent.

    Throws ET_LOAD_STORE:

      Raised if `x` or `y` is not double word-aligned (See Note: Vector Alignment)

  headroom_t dct8x8_inverse(int8_t y[8][8], const int8_t x[8][8], const right_shift_t sat)

    8-by-8 2D 8-bit inverse DCT.

    This function performs a 2-dimensional 8-by-8 type-III (inverse) DCT on 8-bit input tensor \(\bar x\) (with elements \(x_{rc}\) ). Output tensor \(\bar y\) (with elements \(y_{rc}\) ) is populated with the result.

    This 2D DCT is performed by first applying a 1D 8-point DCT across each row of \(\bar x\) , and then applying a 1D 8-point DCT to each column of that intermediate tensor.

    The output is scaled by a factor of \(2^{-\mathtt{sat}}\) . With \(\mathtt{sat}=0\) this scaling is just enough to avoid any possible saturation. If saturation is considered acceptable, or known a priori to not be possible, negative values for \(\mathtt{sat}\) can be used to increase precision on the output.

    This operation may be safely performed in-place if `x` and `y` point to the same vector.

    `x` and `y` must point to 8-byte-aligned addresses.

    Operation Performed

      \[\begin{split}\begin{aligned} & y_{rc} \leftarrow \frac{ \frac{1}{N^2} \sum_{m=0}^{N-1} \sum_{n=0}^{N-1} \left( \ x_{mn} \cos\left( n\pi\frac{2c+1}{2N} \right) \cos\left(m\pi\frac{2r+1}{2N} \right)\right)}{2^{\mathtt{sat}}}\\ & \\ & \qquad\text{for } r,c \in \{0,1,\dots,(N-1)\} \\ & \qquad\text{with } N = 8 \\ \end{aligned}\end{split}\]
    Parameters:

      - y – [out] Output vector \(\bar y\)

      - x – [in] Input vector \(\bar x\)

      - sat – [in] Additional output scaling exponent.

    Throws ET_LOAD_STORE:

      Raised if `x` or `y` is not double word-aligned (See Note: Vector Alignment)

  Variables

  static const exponent_t dct6_exp = 4

    Scaling exponent associated with `dct6_forward()`

    Let \(\bar x\) be the input to `dct6_forward()` and \(\bar y\) the output. If \(x\_exp\) and \(y\_exp\) are the exponents associated with \(\bar x\) and \(\bar y\) respectively, then the following relation holds: \(y\_exp = x\_exp + dct6\_exp\)

  static const exponent_t dct8_exp = 4

    Scaling exponent associated with `dct8_forward()`

    Let \(\bar x\) be the input to `dct6_forward()` and \(\bar y\) the output. If \(x\_exp\) and \(y\_exp\) are the exponents associated with \(\bar x\) and \(\bar y\) respectively, then the following relation holds: \(y\_exp = x\_exp + dct8_exp\)

  static const exponent_t dct12_exp = 7

    Scaling exponent associated with `dct12_forward()`

    Let \(\bar x\) be the input to `dct12_forward()` and \(\bar y\) the output. If \(x\_exp\) and \(y\_exp\) are the exponents associated with \(\bar x\) and \(\bar y\) respectively, then the following relation holds: \(y\_exp = x\_exp + dct12_exp\)

  static const exponent_t dct16_exp = 7

    Scaling exponent associated with `dct16_forward()`

    Let \(\bar x\) be the input to `dct16_forward()` and \(\bar y\) the output. If \(x\_exp\) and \(y\_exp\) are the exponents associated with \(\bar x\) and \(\bar y\) respectively, then the following relation holds: \(y\_exp = x\_exp + dct16_exp\)

  static const exponent_t dct24_exp = 10

    Scaling exponent associated with `dct24_forward()`

    Let \(\bar x\) be the input to `dct24_forward()` and \(\bar y\) the output. If \(x\_exp\) and \(y\_exp\) are the exponents associated with \(\bar x\) and \(\bar y\) respectively, then the following relation holds: \(y\_exp = x\_exp + dct24_exp\)

  static const exponent_t dct32_exp = 10

    Scaling exponent associated with `dct32_forward()`

    Let \(\bar x\) be the input to `dct32_forward()` and \(\bar y\) the output. If \(x\_exp\) and \(y\_exp\) are the exponents associated with \(\bar x\) and \(\bar y\) respectively, then the following relation holds: \(y\_exp = x\_exp + dct32_exp\)

  static const exponent_t dct48_exp = 13

    Scaling exponent associated with `dct48_forward()`

    Let \(\bar x\) be the input to `dct48_forward()` and \(\bar y\) the output. If \(x\_exp\) and \(y\_exp\) are the exponents associated with \(\bar x\) and \(\bar y\) respectively, then the following relation holds: \(y\_exp = x\_exp + dct48\_exp\)

  static const exponent_t dct64_exp = 13

    Scaling exponent associated with `dct64_forward()`

    Let \(\bar x\) be the input to `dct64_forward()` and \(\bar y\) the output. If \(x\_exp\) and \(y\_exp\) are the exponents associated with \(\bar x\) and \(\bar y\) respectively, then the following relation holds: \(y\_exp = x\_exp + dct64\_exp\)