# DCT

Discrete cosine transform (DCT) of input

## Library

Transforms

`dspxfrm3`

## Description

The DCT block computes the unitary discrete cosine transform (DCT) of each channel in the M-by-N input matrix, u.

```y = dct(u) % Equivalent MATLAB code ```

When the input is a sample-based row vector, the DCT block computes the discrete cosine transform across the vector dimension of the input. For all other N-D input arrays, the block computes the DCT across the first dimension. The size of the first dimension (frame size), must be a power of two. To work with other frame sizes, use the Pad block to pad or truncate the frame size to a power-of-two length.

When the input to the DCT block is an M-by-N matrix, the block treats each input column as an independent channel containing M consecutive samples. The block outputs an M-by-N matrix whose lth column contains the length-M DCT of the corresponding input column.

$y\left(k,l\right)=w\left(k\right)\sum _{m=1}^{M}u\left(m,l\right)\mathrm{cos}\frac{\pi \left(2m-1\right)\left(k-1\right)}{2M},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}k=1,...,M$

where

$w\left(k\right)=\left\{\begin{array}{c}\frac{1}{\sqrt{M}},\\ \sqrt{\frac{2}{M}},\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{c}\begin{array}{l}k=1\\ \end{array}\\ 2\le k\le M\end{array}$

The Sine and cosine computation parameter determines how the block computes the necessary sine and cosine values. This parameter has two settings, each with its advantages and disadvantages, as described in the following table.

Sine and Cosine Computation Parameter Setting

Sine and Cosine Computation Method

Effect on Block Performance

`Table lookup`

The block computes and stores the trigonometric values before the simulation starts, and retrieves them during the simulation. When you generate code from the block, the processor running the generated code stores the trigonometric values computed by the block in a speed-optimized table, and retrieves the values during code execution.

The block usually runs much more quickly, but requires extra memory for storing the precomputed trigonometric values.

`Trigonometric fcn`

The block computes sine and cosine values during the simulation. When you generate code from the block, the processor running the generated code computes the sine and cosine values while the code runs.

The block usually runs more slowly, but does not need extra data memory. For code generation, the block requires a support library to emulate the trigonometric functions, increasing the size of the generated code.

This block supports Simulink® virtual buses.

### Fixed-Point Data Types

The following diagrams show the data types used within the DCT block for fixed-point signals. You can set the sine table, accumulator, product output, and output data types displayed in the diagrams in the DCT block dialog as discussed in Dialog Box.

Inputs to the DCT block are first cast to the output data type and stored in the output buffer. Each butterfly stage processes signals in the accumulator data type, with the final output of the butterfly being cast back into the output data type.

The output of the multiplier is in the product output data type when at least one of the inputs to the multiplier is real. When both of the inputs to the multiplier are complex, the result of the multiplication is in the accumulator data type. For details on the complex multiplication performed, see Multiplication Data Types.

 Note:   When the block input is fixed point, all internal data types are signed fixed point.

## Dialog Box

The Main pane of the DCT block dialog appears as follows.

Sine and cosine computation

Sets the block to compute sines and cosines by either looking up sine and cosine values in a speed-optimized table (```Table lookup```), or by making sine and cosine function calls (`Trigonometric fcn`). See the table in the Description section.

The Data Types pane of the DCT block dialog appears as follows.

Rounding mode

Select the rounding mode for fixed-point operations. The sine table values do not obey this parameter; they always round to `Nearest`.

Overflow mode

Select the overflow mode for fixed-point operations. The sine table values do not obey this parameter; instead, they are always saturated.

Sine table data type

Choose how you specify the word length of the values of the sine table. The fraction length of the sine table values always equals the word length minus one. You can set this parameter to:

• A rule that inherits a data type, for example, ```Inherit: Same word length as input```

• An expression that evaluates to a valid data type, for example, `fixdt(1,16)`

The sine table values do not obey the Rounding mode and Overflow mode parameters; instead, they are always saturated and rounded to `Nearest`.

Product output data type

Specify the product output data type. See Fixed-Point Data Types and Multiplication Data Types for illustrations depicting the use of the product output data type in this block. You can set this parameter to:

• A rule that inherits a data type, for example, ```Inherit: Inherit via internal rule```

• An expression that evaluates to a valid data type, for example, `fixdt(1,16,0)`

Click the Show data type assistant button to display the Data Type Assistant, which helps you set the Product output data type parameter.

See Specify Data Types Using Data Type Assistant for more information.

Accumulator data type

Specify the accumulator data type. See Fixed-Point Data Types for illustrations depicting the use of the accumulator data type in this block. You can set this parameter to:

• A rule that inherits a data type, for example, ```Inherit: Inherit via internal rule```

• An expression that evaluates to a valid data type, for example, `fixdt(1,16,0)`

Click the Show data type assistant button to display the Data Type Assistant, which helps you set the Accumulator data type parameter.

See Specify Data Types Using Data Type Assistant for more information.

Output data type

Specify the output data type. See Fixed-Point Data Types for illustrations depicting the use of the output data type in this block. You can set this parameter to:

• A rule that inherits a data type, for example, ```Inherit: Inherit via internal rule```.

When you select `Inherit: Inherit via internal rule`, the block calculates the output word length and fraction length automatically. The internal rule first calculates an ideal output word length and fraction length using the following equations:

$W{L}_{idealoutput}=W{L}_{input}+floor\left({\mathrm{log}}_{2}\left(DCTlength-1\right)\right)+1$

$F{L}_{idealoutput}=F{L}_{input}$

Using these ideal results, the internal rule then selects word lengths and fraction lengths that are appropriate for your hardware. For more information, see Inherit via Internal Rule.

• An expression that evaluates to a valid data type, for example, `fixdt(1,16,0)`

Click the Show data type assistant button to display the Data Type Assistant, which helps you set the Output data type parameter.

See Specify Block Output Data Types for more information.

Lock data type settings against changes by the fixed-point tools

Select this parameter to prevent the fixed-point tools from overriding the data types you specify on the block mask.

## Supported Data Types

PortSupported Data Types

Input

• Double-precision floating point

• Single-precision floating point

• Fixed point (signed and unsigned)

• 8-, 16-, and 32-bit signed integers

• 8-, 16-, and 32-bit unsigned integers

Output

• Double-precision floating point

• Single-precision floating point

• Fixed point (signed only)

• 8-, 16-, and 32-bit signed integers

 Complex Cepstrum DSP System Toolbox FFT DSP System Toolbox IDCT DSP System Toolbox Real Cepstrum DSP System Toolbox `dct` Signal Processing Toolbox