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	DCT Type IV Tables

	DCT4 F32

	DCT4 Q15

	DCT4 Q31

Description

Representation of signals by minimum number of values is important for storage and transmission. The possibility of large discontinuity between the beginning and end of a period of a signal in DFT can be avoided by extending the signal so that it is even-symmetric. Discrete Cosine Transform (DCT) is constructed such that its energy is heavily concentrated in the lower part of the spectrum and is very widely used in signal and image coding applications. The family of DCTs (DCT type- 1,2,3,4) is the outcome of different combinations of homogeneous boundary conditions. DCT has an excellent energy-packing capability, hence has many applications and in data compression in particular.

DCT is essentially the Discrete Fourier Transform(DFT) of an even-extended real signal. Reordering of the input data makes the computation of DCT just a problem of computing the DFT of a real signal with a few additional operations. This approach provides regular, simple, and very efficient DCT algorithms for practical hardware and software implementations.

DCT type-II can be implemented using Fast fourier transform (FFT) internally, as the transform is applied on real values, Real FFT can be used. DCT4 is implemented using DCT2 as their implementations are similar except with some added pre-processing and post-processing. DCT2 implementation can be described in the following steps:

Re-ordering input
Calculating Real FFT
Multiplication of weights and Real FFT output and getting real part from the product.

This process is explained by the block diagram below:

Discrete Cosine Transform - type-IV

Algorithm: The N-point type-IV DCT is defined as a real, linear transformation by the formula:
\[ X_c(k) = \sqrt{\frac{2}{N}}\sum_{n=0}^{N-1} x(n)cos\Big[\Big(n+\frac{1}{2}\Big)\Big(k+\frac{1}{2}\Big)\frac{\pi}{N}\Big] \]
where k = 0, 1, 2, ..., N-1

: Its inverse is defined as follows:
\[ x(n) = \sqrt{\frac{2}{N}}\sum_{k=0}^{N-1} X_c(k)cos\Big[\Big(n+\frac{1}{2}\Big)\Big(k+\frac{1}{2}\Big)\frac{\pi}{N}\Big] \]
where n = 0, 1, 2, ..., N-1

: The DCT4 matrices become involutory (i.e. they are self-inverse) by multiplying with an overall scale factor of sqrt(2/N). The symmetry of the transform matrix indicates that the fast algorithms for the forward and inverse transform computation are identical. Note that the implementation of Inverse DCT4 and DCT4 is same, hence same process function can be used for both.

Lengths supported by the transform:: As DCT4 internally uses Real FFT, it supports all the lengths 128, 512, 2048 and 8192. The library provides separate functions for Q15, Q31, and floating-point data types.

Instance Structure: The instances for Real FFT and FFT, cosine values table and twiddle factor table are stored in an instance data structure. A separate instance structure must be defined for each transform. There are separate instance structure declarations for each of the 3 supported data types.

Initialization Functions

There is also an associated initialization function for each data type. The initialization function performs the following operations:

Sets the values of the internal structure fields.
Initializes Real FFT as its process function is used internally in DCT4, by calling arm_rfft_init_f32().

Use of the initialization function is optional. However, if the initialization function is used, then the instance structure cannot be placed into a const data section. To place an instance structure into a const data section, the instance structure must be manually initialized. Manually initialize the instance structure as follows:

    arm_dct4_instance_f32 S = {N, Nby2, normalize, pTwiddle, pCosFactor, pRfft, pCfft};
    arm_dct4_instance_q31 S = {N, Nby2, normalize, pTwiddle, pCosFactor, pRfft, pCfft};
    arm_dct4_instance_q15 S = {N, Nby2, normalize, pTwiddle, pCosFactor, pRfft, pCfft};

where N is the length of the DCT4; Nby2 is half of the length of the DCT4; normalize is normalizing factor used and is equal to sqrt(2/N); pTwiddle points to the twiddle factor table; pCosFactor points to the cosFactor table; pRfft points to the real FFT instance; pCfft points to the complex FFT instance; The CFFT and RFFT structures also needs to be initialized, refer to arm_cfft_radix4_f32() and arm_rfft_f32() respectively for details regarding static initialization.

Fixed-Point Behavior: Care must be taken when using the fixed-point versions of the DCT4 transform functions. In particular, the overflow and saturation behavior of the accumulator used in each function must be considered. Refer to the function specific documentation below for usage guidelines.

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Description