Content | |
DCT Type IV Tables | |
DCT4 F32 | |
DCT4 Q15 | |
DCT4 Q31 | |
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:
This process is explained by the block diagram below:
\[ X_c(k) = \sqrt{\frac{2}{N}}\sum_{n=0}^{N-1} x(n)cos\Big[\Big(n+\frac{1}{2}\Big)\Big(k+\frac{1}{2}\Big)\frac{\pi}{N}\Big] \]
wherek = 0, 1, 2, ..., N-1
\[ x(n) = \sqrt{\frac{2}{N}}\sum_{k=0}^{N-1} X_c(k)cos\Big[\Big(n+\frac{1}{2}\Big)\Big(k+\frac{1}{2}\Big)\frac{\pi}{N}\Big] \]
wheren = 0, 1, 2, ..., N-1
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.