A fast and stable algorithm for locating a desired eigenvalue and its corresponding eigenvector is presented. Its effectiveness is shown through a numerical simulation. The proposed algorithm is fast and numerically accurate enough to be applied to a real application.

This paper shows that a convolution property holds for sixteen members of a sinusoidal unitary transform family (DCTs and DSTs), on condition that an impulse response is an even function. Instead of the periodicity of an input signal assumed in the DFT case, DCTs require the input signal to be even symmetric outside boundaries and DSTs require it to be odd symmetric. The property is obtained by solving the eigenvalue problem of the matrices representing the convolution. The content of the property is that the DCT (or the DST) element of the output signal is the product of the DCT (or the DST) element of the input signal and the DFT element of the impulse response. The result for the well-known DCT is useful for a strongly-correlated signal and two examples demonstrate it.

The matrix whose eigenvectors are the basis vectors of the DCT is introduced. This matrix leads to a convolution-product property using the DCT. Based on the property, the parameter of uniform blur, such as motion blur or out-of-focus blur, is estimated from the local minima of the DCT energy spectrum of a blurred image. Computer experiments confirmed that the DCT is superior to the DFT for estimating the parameter.