Frame bounds for biorthogonal cosine-modulated filter banks

We derive explicit expressions for the eigenvalues of the frame operator for cosine-modulated filter banks. The filter banks may be critically sampled or oversampled by an integer factor. The analysis of low-delay, biorthogonal filter banks shows that prototypes solely designed to minimize the stopband energy may lead to wide open frames and thus to an undesirable numerical behavior. Because the computational cost of determining the frame bounds with the proposed method is very low, we can directly use the bounds during prototype optimization and obtain prototypes with minimum stopband energy under the condition of fixed frame bounds.