deriv_qn: Analytic D matrix quantisation noise process
Description
Analytic D matrix quantisation noise process
Usage
deriv_qn(tau)
Arguments
tau
A vec that contains the scales to be processed (e.g. 2^(1:J))
Value
A matrix with the first column containing the partial derivative with respect to $Q[0]^2$.
Details
The haar wavelet variance is given as $nu^2(tau) = 3*Q[0]^2 / 2*tau^2$.
Taking the derivative with respect to $Q[0]^2$ yields: $$\frac{\partial }{{\partial Q_0^2}}{\nu ^2}\left( \tau \right) = \frac{3}{{2{\tau ^2}}}$$.
The second derivative derivative with respect to $Q[0]^2$ is then: $$\frac{{{\partial ^2}}}{{\partial Q_0^4}}{\nu ^2}\left( \tau \right) = 0$$.