Princen-Bradley Filter Bank

• Alternate DCT and DST using 50% OLA, constant-OLA window, and quarter-frame rotation
• $\hbox{\sc DCT}(x) \approx$   re$\left\{\hbox{\sc FFT}(x)\right\} = \frac{X(\omega_k ) + X^*(\omega_k )}{2} \;\longleftrightarrow\;\frac{x + \hbox{\sc Flip}(x)}{2}$
• Thus, DCT data is time aliased with its flip
• Similarly, $\hbox{\sc DST}(x) \approx$   im$\left\{\hbox{\sc FFT}(x)\right\} = \frac{X(\omega_k ) - X^*(\omega_k )}{2j} \;\longleftrightarrow\;\frac{x - \hbox{\sc Flip}(x)}{2}$
• Thus, DST data is time aliased with minus its flip
• Alternating DCT and DST in this way cancels aliasing
• This is ``time-domain aliasing cancellation''
• Princen-Bradley filter bank = special case of ``Lapped Orthogonal Transforms (LOT)'' (see Malvar)
  • Let number of filter bank channels = $N$
  • Let length of each channel analysis filter be $M$
  • LOT = Critically sampled FIR filter bank with $M = 2N$