5.3 Filter design for multirate systems

Design a factor-of- $L$ interpolator for use in a CD player, we might wish that the out-of-band error be below the least significant bit, or 96dBdown, and $<0.05\%$ error in the passband, so these specifications could be used for optimal $L_{\infty }$ filter design.

Least-squares polyphase filter design

• Deterministic x(n) Minimize $$\sum_{n=()}$$∞ ∞ x n x d n 2 Given $x(n)=(x(n), h(n))$ and $x_d(n)=(x(n), h_d(n))$ . Using Parseval's theorem, this becomes
  $\min\{\sum_{n=()} \}$∞ ∞ x n x d n 2 1 2 ω X ω H ω X ω H d ω 2 1 2 ω H ω H d ω X ω 2
  This is simply weighted least squares design, with $\left|X(\omega )\right|^2$ as the weighting function.
• stochastic X(ω)
  $\min\{(\left|x(n)-x_d(n)\right|^2)\}=(\left|(x(n), h(n)-h_d(n))\right|^2)=\min\{\frac{1}{2\pi }\int_{-\pi }^{\pi } \left|H_d(\omega )-H(\omega )\right|^2{S}_{xx}(\omega )\,d \omega \}$
  $S_{xx}(\omega )$ is the power spectral density of $x$ . $$S_{xx}(\omega )=\mathrm{DTFT}(r_{xx}(k))$$ $$r_{xx}(k)=(x(k+l)\overline{x(l)})$$ Again, a weighted least squares filter design problem.