General Causal FIR Filters

The most general case of a TDL having a tap after every delay element is the general causal finite impulse response (FIR) filter,

• Causal
  ($y(n)$ may not depend on ``future'' input samples $x(n+1)$, $x(n+2)$, etc.)
• Also called a transversal filter.

Difference Equation:

$$y(n) = b_0 x(n) + b_1 x(n-1) + b_2 x(n-2) + b_3 x(n-3) + \cdots + b_M x(n-M)$$

Transfer Function:

$$H(z) = b_0 + b_1 z^{-1} + b_2 z^{-2} + b_3 z^{-3} + \cdots + b_M z^{-M} = \sum_{m=0}^M b_m z^{-m} \mathrel{\stackrel{\mathrm{\Delta}}{=}}B(z)$$