The Running-Sum Lowpass Filter

Impulse Response:

$$h(n) \isdef \left\{\begin{array}{ll} 1, & n=0,1,2,...,N-1 \\ [5pt] 0, & \hbox{otherwise} \\ \end{array} \right.$$

Implementation:

$$\begin{eqnarray*} y(n) &=& (h\ast x)(n) \isdef \sum_{m=-\infty}^{\infty} h(m) x(n-m) = \sum_{m=0}^{N-1} x(n-m)\\ &=& x(n) + x(n-1) + \cdots + x(n-N+1) \end{eqnarray*}$$

(``Unnormalized moving average'' FIR filter)

Transfer Function:

$$H(z) = 1 + z^{-1}+ \cdots + z^{-N+1} = \frac{1-z^{-N}}{1-z^{-1}}$$