Coefficients as Impulse Response

The impulse response $y(n)=h(n)$ is equivalent to coefficients of our FIR filter $b_k$.

This can be shown using the general FIR equation, with input $x(n) = \delta(n)$ (recall $\delta$ only has a nonzero value when $n=0$).

$$h(0) = b_0\delta(0) + b_1\cancelto{0}{\delta(0-1)} + b_2\cancelto{0}{\delta(0-2)} + ...$$

$$= b_0,$$

$$h(1) = b_0\cancelto{0}{\delta(1)} + b_1\delta(1-1) + b_2\cancelto{0}{\delta(1-2)} + ...$$

$$= b_1,$$

$$h(2) = b_0\delta(2) + b_1\delta(2-1) + b_2\delta(2-2) + ...$$

$$= b_2,$$

$$...$$

When the relation between the input $x(n)$ and the output $y(n)$ of the FIR filter is expressed in terms of the input and impulse response, we say the output is obtained by convolving the sequences $x(n)$ and $h(n)$.