Signal Processing Stack Exchange is a question and answer site for practitioners of the art and science of signal, image and video processing. It only takes a minute to sign up.
Sign up to join this community
I'm getting started with signal processing, and I'm stuck on a problem that asks me to take the $\mathcal Z$-transform of the following causal, DT-LTI system: $$\sum_{k=0}^M b_k x[n-k] $$
I'm not really sure how to take the $\mathcal Z$-transform with the summation in there. Can anyone point me in the right direction?
You can use the linearity property of the $\mathcal Z$-transform:
$$a_1x_1[n] + a_2x_2[n] \overset{z}\longleftrightarrow a_1X_1(z) + a_2X_2(z)$$
together with the time shifting property:
$$x[n-k] \overset{z}\longleftrightarrow z^{-k}X(z)$$
Expand the summation like below and apply the two properties.
$$\sum_{k=0}^M b_k x[n-k]=b_0x[n]+b_1x[n-1]+\ldots+b_Mx[n-M]$$
