Cascading systems

The commutative and associative properties of an LTI system results into the following so called cascade equivalences, which imply that there are different representations of a cascade of LTI systems. Referring to Fig. 1 this can be shown as follows:

The upper left hand figure of Fig. 1 shows the cascading of two LTI systems, one with impulse response $h_1\left[n\right]$ and the other one with impulse response $h_2\left[n\right]$. When applying a delta pulse $\delta \left[n\right]$ to the first LTI system, the output equals the impulse response $h_1\left[n\right]$. When applying this sequence to the second LTI system, the result is the convolution sum result $h_1\left[n\right]\star h_2\left[n\right]$.

The lower left hand figure shows one LTI system with impulse response $h\left[n\right]=h_1\left[n\right]\star h_2\left[n\right]$. When applying a delta pulse $\delta \left[n\right]$ to this system the result is the same as above.

The cascading of two LTI systems, one with impulse response $h_1\left[n\right]$ and the other one with impulse response $h_2\left[n\right]$, can be combined to one LTI system with impulse response $h\left[n\right]=h_1\left[n\right]\star h_2\left[n\right]$.}

Finally it follows from the commutative property of the convolution sum operator that: $$h\left[n\right]=h_1\left[n\right]\star h_2\left[n\right]=h_2\left[n\right]\star h_1\left[n\right]$$ which is shown in the lower right part of Fig. 1. This combined impulse response can be split again into the cascading of two LTI systems, as shown in the upper right figure. In this last step however the LTI systems with impulse response $h_1\left[n\right]$ and $h_2\left[n\right]$ have interchanged from order.

The order of two LTI systems, one with impulse response $h_1\left[n\right]$ and the other one with $h_2\left[n\right]$, can be interchanged.