Cascade of LTI Systems is LTI

Consider the cascade of two LTI systems:

We can show that the composite system is LTI.

Time invariance:

We must show that for any real t, S · Dt = Dt · S. This follows because S1 and S2 are time invariant,

S · Dt = S2 · S1 · Dt = S2 · Dt · S1 = Dt · S2 · S1 = Dt · S.

Linearity:

We must show that S(ax1 + bx2) = aS(x1) + bS(x2). This follows because S1 and S2 are linear,

S(ax1 + bx2) = S2(S1(ax1 + bx2))

= S2(aS1(x1) + bS1(x2))

= aS2(S1(x1)) + bS2(S1(x2))

= aS(x1) + bS(x2)