The Convolution Integral

The response of any linear system to any arbitrary input f(t) can be obtained from in­tegrals of the two basic response functions h(t) and A(t). h(t) is the response to the unit impulse 5(f), and A(t) is the response to the unit step 1(f). The system is assumed to be initially quiescent. If not, the transient associated with nonzero initial condi­tions must be added to the following integrals. The response to /(f) is then given by Duhamel’s integral, or the convolution integral:

(a)

t (A.3,1)

x[t) = f A{t – t)/(r) dr (/(0) = 0) ib)

■4=0

When /(0) is not zero, then there must be added to (A.3,15) a term to allow for the initial step in /(f); i. e.,

xit) = f(0)A(t) + f Ait – t)/(t) dr (A.3,2)

Jt=0

The physical significance of these integrals is brought out by considering them as the limits of the following sums

xit) = 2/i(f – t)/(t) At ia) ^

xit) = Ait)fi0) + b4(f – t)/(t) At (5) C ’

Typical terms of the summations are illustrated in Figs. A. l and A.2. The summation forms are quite convenient for computation, especially when the interval At is kept constant.

V* = Cbaya (a) (A.4,3) Ua – Щ (Ь)

where (A.4,4)

I = T 1

^ab — *-*ba

When a vector is successively transformed through several frames of reference, for example, Fa, Fb, Fc. . . then

vfc = L haa

and c = Lcbb = LJLh(ya)

Since also vc = Ltav(„ then it follows that

Lea = Lc/)Lfca

and similarly for additional transformations.

The sequence of subscripts in the preceding expression should be noted, as it provides a convenient mnemonic for remembering these relations.