Signals and Systems - Electrical Engineering

138 C H A P T E R 2: Continuous-Time Systems

Next we define the impulse response of an LTI and find the response due tox(t). The impulse response of

an analog LTI system,h(t), is the output of the system corresponding to an impulseδ(t)as input, and initial

conditions equal to zero.

If the inputx(t)in Equation (2.17) is seen as an infinite sum of weighted and shifted impulses

x(τ)δ(t−τ)then the output of an LTI system is the superposition of the responses to each of these

terms.

The response of an LTI systemSrepresented by its impulse responseh(t)=S[δ(t)](i.e., the output of the

system to an impulse signalδ(t)and zero initial conditions) to any signalx(t)is theconvolution integral

y(t)=

∫∞

−∞

x(τ)h(t−τ)dτ=

∫∞

−∞

x(t−τ)h(τ)dτ

=[x∗h](t)=[h∗x](t) (2.18)

where the symbol∗stands for the convolution integral of the input signal and the impulse response of the

system.

The above can be seen as follows:

n Assuming no energy is initially stored in the system (i.e., initial conditions are zero) the response

toδ(t)is the impulse responseh(t).

n Given that the system is time invariant, the response toδ(t−τ)ish(t−τ)and by linearity the

response tox(τ)δ(t−τ)isx(τ)h(t−τ)sincex(τ)is not a function of timet.

n Thus, the response of the system to the generic representation Equation (2.17)

x(t)=

∫∞

−∞

x(τ)δ(t−τ)dτ

is by superposition

y(t)=

∫∞

−∞

x(τ)h(t−τ)dτ

or equivalently

y(t)=

∫∞

−∞

x(t−σ)h(σ)dσ

after lettingσ=t−τ. The two integrals are identical—each gives the response of the LTI system.

The impulse responseh(t)represents the system. Notice that in the convolution integral the input

and the impulse response commute (i.e., are interchangeable).