National Instruments 370753C-01 Impulse Response of a System,

Chapter 4 System Analysis

Impulse Response of a System

An impulse input to a system is defined somewhat differently depending on

whether the system is discrete or continuous. For a continuous-time system,

an impulse is a unit-area signal of infinite amplitude and infinitely small

duration occurring at time t= 0, and having value zero at all other times.

For a discrete system, an impulse can be thought of as a physical pulse

which has unit amplitude at the first sample period and zero amplitude for

all other times.

The Laplace transform of the continuous-time impulse—often referred to

as δ(t)—is 1. Thus, the Laplace transform of a output of a system to a unit

impulse is merely its transfer function H(s), as discussed in the

Time-Domain Solution of System Equations section.

A similar definition, using the z-transform, can be made for the

discrete-time impulse response. However, the values of the impulse

response of a discrete system also have the property that they define the

Markov parameters for that system. Based on the state-space representation

of the system, these parameters are defined as the values

These parameters are uniquely determined by the transfer function of the

system [Kai80]:

and they also are the terms of the discrete impulse response.

y = impulse(Sys,t)

The impulse( ) function computes the impulse response of a dynamic

system. The time vector, t, is an optional input. If not specified, a default

time range will be computed using deftimerange( ). Refer to the

deftimerange( ) section. For a continuous-time system, the impulse

response is calculated at each point in the time vector. For a discrete

system, the first n Markov parameters are returned, where n is the length

of the time vector (which must be regularly spaced).

hiCAi1–Bi,12,…},={=

Hz() CzI A–()

1–Bh

izi–

i1=

∞

∑