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8.9: Unit-Step-Response Function and IRF

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  • From Chapter 4, Equation 4.6.4, the general equation that relates the output Laplace transform to the input transform through the system transfer function is

    \[\left.L[x(t)]\right|_{I C s=0}=T F(s) \times L[u(t)]\label{eqn:8.31}\]

    Equation \(\ref{eqn:8.31}\) is valid for any LTI SISO system. Suppose that the input to such a system is the unit-step function, \(u(t)=U H(t)\) with step magnitude \(U\) = 1:

    \[u(t)=H(t) \quad \Rightarrow \quad \text { from Equation }2.4.5, \quad L[u(t)]=L[H(t)]=\frac{1}{s}\label{eqn:8.32}\]

    We denote the unit-step-response function as \(x_{H}(t)\): as usual for unit-step response and unit-impulse response, we specify zero ICs. Substituting Equation \(\ref{eqn:8.32}\) into Equation \(\ref{eqn:8.31}\) gives

    \[L\left[x_{H}(t)\right]=T F(s) \times \frac{1}{s}\label{eqn:8.33}\]

    Next, suppose that the input is the ideal unit-impulse function, \(u(t)=I_{U} \delta(t)\) with impulse magnitude \(I_{U}=1\):

    \[u(t)=\delta(t) \quad \Rightarrow \quad \text { from Eq. }(8.4.6), \quad L[u(t)]=L[\delta(t)]=1\label{eqn:8.34}\]

    We denote the unit-impulse-response function, with zero ICs, as \(h(t)\), and we abbreviate this important function in text as IRF1. Substituting Equation \(\ref{eqn:8.34}\) into Equation \(\ref{eqn:8.31}\) gives

    \[L[h(t)]=T F(s) \times 1 \equiv T F(s)\label{eqn:8.35}\]

    Equation \(\ref{eqn:8.35}\) is an important relationship in linear-system theory: The Laplace transform of the unit-impulse-response function (IRF) equals the transfer function (TF).

    Comparing Equations \(\ref{eqn:8.32}\) and \(\ref{eqn:8.34}\) shows that

    \[s L[H(t)]=L[\delta(t)]\label{eqn:8.36}\]

    Next, applying Equation 8.6.3, with \(H\left(0^{-}\right)=0\), then Equation \(\ref{eqn:8.36}\), gives

    \[s L[H(t)]=L\left[\frac{d H}{d t}(t)\right]+H\left(0^{-}\right)=L\left[\frac{d H}{d t}(t)\right]=L[\delta(t)]\label{eqn:8.37}\]

    Thus, we infer from Equation \(\ref{eqn:8.37}\) that

    \[\frac{d H}{d t}(t)=\delta(t)\label{eqn:8.38}\]

    We can also derive Equation \(\ref{eqn:8.38}\) formally by differentiating with respect to time \(t\) the following specific form of Equation 8.4.4: \(H(t-0)=\int_{\tau=0^{-}}^{\tau=t>0} \delta(\tau-0) d \tau\). Expressed in words, the time derivative of the unit-step function is the unit-impulse function. These derivations of Equation \(\ref{eqn:8.38}\) are not mathematically rigorous, and the result might seem implausible since both \(H(t)\) and \(\delta(t)\) are strongly discontinuous functions. However, Equation \(\ref{eqn:8.38}\) can be proved with the theory of generalized functions (Lighthill, 1958), and it can also be demonstrated plausibility with the use of limiting processes on functions other than the flat impulse of Section 8.3 (e.g., homework Problem 8.6).

    Comparing Equations \(\ref{eqn:8.33}\) and \(\ref{eqn:8.35}\) shows that

    \[T F(s)=s L\left[x_{H}(t)\right]=L[h(t)]\label{eqn:8.39}\]

    Applying Equation 8.6.3 again, with \(x_{H}\left(0^{-}\right)=0\) by definition, gives

    \[\frac{d x_{H}}{d t}(t)=h(t)\label{eqn:8.40}\]

    In words, the velocity of unit-step response equals the unit-impulse response. Equation \(\ref{eqn:8.40}\) for responses (outputs) is directly analogous to Equation \(\ref{eqn:8.38}\) for excitations (inputs). The identical form of the two equations is a consequence of system linearity. Equation \(\ref{eqn:8.40}\) is another important relationship in linear-system theory, for which we shall have a convenient application in Section 9.8.