Step Functions, Delta Distributions, and Causal Convolution

1Introduction

This lecture distinguishes a step input, whose finite amplitude switches at a specified time, from a delta input, which supplies a finite total amount instantaneously. A finite step joins a first-order solution continuously, whereas a delta input can produce a jump. For a causal linear time-invariant system, convolution with the impulse response represents the zero-state response.

2Definitions of the Step and Delta

Let $a>0$a>0. The Heaviside step function is

Its value at $t=a$t=a does not affect ordinary integrals or the continuous solutions below. The Dirac delta is a distribution defined by

for every smooth compactly supported test function $\phi$\varphi. The integral notation abbreviates this action. The delta has total mass one, and distributionally

3A Finite Step Does Not Make the Solution Jump

For $\lambda >0$\lambda>0, consider

For $0\text{[PARSE ERROR: Undefined("Command("le")")]}t<a$0\le t<a, $y=0$y=0. Integrating across $a$a gives

If $y$y is locally bounded, the integral terms vanish as $\epsilon \to 0$\varepsilon\to0, so $y(a+)=y(a-)$y(a+)=y(a-). The matching condition yields

4A Delta Input Produces a Jump

For

distributional integration across $a$a gives $y(a+)-y(a-)=A$y(a+)-y(a-)=A and

More generally, consider

Assume that $p\in C^1$p\in C^1, that $q,r$q,r are continuous near $a$a, and that $p\left(a\right)\ne 0$p(a)\ne0. Let $y$y be a piecewise classical solution that is $C^2$C^2 on either side of $a$a and for which $y,y^{\prime }$y,y' have finite one-sided limits. With $[y{]}_a=y(a+)-y(a-)$[y]_a=y(a+)-y(a-), the singular part of its distributional second derivative is

Because $p\left(t\right){\delta }^{\prime }(t-a)=p\left(a\right){\delta }^{\prime }(t-a)-p^{\prime }\left(a\right)\delta (t-a)$p(t)\delta'(t-a)=p(a)\delta'(t-a)-p'(a)\delta(t-a), the ${\delta }^{\prime }$\delta' coefficient is $p\left(a\right)[y{]}_a$p(a)[y]_a, while the $\delta$\delta coefficient is $p\left(a\right)[y^{\prime }{]}_a+(q\left(a\right)-p^{\prime }\left(a\right)\left)\right[y{]}_a$p(a)[y']_a+(q(a)-p'(a))[y]_a. Comparing the ${\delta }^{\prime }$\delta' coefficients first gives $[y{]}_a=0$[y]_a=0; the subsequent $\delta$\delta comparison then gives $p\left(a\right)[y^{\prime }{]}_a=A$p(a)[y']_a=A. Hence $y$y is continuous and

This conclusion depends on the stated piecewise regularity and coefficient hypotheses.

5Causal Convolution and the Laplace Transform

Extend functions on $t\text{[PARSE ERROR: Undefined("Command("ge")")]}0$t\ge0 by zero for $t<0$t<0. Define

If $f,g$f,g are piecewise continuous on finite intervals and of exponential order, then on a common right half-plane where absolute convergence justifies exchanging integrals,

For the unilateral transform and $a>0$a>0,

The first transform has region of convergence $\mathrm{Re}s>0$\operatorname{Re}s>0. The second uses the distributional extension and is defined on the entire complex $s$s-plane for fixed $a>0$a>0. Delta convolution implements the causal shift

6Decomposition into Zero-Input and Zero-State Responses

An LTI system preserves linear combinations and commutes with time shifts. It is causal if output at time $t$t does not depend on future input. For $y^{\prime }+\lambda y=f\left(t\right)$y'+\lambda y=f(t), the causal impulse response is $h\left(t\right)=H\left(t\right)e^{-\lambda t}$h(t)=H(t)e^{-\lambda t} and satisfies $h^{\prime }+\lambda h=\delta \left(t\right)$h'+\lambda h=\delta(t) distributionally. The zero-state response is

For $y\left(0\right)=y_0$y(0)=y_0, add the zero-input response:

Thus the statement that the response is $h*f$h*f applies to the zero-state response of a causal LTI system.

7Dimensional Contract

With $\left[t\right]=\mathrm{s}$[t]=\mathrm s and $\left[y\right]=\mathrm{Y}$[y]=\mathrm Y, consistency requires

Since $\left[\delta \right(t-a\left)\right]=s^{-1}$[\delta(t-a)]=\mathrm{s^{-1}}, $\left[A\right]=\mathrm{Y}$[A]=\mathrm Y in $A\delta (t-a)$A\delta(t-a). The coefficient is the time-integrated input strength

and has the same dimension as the jump in $y$y.

8Scope

Equations containing delta distributions are interpreted distributionally. Representation by one impulse response requires linearity, time invariance, causality, and the zero-state contract. It does not generally apply to nonlinear or time-varying systems.

9Exercise and Synthesis Lecture

The following synthesis lecture reviews method selection across Laplace transforms, series methods, qualitative analysis, and numerical methods.

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