Singularity Functions

This is a continuation from the previous tutorial - causal LTI systems described by differential and difference equations.

In this tutorial, we take another look at the continuous-time unit impulse function in order to gain additional intuitions about this important idealized signal and to introduce a set of related signals known collectively as singularity functions.

In particular, in the unit impulse and unit step functions tutorial we suggested that a continuous-time unit impulse could be viewed as the idealization of a pulse that is "short enough" so that its shape and duration is of no practical consequence—i.e., so that as far as the response of any particular LTI system is concerned, all of the area under the pulse can be thought of as having been applied instantaneously.

In this tutorial, we would first like to provide a concrete example of what this means and then use the interpretation embodied within the example to show that the key to the use of unit impulses and other singularity functions is in the specification of how LTI systems respond to these idealized signals; i.e., the signals are in essence defined in terms of how they behave under convolution with other signals.

1. The Unit Impulse as an Idealized Short Pulse

From the sifting property, eq. (2.27) [refer to the continuous-time LTI systems and convolution integral tutorial], the unit impulse \(\delta(t)\) is the impulse response of the identity system. That is,

\[\tag{2.133}x(t)=x(t)*\delta(t)\]

for any signal \(x(t)\).

Therefore, if we take \(x(t)=\delta(t)\), we have

\[\tag{2.134}\delta(t)=\delta(t)*\delta(t)\]

Equation (2.134) is a basic property of the unit impulse, and it also has a significant implication for our interpretation of the unit impulse as an idealized pulse.

For example, as in the unit impulse and unit step functions tutorial, suppose that we think of \(\delta(t)\) as the limiting form of a rectangular pulse. Specifically, let \(\delta_\Delta(t)\) correspond to the rectangular pulse defined in Figure 1.34 [refer to the unit impulse and unit step functions tutorial], and let

\[\tag{2.135}r_\Delta(t)=\delta_\Delta(t)*\delta_\Delta(t)\]

Then \(r_\Delta(t)\) is as sketched in Figure 2.33.

If we wish to interpret \(\delta(t)\) as the limit as \(\Delta\rightarrow0\) of \(\delta_\Delta(t)\), then, by virtue of eq. (2.134), the limit as \(\Delta\rightarrow0\) for \(r_\Delta(t)\) must also be a unit impulse.

In a similar manner, we can argue that the limits as \(\Delta\rightarrow0\) of \(r_\Delta(t)*r_\Delta(t)\) or \(r_\Delta(t)*\delta_\Delta(t)\) must be unit impulses, and so on.

Thus, we see that for consistency, if we define the unit impulse as the limiting form of some signal, then in fact, there is an unlimited number of very dissimilar-looking signals, all of which behave like an impulse in the limit.

The key words in the preceding paragraph are "behave like an impulse," where, as
we have indicated, what we mean by this is that the response of an LTI system to all of these signals is essentially identical, as long as the pulse is "short enough," i.e., \(\Delta\) is "small enough."

The following example illustrates this idea:

Example 2.16

Consider the LTI system described by the first-order differential equation

\[\tag{2.136}\frac{\text{d}y(t)}{\text{d}t}+2y(t)=x(t)\]

together with the condition of initial rest.

Figure 2.34 depicts the response of this system to \(\delta_\Delta(t)\), \(r_\Delta(t)\), \(r_\Delta(t)*\delta_\Delta(t)\), and \(r_\Delta(t)*r_\Delta(t)\) for several values of \(\Delta\).

For \(\Delta\) large enough, the responses to these input signals differ noticeably. However, for \(\Delta\) sufficiently small, the responses are essentially indistinguishable, so that all of the input signals "behave" in the same way.

Furthermore, as suggested by the figure, the limiting form of all of these responses is precisely \(e^{-2t}u(t)\). Since the limit of each of these signals as \(\Delta\rightarrow0\) is the unit impulse, we conclude that \(e^{-2t}u(t)\) is the impulse response for this system.

One important point to be emphasized is that what we mean by "A small enough" depends on the particular LTI system to which the preceding pulses are applied. For example, in Figure 2.35, we have illustrated the responses to these pulses for different values of \(\Delta\) for the causal LTI system described by the first-order differential equation

\[\tag{2.137}\frac{\text{d}y(t)}{\text{d}t}+20y(t)=x(t)\]

As seen in the figure, we need a smaller value of \(\Delta\) in this case in order for the responses to be indistinguishable from each other and from the impulse response \(h(t)=e^{-20t}u(t)\) for the system.

Thus, while what we mean by "\(\Delta\) small enough" is different for these two systems, we can find values of \(\Delta\) small enough for both. The unit impulse is then the idealization of a short pulse whose duration is short enough for all systems.

2. Defining the Unit Impulse through Convolution

As the preceding example illustrates, for \(\Delta\) small enough, the signals \(\delta_\Delta(t)\), \(r_\Delta(t)\), \(r_\Delta(t)*\delta_\Delta(t)\), and \(r_\Delta(t)*r_\Delta(t)\) all act like impulses when applied to an LTI system.

In fact, there are many other signals for which this is true as well. What it suggests is that we should think of a unit impulse in terms of how an LTI system responds to it.

While usually a function or signal is defined by what it is at each value of the independent variable, the primary importance of the unit impulse is not what it is at each value of \(t\), but rather what it does under convolution.

Thus, from the point of view of linear systems analysis, we may alternatively define the unit impulse as that signal which, when applied to an LTI system, yields the impulse response.

That is, we define \(\delta(t)\) as the signal for which

\[\tag{2.138}x(t)=x(t)*\delta(t)\]

for any \(x(t)\).

In this sense, signals, such as \(\delta_\Delta(t)\), \(r_\Delta(t)\), etc., which correspond to short pulses with vanishingly small duration as \(\Delta\rightarrow0\), all behave like a unit impulse in the limit because, if we replace \(\delta(t)\) by any of these signals, then eq. (2.138) is satisfied in the limit.

All the properties of the unit impulse that we need can be obtained from the operational definition given by eq. (2.138).

For example, if we let \(x(t)=1\) for all \(t\), then

\[\begin{align}1=x(t)&=x(t)*\delta(t)=\delta(t)*x(t)=\displaystyle\int\limits_{-\infty}^{+\infty}\delta(\tau)x(t-\tau)\text{d}\tau\\&=\displaystyle\int\limits_{-\infty}^{+\infty}\delta(\tau)\text{d}\tau\end{align}\]

so that the unit impulse has unit area.

It is sometimes useful to use another completely equivalent operational definition of \(\delta(t)\). To obtain this alternative form, consider taking an arbitrary signal \(g(t)\), reversing it in time to obtain \(g(-t)\), and then convolving this with \(\delta(t)\). Using eq. (2.138), we obtain

\[g(-t)=g(-t)*\delta(t)=\delta(t)*g(-t)=\displaystyle\int\limits_{-\infty}^{+\infty}\delta(\tau)g(-(t-\tau))\text{d}\tau=\int\limits_{-\infty}^{+\infty}g(\tau-t)\delta(\tau)\text{d}\tau\]

which, for \(t=0\), yields

\[\tag{2.139}g(0)=\displaystyle\int\limits_{-\infty}^{+\infty}g(\tau)\delta(\tau)\text{d}\tau\]

Therefore, the operational definition of \(\delta(t)\) given by eq. (2.138) implies eq. (2.139).

On the other hand, eq. (2.139) implies eq. (2.138). To see this, let \(x(t)\) be a given signal, fix a time \(t\), and define

\[g(\tau)=x(t-\tau)\]

Then, using eq. (2.139), we have

\[x(t)=g(0)=\displaystyle\int\limits_{-\infty}^{+\infty}g(\tau)\delta(\tau)\text{d}\tau=\int\limits_{-\infty}^{+\infty}x(t-\tau)\delta(\tau)\text{d}\tau=\int\limits_{-\infty}^{+\infty}\delta(\tau)x(t-\tau)\text{d}\tau=\delta(t)*x(t)=x(t)*\delta(t)\]

which is precisely eq. (2.138).

Therefore, eq. (2.139) is an equivalent operational definition of the unit impulse. That is, the unit impulse is the signal which, when multiplied by a signal \(g(t)\) and then integrated from \(-\infty\) to \(+\infty\), produces the value \(g(0)\).

Since we will be concerned principally with LTI systems, and thus with convolution, the characterization of \(\delta(t)\) given in eq. (2.138) will be the one to which we will refer most often.

However, eq. (2.139) is useful in determining some of the other properties of the unit impulse.

For example, consider the signal \(f(t)\delta(t)\), where \(f(t)\) is another signal. Then, from eq. (2.139),

\[\tag{2.140}\displaystyle\int\limits_{-\infty}^{+\infty}g(\tau)f(\tau)\delta(\tau)\text{d}\tau=g(0)f(0)\]

On the other hand, if we consider the signal \(f(0)\delta(t)\), we see that

\[\tag{2.141}\displaystyle\int\limits_{-\infty}^{+\infty}g(\tau)f(0)\delta(\tau)\text{d}\tau=g(0)f(0)\]

Comparing eqs. (2.140) and (2.141), we find that the two signals \(f(t)\delta(t)\) and \(f(0)\delta(t)\) behave identically when they are multiplied by any signal \(g(t)\) and then integrated from \(-\infty\) to \(+\infty\).

Consequently, using this form of the operational definition of signals, we conclude that

\[\tag{2.142}f(t)\delta(t)=f(0)\delta(t)\]

which is a property that we derived by alternative means in the unit impulse and unit step functions tutorial. [See eq. (1.76).]

3. Unit Doublets and Other Singularity Functions

The unit impulse is one of a class of signals known as singularity functions, each of which can be defined operationally in terms of its behavior under convolution.

Consider the LTI system for which the output is the derivative of the input, i.e.,

\[\tag{2.143}y(t)=\frac{\text{d}x(t)}{\text{d}t}\]

The unit impulse response of this system is the derivative of the unit impulse, which is called the unit doublet \(u_1(t)\).

From the convolution representation for LTI systems, we have

\[\tag{2.144}\frac{\text{d}x(t)}{\text{d}t}=x(t)*u_1(t)\]

for any signal \(x(t)\).

Just as eq. (2.138) serves as the operational definition of \(\delta(t)\), we will take eq. (2.144) as the operational definition of \(u_1(t)\).

Similarly, we can define \(u_2(t)\), the second derivative of \(\delta(t)\), as the impulse response of an LTI system that takes the second derivative of the input, i.e.,

\[\tag{2.145}\frac{\text{d}^2x(t)}{\text{d}t^2}=x(t)*u_2(t)\]

From eq. (2.144), we see that

\[\tag{2.146}\frac{\text{d}^2x(t)}{\text{d}t^2}=\frac{\text{d}}{\text{d}t}\left(\frac{\text{d}x(t)}{\text{d}t}\right)=x(t)*u_1(t)*u_1(t)\]

and therefore,

\[\tag{2.147}u_2(t)=u_1(t)*u_1(t)\]

In general, \(u_k(t)\), \(k\gt0\), is the \(k\)th derivative of \(\delta(t)\) and thus is the impulse response of a system that takes the \(k\)th derivative of the input. Since this system can be obtained as the cascade of \(k\) differentiators, we have

\[\tag{2.148}u_k(t)=\underbrace{u_1(t)*\cdots*u_1(t)}_{k \text{ times}}\]

As with the unit impulse, each of these singularity functions has properties that can be derived from its operational definition.

For example, if we consider the constant signal \(x(t)=1\), we find that

\[\begin{align}0=\frac{\text{d}x(t)}{\text{d}t}&=x(t)*u_1(t)=u_1(t)*x(t)=\displaystyle\int\limits_{-\infty}^{+\infty}u_1(\tau)x(t-\tau)\text{d}\tau\\&=\displaystyle\int\limits_{-\infty}^{+\infty}u_1(\tau)\text{d}\tau\end{align}\]

so that the unit doublet has zero area.

Moreover, if we convolve the signal \(g(-t)\) with \(u_1(t)\), we obtain

\[g(-t)*u_1(t)=\frac{\text{d}g(-t)}{\text{d}t}=-g'(-t)\]

and

\[g(-t)*u_1(t)=u_1(t)*g(-t)=\displaystyle\int\limits_{-\infty}^{+\infty}u_1(\tau)g(-(t-\tau))\text{d}\tau=\int\limits_{-\infty}^{+\infty}g(\tau-t)u_1(\tau)\text{d}\tau\]

so that

\[-g'(-t)=\int\limits_{-\infty}^{+\infty}g(\tau-t)u_1(\tau)\text{d}\tau\]

which, for \(t=0\), yields

\[\tag{2.149}-g'(0)=\int\limits_{-\infty}^{+\infty}g(\tau)u_1(\tau)\text{d}\tau\]

In an analogous manner, we can derive related properties of \(u_1(t)\) and higher order singularity functions.

As with the unit impulse, each of these singularity functions can be informally related to short pulses.

For example, since the unit doublet is formally the derivative of the unit impulse, we can think of the doublet as the idealization of the derivative of a short pulse with unit area.

For instance, consider the short pulse \(\delta_\Delta(t)\) in Figure 1.34 [refer to the unit impulse and unit step functions tutorial]. This pulse behaves like an impulse as \(\Delta\rightarrow0\). Consequently, we would expect its derivative to behave like a doublet as \(\Delta\rightarrow0\).

\(\text{d}\delta_\Delta(t)/\text{d}t\) is as depicted in Figure 2.36: It consists of a unit impulse at \(t=0\) with area \(+1/\Delta\), followed by a unit impulse of area \(-1/\Delta\) at \(t=\Delta\), i.e.,

\[\tag{2.150}\frac{\text{d}\delta_\Delta(t)}{\text{d}t}=\frac{1}{\Delta}[\delta(t)-\delta(t-\Delta)]\]

Consequently, using the fact that \(x(t)*\delta(t-t_0)=x(t-t_0)\) [see eq. (2.70) in the properties of LTI systems tutorial], we find that

\[\tag{2.151}x(t)*\frac{\text{d}\delta_\Delta(t)}{\text{d}t}=\frac{x(t)-x(t-\Delta)}{\Delta}\approx\frac{\text{d}x(t)}{\text{d}t}\]

where the approximation becomes increasingly accurate as \(\Delta\rightarrow0\). Comparing eq. (2.151) with eq. (2.144), we see that \(\text{d}\delta_\Delta(t)/\text{d}t\) does indeed behave like a unit doublet as \(\Delta\rightarrow0\).

In addition to singularity functions that are derivatives of different orders of the unit impulse, we can also define signals that represent successive integrals of the unit impulse function.

As we saw in Example 2.13 [refer to the properties of LTI systems tutorial], the unit step is the impulse response of an integrator:

\[y(t)=\int\limits_{-\infty}^tx(\tau)\text{d}\tau\]

Therefore,

\[\tag{2.152}u(t)=\int\limits_{-\infty}^t\delta(\tau)\text{d}\tau\]

and we also have the following operational definition of \(u(t)\):

\[\tag{2.153}x(t)*u(t)=\int\limits_{-\infty}^tx(\tau)\text{d}\tau\]

Similarly, we can define the system that consists of a cascade of two integrators. Its impulse response is denoted by \(u_{-2}(t)\), which is simply the convolution of \(u(t)\), the impulse response of one integrator, with itself:

\[\tag{2.154}u_{-2}(t)=u(t)*u(t)=\int\limits_{-\infty}^tu(\tau)\text{d}\tau\]

Since \(u(t)\) equals 0 for \(t\lt0\) and equals 1 for \(t\gt0\), it follows that

\[\tag{2.155}u_{-2}(t)=tu(t)\]

This signal, which is referred to as the unit ramp function, is shown in Figure 2.37.

Also, we can obtain an operational definition for the behavior of \(u_{-2}(t)\) under convolution from eqs. (2.153) and (2.154): 

\[\tag{2.156}\begin{align}x(t)*u_{-2}(t)&=x(t)*u(t)*u(t)\\&=\left(\int\limits_{-\infty}^tx(\sigma)\text{d}\sigma\right)*u(t)\\&=\int\limits_{-\infty}^t\left(\int\limits_{-\infty}^\tau{x(\sigma)\text{d}\sigma}\right)\text{d}\tau\end{align}\]

In an analogous fashion, we can define higher order integrals of \(\delta(t)\) as the impulse responses of cascades of integrators:

\[\tag{2.157}u_{-k}(t)=\underbrace{u(t)*\cdots*u(t)}_{k\text{ times}}=\int\limits_{-\infty}^tu_{-(k-1)}(\tau)\text{d}\tau\]

The convolution of \(x(t)\) with \(u_{-3}(t),u_{-4}(t),\ldots\) generate correspondingly higher order integrals of \(x(t)\).

Also, note that the integrals in eq. (2.157) can be evaluated directly, as was done in eq. (2.155), to obtain

\[\tag{2.158}u_{-k}(t)=\frac{t^{k-1}}{(k-1)!}u(t)\]

Thus, unlike the derivatives of \(\delta(t)\), the successive integrals of the unit impulse are functions that can be defined for each value of \(t\) [eq. (2.158)], as well as by their behavior under convolution.

At times it will be worthwhile to use an alternative notation for \(\delta(t)\) and \(u(t)\), namely,

\[\tag{2.159}\delta(t)=u_0(t)\]

\[\tag{2.160}u(t)=u_{-1}(t)\]

With this notation, \(u_k(t)\) for \(k\gt0\) denotes the impulse response of a cascade of \(k\) differentiators, \(u_0(t)\) is the impulse response of the identity system, and, for \(k\lt0\), \(u_k(t)\) is the impulse response of a cascade of \(|k|\) integrators.

Furthermore, since a differentiator is the inverse system of an integrator,

\[u(t)*u_1(t)=\delta(t)\]

or, in our alternative notation,

\[\tag{2.161}u_{-1}(t)*u_1(t)=u_0(t)\]

More generally, from eqs. (2.148), (2.157), and (2.161), we see that for any integers \(k\) and \(r\),

\[\tag{2.162}u_k(t)*u_r(t)=u_{k+r}(t)\]

If \(k\) and \(r\) are both positive, eq. (2.162) states that a cascade of \(k\) differentiators followed by \(r\) more differentiators yields an output that is the \((k+r)\)th derivative of the input.

Similarly, if \(k\) is negative and \(r\) is negative, we have a cascade of \(|k|\) integrators followed by another \(|r|\) integrators.

Also, if \(k\) is negative and \(r\) is positive, we have a cascade of \(|k|\) integrators followed by \(r\) differentiators, and the overall system is equivalent to a cascade of \(|k+r|\) integrators if \(k+r\lt0\), a cascade of \(k+r\) differentiators if \(k+r\gt0\), or the identity system if \(k+r=0\).

Therefore, by defining singularity functions in terms of their behavior under convolution, we obtain a characterization that allows us to manipulate them with relative ease and to interpret them directly in terms of their significance for LTI systems.

Since this is our primary concern in this tutorial series, the operational definition for singularity functions that we have given in this tutorial will suffice for our purposes.

The next tutorial discusses about the response of LTI systems to complex exponentials.