# Properties of Linear Time-Invariant Systems

This is a continuation from the previous tutorial - continuous-time LTI systems and convolution integral.

In the preceding two tutorials, we developed the extremely important representations of continuous-time and discrete-time LTI systems in terms of their unit impulse responses. In discrete time the representation takes the form of the convolution sum, while its continuous-time counterpart is the convolution integral, both of which we repeat here for convenience:

$\tag{2.39}y[n]=\sum_{k=-\infty}^{+\infty}x[k]h[n-k]=x[n]*h[n]$

$\tag{2.40}y(t)=\displaystyle\int\limits_{-\infty}^{+\infty}x(\tau)h(t-\tau)\text{d}\tau=x(t)*h(t)$

As we have pointed out, one consequence of these representations is that the characteristics of an LTI system are completely determined by its impulse response.

It is important to emphasize that this property holds in general only for LTI systems. In particular, as illustrated in the following example, the unit impulse response of a nonlinear system does not completely characterize the behavior of the system.

Example 2.9

Consider a discrete-time system with unit impulse response

$\tag{2.41}h[n]=\begin{cases}1,\qquad{n=0,1}\\0,\qquad\text{otherwise}\end{cases}$

If the system is LTI, then eq. (2.41) completely determines its input-output behavior. In particular, by substituting eq. (2.41) into the convolution sum, eq. (2.39), we find the following explicit equation describing how the input and output of this LTI system are related:

$\tag{2.42}y[n]=x[n]+x[n-1]$

On the other hand, there are many nonlinear systems with the same response—i.e., that given in eq. (2.41)—to the input $$\delta[n]$$. For example, both of the following systems have this property:

$y[n]=(x[n]+x[n-1])^2$

$y[n]=\max(x[n],x[n-1])$

Consequently, if the system is nonlinear it is not completely characterized by the impulse response in eq. (2.41).

The preceding example illustrates the fact that LTI systems have a number of properties not possessed by other systems, beginning with the very special representations that they have in terms of convolution sums and integrals. In the remainder of this tutorial, we explore some of the most basic and important of these properties.

## 1. The Commutative Property

A basic property of convolution in both continuous and discrete time is that it is a commutative operation. That is, in discrete time

$\tag{2.43}x[h]*h[n]=h[n]*x[n]=\sum_{k=-\infty}^{+\infty}h[k]x[n-k]$

and in continuous time

$\tag{2.44}x(t)*h(t)=h(t)*x(t)=\displaystyle\int\limits_{-\infty}^{+\infty}h(\tau)x(t-\tau)\text{d}\tau$

These expressions can be verified in a straightforward manner by means of a substitution of variables in eqs. (2.39) and (2.40).

For example, in the discrete-time case, if we let $$r=n-k$$ or, equivalently, $$k=n-r$$, eq. (2.39) becomes

$\tag{2.45}x[n]*h[n]=\sum_{k=-\infty}^{+\infty}x[k]h[n-k]=\sum_{r=-\infty}^{+\infty}x[n-r]h[r]=h[n]*x[n]$

With this substitution of variables, the roles of $$x[n]$$ and $$h[n]$$ are interchanged. According to eq. (2.45), the output of an LTI system with input $$x[n]$$ and unit impulse response $$h[n]$$ is identical to the output of an LTI system with input $$h[n]$$ and unit impulse response $$x[n]$$.

For example, we could have calculated the convolution in Example 2.4 in the discrete-time LTI systems and convolution sum tutorial by first reflecting and shifting $$x[n]$$, then multiplying the signals $$x[n-k]$$ and $$h[k]$$, and finally summing the products for all values of $$k$$.

Similarly, eq. (2.44) can be verified by a change of variables, and the implications of this result in continuous time are the same: The output of an LTI system with input $$x(t)$$ and unit impulse response $$h(t)$$ is identical to the output of an LTI system with input $$h(t)$$ and unit impulse response $$x(t)$$.

Thus, we could have calculated the convolution in Example 2.7 in the continuous-time LTI systems and convolution integral tutorial by reflecting and shifting $$x(t)$$, multiplying the signals $$x(t-\tau)$$ and $$h(\tau)$$, and integrating over $$-\infty\lt\tau\lt+\infty$$.

In specific cases, one of the two forms for computing convolutions [i.e., eq. (2.39) or (2.43) in discrete time and eq. (2.40) or (2.44) in continuous time] may be easier to visualize, but both forms always result in the same answer.

## 2. The Distributive Property

Another basic property of convolution is the distributive property. Specifically, convolution distributes over addition, so that in discrete time

$\tag{2.46}x[n]*(h_1[n]+h_2[n])=x[n]*h_1[n]+x[n]*h_2[n]$

and in continuous time

$\tag{2.47}x(t)*[h_1(t)+h_2(t)]=x(t)*h_1(t)+x(t)*h_2(t)$

This property can be verified in a straightforward manner.

The distributive property has a useful interpretation in terms of system interconnections. Consider two continuous-time LTI systems in parallel, as indicated in Figure 2.23(a). The systems shown in the block diagram are LTI systems with the indicated unit impulse responses. This pictorial representation is a particularly convenient way in which to denote LTI systems in block diagrams, and it also reemphasizes the fact that the impulse response of an LTI system completely characterizes its behavior.

The two systems, with impulse responses $$h_1(t)$$ and $$h_2(t)$$, have identical inputs, and their outputs are added. Since

$y_1(t)=x(t)*h_1(t)$

and

$y_2(t)=x(t)*h_2(t)$

the system of Figure 2.23(a) has output

$\tag{2.48}y(t)=x(t)*h_1(t)+x(t)*h_2(t)$

corresponding to the right-hand side of eq. (2.47).

The system of Figure 2.23(b) has output

$\tag{2.49}y(t)=x(t)*[h_1(t)+h_2(t)]$

corresponding to the left-hand side of eq. (2.47).

Applying eq. (2.47) to eq. (2.49) and comparing the result with eq. (2.48), we see that the systems in Figures 2.23(a) and (b) are identical.

There is an identical interpretation in discrete time, in which each of the signals in Figure 2.23 is replaced by a discrete-time counterpart (i.e., $$x(t)$$, $$h_1(t)$$, $$h_2(t)$$, $$y_1(t)$$, $$y_2(t)$$, and $$y(t)$$ are replaced by $$x[n]$$, $$h_1[n]$$, $$h_2[n]$$, $$y_1[n]$$, $$y_2[n]$$, and $$y[n]$$, respectively).

In summary, then, by virtue of the distributive property of convolution, a parallel combination of LTI systems can be replaced by a single LTI system whose unit impulse response is the sum of the individual unit impulse responses in the parallel combination.

Also, as a consequence of both the commutative and distributive properties, we have

$\tag{2.50}[x_1[n]+x_2[n]]*h[n]=x_1[n]*h[n]+x_2[n]*h[n]$

and

$\tag{2.51}[x_1(t)+x_2(t)]*h(t)=x_1(t)*h(t)+x_2(t)*h(t)$

which simply state that the response of an LTI system to the sum of two inputs must equal the sum of the responses to these signals individually.

As illustrated in the next example, the distributive property of convolution can also be exploited to break a complicated convolution into several simpler ones.

Example 2.10

Let $$y[n]$$ denote the convolution of the following two sequences:

$\tag{2.52}x[n]=\left(\frac{1}{2}\right)^nu[n]+2^nu[-n]$

$\tag{2.53}h[n]=u[n]$

Note that the sequence $$x[n]$$ is nonzero along the entire time axis. Direct evaluation of such a convolution is somewhat tedious. Instead, we may use the distributive property to express $$y[n]$$ as the sum of the results of two simpler convolution problems. In particular, if we let $$x_1[n]=(1/2)^nu[n]$$ and $$x_2[n]=2^nu[-n]$$, it follows that

$\tag{2.54}y[n]=(x_1[n]+x_2[n])*h[n]$

Using the distributive property of convolution, we may rewrite eq. (2.54) as

$\tag{2.55}y[n]=y_1[n]+y_2[n]$

where

$\tag{2.56}y_1[n]=x_1[n]*h[n]$

and

$\tag{2.57}y_2[n]=x_2[n]*h[n]$

The convolution in eq. (2.56) for $$y_1[n]$$ can be obtained from Example 2.3 (with $$\alpha=1/2$$) [refer to the discrete-time LTI systems tutorial], while $$y_2[n]$$ was evaluated in Example 2.5 [refer to the discrete-time LTI systems tutorial]. Their sum is $$y[n]$$, which is shown in Figure 2.24.

## 3. The Associative Property

Another important and useful property of convolution is that it is associative. That is, in discrete time

$\tag{2.58}x[n]*(h_1[n]*h_2[n])=(x[n]*h_1[n])*h_2[n]$

and in continuous time

$\tag{2.59}x(t)*[h_1(t)*h_2(t)]=[x(t)*h_1(t)]*h_2(t)$

This property is proven by straightforward manipulations of the summations and integrals involved.

As a consequence of the associative property, the expressions

$\tag{2.60}y[n]=x[n]*h_1[n]*h_2[n]$

and

$\tag{2.61}y(t)=x(t)*h_1(t)*h_2(t)$

are unambiguous. That is, according to eqs. (2.58) and (2.59), it does not matter in which order we convolve these signals.

An interpretation of the associative property is illustrated for discrete-time systems in Figures 2.25(a) and (b).

In Figure 2.25(a),

\begin{align}y[n]&=w[n]*h_2[n]\\&=(x[n]*h_1[n])*h_2[n]\end{align}

In Figure 2.25(b),

\begin{align}y[n]&=x[n]*h[n]\\&=x[n]*(h_1[n]*h_2[n])\end{align}

According to the associative property, the series interconnection of the two systems in Figure 2.25(a) is equivalent to the single system in Figure 2.25(b). This can be generalized to an arbitrary number of LTI systems in cascade, and the analogous interpretation and conclusion also hold in continuous time.

By using the commutative property together with the associative property, we find another very important property of LTI systems.

Specifically, from Figures 2.25(a) and (b), we can conclude that the impulse response of the cascade of two LTI systems is the convolution of their individual impulse responses.

Since convolution is commutative, we can compute this convolution of $$h_1[n]$$ and $$h_2[n]$$ in either order. Thus, Figures 2.25(b) and (c) are equivalent, and from the associative property, these are in tum equivalent to the system of Figure 2.25(d), which we note is a cascade combination of two systems as in Figure 2.25(a), but with the order of the cascade reversed.

Consequently, the unit impulse response of a cascade of two LTI systems does not depend on the order in which they are cascaded. In fact, this holds for an arbitrary number of LTI systems in cascade: The order in which they are cascaded does not matter as far as the overall system impulse response is concerned. The same conclusions hold in continuous time as well.

It is important to emphasize that the behavior of LTI systems in cascade—and, in particular, the fact that the overall system response does not depend upon the order of the systems in the cascade—is very special to such systems.

In contrast, the order in which nonlinear systems are cascaded cannot be changed, in general, without changing the overall response. For instance, if we have two memoryless systems, one being multiplication by 2 and the other squaring the input, then if we multiply first and square second, we obtain

$y[n]=4x^2[n]$

However, if we multiply by 2 after squaring, we have

$y[n]=2x^2[n]$

Thus, being able to interchange the order of systems in a cascade is a characteristic particular to LTI systems. In fact, we need both linearity and time invariance in order for this property to be true in general.

## 4. LTI Systems with and without Memory

As specified in the continuous-time and discrete-time systems tutorial, a system is memoryless if its output at any time depends only on the value of the input at that same time.

From eq. (2.39), we see that the only way that this can be true for a discrete-time LTI system is if $$h[n]=0$$ for $$n\ne0$$. In this case the impulse response has the form

$\tag{2.62}h[n]=K\delta[n]$

where $$K=h[0]$$ is a constant, and the convolution sum reduces to the relation

$\tag{2.63}y[n]=Kx[n]$

If a discrete-time LTI system has an impulse response $$h[n]$$ that is not identically zero for $$n\ne0$$, then the system has memory. An example of an LTI system with memory is the system given by eq. (2.42). The impulse response for this system, given in eq. (2.41), is nonzero for $$n=1$$.

From eq. (2.40), we can deduce similar properties for continuous-time LTI systems with and without memory. In particular, a continuous-time LTI system is memoryless if $$h(t)=0$$ for $$t\ne0$$, and such a memoryless LTI system has the form

$\tag{2.64}y(t)=Kx(t)$

for some constant $$K$$ and has the impulse response

$\tag{2.65}h(t)=K\delta(t)$

Note that if $$K=1$$ in eqs. (2.62) and (2.65), then these systems become identity systems, with output equal to the input and with unit impulse response equal to the unit impulse. In this case, the convolution sum and integral formulas imply that

$x[n]=x[n]*\delta[n]$

and

$x(t)=x(t)*\delta(t)$

which reduce to the sifting properties of the discrete-time and continuous-time unit impulses:

$x[n]=\sum_{k=-\infty}^{+\infty}x[k]\delta[n-k]$

$x(t)=\displaystyle\int\limits_{-\infty}^{+\infty}x(\tau)\delta(t-\tau)\text{d}\tau$

## 5. Invertibility of LTI Systems

Consider a continuous-time LTI system with impulse response $$h(t)$$. Based on the discussion in the continuous-time and discrete-time systems tutorial, this system is invertible only if an inverse system exists that, when connected in series with the original system, produces an output equal to the input to the first system. Furthermore, if an LTI system is invertible, then it has an LTI inverse.

Therefore, we have the picture shown in Figure 2.26.

We are given a system with impulse response $$h(t)$$. The inverse system, with impulse response $$h_1(t)$$, results in $$w(t)=x(t)$$—such that the series interconnection in Figure 2.26(a) is identical to the identity system in Figure 2.26(b).

Since the overall impulse response in Figure 2.26(a) is $$h(t)*h_1(t)$$, we have the condition that $$h_1(t)$$ must satisfy for it to be the impulse response of the inverse system, namely,

$\tag{2.66}h(t)*h_1(t)=\delta(t)$

Similarly, in discrete time, the impulse response $$h_1[n]$$ of the inverse system for an LTI system with impulse response $$h[n]$$ must satisfy

$\tag{2.67}h[n]*h_1[n]=\delta[n]$

The following two examples illustrate invertibility and the construction of an inverse system.

Example 2.11

Consider the LTI system consisting of a pure time shift

$\tag{2.68}y(t)=x(t-t_0)$

Such a system is a delay if $$t_0\gt0$$ and an advance if $$t_0\lt0$$. For example, if $$t_0\gt0$$, then the output at time $$t$$ equals the value of the input at the earlier time $$t-t_0$$.

If $$t_0=0$$, the system in eq. (2.68) is the identity system and thus is memoryless. For any other value of $$t_0$$, this system has memory, as it responds to the value of the input at a time other than the current time.

The impulse response for the system can be obtained from eq. (2.68) by taking the input equal to $$\delta(t)$$, i.e.,

$\tag{2.69}h(t)=\delta(t-t_0)$

Therefore,

$\tag{2.70}x(t-t_0)=x(t)*\delta(t-t_0)$

That is, the convolution of a signal with a shifted impulse simply shifts the signal.

To recover the input from the output, i.e., to invert the system, all that is required is to shift the output back. The system with this compensating time shift is then the inverse system. That is, if we take

$h_1(t)=\delta(t+t_0)$

then

$h(t)*h_1(t)=\delta(t-t_0)*\delta(t+t_0)=\delta(t)$

Similarly, a pure time shift in discrete time has the unit impulse response $$\delta[n-n_0]$$, so that convolving a signal with a shifted impulse is the same as shifting the signal.

Furthermore, the inverse of the LTI system with impulse response $$\delta[n-n_0]$$ is the LTI system that shifts the signal in the opposite direction by the same amount—i.e., the LTI system with impulse response $$\delta[n+n_0]$$.

Example 2.12

Consider an LTI system with impulse response

$\tag{2.71}h[n]=u[n]$

Using the convolution sum, we can calculate the response of this system to an arbitrary input:

$\tag{2.72}y[n]=\sum_{k=-\infty}^{+\infty}x[k]u[n-k]$

Since $$u[n-k]$$ is 0 for $$n-k\lt0$$ and 1 for $$n-k\ge0$$, eq. (2.72) becomes

$\tag{2.73}y[n]=\sum_{k=-\infty}^nx[k]$

That is, this system, which we first encountered in the continuous-time and discrete-time systems tutorial [see eq. (1.92)], is a summer or accumulator that computes the running sum of all the values of the input up to the present time.

As we saw in the continuous-time and discrete-time systems tutorial, such a system is invertible, and its inverse, as given by eq. (1.99), is

$\tag{2.74}y[n]=x[n]-x[n-1]$

which is simply a first difference operation.

Choosing $$x[n]=\delta[n]$$, we find that the impulse response of the inverse system is

$\tag{2.75}h_1[n]=\delta[n]-\delta[n-1]$

As a check that $$h[n]$$ in eq. (2.71) and $$h_1[n]$$ in eq. (2.75) are indeed the impulse responses of LTI systems that are inverses of each other, we can verify eq. (2.67) by direct calculation:

\tag{2.76}\begin{align}h[n]*h_1[n]&=u[n]*\{\delta[n]-\delta[n-1]\}\\&=u[n]*\delta[n]-u[n]*\delta[n-1]\\&=u[n]-u[n-1]\\&=\delta[n]\end{align}

## 6. Causality for LTI Systems

In the continuous-time and discrete-time systems tutorial, we introduced the property of causality: The output of a causal system depends only on the present and past values of the input to the system.

By using the convolution sum and integral, we can relate this property to a corresponding property of the impulse response of an LTI system.

Specifically, in order for a discrete-time LTI system to be causal, $$y[n]$$ must not depend on $$x[k]$$ for $$k\gt{n}$$. From eq. (2.39), we see that for this to be true, all of the coefficients $$h[n-k]$$ that multiply values of $$x[k]$$ for $$k\gt{n}$$ must be zero. This then requires that the impulse response of a causal discrete-time LTI system satisfy the condition

$\tag{2.77}h[n]=0\qquad\text{for }n\lt0$

According to eq. (2.77), the impulse response of a causal LTI system must be zero before the impulse occurs, which is consistent with the intuitive concept of causality.

More generally, causality for a linear system is equivalent to the condition of initial rest; i.e., if the input to a causal system is 0 up to some point in time, then the output must also be 0 up to that time.

It is important to emphasize that the equivalence of causality and the condition of initial rest applies only to linear systems.

For example, as discussed in Example 1.20 in the continuous-time and discrete-time systems tutorial, the system $$y[n]=2x[n]+3$$ is not linear. However, it is causal and, in fact, memoryless. On the other hand, if $$x[n]=0$$, $$y[n]=3\ne0$$, so it does not satisfy the condition of initial rest.

For a causal discrete-time LTI system, the condition in eq. (2.77) implies that the convolution sum representation in eq. (2.39) becomes

$\tag{2.78}y[n]=\sum_{k=-\infty}^nx[k]h[n-k]$

and the alternative equivalent form, eq. (2.43), becomes

$\tag{2.79}y[n]=\sum_{k=0}^{\infty}h[k]x[n-k]$

Similarly, a continuous-time LTI system is causal if

$\tag{2.80}h(t)=0\qquad\text{for }t\lt0$

and in this case the convolution integral is given by

$\tag{2.81}y(t)=\displaystyle\int\limits_{-\infty}^{t}x(\tau)h(t-\tau)\text{d}\tau=\int\limits_0^{\infty}h(\tau)x(t-\tau)\text{d}\tau$

Both the accumulator ($$h[n]=u[n]$$) and its inverse ($$h[n]=\delta[n]-\delta[n-1]$$), described in Example 2.12, satisfy eq. (2.77) and therefore are causal.

The pure time shift with impulse response $$h(t)=\delta(t-t_0)$$ is causal for $$t_0\ge0$$ (when the time shift is a delay), but is noncausal for $$t_0\lt0$$ (in which case the time shift is an advance, so that the output anticipates future values of the input).

Finally, while causality is a property of systems, it is common terminology to refer to a signal as being causal if it is zero for $$n\lt0$$ or $$t\lt0$$. The motivation for this terminology comes from eqs. (2.77) and (2.80): Causality of an LTI system is equivalent to its impulse response being a causal signal.

## 7. Stability of LTI Systems

Recall from the continuous-time and discrete-time systems tutorial that a system is stable if every bounded input produces a bounded output.

In order to determine conditions under which LTI systems are stable, consider an input $$x[n]$$ that is bounded in magnitude:

$\tag{2.82}|x[n]|\lt{B}\qquad\text{for all }n$

Suppose that we apply this input to an LTI system with unit impulse response $$h[n]$$. Then, using the convolution sum, we obtain an expression for the magnitude of the output:

$\tag{2.83}|y[n]|=\left|\sum_{k=-\infty}^{+\infty}h[k]x[n-k]\right|$

Since the magnitude of the sum of a set of numbers is no larger than the sum of the magnitudes of the numbers, it follows from eq. (2.83) that

$\tag{2.84}|y[n]|\le\sum_{k=-\infty}^{+\infty}|h[k]||x[n-k]|$

From eq. (2.82), $$|x[n-k]|\lt{B}$$ for all values of $$k$$ and $$n$$. Together with eq. (2.84), this implies that

$\tag{2.85}|y[n]|\le{B}\sum_{k=-\infty}^{+\infty}|h[k]|\qquad\text{for all }n$

From eq. (2.85), we can conclude that if the impulse response is absolutely summable, that is, if

$\tag{2.86}\sum_{k=-\infty}^{+\infty}|h[k]|\lt\infty$

then $$y[n]$$ is bounded in magnitude, and hence, the system is stable.

Therefore, eq. (2.86) is a sufficient condition to guarantee the stability of a discrete-time LTI system.

In fact, this condition is also a necessary condition, since if eq. (2.86) is not satisfied, there are bounded inputs that result in unbounded outputs. Thus, the stability of a discrete-time LTI system is completely equivalent to eq. (2.86).

In continuous time, we obtain an analogous characterization of stability in terms of the impulse response of an LTI system.

Specifically, if $$|x(t)|\lt{B}$$ for all $$t$$, then, in analogy with eqs. (2.83)—(2.85), it follows that

\begin{align}|y(t)|&=\left|\displaystyle\int\limits_{-\infty}^{+\infty}h(\tau)x(t-\tau)\text{d}\tau\right|\\&\le\displaystyle\int\limits_{-\infty}^{+\infty}|h(\tau)||x(t-\tau)|\text{d}\tau\\&\le{B}\displaystyle\int\limits_{-\infty}^{+\infty}|h(\tau)|\text{d}\tau\end{align}

Therefore, the system is stable if the impulse response is absolutely integrable, i.e., if

$\tag{2.87}\displaystyle\int\limits_{-\infty}^{+\infty}|h(\tau)|\text{d}\tau\lt\infty$

As in discrete time, if eq. (2.87) is not satisfied, there are bounded inputs that produce unbounded outputs; therefore, the stability of a continuous-time LTI system is equivalent to eq. (2.87).

The use of eqs (2.86) and (2.87) to test for stability is illustrated in the next two examples.

Example 2.13

Consider a system that is a pure time shift in either continuous time or discrete time. Then, in discrete time

$\tag{2.88}\sum_{n=-\infty}^{+\infty}|h[n]|=\sum_{n=-\infty}^{+\infty}|\delta[n-n_0]|=1$

while in continuous time

$\tag{2.89}\displaystyle\int\limits_{-\infty}^{+\infty}|h(\tau)|\text{d}\tau=\int\limits_{-\infty}^{+\infty}|\delta(\tau-t_0)\text{d}\tau=1$

and we conclude that both of these systems are stable. This should not be surprising, since if a signal is bounded in magnitude, so is any time-shifted version of that signal.

Now consider the accumulator described in Example 2.12. As we discussed in the continuous-time and discrete-time systems tutorial, this is an unstable system, since, if we apply a constant input to an accumulator, the output grows without bound.

That this system is unstable can also be seen from the fact that its impulse response $$u[n]$$ is not absolutely summable:

$\sum_{n=-\infty}^{\infty}|u[n]|=\sum_0^{\infty}u[n]=\infty$

Similarly, consider the integrator, the continuous-time counterpart of the accumulator:

$\tag{2.90}y(t)=\displaystyle\int\limits_{-\infty}^tx(\tau)\text{d}\tau$

This is an unstable system for precisely the same reason as that given for the accumulator; i.e., a constant input gives rise to an output that grows without bound.

The impulse response for the integrator can be found by letting $$x(t)=\delta(t)$$, in which case

$h(t)=\displaystyle\int\limits_{-\infty}^t\delta(\tau)\text{d}\tau=u(t)$

and

$\displaystyle\int\limits_{-\infty}^{+\infty}|u(\tau)|\text{d}\tau=\int\limits_0^{+\infty}\text{d}\tau=\infty$

Since the impulse response is not absolutely integrable, the system is not stable.

## 8. The Unit Step Response of an LTI System

Up to now, we have seen that the representation of an LTI system in terms of its unit impulse response allows us to obtain very explicit characterizations of system properties.

Specifically, since $$h[n]$$ or $$h(t)$$ completely determines the behavior of an LTI system, we have been able to relate system properties such as stability and causality to properties of the impulse response.

There is another signal that is also used quite often in describing the behavior of LTI systems: the unit step response, $$s[n]$$ or $$s(t)$$, corresponding to the output when $$x[n]=u[n]$$ or $$x(t)=u(t)$$.

We will find it useful on occasion to refer to the step response, and therefore, it is worthwhile relating it to the impulse response.

From the convolution-sum representation, the step response of a discrete-time LTI system is the convolution of the unit step with the impulse response; that is,

$s[n]=u[n]*h[n]$

However, by the commutative property of convolution, $$s[n]=h[n]*u[n]$$, and therefore, $$s[n]$$ can be viewed as the response to the input $$h[n]$$ of a discrete-time LTI system with unit impulse response $$u[n]$$.

As we have seen in Example 2.12, $$u[n]$$ is the unit impulse response of the accumulator. Therefore,

$\tag{2.91}s[n]=\sum_{k=-\infty}^nh[k]$

From this equation and from Example 2.12, it is clear that $$h[n]$$ can be recovered from $$s[n]$$ using the relation

$\tag{2.92}h[n]=s[n]-s[n-1]$

That is, the step response of a discrete-time LTI system is the running sum of its impulse response [eq. (2.91)]. Conversely, the impulse response of a discrete-time LTI system is the first difference of its step response [eq. (2.92)].

Similarly, in continuous time, the step response of an LTI system with impulse response $$h(t)$$ is given by $$s(t)=u(t)*h(t)$$, which also equals the response of an integrator [with impulse response $$u(t)$$] to the input $$h(t)$$.

That is, the unit step response of a continuous-time LTI system is the running integral of its impulse response, or

$\tag{2.93}s(t)=\displaystyle\int\limits_{-\infty}^th(\tau)\text{d}\tau$

and from eq. (2.93), the unit impulse response is the first derivative of the unit step response, or

$\tag{2.94}h(t)=\frac{\text{d}s(t)}{\text{d}t}=s'(t)$

Therefore, in both continuous and discrete time, the unit step response can also be used to characterize an LTI system, since we can calculate the unit impulse response from it.

The next tutorial introduces causal LTI systems described by differential and difference equations