Chapter 3. Time Domain Analysis of Continuous Time Systems
3.1. Continuous Time Systems*
Introduction
As you already now know, a continuous time system operates on a continuous time signal input and produces a continuous time signal output. There are numerous examples of useful continuous time systems in signal processing as they essentially describe the world around us. The class of continuous time systems that are both linear and time invariant, known as continuous time LTI systems, is of particular interest as the properties of linearity and time invariance together allow the use of some of the most important and powerful tools in signal processing.
Continuous Time Systems
Linearity and Time Invariance
A system H is said to be linear if it satisfies two important conditions. The first, additivity, states for every pair of signals x,y that H(x + y) = H(x) + H(y). The second, homogeneity of degree one, states for every signal x and scalar a we have H(a x) = a H(x). It is clear that these conditions can be combined together into a single condition for linearity. Thus, a system is said to be linear if for every signals x,y and scalars a,b we have that
Linearity is a particularly important property of systems as it allows us to leverage the powerful tools of linear algebra, such as bases, eigenvectors, and eigenvalues, in their study.
A system H is said to be time invariant if a time shift of an input produces the corresponding shifted output. In other, more precise words, the system H commutes with the time shift operator S T for every T ∈ R. That is,
Time invariance is desirable because it eases computation while mirroring our intuition that, all else equal, physical systems should react the same to identical inputs at different times.
When a system exhibits both of these important properties it allows for a more straigtforward analysis than would otherwise be possible. As will be explained and proven in subsequent modules, computation of the system output for a given input becomes a simple matter of convolving the input with the system's impulse response signal. Also proven later, the fact that complex exponential are eigenvectors of linear time invariant systems will enable the use of frequency domain tools such as the various Fouier transforms and associated transfer functions, to describe the behavior of linear time invariant systems.
Example 3.1.
Consider the system H in which
for all signals f . Given any two signals f,g and scalars a,b
for all real t . Thus, H is a linear system. For all real T and signals f ,
for all real t . Thus, H is a time invariant system. Therefore, H is a linear time invariant system.
Differential Equation Representation
It is often useful to to describe systems using equations involving the rate of change in some quantity. For continuous time systems, such equations are called differential equations. One important class of differential equations is the set of linear constant coefficient ordinary differential equations, which are described in more detail in subsequent modules.
Example 3.2.
Consider the series RLC circuit shown in Figure 3.1. This system can be modeled using differential equations. We can use the voltage equations for each circuit element and Kirchoff's voltage law to write a second order linear constant coefficient differential equation describing the charge on the capacitor.
The voltage across the battery is simply
V
. The voltage across the capacitor is 
. The voltage across the resistor is 
. Finally, the voltage across the inductor is 
. Therefore, by Kirchoff's voltage law, it follows that
Figure 3.1.
Continuous Time Systems Summary
Many useful continuous time systems will be encountered in a study of signals and systems. This course is most interested in those that demonstrate both the linearity property and the time invariance property, which together enable the use of some of the most powerful tools of signal processing. It is often useful to describe them in terms of rates of change through linear constant coefficient ordinary differential equations.
3.2. Continuous Time Impulse Response*
Introduction
| The output of an LTI system is completely determined by the input and the system's response to a unit impulse.
Figure 3.2. System Output  We can determine the system's output, y(t), if we know the system's impulse response, h(t), and the input, f(t). |
| The output for a unit impulse input is called the impulse response.
Figure 3.3.  |
Example Approximate Impulses
Hammer blow to a structure
Hand clap or gun blast in a room
Air gun blast underwater
LTI Systems and Impulse Responses
Finding System Outputs
By the sifting property of impulses, any signal can be decomposed in terms of an integral of shifted, scaled impulses.
δ(t − τ) peaks up where t = τ .
Figure 3.4.
Since we know the response of the system to an impulse and any signal can be decomposed into impulses, all we need to do to find the response of the system to any signal is to decompose the signal into impulses, calculate the system's output for every impulse and add the outputs back together. This is the process known as Convolution. Since we are in Continuous Time, this is the Continuous Time Convolution Integral.
Finding Impulse Responses
Theory:
| |
Practice:
| |
We will assume that h(t) is given for now.
|
Impulse Response Summary
When a system is "shocked" by a delta function, it produces an output known as its impulse response. For an LTI system, the impulse response completely determines the output of the system given any arbitrary input. The output can be found using continuous time convolution.
3.3. Continuous Time Convolution*
Introduction
Convolution, one of the most important concepts in electrical engineering, can be used to determine the output a system produces for a given input signal. It can be shown that a linear time invariant system is completely characterized by its impulse response. The sifting property of the continuous time impulse function tells us that the input signal to a system can be represented as an integral of scaled and shifted impulses and, therefore, as the limit of a sum of scaled and shifted approximate unit impulses. Thus, by linearity, it would seem reasonable to compute of the output signal as the limit of a sum of scaled and shifted unit impulse responses and, therefore, as the integral of a scaled and shifted impulse response. That is exactly what the operation of convolution accomplishes. Hence, convolution can be used to determine a linear time invariant system's output from knowledge of the input and the impulse response.
Convolution and Circular Convolution
Convolution
Operation Definition
Continuous time convolution is an operation on two continuous time signals defined by the integral
for all signals f,g defined on R. It is important to note that the operation of convolution is commutative, meaning that
for all signals f,g defined on R. Thus, the convolution operation could have been just as easily stated using the equivalent definition
for all signals f,g defined on R. Convolution has several other important properties not listed here but explained and derived in a later module.
Definition Motivation
The above operation definition has been chosen to be particularly useful in the study of linear time invariant systems. In order to see this, consider a linear time invariant system H with unit impulse response h . Given a system input signal x we would like to compute the system output signal H(x). First, we note that the input can be expressed as the convolution
by the sifting property of the unit impulse function. Writing this integral as the limit of a summation,
where
approximates the properties of δ(t). By linearity
which evaluated as an integral gives
Since H δ(t – τ) is the shifted unit impulse response h(t – τ), this gives the result
Hence, convolution has been defined such that the output of a linear time invariant system is given by the convolution of the system input with the system unit impulse response.
Graphical Intuition
It is often helpful to be able to visualize the computation of a convolution in terms of graphical processes. Consider the convolution of two functions f,g given by
The first step in graphically understanding the operation of convolution is to plot each of the functions. Next, one of the functions must be selected, and its plot reflected across the τ = 0 axis. For each real t , that same function must be shifted left by t . The product of the two resulting plots is then constructed. Finally, the area under the resulting curve is computed.
Example 3.3.
Recall that the impulse response for the capacitor voltage in a series RC circuit is given by
and consider the response to the input voltage
We know that the output for this input voltage is given by the convolution of the impulse response with the input signal
We would like to compute this operation by beginning in a way that minimizes the algebraic complexity of the expression. Thus, since x(t) = u(t) is the simpler of the two signals, it is desirable to select it for time reversal and shifting. Thus, we would like to compute
The step functions can be used to further simplify this integral by narrowing the region of integration to the nonzero region of the integrand. Therefore,
Hence, the output is
which can also be written as
Circular Convolution
Continuous time circular convolution is an operation on two finite length or periodic continuous time signals defined by the integral
for all signals
f,g
defined on R[0,T] where 
are periodic extensions of
f
and
g
. It is important to note that the operation of circular convolution is commutative, meaning that
for all signals f,g defined on R[0,T]. Thus, the circular convolution operation could have been just as easily stated using the equivalent definition
for all signals
f,g
defined on R[0,T] where 
are periodic extensions of
f
and
g
. Circular convolution has several other important properties not listed here but explained and derived in a later module.
Alternatively, continuous time circular convolution can be expressed as the sum of two integrals given by
for all signals f,g defined on R[0,T].
Meaningful examples of computing continuous time circular convolutions in the time domain would involve complicated algebraic manipulations dealing with the wrap around behavior, which would ultimately be more confusing than helpful. Thus, none will be provided in this section. However, continuous time circular convolutions are more easily computed using frequency domain tools as will be shown in the continuous time Fourier series section.
Definition Motivation
The above operation definition has been chosen to be particularly useful in the study of linear time invariant systems. In order to see this, consider a linear time invariant system H with unit impulse response h . Given a finite or periodic system input signal x we would like to compute the system output signal H(x). First, we note that the input can be expressed as the circular convolution
by the sifting property of the unit impulse function. Writing this integral as the limit of a summation,
where
approximates the properties of δ(t). By linearity
which evaluated as an integral gives
Since H δ(t – τ) is the shifted unit impulse response h(t – τ), this gives the result
Hence, circular convolution has been defined such that the output of a linear time invariant system is given by the convolution of the system input with the system unit impulse response.
Graphical Intuition
It is often helpful to be able to visualize the computation of a circular convolution in terms of graphical processes. Consider the circular convolution of two finite length functions f,g given by
The first step in graphically understanding the operation of convolution is to plot each of the periodic extensions of the functions. Next, one of the functions must be selected, and its plot reflected across the τ = 0 axis. For each t ∈ R[0,T], that same function must be shifted left by t . The product of the two resulting plots is then constructed. Finally, the area under the resulting curve on R[0,T] is computed.
Convolution Demonstration
Figure 3.6.
Convolution Summary
Convolution, one of the most important concepts in electrical engineering, can be used to determine the output signal of a linear time invariant system for a given input signal with knowledge of the system's unit impulse response. The operation of continuous time convolution is defined such that it performs this function for infinite length continuous time signals and systems. The operation of continuous time circular convolution is defined such that it performs this function for finite length and periodic continuous time signals. In each case, the output of the system is the convolution or circular convolution of the input signal with the unit impulse response.
3.4. Properties of Continuous Time Convolution*
Introduction
We have already shown the important role that continuous time convolution plays in signal processing. This section provides discussion and proof of some of the important properties of continuous time convolution. Analogous properties can be shown for continuous time circular convolution with trivial modification of the proofs provided except where explicitly noted otherwise.
Continuous Time Convolution Properties
Associativity
The operation of convolution is associative. That is, for all continuous time signals f 1,f 2,f 3 the following relationship holds.
In order to show this, note that
proving the relationship as desired through the substitution τ 3 = τ 1 + τ 2 .
Commutativity
The operation of convolution is commutative. That is, for all continuous time signals f 1,f 2 the following relationship holds.
In order to show this, note that
proving the relationship as desired through the substitution τ 2 = t – τ 1 .
Distribitivity
The operation of convolution is distributive over the operation of addition. That is, for all continuous time signals f 1,f 2,f 3 the following relationship holds.
In order to show this, note that
proving the relationship as desired.
Multilinearity
The operation of convolution is linear in each of the two function variables. Additivity in each variable results from distributivity of convolution over addition. Homogenity of order one in each varible results from the fact that for all continuous time signals f 1,f 2 and scalars a the following relationship holds.
In order to show this, note that
proving the relationship as desired.
Conjugation
The operation of convolution has the following property for all continuous time signals f 1,f 2 .
In order to show this, note that
proving the relationship as desired.
Time Shift
The operation of convolution has the following property for all continuous time signals f 1,f 2 where S T is the time shift operator.
In order to show this, note that
proving the relationship as desired.
Differentiation
The operation of convolution has the following property for all continuous time signals f 1,f 2 .
In order to show this, note that
proving the relationship as desired.
Impulse Convolution
The operation of convolution has the following property for all continuous time signals f where δ is the Dirac delta funciton.
In order to show this, note that
proving the relationship as desired.
Width
The operation of convolution has the following property for all continuous time signals f 1,f 2 where Duration(f) gives the duration of a signal f .
.
In order to show this informally, note that 
is nonzero for all
t
for which there is a
τ
such that
f
1(τ)f
2(t – τ) is nonzero. When viewing one function as reversed and sliding past the other, it is easy to see that such a
τ
exists for all
t
on an interval of length 
. Note that this is not always true of circular convolution of finite length and periodic signals as there is then a maximum possible duration within a period.
Convolution Properties Summary
As can be seen the operation of continuous time convolution has several important properties that have been listed and proven in this module. With slight modifications to proofs, most of these also extend to continuous time circular convolution as well and the cases in which exceptions occur have been noted above. These identities will be useful to keep in mind as the reader continues to study signals and systems.
3.5. Eigenfunctions of Continuous Time LTI Systems*
Introduction
Prior to reading this module, the reader should already have some experience with linear algebra and should specifically be familiar with the eigenvectors and eigenvalues of linear operators. A linear time invariant system is a linear operator defined on a function space that commutes with every time shift operator on that function space. Thus, we can also consider the eigenvector functions, or eigenfunctions, of a system. It is particularly easy to calculate the output of a system when an eigenfunction is the input as the output is simply the eigenfunction scaled by the associated eigenvalue. As will be shown, continuous time complex exponentials serve as eigenfunctions of linear time invariant systems operating on continuous time signals.
Eigenfunctions of LTI Systems
Consider a linear time invariant system H with impulse response h operating on some space of infinite length continuous time signals. Recall that the output H(x(t)) of the system for a given input x(t) is given by the continuous time convolution of the impulse response with the input
Now consider the input x(t) = e st where s ∈ C. Computing the output for this input,
Thus,
where
is the eigenvalue corresponding to the eigenvector e st .
There are some additional points that should be mentioned. Note that, there still may be additional eigenvalues of a linear time invariant system not described by e st for some s ∈ C. Furthermore, the above discussion has been somewhat formally loose as e st may or may not belong to the space on which the system operates. However, for our purposes, complex exponentials will be accepted as eigenvectors of linear time invariant systems. A similar argument using continuous time circular convolution would also hold for spaces finite length signals.
Eigenfunction of LTI Systems Summary
As has been shown, continuous time complex exponential are eigenfunctions of linear time invariant systems operating on continuous time signals. Thus, it is particularly simple to calculate the output of a linear time invariant system for a complex exponential input as the result is a complex exponential output scaled by the associated eigenvalue. Consequently, representations of continuous time signals in terms of continuous time complex exponentials provide an advantage when studying signals. As will be explained later, this is what is accomplished by the continuous time Fourier transform and continuous time Fourier series, which apply to aperiodic and periodic signals respectively.
3.6. BIBO Stability of Continuous Time Systems*
Introduction
BIBO stability stands for bounded input, bounded output stability. BIBO stablity is the system property that any bounded input yields a bounded output. This is to say that as long as we input a signal with absolute value less than some constant, we are guaranteed to have an output with absolute value less than some other constant.
Continuous Time BIBO Stability
In order to understand this concept, we must first look more closely into exactly what we mean by bounded. A bounded signal is any signal such that there exists a value such that the absolute value of the signal is never greater than some value. Since this value is arbitrary, what we mean is that at no point can the signal tend to infinity, including the end behavior.
Figure 3.7.
Time Domain Conditions
Now that we have identified what it means for a signal to be bounded, we must turn our attention to the condition a system must possess in order to guarantee that if any bounded signal is passed through the system, a bounded signal will arise on the output. It turns out that a continuous time LTI system with impulse response h(t) is BIBO stable if and only if
This is to say that the impulse response is absolutely integrable.
Laplace Domain Conditions
Stability is very easy to infer from the pole-zero plot of a transfer function. The only condition necessary to demonstrate stability is to show that the ⅈω -axis is in the region of convergence. Consequently, for stable causal systems, all poles must be to the left of the imaginary axis.
Figure 3.8.
BIBO Stability Summary
Bounded input bounded output stability, also known as BIBO stability, is an important and generally desirable system characteristic. A system is BIBO stable if every bounded input signal results in a bounded output signal, where boundedness is the property that the absolute value of a signal does not exceed some finite constant. In terms of time domain features, a continuous time system is BIBO stable if and only if its impulse response is absolutely integrable. Equivalently, in terms of Laplace domain features, a continuous time system is BIBO stable if and only if the region of convergence of the transfer function includes the imaginary axis.
3.7. Linear Constant Coefficient Differential Equations*
Introduction: Ordinary Differential Equations
In our study of signals and systems, it will often be useful to describe systems using equations involving the rate of change in some quantity. Such equations are called differential equations. For instance, you may remember from a past physics course that an object experiences simple harmonic motion when it has an acceleration that is proportional to the magnitude of its displacement and opposite in direction. Thus, this system is described as the differential equation shown in Equation 3.58.
Because the differential equation in Equation 3.58 has only one independent variable and only has derivatives with respect to that variable, it is called an ordinary differential equation. There are more complicated differential equations, such as the Schrodinger equation, which involve derivatives with respect to multiple independent variables. These are called partial differential equations, but they are not within the scope of this module.
Given a sufficiently descriptive set of initial conditions or boundary conditions, if there is a solution to the differential equation, that solution is unique and describes the behavior of the system. Of course, the results are only accurate to the degree that the model mirrors reality.
Linear Constant Coefficient Ordinary Differential Equations
An important subclass of ordinary differential equations is the set of linear constant coefficient ordinary differential equations. These equations are of the form
where A is a differential operator of the form given in Equation 3.60.
Note that operators of this type satisfy the linearity conditions, and a 1,...,a n are real constants. Furthermore, Equation Equation 3.59 with these operators has derivatives with respect to only one variable, making it an ordinary differential equation.
A similar concept for a discrete time setting, difference equations, is discussed in the chapter on time domain analysis of discrete time systems. There are many parallels between the discussion of linear constant coefficient ordinary differential equations and linear constant coefficient differece equations.
Applications of Differential Equations
Consider the decay model in which a quantity of an unstable isotope decreases at a rate proportional to the quanity of unstable isotope remaining. Thus, the decay of the isotope is modeled by the first order linear constant coefficient differential equation
where r is some real rate.
Now consider the series RLC circuit shown in Figure 3.9. This system can be modeled using differential equations. We can use the voltage equations for each circuit element and Kirchoff's voltage law to write a second order linear constant coefficient differential equation describing the charge on the capacitor.
The voltage across the battery is simply
V
. The voltage across the capacitor is 
. The voltage across the resistor is 
. Finally, the voltage across the inductor is 
. Therefore, by Kirchoff's voltage law, it follows that
Figure 3.9.
The section Solving Linear Constant Coefficient Differential Equations will describe in depth how solutions to differential equations like those in the examples may be obtained.
Linear Constant Coefficient Oridinary Differential Equations Summary
Differential equations are an important mathematical tool for modeling continuous time systems. An important subclass of these is the class of linear constant coefficient ordinary differential equations. Linear constant coefficient ordinary differential equations are often particularly easy to solve as will be described in the module on solutions to linear constant coefficient ordinary differential equations and are useful in describing a wide range of situations that arise in electrical engineering and in other fields.
3.8. Solving Linear Constant Coefficient Differential Equations*
Introduction
The approach to solving linear constant coefficient ordinary differential equations is to find the general form of all possible solutions to the equation and then apply a number of conditions to find the appropriate solution. The two main types of problems are initial value problems, which involve constraints on the solution and its derivatives at a single point, and boundary value problems, which involve constraints on the solution or its derivatives at several points.
The number of initial conditions needed for an N th order differential equation, which is the order of the highest order derivative, is N , and a unique solution is always guaranteed if these are supplied. Boundary value problems can be slightly more complicated and will not necessarily have a unique solution or even a solution at all for a given set of conditions. Thus, this module will focus exclusively on initial value problems.
Solving Linear Constant Coefficient Ordinary Differential Equations
Consider some linear constant coefficient ordinary differential equation given by A x(t) = f(t), where A is a differential operator of the form
Let
x
h
(t) and
x
p
(t) be two functions such that
A
x
h
(t) = 0 and
A
x
p
(t) = f(t). By the linearity of
A
, note that 
. Thus, the form of the general solution
x
g
(t) to any linear constant coefficient ordinary differential equation is the sum of a homogeneous solution
x
h
(t) to the equation
A
x = 0 and a particular solution
x
p
(t) that is specific to the forcing function
f(t).
We wish to determine the forms of the homogeneous and nonhomogeneous solutions in full generality in order to avoid incorrectly restricting the form of the solution before applying any conditions. Otherwise, a valid set of initial or boundary conditions might appear to have no corresponding solution trajectory. The following discussion shows how to accomplish this for linear constant coefficient ordinary differential equations.
Finding the Homogeneous Solution
In order to find the homogeneous solution to A x(t) = f(t), consider the differential equation A x(t) = 0. We know that the solutions have the form c e λt for some complex constants c,λ . Since A c e λt = 0 for a solution, it follows that
so it also follows that
Therefore, the parameters of the solution exponents are the roots of the above polynomial, called the characteristic polynomial.
For equations of order two or more, there will be several roots. If all of the roots are distinct, then the the general form of the homogeneous solution is simply
If a root has multiplicity that is greater than one, the repeated solutions must be multiplied by each powers of t from 0 to one less than the root multiplicity (in order to ensure linearly independent solutions). For instance, if λ 1 had multiplicity 2 and λ 2 had multiplicity 3, the homogeneous solution would be
Example 3.4.
Consider the decay model in which a quantity of an unstable isotope decreases at a rate proportional to the quanity of unstable isotope remaining. Thus, the decay of the isotope is modeled by the first order linear constant coefficient differential equation
where r is some real rate. This differential equation could easily be solved through straightforward integration. However, the methods described above will be used instead. Note that the forcing function is zero, so only a homogenous solution is needed. It is easy to see that the characteristic polynomial is λ + r = 0, so there is one root λ 1 = r . Thus the solution is of the form
Given a rate and an initial condition, this can be applied to a specific situation. For instance, we know that carbon-14 decays at a rate of approximately r = 1.21x10 – 4 year – 1 , and if we normalize the natural concentration of carbon-14 to x(0) = 1 the solution becomes x(t) = e – 1.21x10 – 4 t . Knowledge of this curve would be useful for radioisotope based dating.
Finding the Particular Solution
Finding the particular solution is slightly more complicated task than finding the homogeneous solution. A formal method, called variation of parameters accomplishes this, and there are also several heuristics that can be used. It can also be found through convolution of the input with the unit impulse response, once the unit impulse response is known. Finding the particular solution to a differential equation is discussed further in the chapter concerning the Laplace transform, which greatly simplifies the procedure for solving linear constant coefficient ordinary differential equations using frequency domain tools.
Example 3.5.
Consider the series RLC circuit shown in Figure 3.10. This system can be modeled using differential equations. We can use the voltage equations for each circuit element and Kirchoff's voltage law to write a second order linear constant coefficient differential equation describing the charge on the capacitor.
The voltage across the battery is simply
V
. The voltage across the capacitor is 
. The voltage across the resistor is 
. Finally, the voltage across the inductor is 
. Therefore, by Kirchoff's voltage law, it follows that
Figure 3.10.
First, the homogeneous solution is found. It is easy to see that the characteristic polynomial is 
. Therefore, the two roots are 
and 
. Often, these are stated in terms of the attenuation factor 
and the resonant frequency 
. Thus, 
and 
.
Thus, the homogeneous equation is of the form
It turns out that the response to the constant voltage source forcing function is a constant, so
Hence, the general solution is
where c 1 and c 2 depend on the initial conditions. The system demonstrates a rich array of behaviors based on the relative values of α and ω 0 , which the reader is encouraged to explore.
Solving Differential Equations Summary
Linear constant coefficient ordinary differential equations are useful for modeling a wide variety of continuous time systems. The approach to solving them is to find the general form of all possible solutions to the equation and then apply a number of conditions to find the appropriate solution. This is done by finding the homogeneous solution to the differential equation that does not depend on the forcing function input and a particular solution to the differential equation that does depend on the forcing function input.