2/5/2016
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The Linear, TimeInvariant System
Most of the microwave devices and networks that we will
study in this course are both linear and time invariant (or
approximately so).
Let’s make sure that we understand what these terms
mean, as linear, time-invariant systems allow us to
apply a large and helpful mathematical toolbox!
LIIN
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Mathematicians often speak of operators,
which is “mathspeak” for any mathematical
operation that can be applied to a single
element (e.g., value, variable, vector, matrix, or
function).
...operators, operators, operators!!
For example, a function f  x  describes an operation on
variable x (i.e., f  x  is operator on x ). E.G.:
f y   y 2  3
Jim Stiles
g t   2t
The Univ. of Kansas
y x   x
Dept. of EECS
2/5/2016
687292118
2/13
Moreover, we find that functions can likewise be operated on!
For example, integration and differentiation are likewise
mathematical operations—operators that operate on
functions. E.G.,:
 f  y  dy
d g t 
dt


y  x  dx

A special and very important class of operators are
linear operators.
Linear operators are denoted as L  y  , where:
* L symbolically denotes the mathematical operation;
* And y denotes the element (e.g., function, variable,
vector) being operated on.
A linear operator is any operator that satisfies the following
two statements for any and all y :
1. L y1  y2   L y1   L  y2 
2. L a y   a L y  , where a is any constant.
Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/5/2016
687292118
3/13
From these two statements we can likewise conclude that a
linear operator has the property:
L a y1  b y2   a L y1   b L y2 
where both a and b are constants.
Essentially, a linear operator has the property that
any weighted sum of solutions is also a solution!
For example, consider the function:
L t   g t   2t
At t  1 :
and at t  2 :
g t  1  2 1  2
g t  2  2 2  4
Now at t  1  2  3 we find:
g 1  2   2  3 
6
24
 g 1   g  2 
Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/5/2016
687292118
4/13
More generally, we find that:
g t1  t2   2 t1  t2 
 2t1  2t2
 g t1   g t2 
and
g a t   2 a t
 a 2t
 a g t 
Thus, we conclude that the function g t   2t is indeed a
linear function!
Now consider this function:
y x   m x  b
Q: But that’s the equation of a line! That
must be a linear function, right?
A: I’m not sure—let’s find out!
We find that:
y a x   m ax   b
 a mx  b
but:
a y  x   a m x  b 
 a mx  ab
Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/5/2016
therefore:
Likewise:
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y a x   a y  x  !!!
y  x1  x 2   m  x 1  x 2   b
 m x1  m x 2  b
but:
y  x1   y  x2   m x1  b   m x2  b 
 m x1  m x2  2b
therefore:
y  x1  x2   y  x1   y  x2  !!!
The equation of a line is not a linear function!
Moreover, you can show that the functions:
f y   y 2  3
y x   x
are likewise non-linear.
Remember, linear operators need not be
functions. Consider the derivative
operator, which operates on functions.
d
dx
d f x 
dx
f x 
Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/5/2016
687292118
6/13
Note that:
d
d f x  d g x 

f  x   g  x  
dx
dx
dx
and also:
d
d f x 
a f  x   a
dx
dx
We thus can conclude that the derivative operation is a linear
operator on function f  x  :
d f x 
 L f  x 
dx
You can likewise show that the integration operation is
likewise a linear operator:
 f  y  dy
 L f  y  
But, you will find that operations such as:
d g 2 t 
dt


y  x  dx

are not linear operators (i.e., they are non-linear operators).
Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/5/2016
687292118
7/13
We find that most mathematical operations are in fact nonlinear! Linear operators are thus form a small subset of all
possible mathematical operations.
Q: Yikes! If linear operators are so rare, we are we wasting
our time learning about them??
A: Two reasons!
Reason 1: In electrical engineering, the behavior of most of
our fundamental circuit elements are described by linear
operators—linear operations are prevalent in circuit analysis!
Reason 2: To our great relief, the two characteristics of
linear operators allow us to perform these mathematical
operations with relative ease!
Q: How is performing a linear operation easier than
performing a non-linear one??
A: The “secret” lies is the result:
L a y1  b y2   a L y1   b L y2 
Note here that the linear operation performed on a relatively
complex element a y1  b y2 can be determined immediately
from the result of operating on the “simple” elements y1 and
y2 .
Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/5/2016
687292118
8/13
To see how this might work, let’s consider some arbitrary
function of time v t  , a function that exists over some finite
amount of time T (i.e., v t   0 for t  0 and t T ).
Say we wish to perform some linear operation on this
function:
L v t   ??
Depending on the difficulty of the
operation L , and/or the complexity of the
function v t  , directly performing this
operation could be very painful (i.e.,
approaching impossible).
Instead, we find that we can often expand a very complex and
stressful function in the following way:
v t   a0  0 t   a1  1 t   a2  2 t  


a
n 
n
 n t 
where the values an are constants (i.e.,
coefficients), and the functions  n t  are known
as basis functions.
For example, we could choose the basis functions:
 n t   t n
for
n 0
Resulting in a polynomial of variable t:
Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/5/2016
687292118
9/13
v t   a0  a1 t  a2 t  a3 t 
2
3

  an t n
n 0
This signal expansion is of course know as the Taylor Series
expansion. However, there are many other useful expansions
(i.e., many other useful basis  n t  ).
* The key thing is that the basis functions  n t  are
independent of the function v t  . That is to say, the
basis functions are selected by the engineer (i.e., you)
doing the analysis.
* The set of selected basis functions form what’s known as
a basis. With this basis we can analyze the function
v t  .
* The result of this analysis provides the coefficients an
of the signal expansion. Thus, the coefficients are
directly dependent on the form of function v t  (as well
as the basis used for the analysis). As a result, the set
of coefficients a1 , a2 , a3,  completely describe the
function v t  !
Q: I don’t see why this “expansion” of function of v t  is
helpful, it just looks like a lot more work to me.
A: Consider what happens when we wish to perform a linear
operation on this function:
Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/5/2016
687292118
10/13

 

L v t   L   an  n t     an L  n t 
n 
 n 
Look what happened! Instead of performing the linear
operation on the arbitrary and difficult function v t  , we can
apply the operation to each of the individual basis functions
 n t  .
Q: And that’s supposed to be easier??
A: It depends on the linear operation and on the basis
functions  n t  . Hopefully, the operation L  n t  is simple
and straightforward. Ideally, the solution to L  n t  is
already known!
Q: Oh yeah, like I’m going to get so lucky.
I’m sure in all my circuit analysis problems
evaluating L  n t  will be long, frustrating,
and painful.
A: Remember, you get to choose the basis over which the
function v t  is analyzed. A smart engineer will choose a
basis for which the operations L  n t  are simple and
straightforward!
Q: But I’m still confused. How do I choose what basis  n t 
to use, and how do I analyze the function v t  to determine
the coefficients an ??
Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/5/2016
687292118
11/13
A: Perhaps an example would help.
Among the most popular basis is this one:
 j  2T n  t
e

n  
0


0  t T
t  0,t T
and:
T
 2 n 
t

T
j 
1
1
an  v t   n t  dt  v t  e  T
T 0
T 0
dt
So therefore:
v t  

a
n 
n
e
 2 n 
j
t
 T 
for 0  t T
The astute among you will recognize this signal
expansion as the Fourier Series!
Q: Yes, just why is Fourier analysis so
prevalent?
A: The answer reveals itself when we apply a linear operator
to the signal expansion:
 2 n 

 
  j  2T n  t 
j 
t 
 T 
L v t   L   an e
   an L e   
n 
 n 


Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/5/2016
687292118
12/13
Note then that we must simply evaluate:
  j  2T n  t 
L e   


for all n.
We will find that performing almost any
linear operation L on basis functions of
this type to be exceeding simple (more on
this later)!
TIIM
ME
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NV
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Q: That’s right! You said that most of the microwave devices
that we will study are (approximately) linear, time-invariant
devices. What does time invariance mean?
A: From the standpoint of a linear operator, it means that
that the operation is independent of time—the result does
not depend on when the operation is applied. I.E., if:
L x t   y t 
then:
L x t     y t   
where  is a delay of any value.
Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/5/2016
687292118
13/13
The devices and networks that you are
about to study in EECS 723 are in fact
fixed and unchanging with respect to time
(or at least approximately so).
As a result, the mathematical operations
that describe most (but not all!) of our
circuit devices are both linear and timeinvariant operators. We therefore refer
to these devices and networks as linear,
time-invariant systems.
Jim Stiles
The Univ. of Kansas
Dept. of EECS