Convolution Integrals with Nspire CAS

ACA 2014

Applications of Computer Algebra

Session: Integration: implementation and applications

Fordham University

New York, NY, USA, July 9-12

2

• Introduction

• Convolution of two functions :

• Case of Laplace transforms

• Continuous LTI systems

• Computing the convolution

• Symbolic Convolution in Nspire CAS

• Conclusion

Overview

3

Why this subject?

At the last ACA conference, we showed how

useful it was for a Computer Algebra System

(CAS) to perform symbolic integration of

Piecewise Continuous Functions (PCF) and

expressions containing such functions.

Sadly, Texas Instruments computer algebra

system Nspire CAS is able to deal with PCF but

presents some limitations.

Introduction

4

Let a PCF be defined over an interval (or the

entire real line) and be continuous over each sub-

interval. Here is what Nspire CAS does well :

• The symbolic derivative, the symbolic

indefinite integral and the symbolic definite

integral.

• The system correctly checks the endpoints

for the derivative and adjusts the constants

of integration for each sub-interval for the

integral.

Introduction

5

Introduction

Constants are added

to obtain a continuous

antiderivative.

Exact value.

Here is an example.

6

Function f(x) and definite integral:

Introduction

7

Important: a continuous antiderivative!

Introduction

( )y f x=

( )f x dx∫

8

We just saw that Nspire CAS uses templates to

define piecewise functions.

These templates are attractive but some

limitations appear if one wants to perform more

complicated symbolic operations.

Introduction

As an example, we continue with our earlier

function y = f(x) and try to compute .

9

Introduction

2 ( )x f x dx∫

Unable to compute!

No exact value!

10

Nspire CAS built-in integrator is unable to

perform the symbolic integration of a product of

a single expression with a piecewise function.

Furthermore, it only returns a floating point

approximation for the definite integral.

Introduction

11

This is why Frédérick Henri has programmed

some functions (showed last year at ACA).

With these functions (included in a special

library used by us at ETS), Nspire CAS is now

able to do all of this correctly!

Introduction

Here is a brief recap of how it works:

1. We group into a single piecewise function

an expression using the function

grouper_fct(ex, var) where ex is

an expression in variable var.

12

Introduction

Example of grouper_fct :

13

Introduction

≤<∞−

≤<

∞<≤−

<<−

0),cos(

30),sin(

2,)cos(

23,)sin(

xx

xx

xxx

xxx

π

π

<<

+ >−

× >

Elsex

xx

Elsex

xx

Else

x

),cos(

20),sin(

,

0,

,0

3,1 π

14

2. We use the built-in integrator of Nspire

CAS to perform the integral.

Note that in the upcoming slide:

• kit_ETS_fh\integral_mcx(ex,

var) stands for the indefinite integral of

ex wrt to var.

• kit_ETS_fh\integral_mcx_d(ex,

var, lo, up) stands for the definite

integral of ex wrt var from lo to up.

Introduction

15

Introduction

Nspire CAS built-in integrator.

Only a floating point approximation.

Symbolic piecewise antiderative!

Exact value!

16

� Case of Laplace transforms

Usually, students are introduced to Laplace

transforms inside an ODE course.

Functions f(t) are defined for and the

Laplace transform of f is the function F defined

by the improper integral

Convolution of Two Functions

0

( ) ( ) stF s f t e dt

∞−= ∫

0t ≥

17


Let’s use the notation for the

correspondence between the function f(t) and its

transform F(s).

Note that f(t) is in fact f(t) u(t) where u(t) is the

unit-step function (Heaviside function):


( ) ( )f t F s↔

0, 0( )

1, 0

tu t

t

<=

>

18


Then we have the “convolution property”

This last integral is called the convolution (in the

sense of Laplace transforms) of f and g.


0

( ) ( ) ( ) ( ) ( ) ( )

t

F s G s f t g t f g t dτ τ τ↔ ∗ ≡ −∫

19


Note that it can be written in the more general

form

if x(t) = f(t)u(t) and h(t) = g(t)u(t).


( ) ( )x h t dτ τ τ∞

−∞

−∫

20


Unfortunately, in a classical ODE course, few

words are said about convolution or the reasons

why it is important.

For simplicity, let’s take a linear second order,

constant coefficients ODE


( ) ( ) ( ) ( ) ( ), (0) 0, (0) 0a y t b y t c y t x t u t y y′′ ′ ′+ + = ⋅ = =

21


Then the solution of

is given by the convolution

where

h(t) is known as the impulse response and H(s)

as the transfer function.


( ) ( ) ( ) ( ) ( ), (0) 0, (0) 0a y t b y t c y t x t u t y y′′ ′ ′+ + = ⋅ = =

( ) ( ) ( )y t x t h t= ∗

2

1( ) ( )h t H s

a s b s c↔ ≡

+ +

22


Usually, the ODE represents a damped mass-

spring problem or a RLC circuit problem.

In this case, the coefficients a, b and c are

positive and there is no loss of generality taking

zero initial conditions since the transient solution

dies out.

So, the convolution solves the ODE.


23

� Continuous LTI systems

We now consider a (continuous) system

where an input x(t) enters the system and an

output y(t) is produced. Some examples:


( )

4( ) ( ), ( ) 5 (2 1) 10

( ) cos( ( )), ( ) ( )

( ) ( )

t

y t x t y t x t

dy t x t y t x t

dt

y t x dτ τ−∞

= = + +

= =

= ∫

x(t) SYSTEM y(t)

24


Another example is a mass-spring system: the

external force f(t) is the input (f(t) = 0 if t < 0)

and the position y(t) of the object at time t is the

output.

The following ODE is the model used:


( ) ( ) ( ) ( ), (0) 0, (0) 0m y t b y t k y t f t y y′′ ′ ′⋅ + ⋅ + ⋅ = = =

Mass of the object

Constant of friction

Spring constant

25


Let the input produce the output and let

the input produce the output .

The system is linear if the linearly combined

input

produces the linear combined output


1( )x t 1( )y t

2 ( )x t 2 ( )y t

1 2( ) ( ) ( )x t a x t b x t= ⋅ + ⋅

1 2( ) ( ) ( )y t a y t b y t= ⋅ + ⋅

26


Let the input x(t) produces the output y(t), let a

be any real number.

The system is time-invariant if the shifted output

y(t − a) is the same as the output produced by the

shifted input x(t − a).

That is:

S(x(t)) = y(t) S(x(t − a)) = y(t − a)


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The system is said linear, time-invariant (LTI) if

both conditions are satisfied.

Example: consider again the capacitor as a

system. When a current i(t) passes through the

capacitor C, the voltage across the capacitor is


1( ) ( )

t

v t i dC

τ τ−∞

= ∫

28


This is a linear system because of the linearity

of the integral. The system is also time-invariant

as a change of variable shows it.


1 1( ) ( ) ( )

t t a

i a d i d v t aC C

τ τ τ τ−

−∞ −∞

− = = −∫ ∫

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� Computing the convolution

In signal analysis courses, students learn how to

compute by hand the convolution of two signals.

They are introduced to important functions such

as unit step function u(t) and unit-impulse (Dirac

delta) “function” δ(t). But there is no need to

deal with the theory of generalized functions.


30


Instead, they are using limiting process. The

Dirac delta “function” is replaced by an

approximate unit-impulse function:

In Derive, this is nothing else than


( )1

( ) ( ) ( )t u t u tδ = − − ∆∆

�

( ) ( )1 1

STEP( ) STEP( ) CHI 0, ,t t t− − ∆ = ∆∆ ∆

31


Given a system, the output to the unit-impulse

δ(t) is called the system impulse response h(t).

Fact: in a continuous LTI system, the output y(t)

to the input x(t) is given by the convolution


( ) ( ) ( )y t x h t dτ τ τ∞

−∞

= −∫

32


This is easy to justify. The signal x(t) is replaced

by a staircase approximation and a limit:

If is the output to , then the linearity of

the integral and time invariance yields the result.


( )h t� ( )tδ�

0

( ) ( ) ( )

( ) lim ( ) ( ) ( )

k

x t x k t k

x t x t x t d

δ

τ δ τ τ

∞

=−∞

∞

∆→−∞

= ∆ − ∆ ∆

= = −

∑

∫

��

�

33


Suppose you want to perform a convolution:

Then, 4 steps must be completed :

• reverse the time in the signal h;

• shift the variable;

• multiply by the signal x;

• integrate over all values of τ, doing this for

every value of t.


( ) ( ) ( )y t x h t dτ τ τ∞

−∞

= −∫

34


For example, the convolution of two rectangular

pulses of finite (but different) duration is a

trapezoidal signal:

Let’s switch to Nspire CAS and show an

animation of this.


Convolution of

x(t) and h(t).

35


Derive can easily perform the last convolution

because of its ability to integrate piecewise

functions.

Even though the Dirac delta function has never

been implemented into Derive, we can compute

symbolic limits involving indicator functions.

So we can use impulse functions!


36


Let us show a Derive screen where the

convolution is defined.

Then we will show the convolution of the two

rectangular pulses x(t) = CHI(0, t, 1) and h(t) =

1.5CHI(0, t, 2).

In Derive, CHI(a, x, b) is the indicator function

of the open interval a < x < b. The notation is

χ(a, x, b).


37


Finally, we will illustrate the fact that the

convolution with the shifted Dirac delta function

δ(t − a) produces a translation on a signal x(t):

The next slide shows this with a = 1.


( ) ( ) ( )x t t a x t aδ∗ − = −

38


It is so easy to define a convolution of 2

signals in Derive!

We take 2 rectangular pulses, using the

indicator function χ of Derive.

Here is the

result and the

graph.

Now we convolve the « output » with δ(t − 1),

using a limit of indicator function: this produces, as

expected, a translation of the signal « output ».

39

We know Nspire CAS can’t simplify the product

of a piecewise expression with another

expression.

So, if x(t) is piecewise with compact support, the

integral of x(t) with another expression only

yields a floating point value. The next slide

illustrates this.

Symbolic Convolution in Nspire CAS

40

Here is, again, an example.

Now let’s try to perform the convolution of x(t)

and h(t).


No exact value.

Exact value!

41

Nspire CAS built-in integrator won’t succeed …


Oups…!

42

… neither will Frédérick’s function!


43

So, what can we do?

Our goal is to find a way for Nspire CAS to

compute the symbolic convolution without

giving up the use of templates.

As far as integration is concerned, endpoints of

each subinterval are irrelevant.


44

Here is what we will do:

1. Convert a piecewise function into a linear

combination of signum functions.

2. Import from Derive a very important rule:


45

3. Perform the symbolic integration of each

term.

4. Convert the result into a piecewise

expression (if applicable).

5. Moreover, if one needs to use a Dirac delta

function, it should be possible to achieve,

using limit of indicator functions.


Note : the rule

yields a continuous antiderivative. Here is why.

Let G(x) be an antiderivative of F(x). Consider

the function “Albert”:

This function is everywhere continuous and

differentiable except at the point x = −b/a.


Albert( ) : SIGN( ) ( )b

x ax b G x Ga

= + − −

46

The derivative of SIGN(ax + b) is 0 except at x =

−b/a. So, for x ≠ −b/a,

For x ≠ −b/a, Albert(x) is continuous

everywhere because SIGN and G are continuous.

If x = −b/a, it is also continuous because SIGN is

a bounded function:


( )Albert( ) 0 ( ) SIGN( ) ( ) 0 SIGN( ) ( ).d b

x G x G ax b F x ax b F xdx a

= ⋅ − − + + − = +

47

lim SIGN( ) ( ) 0b

xa

bax b G x G

a→−

+ − − =

48

New functions were defined and saved in an

updated version of the library Kit_ETS_FH.

Frédérick Henri took care of the programming

side of the job and Michel Beaudin took care of

the mathematical requests.

Here is a description of the principal functions.


49

The sign function is already implemented into Nspire CAS. We have defined the unit step functionand the piecewise indicator function of the interval ]a, b[:

Frédérick’s function unpiece transforms a piecewise function into a linear combination of indicator functions. The inverse is achieved with the function signtopiece.


1 sign( )step( ) :

2

chi( , , ) : step( ) step( )

xx

a x b x a x b

+=

= − − −

Here are some examples:


We add the « step » function.

We add the indicator function and

call it « chi ».

We « unpiece ».

We« convert »

to piecewise.

« grouper_fct »

finishes the job.

50

51

Nspire CAS is now able to do as Derive when

comes time to integrate a product of sign with

another expression. In fact, Frédérick has

programmed the function integral_sign in

order to compute the integral of expressions as

sign(ax + b)·f(x).

For more complicated examples, expansion is

used and the function integral2 does the job.

Let’s take a look at an example.


52

The answer would be different if sign(2x−5) was

factored out first; the 2 answers would differ by

a constant, as for primitives.


Nspire built-in integrator.

Frédérick’s function!

Derive built-in integrator.

It is the same!

53

Now, let be given 2 piecewise continuous

signals, say x(t) and h(t). We “unpiece” the

product x(τ)h(t − τ). This become a linear combination of sign functions and we integrate

it using the new integral2 function. The

antiderivative is then evaluated between ∞ and

−∞. This yields the convolution of x and h. The function convol_gen does the job.


54

If the answer is a linear combination of absolute

values expressions, our function

conv_abs_to_p converts it into a piecewise

function. Let’s give a concrete example, taking

the two earlier rectangular pulses:


0, 00, 0

3( ) 1, 0 1, ( ) , 0 2

20, 1

0, 2

tt

x t t h t t

tt

<<

= < < = < < > >

55

Definition of

both signals.

Convolution of

both signals.

Conversion to a

piecewise function.

Simplification of the

answer and its graph.

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Finally, the convolution of this output

trapezoidal signal with δ(t − 1) should produce a

translation of one unit on the right: that is the last

trapezoidal output should move one unit to the

right.

Again, we will use a limit of indicator function

in order to use a Dirac delta function.


57


The 2 signals.

We call the convolution of

the 2 signals « output ».

We take an approximate

unit-impulse.

The convolution of « output » with the

approximate unit-impulse is a huge expression!

But we know we will take the limit of this.

58


And the limit is exactly what we are expecting.

The graph is the trapezoidal signal moved one

unit to the right.

59

In 2006, at the beginning of Nspire CAS,

Bernhard Kutzler proclaimed it as the “true

successor of Derive”. Albert Rich would have

disagreed…

Michel Beaudin, with the help of Frédérick

Henri, wants to make Bernhard’s affirmation

come true. But there is a lot more to be done, as

far as the CAS part of the system is concerned.

Conclusion

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This represents a fantastic opportunity to do

mathematics and to discover, day after day, how

Derive was special.

Helping Nspire CAS to become more powerful

(in the CAS sense) represents my contribution to

a product that gave me a lot of enthusiasm for

the past 15 years!

Conclusion

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Documents

Convolution Integrals with Nspire CAS