Piecewise Functions and Convolution Integrals

Piecewise Functions and

Convolution IntegralsMichel Beaudin, Frédérick Henri, ÉTS, Montréal, Canada

ACA 2014 Applications of Computer Algebra

Session: Integration: implementation and applicationsFordham University

New York, NY, USA, July 9-12

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• Introduction• Convolution of two functions :

• Case of Laplace transforms• Continuous LTI systems• Computing the convolution

• Symbolic Convolution in Nspire CAS• Conclusion

Overview

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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

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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

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Introduction

Constants are added to obtain a continuousantiderivative.

Exact value.

Here is an example.

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Function f(x) and definite integral:

Introduction

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Important: a continuous antiderivative!

Introduction

( )y f x

( )f x dx

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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 .

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Introduction

2 ( )x f x dx

Unable to compute!

No exact value!

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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

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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

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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.

Elsexxx

Elsexxx

Elsex

),cos(20),sin(

,0,

,03,1

Introduction

0),cos(30),sin(

2,)cos(23,)sin(

xxxx

xxxxxx

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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

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Introduction

Nspire CAS built-in integrator.

Only a floating point approximation.

Symbolic piecewise antiderative!

Exact value!

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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

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Case of Laplace transforms 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, 0t

u tt

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Case of Laplace transforms 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 d

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Case of Laplace transforms 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 d

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Case of Laplace transforms 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

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Case of Laplace transforms Then the solution of

is given by the convolutionwhere

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

21( ) ( )h t H s

a s b s c

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Case of Laplace transforms 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.


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Continuous LTI systemsWe 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 tdy t x t y t x tdt

y t x d

x(t) SYSTEM y(t)

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Continuous LTI systemsAnother 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 objectConstant of friction

Spring constant

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Continuous LTI systemsLet 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 t2 ( )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

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Continuous LTI systemsLet 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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Continuous LTI systemsThe 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

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Continuous LTI systemsThis 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

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Computing the convolutionIn 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” d(t). But there is no need to deal with the theory of generalized functions.


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Computing the convolutionInstead, 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 td

1 1STEP( ) STEP( ) CHI 0, ,t t t

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Computing the convolutionGiven a system, the output to the unit-impulse d(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 d

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Computing the convolutionThis 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 ( )td0

( ) ( ) ( )

( ) lim ( ) ( ) ( )

kx t x k t k

x t x t x t d

d

d

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Computing the convolutionSuppose 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 , doing this for

every value of t.


( ) ( ) ( )y t x h t d

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Computing the convolutionFor 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 ofx(t) and h(t).

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Computing the convolutionDerive 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!


35

Computing the convolutionLet 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 c(a, x, b).


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Computing the convolutionFinally, we will illustrate the fact that the convolution with the shifted Dirac delta function d(t a) produces a translation on a signal x(t):

The next slide shows this with a = 1.


( ) ( ) ( )x t t a x t ad

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It is so easy to define a convolution of 2 signals in Derive!

We take 2 rectangular pulses, using theindicator function c of Derive.

Here is the result and the graph.

Now we convolve the « output » with d(t 1),using a limit of indicator function: this produces, as expected, a translation of the signal « output ».

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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

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Here is, again, an example.

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


No exact value.

Exact value!

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Nspire CAS built-in integrator won’t succeed …


Oups…!

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… neither will Frédérick’s function!


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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.


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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:


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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( ) ( ) bx ax b G x Ga

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

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lim SIGN( ) ( ) 0bxa

bax b G x Ga

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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.


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The sign function is already implemented into Nspire CAS. We have defined the unit step function and 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 andcall it « chi ».

We « unpiece ».

We« convert »to piecewise.

« grouper_fct »finishes the job.

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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.


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The answer would be different if sign(2x5) 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!

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Now, let be given 2 piecewise continuous signals, say x(t) and h(t). We “unpiece” the product x()h(t ). 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.


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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, 03( ) 1, 0 1, ( ) , 0 22

0, 1 0, 2

ttx t t h t t

t t

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Definition ofboth signals.

Convolution ofboth signals.

Conversion to apiecewise function.

Simplification of theanswer and its graph.

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Finally, the convolution of this output trapezoidal signal with d(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.


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The 2 signals.

We call the convolution ofthe 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.

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And the limit is exactly what we are expecting.The graph is the trapezoidal signal moved one unit to the right.

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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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Thank You!

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Piecewise Functions and Convolution Integrals

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