Usage of heaviside command valid/legal for discrete time ?

I have been using heaviside function for continuous time signals and system but now i want to use this command in discrete time signals & systems but i read in a book(Signals and Systems Laboratory with MATLAB,pg 102) that this command cannot be used for discrete time

I have tried heaviside command and it gives expected true result but i am not sure whether its use is legal in programming terms

clc
clear all
close all
n=0:5
x=heaviside(n)
stem(n,x)

Both expected and actual results are same but i am not sure whether its use is legal or not?

Answer 1

[Walter Roberson]

heaviside() is part of the Symbolic Toolbox. It is implemented by two different functions, one of which is used when the input is symbolic and the other when the input is not symbolic. When the input is symbolic, the symbolic engine is called. When the input is not symbolic, it effectively use Y(X > 0) = 1 and then does fixing up to account for sympref HeavisideAtOrigin, which was functionality added in your release.

Heaviside might possibly in some theoretical sense not be a function, especially since the value at 0 is not completely defined, but it is so close to being a function that it makes sense to generalize it to handle numeric input.

Heaviside is intended to represent something that turns on and stays on, which is in a way incompatible with the idea of discrete time because discrete time says that anything in-between the defined points is unknowable. But it is too useful to refuse to define Heaviside at specific numeric values.

[ABTJ]

Please elaborate your phrase"But it is too useful to refuse to define Heaviside at specific numeric values"

[Walter Roberson]

Suppose I want to evaluate Heaviside at time t=2.1 . Your text would say that if that t is drawn from continuous time then Heaviside is defined for it and that Heaviside should be 1 there. So if I do

t = 2.1;
heaviside(t)

then according to your book, I should get 1.0 as output provided that t represents continuous time. But at the same time, your book would suggest that

t = 2.1;
heaviside(t)

should somehow fail if t is drawn from discrete time. But how is heaviside to know whether t = 2.1 is continuous or discrete? Is there a MATLAB,

t = discete_double(2.1)

and

t = continuous_double(2.1)

that Heaviside should detect the two different data types and act differently for?

Indeed, MATLAB itself represents all double precision values according to IEEE 754 Floating Point, which uses finite precision. Each value is IEEE 754 floating point is represented as an integer in the range 2^52 to 2^53-1, divided by 2^53, multiplied by 2 to some power. All double precision values are discrete in MATLAB. Therefore there are theoretical justifications for saying that heaviside() applied to any double precision number should be refused as being discrete time.

What would be left? Well, there are theoretical reasons for saying that the Symbolic Toolbox representation of values as rationals is "continuous", and that therefor it should be valid to apply heaviside(sym(21)/sym(10)) on the grounds that the ratio 21/10 is in some sense continuous where 2.1 is not. But what about heaviside(sym(2.1)) ? Should it be accepted or refused? It turns out that sym(2.1) is converted to sym(21)/sym(10) . And sym('2') is converted to a rational, but sym('2.1') is not converted to a rational; neither is sym(2.1, 'd') . So sym() would have to check whether it was working with symbolic rationals or not and perhaps refuse heaviside applied to symbolic floating point...

In practice is a nuisance to be forced to code

t = 2.1;
double(heaviside(sym(t)))

Heaviside occurs often. It would be a distinct inconvenience to force people to write their uses of heaviside in that way, just for some kind of theoretical purity about whether the same value 2.1 has been drawn from continuous or discrete time: it would be like forcing everyone to play "Simon Says" -- "Simon says calculate Heaviside of 2.1!", "Simon says calculate Heaviside of -3!", "Simon says calculate Heaviside of 0!", "calculate Heaviside of 0!" -- Hah, Hah, got you, 0 is not continuous when I don't say Simon Says!

[ABTJ]

Thank you so very much for detail.

But Please kindly also give one word/one phrase answer for this question so if a beginner level user visits this question he can find answer easily withhout having to understand the in-depth technical discussion you wrote

I mean please write as answer : yes or no or depends on context/condition

to make answer simple and straight forward

[Walter Roberson]

What does "legal" mean in this question?

[ABTJ]

By "legal" i mean that MATLAB language and Compiler allows its use

As example the case you might have seen with C++ programing in relevant compilers forexample Turbo C++

[Walter Roberson]

Pure MATLAB without toolboxes does not permit heaviside() to be used at all, as heaviside is part of the Symbolic Toolbox.

In the Symbolic Toolbox, Mathworks deliberately implemented heaviside for numeric data types. Numeric examples are given as far back as I looked in the documentation (I looked back to R2006a.) The fact that it ran for you is no accident: Mathworks specifically choose to enable it.

What your book is talking about is whether using Heaviside on discrete time can be justified in theoretical terms, which is a different matter than whether Mathworks permits the operation.

The theory question is much like the question of the square root of 2 is a valid operation or not, since it cannot be expressed in any discrete system based upon rational numbers.

You mention MATLAB Compiler. MATLAB Compiler has no support for anything in the Symbolic Toolbox, not even for the routines that have easy numeric translations.