System Design, Modeling, and Simulation using Ptolemy...

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System Design, Modeling, and Simulation using Ptolemy II

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13Expressions

Christopher Brooks, Thomas Huining Feng, Edward A. Lee, Xiaojun Liu,Stephen Neuendorffer, Neil Smyth, Yuhong Xiong

Contents13.1 Simple Arithmetic Expressions . . . . . . . . . . . . . . . . . . . . 449

13.1.1 Constants and Literals . . . . . . . . . . . . . . . . . . . . . 44913.1.2 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45313.1.3 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45413.1.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 456

13.2 Uses of Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . 45613.2.1 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 45713.2.2 Port Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 45713.2.3 String Parameters . . . . . . . . . . . . . . . . . . . . . . . . 45813.2.4 Expression Actor . . . . . . . . . . . . . . . . . . . . . . . . 46013.2.5 State Machines . . . . . . . . . . . . . . . . . . . . . . . . . 460

13.3 Composite Data Types . . . . . . . . . . . . . . . . . . . . . . . . . 46113.3.1 Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46113.3.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46613.3.3 Records . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46813.3.4 Union Types . . . . . . . . . . . . . . . . . . . . . . . . . . 471

13.4 Operations on Tokens . . . . . . . . . . . . . . . . . . . . . . . . . 47213.4.1 Invoking Methods . . . . . . . . . . . . . . . . . . . . . . . 47213.4.2 Accessing Model Elements . . . . . . . . . . . . . . . . . . . 473

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13. EXPRESSIONS

13.4.3 Casting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47513.4.4 Defining Functions . . . . . . . . . . . . . . . . . . . . . . . 47613.4.5 Higher-Order Functions . . . . . . . . . . . . . . . . . . . . 47713.4.6 Using Functions in a Model . . . . . . . . . . . . . . . . . . 47913.4.7 Recursive Functions . . . . . . . . . . . . . . . . . . . . . . 48113.4.8 Built-In Functions . . . . . . . . . . . . . . . . . . . . . . . 481

13.5 Nil Tokens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48713.6 Fixed Point Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 48713.7 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48913.8 Tables of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 493

In Ptolemy II, models specify computations by composing actors. Many computations,however, are awkward to specify this way. A common situation is where we wish toevaluate a simple algebraic expression, such as sin(2π(x − 1)). It is possible to expressthis computation by composing actors in a block diagram, but it is far more convenient togive it textually.

The Ptolemy II expression language provides infrastructure for specifying algebraic ex-pressions textually and for evaluating them. The expression language is used to specifythe values of parameters, guards and actions in state machines, and for the calculationperformed by the Expression actor. In fact, the expression language is part of the genericinfrastructure in Ptolemy II, and it can be used by programmers extending the Ptolemy IIsystem. In this chapter, we describe how to use expressions from the perspective of a userrather than a programmer.

Vergil provides an interactive expression evaluator, which is accessed through the menucommand [File→New→Expression Evaluator]. This operates like an interactivecommand shell, and is shown in Figure 13.1. It supports a command history. To ac-cess the previously entered expression, type the up arrow or Control-P. To go back, typethe down arrow or Control-N. The expression evaluator is useful for experimenting withexpressions.

13.1 Simple Arithmetic Expressions

13.1.1 Constants and Literals

The simplest expression is a constant, which can be given either by the symbolic name ofthe constant, or by a literal. By default, the symbolic names of constants supported are:

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13.1. SIMPLE ARITHMETIC EXPRESSIONS

PI, pi, E, e, true, false, i, j, NaN, Infinity, PositiveInfinity,NegativeInfinity, MaxUnsignedByte, MinUnsignedByte, MaxShort,MinShort, MaxInt, MinInt, MaxLong, MinLong, MaxFloat, MinFloat,MaxDouble, and MinDouble. For example,

PI/2.0

is a valid expression that refers to the symbolic name “PI” and the literal “2.0.” Theconstants i and j are the imaginary number with value equal to

√−1. The constant NaN

is “not a number,” which for example is the result of dividing 0.0/0.0. The constantInfinity is the result of dividing 1.0/0.0. The constants that start with “Max” and“Min” are the maximum and minimum values for their corresponding types.

Numerical values without decimal points, such as “10” or “-3” are integers (type int).Numerical values with decimal points, such as “10.0” or “3.14159” are of type double.Numerical values followed by “f” or “F” are of type float. Numerical values withoutdecimal points followed by the character “l” (el) or “L” are of type long. long. Numericalvalues without decimal points followed by the character “s” or “S” are of type short.Unsigned integers followed by “ub” or “UB” are of type unsignedByte, as in “5ub”. AnunsignedByte has a value between 0 and 255; note that it is not quite the same as theJava byte, which has a value between -128 and 127. Numbers of type int, long, short orunsignedByte can be specified in decimal, octal, or hexadecimal. Numbers beginning witha leading “0” are octal numbers. Numbers beginning with a leading “0x” are hexadecimalnumbers. For example, “012” and “0xA” are both equal to the integer 10.

Figure 13.1: The Expression Evaluator.

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13. EXPRESSIONS

A complex is defined by appending an “i” or a “j” to a double for the imaginary part.This gives a purely imaginary complex number which can then leverage the polymorphicoperations in the Token classes to create a general complex number. Thus 2 + 3i willresult in the expected complex number. You can optionally write this 2 + 3*i.

Literal string constants are also supported. Anything between double quotation marks,"...", is interpreted as a string constant. The following built-in string-valued constantsare defined:

variablename

meaning JVM property name example value

PTII The directoryin whichPtolemy II isinstalled

ptolemy.ptII.dir c:\tmp

HOME The user homedirectory

user.home c:\Documents and Settings\you

CWD The currentworkingdirectory

user.dir c:\ptII

TMPDIR The temporarydirectory

java.io.tmpdir c:\Documents and Settings\you\Local Settings\Temp\

USERNAME The useraccount name

user.name ptolemy

The value of these variables is the value given by the corresponding Java virtual ma-chine (JVM) property, such as user.home for HOME. The properties user.dir anduser.home are standard in Java. Their values are platform dependent; see the documen-tation for the method getProperties in the java.lang.System class for details.∗ Vergilwill display all the Java properties if you invoke [View→JVM Properties] in the menuof a Graph Editor.

The ptolemy.ptII.dir property is set automatically when Vergil or any other PtolemyII executable is started up. You can also set it when you start a Ptolemy II process usingthe java command by a syntax like the following:

java -Dptolemy.ptII.dir=${PTII} classname

where classname is the full class name of a Java application. You can similarly set theother variables in the table. For example, to invoke Vergil in a particular directory, use∗Note that user.dir and user.home are usually not readable in unsigned applets, in which case,

attempts to use these variables in an expression will result in an exception.


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variable name value variable name valueCLASSPATH ”xxxxxxCLASSPATHxxxxxx” CWD ”/Users/eal”E 2.718281828459 HOME ”/Users/eal”Infinity Infinity MaxDouble 1.797693134862316E308MaxFloat 3.402823466385289E38 MaxInt 2147483647MaxLong 9223372036854775807L MaxShort 32767sMaxUnsignedByte 255ub MinDouble 4.9E-324MinFloat 1.401298464324817E-45 MinInt -2147483648MinLong -9223372036854775808L MinShort -32768sMinUnsignedByte 0ub NaN NaNNegativeInfinity -Infinity PI 3.1415926535898PTII ”/ptII” PositiveInfinity InfinityTMPDIR ”/tmp” USERNAME ”eal”backgroundColor {0.9, 0.9, 0.9, 1.0} boolean falsecomplex 0.0 + 0.0i double 0.0e 2.718281828459 false falsefixedpoint fix(0, 2, 2) float 0.0fgeneral present i 0.0 + 1.0iint 0 j 0.0 + 1.0ilong 0L matrix []nil nil null object(null)object object(null) pi 3.1415926535898scalar present short 0sstring ”” true trueunknown present unsignedByte 0ubxmltoken null

Table 13.1: An example of the values returned by the constants function

java -cp /ptII -Duser.dir=/Users/eal \ptolemy.vergil.VergilApplication

The -cp option specifies the classpath (which has to include the root directory of PtolemyII), the -D option specifies the property to set, and the final argument is the class thatincludes a main method that invokes Vergil.

The constants utility function returns a record with all the globally defined constants.If you open the expression evaluator and invoke this function, you will see that its valueis similar to what is shown in Figure 13.1.


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13.1.2 Variables

Expressions can contain identifiers that are references to variables within the scope of theexpression. For example,

PI*x/2.0

is valid if “x” a variable in scope. In the expression evaluator, the variables that are inscope include the built-in constants plus any assignments that have been previously made.For example,

>> x = pi/21.5707963267949>> sin(x)1.0

In the context of Ptolemy II models, the variables in scope include all parameters definedat the same level of the hierarchy or higher. So for example, if an actor has a param-eter named “x” with value 1.0, then another parameter of the same actor can have anexpression with value “PI*x/2.0”, which will evaluate to π/2.

Consider a parameter P in actor X which is in turn contained by composite actor Y. Thescope of an expression for P includes all the parameters contained by X and Y, plus thoseof the container of Y, its container, etc. That is, the scope includes any parameters definedabove in the hierarchy.

You can add parameters to actors (composite or not) by right clicking on the actor, se-lecting [Customize→Configure] and then clicking on “Add,” or by dragging in a pa-rameter from the Utilities library. Thus, you can add variables to any scope, a ca-pability that serves the same role as the “let” construct in many functional programminglanguages.

Occasionally, it is desirable to access parameters that are not in scope. The expressionlanguage supports a limited syntax that permits access to certain variables out of scope.In particular, if in place of a variable name x in an expression you write A::x, then insteadof looking for x in scope, the interpreter looks for a container named A in the scope and aparameter named x in A. This allows reaching down one level in the hierarchy from eitherthe current container or any of its containers.


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13.1.3 Operators

The arithmetic operators are +, -, *, /, ˆ, and %. Most of these operators operate onmost data types, including arrays, records, and matrices. The ˆ operator computes “tothe power of” or exponentiation, where the exponent can only be a type that losslesslyconverts to an integer such as an int, short, or an unsignedByte.

The unsignedByte, short, int and long types can only represent integer numbers. Opera-tions on these types are integer operations, which can sometimes lead to unexpected re-sults. For instance, 1/2 yields 0, since 1 and 2 are integers, whereas 1.0/2.0 yields 0.5.The exponentiation operator “ˆ” when used with negative exponents can similarly yieldunexpected results. For example, 2ˆ-1 is 0 because the result is computed as 1/(2ˆ1).

The % operation is a modulo or remainder operation. The result is the remainder afterdivision. The sign of the result is the same as that of the dividend (the left argument). Forexample,

>> 3.0 % 2.01.0>> -3.0 % 2.0-1.0>> -3.0 % -2.0-1.0>> 3.0 % -2.01.0

The magnitude of the result is always less than the magnitude of the divisor (the rightargument). Note that when this operator is used on doubles, the result is not the same asthat produced by the remainder function (see Table 13.6). For instance,

>> remainder(-3.0, 2.0)1.0

The remainder function calculates the IEEE 754 standard remainder operation. It usesa rounding division rather than a truncating division, and hence the sign can be positiveor negative, depending on complicated rules (see Section 13.4.8). For example, counter-intuitively,


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13. EXPRESSIONS

>> remainder(3.0, 2.0)-1.0

When an operator involves two distinct types, the expression language has to make adecision about which type to use to implement the operation. If one of the two types canbe converted without loss into the other, then it will be. For instance, int can be convertedlosslessly to double, so 1.0/2 will result in 2 being first converted to 2.0, so the result willbe 0.5. Among the scalar types, unsignedByte can be converted to anything else, short canbe converted to int, int can be converted to double, float can be converted to double anddouble can be converted to complex. Note that long cannot be converted to double withoutloss, nor vice versa, so an expression like 2.0/2L yields the following error message:

Error evaluating expression "2.0/2L"in .Expression.evaluatorBecause:divide method not supported between ptolemy.data.DoubleToken’2.0’ and ptolemy.data.LongToken ’2L’ because the types areincomparable.

Just as long cannot be cast to double, int cannot be cast to float and vice versa.

All scalar types have limited precision and magnitude. As a result of this, arithmeticoperations are subject to underflow and overflow.

• For double numbers, overflow results in the corresponding positive or negative in-finity. Underflow (i.e. the precision does not suffice to represent the result) willyield zero.

• For int and fixedpoint types, overflow results in wraparound. For instance, the valueof MaxInt is 2147483647, but the expression MaxInt + 1 yields -2147483648.Similarly, while MaxUnsignedByte has value 255ub, MaxUnsignedByte + 1ub

has value 0ub. Note, however, that MaxUnsignedByte + 1 yields 256, which isan int, not an unsignedByte. This is because MaxUnsignedByte can be losslesslyconverted to an int, so the addition is int addition, not unsignedByte addition.

The bitwise operators are &, |, # and ˜. They operate on boolean, unsignedByte, short,int and long (but not fixedpoint, float, double or complex). The operator & is bitwiseAND, ˜ is bitwise NOT, and | is bitwise OR, and # is bitwise XOR ( exclusive or, afterMATLAB).


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13.2. USES OF EXPRESSIONS

The relational operators are <, <=, >, >=, == and !=. They return type boolean. Note thatthese relational operators check the values when possible, irrespective of type. So, forexample,

1 == 1.0

returns true. If you wish to check for equality of both type and value, use the equals

method, as in

>> 1.equals(1.0)false

Boolean-valued expressions can be used to give conditional values. The syntax for this is

boolean ? value1 : value2

If the boolean is true, the value of the expression is value1; otherwise, it is value2. Thelogical boolean operators are &&, ||, !, & and |. They operate on type boolean and returntype boolean. The difference between logical && and logical & is that & evaluates all theoperands regardless of whether their value is now irrelevant. Similarly for logical || and|. This approach is borrowed from Java. Thus, for example, the expression false && x

will evaluate to false irrespective of whether x is defined. On the other hand, false & x

will throw an exception if x is undefined.

The << and >> operators performs arithmetic left and right shifts respectively. The >>>operator performs a logical right shift, which does not preserve the sign. They operate onunsignedByte, short, int, and long.

13.1.4 Comments

In expressions, anything inside /*...*/ is ignored, so you can insert comments.

13.2 Uses of Expressions

Expressions are used in Ptolemy II to assign values to parameters, to specify the input-output function realized by an Expression actor, and to specify guards and actions in statemachines.


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13.2.1 Parameters

The values of most parameters of actors can be given as expressions†. The variablesin the expression refer to other parameters that are in scope, which are those containedby the same container or some container above in the hierarchy. They can also refer-ence variables in a scope-extending attribute, which includes variables defining units,as explained below in section 13.7. Adding parameters to actors is straightforward, asexplained in chapter 2.

13.2.2 Port Parameters

It is possible to define a parameter that is also a port. Such a PortParameter providesa default value, which is specified like the value of any other parameter. When the cor-responding port receives data, however, the default value is overridden with the valueprovided at the port. Thus, this object functions like a parameter and a port. The currentvalue of the PortParameter is accessed like that of any other parameter. Its current valuewill be either the default or the value most recently received on the port.

A PortParameter might be contained by an atomic actor or a composite actor. To put onein a composite actor, drag it into a model from the Utilities library, as shown in Figure13.2.

To be useful, a PortParameter has to be given a name (the default name, “portParam-eter,” is not very compelling). To change the name, right click on the icon and select[Customize→Rename], as shown in Figure 13.2. In the figure, the name is set to “noise-Level.” Then set the default value by double clicking. In the figure, the default value isset to 10.0.

An example of a library actor that uses a PortParameter is the Sinewave actor, which isfound in the Sources→SequenceSources library in Vergil. It is shown in Figure 13.3.If you double click on this actor, you can set the default values for frequency and phase.But both of these values can also be set by the corresponding ports, which are shown withgrey fill.

† The exceptions are parameters that are strictly string parameters, in which case the value of the parameteris the literal string, not the string interpreted as an expression, as for example the function parameter of theTrigFunction actor, which can take on only “sin,” “cos,” “tan”, “asin”, “acos”, and “atan” as values.


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13.2.3 String Parameters

Some parameters have values that are always strings of characters. Such parameters sup-port a simple string substitution mechanism where the value of the string can referenceother parameters in scope by name using the syntax $name or ${name} where name is thename of the parameter in scope. For example, the StringCompare actor in Figure 13.4 hasas the value of firstString “The answer is $PI”. This references the built-in constant PI.The value of secondString is “The answer is 3.1415926535898”. As shown in the figure,these two strings are deemed to be equal because $PI is replaced with the value of PI.

Customize the name:

Figure 13.2: A PortParameter is both a port and a parameter. To use it in acomposite actor, drag it into the actor, change its name to something meaningfuland set its default value.


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13. EXPRESSIONS

Figure 13.3: Sinewave actor, showing its port parameters, and their use at thelower level of hierarchy.


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Figure 13.4: String parameters are indicated in the parameter editor boxes by alight blue background. A string parameter can include references to variables inscope with $name, where name is the name of the variable. In this example, thebuilt-in constant $PI is referenced by name in the first parameter.

13.2.4 Expression Actor

The Expression actor is a particularly useful actor found in the Math. By default, it hasone output and no inputs, as shown in Figure 13.5(a). The first step in using it is to addports, as shown in (b) and (c). Click on Add to add a port, and then type in a unique namefor the port. You then specify an expression using the port names as variables, as shownin (d), resulting in the icon shown in (e).

13.2.5 State Machines

Expressions give the guards for state transitions, as well as the values used in actions thatproduce outputs and actions that set values of parameters in the refinements of destinationstates. This mechanism was explained in the previous chapter.


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13. EXPRESSIONS

(a)

(b)

(c)

(e)

(d)

Figure 13.5: Illustration of the Expression actor.

13.3 Composite Data Types

A composite data type is a data type that aggregates some collection of other data types.The composite data types in the expression language include arrays, matrices, records,and union types.

13.3.1 Arrays

Arrays are specified with curly brackets, e.g., {1, 2, 3} is an array of int, while {"x","y", "z"} is an array of type string. The types are denoted arrayType(int,3) andarrayType(string,3) respectively. An array is an ordered list of tokens of any type,with the primary constraint being that the elements all have the same type. If an arrayis given with mixed types, the expression evaluator will attempt to losslessly convert theelements to a common type. Thus, for example,

{1, 2.3}


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has value

{1.0, 2.3}

Its type is arrayType(double,2). The common type might be arrayType(scalar),which is a union type (a type that can contain multiple distinct types). For example,

{1, 2.3, true}

has value

{1, 2.3, true}

The value is unchanged, but the type of the array is now arrayType(scalar,3).

In Figure 13.5(c), the “Type” column may be used to specify the type of a port. Usually,it is not necessary to set the type, since the type inference mechanism will determine thetype from the connections (see Chapter 14). Occasionally, however, it is necessary orhelpful to force a port to have a particular type.

The Type column accepts expressions like arrayType(int), which specifies an arraywith an unknown length. It is better, however, to specify an array length, if it is known.To do that, use an expression like arrayType(int,n), where n is a positive integer thatis the length of the array that is expected on the port.

The elements of the array can be given by expressions, as in the example {2*pi, 3*pi}.Arrays can be nested; for example, {{1, 2}, {3, 4, 5}} is an array of arrays of inte-gers. The elements of an array can be accessed as follows:

>> {1.0, 2.3}(1)2.3

Note that indexing begins at 0. Of course, if name is the name of a variable in scopewhose value is an array, then its elements may be accessed similarly, as shown in thisexample:

>> x = {1.0, 2.3}


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{1.0, 2.3}>> x(0)1.0

Arithmetic operations on arrays are carried out element-by-element, as shown by the fol-lowing examples:

>> {1, 2}*{2, 2}{2, 4}>> {1, 2}+{2, 2}{3, 4}>> {1, 2}-{2, 2}{-1, 0}>> {1, 2}ˆ2{1, 4}>> {1, 2}%{2, 2}{1, 0}

Addition, subtraction, multiplication, division, and modulo of arrays by scalars is alsosupported, as in the following examples:

>> {1.0, 2.0} / 2.0{0.5, 1.0}>> 1.0 / {2.0, 4.0}{0.5, 0.25}>> 3 * {2, 3}{6, 9}>> 12 / {3, 4}{4, 3}

Arrays of length 1 are equivalent to scalars, as illustrated below:

>> {1.0, 2.0} / {2.0}{0.5, 1.0}>> {1.0} / {2.0, 4.0}{0.5, 0.25}>> {3} * {2, 3}


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{6, 9}>> {12} / {3, 4}{4, 3}

A significant subtlety arises when using nested arrays. Note the following example:

>> {{1.0, 2.0}, {3.0, 1.0}} / {0.5, 2.0}{{2.0, 4.0}, {1.5, 0.5}}

In this example, the left argument of the divide is an array with two elements, and the rightargument is also an array with two elements. The divide is thus element wise. However,each division is the division of an array by a scalar.

An array can be checked for equality with another array as follows:

>> {1, 2}=={2, 2}false>> {1, 2}!={2, 2}true

For other comparisons of arrays, use the compare function (see Table 13.5). As withscalars, testing for equality using the == or != operators tests the values, independent oftype. For example,

>> {1, 2}=={1.0, 2.0}true

You can obtain the length of an array as follows,

>> {1, 2, 3}.length()3

You can extract a subarray by invoking the subarray method as follows:

>> {1, 2, 3, 4}.subarray(2, 2){3, 4}


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The first argument is the starting index of the subarray, and the second argument is thelength.

You can also extract non-contiguous elements from an array using the extract method.This method has two forms. The first form takes a boolean array of the same length asthe original array which indicates which elements to extract, as in the following example:

>> {"red", "green", "blue"}.extract({true,false,true}){"red", "blue"}

The second form takes an array of integers giving the indices to extract, as in the followingexample:

>> {"red","green","blue"}.extract({2,0,1,1}){"blue", "red", "green", "green"}

You can create an empty array with a specific element type using the function emptyArray.For example, to create an empty array of integers, use:

>> emptyArray(int){}

You can combine arrays into a single array using the concatenate function. For exam-ple,

>> concatenate({1, 2}, {3}){1, 2, 3}

You can update an element of an array using the update function, for example,

>> {1, 2, 3}.update(0, 4){4, 2, 3}

The update function creates a new array‡

‡Actually, update creates a new token of type UpdatedArrayToken which keeps track of updated el-ements in a token while preserving the unchanged elements. The alternative would be to produce a newArrayToken, which would result allocating memory and copying the entire source array.


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13.3.2 Matrices

In Ptolemy II, arrays are ordered sets of tokens. Ptolemy II also supports matrices, whichare more specialized than arrays. They contain only certain primitive types, currentlyboolean, complex, double, fixedpoint, int, and long. Currently float, short and unsigned-Byte matrices are not supported. Matrices cannot contain arbitrary tokens, so they cannot,for example, contain matrices. They are intended for data intensive computations. Ma-trices are specified with square brackets, using commas to separate row elements andsemicolons to separate rows. E.g., [1, 2, 3; 4, 5, 5+1] gives a two by three inte-ger matrix (2 rows and 3 columns). Note that an array or matrix element can be given byan expression. A row vector can be given as [1, 2, 3] and a column vector as [1; 2;

3]. Some MATLAB-style array constructors are supported. For example, [1:2:9] givesan array of odd numbers from 1 to 9, and is equivalent to [1, 3, 5, 7, 9]. Similarly,[1:2:9; 2:2:10] is equivalent to [1, 3, 5, 7, 9; 2, 4, 6, 8, 10]. In thesyntax [p:q:r], p is the first element, q is the step between elements, and r is an upperbound on the last element. That is, the matrix will not contain an element larger than r.If a matrix with mixed types is specified, then the elements will be converted to a com-mon type, if possible. Thus, for example, [1.0, 1] is equivalent to [1.0, 1.0], but[1.0, 1L] is illegal (because there is no common type to which both elements can beconverted losslessly, see Chapter 14).

Elements of matrices are referenced using matrixname(n, m), where matrixname isthe name of a matrix variable in scope, n is the row index, and m is the column index.Index numbers start with zero, as in Java, not 1, as in MATLAB. For example,

>> [1, 2; 3, 4](0,0)1>> a = [1, 2; 3, 4][1, 2; 3, 4]>> a(1,1)4

Matrix multiplication works as expected. For example,

>> [1, 2; 3, 4]*[2, 2; 2, 2][6, 6; 14, 14]


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13. EXPRESSIONS

Of course, if the dimensions of the matrix don’t match, then you will get an error message.To do element wise multiplication, use the multipyElements function (see Table 13.8).Matrix addition and subtraction are element wise, as expected, but the division operatoris not supported. Element wise division can be accomplished with the divideElementsfunction, and multiplication by a matrix inverse can be accomplished using the inversefunction (see Table 13.8). A matrix can be raised to an int, short or unsignedByte power,which is equivalent to multiplying it by itself some number of times. For instance,

>> [3, 0; 0, 3]ˆ3[27, 0; 0, 27]

A matrix can also be multiplied or divided by a scalar, as follows:

>> [3, 0; 0, 3]*3[9, 0; 0, 9]

A matrix can be added to a scalar. It can also be subtracted from a scalar, or have a scalarsubtracted from it. For instance,

>> 1-[3, 0; 0, 3][-2, 1; 1, -2]

A matrix can be checked for equality with another matrix as follows:

>> [3, 0; 0, 3]!=[3, 0; 0, 6]true>> [3, 0; 0, 3]==[3, 0; 0, 3]true

For other comparisons of matrices, use the compare function (see Table 13.7). As withscalars, testing for equality using the == or != operators tests the values, independent oftype. For example,

>> [1, 2]==[1.0, 2.0]true

To get type-specific equality tests, use the equals method, as in the following examples:


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>> [1, 2].equals([1.0, 2.0])false>> [1.0, 2.0].equals([1.0, 2.0])true

13.3.3 Records

A record token is a composite type containing named fields, where each field has a value.The value of each field can have a distinct type. Records are delimited by curly braces,with each field given a name. For example, {a=1, b="foo"} is a record with two fields,named “a” and “b”, with values 1 (an integer) and “foo” (a string), respectively. The keyof a field can be an arbitrary string, provided that it is quoted. Only strings that qualifyas valid Java identifiers can be used without quotation marks. Note that quotation markswithin a quoted string must be escaped using a backslash. The value of a field can be anarbitrary expression, and records can be nested (a field of a record token may be a recordtoken).

An ordered record is similar to a normal record except that it preserves the originalordering of the labels. Ordered records are delimited using square brackets rather thancurly braces. For example, [b="foo", a=1] is an ordered record token in which “b”will remain the first label.

Fields that are valid Java identifiers may be accessed using the period operator, optionallywith braces—as if it were a method call. For example, the following two expressions:

{a=1,b=2}.a{a=1,b=2}.a()

both yield 1.

An alternative syntax to access fields uses the get() method. Note that this is the onlyway to access fields for which the key demands the use of quotation marks. For example:

{" a"=1, "\"b"=2}.get("\"b")

yields 2.


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13. EXPRESSIONS

The arithmetic operators +, -, *, /, and % can be applied to records. If the records donot have identical fields, then the operator is applied only to the fields that match, and theresult contains only the fields that match. Thus, for example,

{foodCost=40, hotelCost=100}+ {foodCost=20, taxiCost=20}

yields the result

{foodCost=60}

You can think of an operation as a set intersection, where the operation specifies howto merge the values of the intersecting fields. You can also form an intersection withoutapplying an operation. In this case, using the intersect function, you form a recordthat has only the common fields of two specified records, with the values taken from thefirst record. For example,

>> intersect({a=1, c=2}, {a=3, b=4}){a=1}

Records can be joined (think of a set union) without any operation being applied by usingthe merge function. This function takes two arguments, both of which are record tokens.If the two record tokens have common fields, then the field value from the first record isused. For example,

merge({a=1, b=2}, {a=3, c=3})

yields the result {a=1, b=2, c=3}.

Records can be compared, as in the following examples:

>> {a=1, b=2}!={a=1, b=2}false>> {a=1, b=2}!={a=1, c=2}true

Note that two records are equal only if they have the same field labels and the valuesmatch. As with scalars, the values match irrespective of type. For example:


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>> {a=1, b=2}=={a=1.0, b=2.0+0.0i}true

The order of the fields is irrelevant for normal (unordered) records. Hence

>> {a=1, b=2}=={b=2, a=1}true

Moreover, normal record fields are reported in alphabetical order, irrespective of the orderin which they are defined. For example,

>> {b=2, a=1}{a=1, b=2}

Equality comparisons for ordered records respect the original order of the fields. Forexample,

>> [a=1, b=2]==[b=2, a=1]false

Additionally, ordered record fields are always reported in the order in which they aredefined. For example,

>> [b=2, a=1][b=2, a=1]

To get type-specific equality tests, use the equals method, as in the following examples:

>> {a=1, b=2}.equals({a=1.0, b=2.0+0.0i})false>> {a=1, b=2}.equals({b=2, a=1})true

Finally, You can create an empty record using the emptyRecord function:

>> emptyRecord(){}


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13. EXPRESSIONS

13.3.4 Union Types

Occasionally, more than one distinct data type will be sent over the same connection, or avariable may take on values with one of several data types. Ptolemy II provides a uniontype to accommodate this. A union type is designated as in the following example:

{|a = int, b = complex |}

This indicates a port or variable that may have type int or complex. A typical use of uniontypes uses a UnionMerge and/or UnionDisassembler actor.

Figure 13.6: Union types allow types to resolve to more than one type. [online]


http://ptolemy.org/systems/models/expressions/UnionTypes/index.html

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13.4. OPERATIONS ON TOKENS

Example 13.1: Consider the example shown in Figure 13.6. This is a DE modelwith two data sources, a DiscreteClock, which produces outputs of type int, anda PoissonClock, which produces outputs of type boolean. These two streams ofvalues are merged by the UnionMerge, whose output type becomes a union type.The names of the types in the union are determined by the names of the input portsof the UnionMerge, which are added when building the model. The lower Displayactor displays the merged stream, showing that each token displayed has a singlevalue, either an int or a boolean.

Along the upper path, a UnionDisassembler actor is used to extract from the streamthe “b” types, which are boolean. Only those types are passed to the output, andthe type of the output is inferred to be boolean.

13.4 Operations on Tokens

Every element and subexpression in an expression represents an instance of the Tokenclass in Ptolemy II (or more likely, a class derived from Token). The expression languagesupports a number of operations on tokens that give access to the underlying Java code.

13.4.1 Invoking Methods

The expression language supports invocation of any method of a given token, as long asthe arguments of the method are of type Token and the return type is Token (or a classderived from Token, or something that the expression parser can easily convert to a token,such as a string, double, int, etc.). The syntax for this is (token.methodName(args),where methodName is the name of the method and args is a comma-separated set of ar-guments. Each argument can itself be an expression. Note that the parentheses around thetoken are not required, but might be useful for clarity. As an example, the ArrayTokenand RecordToken classes have a length method, illustrated by the following examples:

{1, 2, 3}.length(){a=1, b=2, c=3}.length()


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13. EXPRESSIONS

each of which returns the integer 3.

The MatrixToken classes have three particularly useful methods, illustrated in the follow-ing examples:

[1, 2; 3, 4; 5, 6].getRowCount()

which returns 3, and

[1, 2; 3, 4; 5, 6].getColumnCount()

which returns 2, and

[1, 2; 3, 4; 5, 6].toArray()

which returns 1, 2, 3, 4, 5, 6. The latter function can be particularly useful for creatingarrays using MATLAB-style syntax. For example, to obtain an array with the integersfrom 1 to 100, you can enter:

[1:1:100].toArray()

13.4.2 Accessing Model Elements

Expressions in a model can reference elements of the model and invoke methods on them.The expression giving a parameter its value may reference by name any object that iscontained by the container of the parameter.

Example 13.2: Figure 13.7 shows a model with four parameters P1 through P4.The parameter P1 has expression

Const2.value

The Const2 here refers to the actor named “Const2” that is contained by the con-tainer of P1, and hence Const2.value refers to the value parameter of the actorConst2. The value of the parameter P1 is therefore equal to the value of the valueparameter of Const2.


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The keyword this in a parameter expression refers to the object that contains the param-eter.

Example 13.3: In Figure 13.7, Const2 has a value parameter with the expression(shown in its icon):

this.getName() + ": " + P2

Here, this refers to Const2, so this.getName() returns a string that is the name“Const2.” The rest of the expression performs string concatenation, appending acolon and the value parameter P2.

The parameter P2 has expression

this.entityList().size()

In this case, this refers to the container of P2, which is the top-level model. Hence,this.entityList() returns a list of entities (actors) contained by this top-levelmodel. And finally, this.entityList().size() returns the number of actorscontained by this top-level model, which is 5.

Now, the first two outputs of this model should be easy to understand:

Figure 13.7: Expressions can access elements of the model, as shown here.[online]


http://ptolemy.org/systems/models/expressions/This/index.html

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13. EXPRESSIONS

Const1: Const2: 5Const2: 5

The second output is the value of P2 (namely 5) prepended by the name of the Constactor generating the output and a colon. The first output is the value P1 (which isthe string "Const2: 5") prepended by the name of the Const actor generatingthe output and a colon.

Parameters can use method invocation to traverse the connections in the model, as illus-trated next.

Example 13.4: In Figure 13.7, parameter P3 has the expression:

Const2.output.connectedPortList().get(0).getFullName()

This gets the full name of the first port in the list of ports that the out-put port of actor Const2 is connected to. Specifically, it returns the string".This.Display.input", as displayed by actor Const3.

Similarly, P4 has expression

Display.input.connectedPortList().size()

which returns the number of sources connected to the Display input.

For an overview of some of the methods that can be invoked on actors, parameters, andports, see Chapter 12. For a complete listing, see the code documentation for Ptolemy II.

13.4.3 Casting

The cast function can be used to explicitly cast a value into a type. When the cast functionis invoked with cast(type, value), where type is the target type and value is the


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value to be cast, a new value is returned (if a predefined casting is applicable) that is inthe specified type. For example, cast(long, 1) yields 1L, which is equal to 1 but is inthe long data type, and cast(string, 1) yields "1", which is in the string data type.

13.4.4 Defining Functions

Users can define new functions in the expression language. The syntax is:

function(arg1:Type, arg2:Type...)function body

where function is the keyword for defining a function. The type of an argument canbe left unspecified, in which case the expression language will attempt to infer it. Thefunction body gives an expression that defines the return value of the function. The returntype is always inferred based on the argument type and the expression. For example:

function(x:double) x*5.0

defines a function that takes a double argument, multiplies it by 5.0, and returns a double.The return value of the above expression is the function itself. Thus, for example, theexpression evaluator yields:

>> function(x:double) x*5.0(function(x:double) (x*5.0))

To apply the function to an argument, simply do

>> (function(x:double) x*5.0) (10.0)50.0

Alternatively, in the expression evaluator, you can assign the function to a variable, andthen use the variable name to apply the function. For example,

>> f = function(x:double) x*5.0(function(x:double) (x*5.0))>> f(10)50.0


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13. EXPRESSIONS

13.4.5 Higher-Order Functions

Functions can be passed as arguments to certain higher-order functions that have beendefined (see Table 13.15). For example, the iterate function takes three arguments, afunction, an integer, and an initial value to which to apply the function. It applies thefunction first to the initial value, then to the result of the application, then to that result,collecting the results into an array whose length is given by the second argument. Forexample, to get an array whose values are multiples of 3, try

>> iterate(function(x:int) x+3, 5, 0){0, 3, 6, 9, 12}

The function given as an argument simply adds three to its argument. The result is thespecified initial value (0) followed by the result of applying the function once to that initialvalue, then twice, then three times, etc.

Another useful higher-order function is the map function. This one takes a function andan array as arguments, and simply applies the function to each element of the array toconstruct a result array. For example,

>> map(function(x:int) x+3, {0, 2, 3}){3, 5, 6}

Ptolemy II also supports a fold function, which can be used to program a loop in anexpression. The fold function applies a function to each element of an array, accumulatinga result as it goes. The function that is folded over the array takes two arguments, theaccumulated result so far and an array element. When the function to be folded is appliedto the first element of the array, the accumulated result is an initial value.

Example 13.5:fold(

function(x:int, e:int) x + 1,0, {1, 2, 3}

)

This computes the length of array {1, 2, 3}. The result is 3, which is equal to{1, 2, 3}.length(). Specifically, the function to be folded is function(x:int,


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e:int) x + 1. Given arguments x and e, it returns x + 1, ignoring the secondargument e. It is first applied to the initial value, 0, and the first element of thearray, 1, yielding 1. It is then applied to the accumulated result, 1, and the secondelement of the array, the value of which it ignores, yielding 2. It is invoked thenumber of times equal to the number of elements in array {1, 2, 3}. Therefore, x isincreased 3 times from the starting value 0.

Example 13.6: The following variant does not ignore the values of the arrayelements:fold(

function(x:int, e:int) x + e,0, {1, 2, 3}

)

This computes the sum of all elements in array {1, 2, 3}, yielding 6.

Example 13.7:fold(

function(x:arrayType(int), e:int)e % 2 == 0 ? x : x.append({e}),

{}, {1, 2, 3, 4, 5})

This computes a subarray of array {1, 2, 3, 4, 5} that contains only odd numbers.The result is {1, 3, 5}.

Example 13.8: Let C be an actor.

fold(function(list:arrayType(string),

port:object("ptolemy.kernel.Port"))


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13. EXPRESSIONS

port.connectedPortList().isEmpty() ?list.append({port}) : list,

{}, C.portList())

This returns a list of C’s ports that are not connected to any other port (withconnectedPortList() being empty). Each port in the returned list is encapsu-lated in an ObjectToken.

13.4.6 Using Functions in a Model

A typical use of functions in a Ptolemy II model is to define a parameter in a model whosevalue is a function. Suppose that the parameter named f has value

function(x:double)x*5.0

Then within the scope of that parameter, the expression f(10.0) will yield result 50.0.

Functions can also be passed along connections in a Ptolemy II model.

Example 13.9: Consider the model shown in Figure 13.8. In that example, theConst actor defines a function that simply squares the argument. Its output, there-fore, is a token with type function. That token is fed to the “f” input of the Expres-sion actor. The expression uses this function by applying it to the token providedon the “y” input. That token, in turn, is supplied by the Ramp actor, so the result isthe curve shown in the plot on the right.

Example 13.10: A more elaborate use is shown in Figure 13.9 In that example,the Const actor produces a function, which is then used by the Expression actor tocreate new function, which is then used by Expression2 to perform a calculation.The calculation performed here multiplies the output of the Ramp to the square ofthe output of the Ramp, thus computing the cube.


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0

10

20

30

40

50

60

70

80

0 1 2 3 4 5 6 7 8 9

SequencePlotter

Figure 13.8: Example of a function being passed from one actor to another.

0.00.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

x102

0 1 2 3 4 5 6 7 8

SequencePlotter

Figure 13.9: More elaborate example with functions passed between actors.


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13. EXPRESSIONS

13.4.7 Recursive Functions

Functions can be recursive, as illustrated by the following (rather arcane) example:

>> fact = function(x:int,f:(function(x,f) int)) (x<1?1:x*f(x-1,f))(function(x:int, f:function(a0:general, a1:general) int)

(x<1)?1:(x*f((x-1), f)))>> factorial = function(x:int) fact(x,fact)(function(x:int) (function(x:int, f:function(a0:general, a1:general) int)(x<1)?1:(x*f((x-1), f)))(x, (function(x:int,f:function(a0:general, a1:general) int) (x<1)?1:(x*f((x-1), f)))))

>> map(factorial, [1:1:5].toArray()){1, 2, 6, 24, 120}

The first expression defines a function named “fact” that takes a function as an argument,and if the argument is greater than or equal to 1, uses that function recursively. The secondexpression defines a new function “factorial” using “fact.” The final command applies thefactorial function to an array to compute factorials.

13.4.8 Built-In Functions

The expression language includes a set of functions, such as sin, cos, etc. The functionsthat are built in include all static methods § of the classes shown in Table 13.2, whichtogether provide a rich set ¶.

The functions currently available are shown in the tables at the end of this chapter, whichalso show the argument types and return types. The argument and return types are thewidest type that can be used. For example, acos will take any argument that can belosslessly cast to a double, such as unsigned byte, short, integer, float. long cannot becast losslessly cast to double, so acos(1L) will fail. Trigonometric functions are givenin Table 13.4. Basic mathematical functions are given in Tables 13.5 and 13.6. Func-tions that take or return matrices, arrays, or records are given in Tables 13.7 through 13.9.Utility functions for evaluating expressions are given in Table 13.10. Functions perform-ing signal processing operations are given in Tables 13.11 through 13.13. I/O and othermiscellaneous functions are given in Tables 13.15 and 13.16.

§ Note that calling methods such as String.format() that have an argument of Object [] can be dif-ficult because of problems specifying an array of Java Objects such as java.lang.Double instead ofptolemy.data.type.DoubleToken¶ Moreover, the set of available methods can easily be extended if you are writing Java code by registering

another class that includes static methods (see the PtParser class in the ptolemy.data.expr package).


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In most cases, a function that operates on scalar arguments can also operate on arraysand matrices. Thus, for example, you can fill a row vector with a sine wave using anexpression like

sin([0.0:PI/100:1.0])

Or you can construct an array as follows,

sin({0.0, 0.1, 0.2, 0.3})

Functions that operate on type double will also generally operate on int, short, or un-signedByte, because these can be losslessly converted to double, but not generally on longor complex. Tables of available functions are shown in the appendix. For example, Table13.4 shows trigonometric functions. Note that these operate on double or complex, andhence on int, short and unsignedByte, which can be losslessly converted to double. Theresult will always be double. For example,

>> cos(0)1.0

Table 13.2: The classes whose static methods are available as functions in theexpression language.

java.lang.Math ptolemy.math.IntegerMatrixMathjava.lang.Double ptolemy.math.DoubleMatrixMathjava.lang.Integer ptolemy.math.ComplexMatrixMathjava.lang.Long ptolemy.math.LongMatrixMathjava.lang.String ptolemy.math.IntegerArrayMathptolemy.data.MatrixToken. ptolemy.math.DoubleArrayStatptolemy.data.RecordToken. ptolemy.math.ComplexArrayMathptolemy.data.expr.UtilityFunctions ptolemy.math.LongArrayMathptolemy.data.expr.FixPointFunctions ptolemy.math.SignalProcessingptolemy.math.Complex ptolemy.math.FixPointptolemy.math.ExtendedMath ptolemy.data.ObjectToken


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13. EXPRESSIONS

These functions will also operate on matrices and arrays, in addition to the scalar typesshown in the table, as illustrated above. The result will be a matrix or array of the samesize as the argument, but always containing elements of type double.

Table 13.7 shows other arithmetic functions beyond the trigonometric functions. As withthe trigonometric functions, those that indicate that they operate on double will also workon int, short and unsignedByte, and unless they indicate otherwise, they will return what-ever they return when the argument is double. Those functions in the table that take scalararguments will also operate on matrices and arrays. For example, since the table indicatesthat the max function can take int, int as arguments, then by implication, it can also takeint, int. For example,

>> max({1, 2}, {2, 1}){2, 2}

Notice that the table also indicates that max can take int as an argument. E.g.

>> max({1, 2, 3})3

In the former case, the function is applied pointwise to the two arguments. In the lattercase, the returned value is the maximum over all the contents of the single argument.

Table 13.7 shows functions that only work with matrices, arrays, or records (that is, thereis no corresponding scalar operation). Recall that most functions that operate on scalarswill also operate on arrays and matrices. Table 13.10 shows utility functions for evaluat-ing expressions given as strings or representing numbers as strings. Of these, the eval

function is the most flexible.

A few of the functions have sufficiently subtle properties that they require further expla-nation. That explanation is here.

eval() and traceEvaluation()

The built-in function eval will evaluate a string as an expression in the expression lan-guage. For example,

eval("[1.0, 2.0; 3.0, 4.0]")


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will return a matrix of doubles. The following combination can be used to read parametersfrom a file:

eval(readFile("filename"))

where the filename can be relative to the current working directory (where Ptolemy IIwas started, as reported by the Java Virtual Machine property user.dir), a user’s homedirectory (as reported by the property user.home), or the classpath, which includes thedirectory tree in which Ptolemy II is installed. Note that if eval is used in an Expressionactor, then it will be impossible for the type system to infer any more specific output typethan general. If you need the output type to be more specific, then you will need to castthe result of eval. For example, to force it to type double:

>> cast(double, eval("pi/2"))1.5707963267949

The traceEvaluation function evaluates an expression given as a string, much likeeval, but instead of reporting the result, reports exactly how the expression was evalu-ated. This can be used to debug expressions, particularly when the expression language isextended by users.

random(), gaussian()

The random and gaussian functions shown in Table 13.5 and Table 13.6 return one ormore random numbers. With the minimum number of arguments (zero or two, respec-tively), they return a single number. With one additional argument, they return an arrayof the specified length. With a second additional argument, they return a matrix with thespecified number of rows and columns.

There is a key subtlety when using these functions in Ptolemy II. In particular, they areevaluated only when the expression within which they appear is evaluated. The result ofthe expression may be used repeatedly without re-evaluating the expression. Thus, forexample, if the value parameter of the Const actor is set to random(), then its outputwill be a random constant, i.e., it will not change on each firing. The output will change,however, on successive runs of the model. In contrast, if this is used in an Expressionactor, then each firing triggers an evaluation of the expression, and consequently willresult in a new random number.


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13. EXPRESSIONS

property()

The property function accesses Java Virtual Machine system properties by name. Somepossibly useful system properties are:

• ptolemy.ptII.dir: The directory in which Ptolemy II is installed.

• ptolemy.ptII.dirAsURL: The directory in which Ptolemy II is installed, but repre-sented as a URL.

• user.dir: The current working directory, which is usually the directory in which thecurrent executable was started.

For a complete list of Java Virtual Machine properties, see the Java documentation forjava.lang.System.getProperties.

remainder()

This function computes the remainder operation on two arguments as prescribed by theIEEE 754 standard, which is not the same as the modulo operation computed by the %operator. The result of remainder(x, y) is x − yn, where n is the integer closest tothe exact value of x/y. If two integers are equally close, then n is the integer that is even.This yields results that may be surprising, as indicated by the following examples:

>> remainder(1,2)1.0>> remainder(3,2)-1.0

Compare this to

>> 3%21

which is different in two ways. The result numerically different and is of type int, whereasremainder always yields a result of type double. The remainder() function is imple-mented by the java.lang.Math class, which calls it IEEEremainder(). The documenta-tion for that class gives the following special cases:


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• If either argument is NaN, or the first argument is infinite, or the second argumentis positive zero or negative zero, then the result is NaN.

• If the first argument is finite and the second argument is infinite, then the result isthe same as the first argument.

DCT() and IDCT()

The discrete cosine transform (DCT) function can take one, two, or three arguments. Inall three cases, the first argument is an array of length N > 0 and the DCT returns an

Xk = sk

N−1∑n=0

xn cos((2n+ 1)kπ

2D) (13.1)

for k from 0 to D − 1, where N is the size of the specified array and D is the size of theDCT. If only one argument is given, then D is set to equal the next power of two largerthan N . If a second argument is given, then its value is the order of the DCT, and the sizeof the DCT is 2order. If a third argument is given, then it specifies the scaling factors skaccording to the following table:

Table 13.3: Normalization options for the DCT function.

Name Third argument Normalization

Normalized 0 sk =

{1√2; k = 0

1, otherwise

Unnormalized 1 sk = 1

Orthonormal 2 sk =

1√D

; k = 0√2D ; otherwise

The default, if a third argument is not given, is “Normalized.” The IDCT function issimilar, and can also take one, two, or three arguments. The formula in this case is

xn =

N−1∑k=0

skXk cos((2n+ 1)kπ

2D). (13.2)


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13. EXPRESSIONS

13.5 Nil Tokens

Null or missing tokens are common in analytical systems like R and SAS where they areused to handle sparsely populated data sources. In database parlance, missing tokens aresometimes called null tokens. Since null is a Java keyword, we use the term “nil.” Niltokens are useful for analyzing real world data such as temperature where the value is notmeasured during every interval. In principle, one may want, for example, a TolerantAver-age actor that does not require all data values to be present — when the TolerantAverageactor sees a nil token, it would ignore it. Note that this can lead to uncertainty. For exam-ple, if average is expecting 30 values and 29 of them are nil, then the average will notbe very accurate.

In Ptolemy II, operations on tokens yield a nil token if any argument is a nil token. Thus,the Average actor is not like TolerantAverage. Upon receiving a nil token, all subsequentresults will be nil. When an operation yields a nil value, the resulting nil token willhave the same type that would normally have resulted from the operation, so type safetyis preserved. Not all data types, however, support nil tokens. In particular, the variousmatrix types cannot have nil values because the underlying matrices are Java native typematrices that do not support nil.

The expression language defines a constant named nil that has value nil and type niltype(see Chapter 14). The cast expression language function can be used to generate nilvalues of other types. For example, “cast(int, nil)” will return a token with valuenil and type int.

13.6 Fixed Point Numbers

Ptolemy II includes a fixed point data type. We represent a fixed point value in the ex-pression language using the following format:

fix(value, totalBits, integerBits)

Thus, a fixed point value of 5.375 that uses 8 bit precision of which 4 bits are used torepresent the (signed) integer part can be represented as:

fix(5.375, 8, 4)


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13.6. FIXED POINT NUMBERS

The value can also be a matrix of doubles. The values are rounded, yielding the nearestvalue representable with the specified precision. If the value to represent is out of range,then it is saturated, meaning that the maximum or minimum fixed point value is returned,depending on the sign of the specified value. For example,

fix(5.375, 8, 3)

will yield 3.968758, the maximum value possible with the (8/3) precision.

In addition to the fix function, the expression language offers a quantize function.The arguments are the same as those of the fix function, but the return type is a Double-Token or DoubleMatrixToken instead of a FixToken or FixMatrixToken. This functioncan therefore be used to quantize double-precision values without ever explicitly workingwith the fixed-point representation.

To make the FixToken accessible within the expression language, the following functionsare available:

• To create a single FixPoint Token using the expression language:

fix(5.34, 10, 4)

This will create a FixToken. In this case, we try to fit the number 5.34 into a 10 bitrepresentation with 4 bits used in the integer part. This may lead to quantizationerrors. By default the round quantizer is used.

• To create a Matrix with FixPoint values using the expression language:

fix([ -.040609, -.001628, .17853 ], 10, 2)

This will create a FixMatrixToken with 1 row and 3 columns, in which each elementis a FixPoint value with precision(10/2). The resulting FixMatrixToken will try tofit each element of the given double matrix into a 10 bit representation with 2 bitsused for the integer part. By default the round quantizer is used.

• To create a single DoubleToken, which is the quantized version of the double valuegiven, using the expression language:

quantize(5.34, 10, 4)


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13. EXPRESSIONS

This will create a DoubleToken. The resulting DoubleToken contains the doublevalue obtained by fitting the number 5.34 into a 10 bit representation with 4 bitsused in the integer part. This may lead to quantization errors. By default the roundquantizer is used.

• To create a Matrix with doubles quantized to a particular precision using the ex-pression language:

quantize([ -.040609, -.001628, .17853 ], 10, 2)

This will create a DoubleMatrixToken with 1 row and 3 columns. The elements ofthe token are obtained by fitting the given matrix elements into a 10 bit representa-tion with 2 bits used for the integer part. Instead of being a fixed point value, thevalues are converted back to their double representation and by default the roundquantizer is used.

13.7 Units

Ptolemy II supports units systems, which are built on top of the expression language.Units systems allow parameter values to be expressed with units, such as “1.0 * cm”,which is equal to “0.01 * meters”. These are expressed this way (with the * for multi-plication) because “cm” and “meters” are actually variables that become in scope whena units system icon is dragged in to a model. A few simple units systems are provided(mainly as examples) in the utilities library.

A model using one of the simple provided units systems is shown in Figure 13.10 Thisunit system is called BasicUnits; the units it defines can be examined by double clickingon its icon, or by invoking “Customize” |“Configure”, as shown in Figure 13.11. Inthat figure, we see that “meters”, “meter”, and “m” are defined, and are all synonymous.Moreover, “cm” is defined, and given value “0.01*meters”, and “in”, “inch” and “inches”are defined, all with value “2.54*cm”.

In the example in Figure 13.10, a constant with value “1.0 * meter” is fed into a Scale actorwith scale factor equal to “2.0/ms”. This produces a result with dimensions of length overtime. If we feed this result directly into a Display actor, then it is displayed as “2000.0meters/seconds”, as shown in Figure 13.12, top display. The canonical units for lengthare meters, and for time are seconds.


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13.7. UNITS

Figure 13.10: Example of a model that includes a unit system. [online]

In Figure 13.10, we also take the result and feed it to the InUnitsOf actor, which dividesits input by its argument, and checks to make sure that the result is unitless. This tells usthat 2 meters/ms is equal to about 78,740 inches/second.

The InUnitsOf actor can be used to ensure that numbers are interpreted correctly in amodel, which can be effective in catching certain kinds of critical errors. For example, if inFigure 13.10, we had entered “seconds/inch” instead of “inches/second” in the InUnitsOfactor, we would have gotten the exception in Figure 13.13 instead of the execution inFigure 13.12.

Units systems are built entirely on the expression language infrastructure in Ptolemy II.The units system icons actually represent instances of scope-extending attributes, whichare attributes whose parameters are in scope as if those parameters were directly containedby the container of the scope extending attribute. That is, scope-extending attributes candefine a collection of variables and constants that can be manipulated as a unit. Two fairlyextensive units systems are provided, CGSUnitBase and ElectronicUnitBase. Nonethe-less, these are intended as examples only, and can no doubt be significantly improved andextended.


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13. EXPRESSIONS

Figure 13.11: Units defined in a units system can be examined by double clickingor by right clicking and selecting Customize and Configure.


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13.7. UNITS

Figure 13.12: Result of running the model in Figure 13.10.

Figure 13.13: Example of an exception resulting from a units mismatch.


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13. EXPRESSIONS

13.8 Tables of Functions

Table 13.4: Trigonometric functions.Name Argument type(s) Return type Description

acos double in the range[-1.0, 1.0] or com-plex

double in the range[0.0, π] or NaNif out of range orcomplex

arc cosinecomplex case:acos(z) = −i log(z + i

√i− z2)

asin double in the range[-1.0, 1.0] or com-plex

double in the range[−π/2, π/2] orNaN if out of rangeor complex

arc sinecomplex case:asin(z) = −i log(iz +

√i− z2)

atan double or complex double in the range[−π/2, π/2] orcomplex

arc tangentcomplex case: atan(z) = − i

2log ( i−z

i+z)

atan2 double, double double in the range[−π, π]

angle of a vector (note: the arguments are (y, x),not (x, y) as one might expect).

acosh double greater than1 or complex

double or complex hyperbolic arc cosine, defined for both dou-ble and complex case by: acosh(z) =log (z +

√z2 − 1)

asinh double or complex double or complex hyperbolic arc sine complex case: asinh(z) =log (z +

√z2 + 1)

cos double or complex double in the range[-1, 1], or complex

cosinecomplex case: cos(z) = exp(iz)+exp(−iz)

2

cosh double or complex double or complex hyperbolic cosine, defined for double or com-plex by: cosh(z) = exp(z)+exp(−z)

2

sin double or complex double or complex sine function complex case:sin(z) = exp(iz)−exp(−iz)

2i

sinh double or complex double or complex hyperbolic sine, defined for double or complexby: sinh(z) = exp(z)−exp(−z)

2

tan double or complex double or complex tangent function, defined for double or complexby: tan(z) = sin(z)

cos(z)

tanh double or complex double or complex hyperbolic tangent, defined for double or com-plex by: tanh(z) = sinh(z)

cosh(z)


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13.8. TABLES OF FUNCTIONS

Table 13.5: Basic mathematical functions, part 1.Function Argument type(s) Return type Description

abs double or complex double or int orlong(complexreturns double

absolute valuecomplex case:abs(a+ ib) = |z| =

√a2 + b2

angle complex double in the range[−π, π]

angle or argument of the complex number: ∠z

ceil double or float double ceiling function, which returns the smallest(closest to negative infinity) double value thatis not less than the argument and is an integer.

compare double, double int compare two numbers, returning -1, 0, or 1if the first argument is less than, equal to, orgreater than the second.

conjugate complex complex complex conjugateexp double or complex double in the range

[0.0, ∞] or com-plex

exponential function (eargument) complexcase: ea+ib = ea(cos (b) + i sin (b))

floor double double floor function, which is the largest (closest topositive infinity) value not greater than the ar-gument that is an integer.

gaussian double, double ordouble, double, int,or double, double,int, int

double or array-Type(double) or[double]

one or more Gaussian random variables withthe specified mean and standard deviation (see13.4.8).

imag complex double imaginary partisInfinite double boolean return true if the argument is infiniteisNaN double boolean return true if the argument is “not a number”log double or complex double or complex natural logarithm complex case: log(z) =

log(abs(z) + i∠z)log10 double double log base 10log2 double double log base 2max double, double or

doublea scalar of the sametype as the argu-ments

maximum

min double, double ordouble

a scalar of the sametype as the argu-ments

minimum

pow double, double orcomplex, complex

double or complex first argument to the power of the second


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13. EXPRESSIONS

Table 13.6: Basic mathematical functions, part 2.Function Argument type(s) Return type Description

random no arguments or intor int, int

double or double or[double]

one or more random numbers between 0.0and 1.0 (see 13.4.8)

real complex double real partremainder double, double double remainder after division, according to the

IEEE 754 floating-point standard (see13.4.8).

round double long round to the nearest long, choosing the nextgreater integer when exactly in between, andthrowing an exception if out of range. If theargument is NaN, the result is 0L. If the argu-ment is out of range, the result is either Max-Long or MinLong, depending on the sign.

roundToInt double int round to the nearest int, choosing the nextgreater integer when exactly in between, andthrowing an exception if out of range. If theargument is NaN, the result is 0. If the argu-ment is out of range, the result is either Max-Int or MinInt, depending on the sign.

sgn double int -1 if the argument is negative, 1 otherwisesqrt double or complex double or complex square root. If the argument is double with

value less than zero, then the result is NaN.complex case: sqrt(z) =

√|z|(cos (∠z

2) +

i sin (∠z2))

toDegrees double double convert radians to degreestoRadians double double convert degrees to radianswithin type, type, double boolean return true if the first argument is in the

neighborhood of the second, meaning that thedistance is less than or equal to the third ar-gument. The first two arguments can be anytype for which such a distance is defined. Forcomposite types, arrays, records, and matri-ces, then return true if the first two argu-ments have the same structure, and each cor-responding element is in the neighborhood.


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Table 13.7: Functions that take or return matrices, arrays, or records, part 1.Function Argument type(s) Return type Description

arrayToMatrix arrayType(type), int, int [type] Create a matrix from the spec-ified array with the specifiednumber of rows and columns

concatenate arrayType(type), array-Type(type)

arrayType(type) Concatenate two arrays.

concatenate arrayType(arrayType(type))

arrayType(type) Concatenate arrays in an arrayof arrays.

conjugateTranspose [complex] [complex] Return the conjugate trans-pose of the specified matrix.

createSequence type, type, int arrayType(type) Create an array with valuesstarting with the first argu-ment, incremented by the sec-ond argument, of length givenby the third argument.

crop [int], int, int, int, int or[double], int, int, int, int or[complex], int, int, int, int or[long], int, int, int, int

[int] or [dou-ble] or [com-plex] or [long]or

Given a matrix of any type,return a submatrix starting atthe specified row and columnwith the specified number ofrows and columns.

determinant [double] or [complex] double or com-plex

Return the determinant of thespecified matrix.

diag arrayType(type) [type] Return a diagonal matrix withthe values along the diagonalgiven by the specified array.

divideElements [type], [type] [type] Return the element-by-element division of twomatrices

emptyArray type arrayType(type) Return an empty array whoseelement type matches thespecified token.

emptyRecord record Return an empty record.find arrayType(type), type arrayType(int) Return an array of the indices

where elements of the speci-fied array match the specifiedtoken.

find arrayType(boolean) arrayType(int) Return an array of the indiceswhere elements of the speci-fied array have value true.

hilbert int [double] Return a square Hilbert ma-trix, where Aij = 1

i+j+1. A

Hilbert matrix is nearly, butnot quite singular.


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13. EXPRESSIONS


identityMatrixComplex int [complex] Return an identity matrix withthe specified dimension.

identityMatrixDouble int [double] Return an identity matrix withthe specified dimension.

identityMatrixInt int [int] Return an identity matrix withthe specified dimension.

identityMatrixLong int [long] Return an identity matrix withthe specified dimension.

intersect record, record record Return a record that containsonly fields that are presentin both arguments, where thevalue of the field is taken fromthe first record.

inverse [double] or [complex] [double] or[complex]

Return the inverse of the spec-ified matrix, or throw an ex-ception if it is singular.

matrixToArray [type] arrayType(type) Create an array containing thevalues in the matrix

merge record, record record Merge two records, givingpriority to the first one whenthey have matching record la-bels.

multiplyElements [type], [type] [type] Multiply element wise the twospecified matrices.

orthonormalizeColumns [double] or [complex] [double] or[complex]

Return a similar matrix withorthonormal columns.

orthonormalizeRows [double] or [complex] [double] or[complex]

Return a similar matrix withorthonormal rows.

repeat int, type arrayType(type) Create an array by repeatingthe specified token the speci-fied number of times.

sort arrayType(string) or ar-rayType(realScalar)

arrayType(string)or array-Type(realScalar)

Return the specified array, butsorted in ascending order. re-alScalar is any scalar tokenexcept complex.

sortAscending arrayType(string) or ar-rayType(realScalar)


Return the specified array, butsorted in ascending order. re-alScalar is any scalar tokenexcept complex.


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sortDescending arrayType(string) or ar-rayType(realScalar)


Return the specified array, butsorted in descending order. re-alScalar is any scalar token ex-cept complex.

subarray arrayType(type), int, int arrayType(type) Extract a subarray starting at thespecified index with the specifiedlength.

sum arrayType(type) or [type] type Sum the elements of the specifiedarray or matrix. This throws anexception if the elements do notsupport addition or if the array isempty (an empty matrix will re-turn zero).

trace [type] type Return the trace of the specifiedmatrix.

transpose [type] [type] Return the transpose of the spec-ified matrix.

update int, arrayType(type) arrayType(type) Update an element in a an array.zeroMatrixComplex int, int [complex] Return a zero matrix with the

specified number of rows andcolumns.

zeroMatrixDouble int, int [double] Return a zero matrix with thespecified number of rows andcolumns.

zeroMatrixInt int, int [int] Return a zero matrix with thespecified number of rows andcolumns.

zeroMatrixLong int, int [long] Return a zero matrix with thespecified number of rows andcolumns.


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13. EXPRESSIONS

Table 13.10: Utility functions for evaluating expressionsFunction Argument type(s) Return type Description

eval string any type evaluate the specified expression(see 13.4.8).

parseInt string or string, int int return an int read from a string,using the given radix if a secondargument is provided.

parseLong string or string, int int return a long read from a string,using the given radix if a secondargument is provided.

toBinaryString int or long string return a binary representation ofthe argument

toOctalString int or long string return an octal representation ofthe argument

toString double or int or int, int orlong

string return a string representation ofthe argument, using the givenradix if a second argument is pro-vided.

traceEvaluation string string evaluate the specified expressionand report details on how it wasevaluated (see 13.4.8).


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Table 13.11: Functions performing signal processing operations, part 1.Function Argument type(s) Return type Description

close double, double boolean Return true if the first argumentis close to the second (withinEPSILON, where EPSILON isa static public variable of thisclass).

convolve arrayType(double),arrayType(double) orarrayType(complex),arrayType(complex)

arrayType(double)or array-Type(complex)

Convolve two arrays and returnan array whose length is sum ofthe lengths of the two argumentsminus one. Convolution of twoarrays is the same as polynomialmultiplication.

DCT arrayType(double) orarrayType(double), int orarrayType(double), int,int

arrayType(double) Return the Discrete CosineTransform of the specified array,using the specified (optional)length and normalization strategy(see 13.4.8).

downsample arrayType(double), int orarrayType(double), int,int

arrayType(double) Return a new array with every n-th element of the argument array,where n is the second argument.If a third argument is given, thenit must be between 0 and n − 1,and it specifies an offset into thearray (by giving the index of thefirst output).

FFT arrayType(double)orarrayType(complex)orarrayType(double), intarrayType(complex), int

arrayType(complex) Return the Fast Fourier Trans-form of the specified array. If thesecond argument is given withvalue n, then the length of thetransform is 2n. Otherwise, thelength is the next power of twogreater than or equal to the lengthof the input array. If the inputlength does not match this length,then input is padded with zeros.

generateBartlettWindow

int arrayType(double) Bartlett (rectangular) windowwith the specified length. Theend points have value 0.0, andif the length is odd, the centerpoint has value 1.0. For lengthM + 1, the formula is: w(n) ={2 nM; if0 ≤ n ≤ M

2

2− 2 nM; if M

2≤ n ≤M


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13. EXPRESSIONS


generateBlackmanWindow

int arrayType(double) Return a Blackman windowwith the specified length. Forlength M + 1, the formula is:w(n) = 0.42 + 0.5 cos ( 2πn

M) +

0.08 cos ( 4πnM

)

generateBlackmanHarrisWindow

int arrayType(double) Return a Blackman-Harriswindow with the specifiedlength. For length M + 1,the formula is: w(n) =0.35875 + 0.48829 cos ( 2πn

M) +

0.14128 cos ( 4πnM

) +0.01168 cos ( 6πn

M)

generateGaussianCurve

arrayType(double), ar-rayType(double), int

arrayType(double) Return a Gaussian curve with thespecified standard deviation, ex-tent, and length. The extent isa multiple of the standard devi-ation. For instance, to get 100samples of a Gaussian curve withstandard deviation 1.0 out to fourstandard deviations, use gener-ateGaussianCurve(1.0, 4.0, 100).

generateHammingWindow

int arrayType(double) Return a Hamming window withthe specified length. For lengthM + 1, the formula is:w(n) = 0.54− 0.46 cos ( 2πn

M)

generateHanningWindow

int arrayType(double) Return a Hanning window withthe specified length. For lengthM + 1, the formula is:w(n) = 0.5− 0.5 cos ( 2πn

/M)

generatePolynomialCurve

arrayType(double),double, double, int

arrayType(double) Return samples of a curve spec-ified by a polynomial. The firstargument is an array with thepolynomial coefficients, begin-ning with the constant term, thelinear term, the squared term,etc. The second argument is thevalue of the polynomial variableat which to begin, and the thirdargument is the increment on thisvariable for each successive sam-ple. The final argument is thelength of the returned array.


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generateRaisedCosinePulse

double, double, int arrayType(double) Return an array containing a symmet-ric raised-cosine pulse. This pulse iswidely used in communication sys-tems, and is called a “raised co-sine pulse” because the magnitude itsFourier transform has a shape thatranges from rectangular (if the ex-cess bandwidth is zero) to a cosinecurved that has been raised to benon-negative (for excess bandwidthof 1.0). The elements of the returnedarray are samples of the function:

h(t) =sin (πt

T)

πtT

× cos ( xπtT

)

1−( 2xtT

)2, where x

is the excess bandwidth (the first ar-gument) and T is the number of sam-ples from the center of the pulse tothe first zero crossing (the second ar-gument). The samples are taken witha sampling interval of 1.0, and thereturned array is symmetric and hasa length equal to the third argument.With an excess Bandwidth of 0.0, thispulse is a sinc pulse.

generateRectangularWindow

int arrayType(double) Return an array filled with 1.0 of thespecified length. This is a rectangularwindow.

IDCT arrayType(double) orarrayType(double), intor arrayType(double),int, int

arrayType(double) Return the inverse discrete cosinetransform of the specified array,using the specified (optional) lengthand normalization strategy (see13.4.8).

IFFT arrayType(double) orarrayType(complex) orarrayType(double), intarrayType(complex),int

arrayType(complex) inverse fast Fourier transform of thespecified array. If the second argu-ment is given with value n, then thelength of the transform is 2n. Other-wise, the length is the next power oftwo greater than or equal to the lengthof the input array. If the input lengthdoes not match this length, then inputis padded with zeros.


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13. EXPRESSIONS


nextPowerOfTwo double int Return the next power of twolarger than or equal to the argu-ment.

poleZeroToFrequency

arrayType(complex),arrayType(complex),complex, int

arrayType(complex) Given an array of pole locations,an array of zero locations, a gainterm, and a size, return an arrayof the specified size represent-ing the frequency response spec-ified by these poles, zeros, andgain. This is calculated by walk-ing around the unit circle andforming the product of the dis-tances to the zeros, dividing bythe product of the distances tothe poles, and multiplying by thegain.

sinc double double Return the sinc function,sin(x)/x, where special careis taken to ensure that 1.0 isreturned if the argument is 0.0.

toDecibels double double Return 20× log10 (z), where z isthe argument.

unwrap arrayType(double) arrayType(double) Modify the specified array to un-wrap the angles. That is, ifthe difference between succes-sive values is greater than π inmagnitude, then the second valueis modified by multiples of 2πuntil the difference is less than orequal to π. In addition, the firstelement is modified so that itsdifference from zero is less thanor equal to π in magnitude.

upsample arrayType(double), int arrayType(double) Return a new array that is the re-sult of inserting n− 1 zeroes be-tween each successive sample inthe input array, where n is thesecond argument. The returnedarray has length nL, where L isthe length of the argument array.It is required that n > 0.


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Table 13.15: Miscellaneous functions, part 1.Function Argument type(s) Return type Description

asURL string string Return a URL representation of the ar-gument.

cast type1, type2 type1 Return the second argument convertedto the type of the first, or throw an ex-ception if the conversion is invalid.

constants none record Return a record identifying all theglobally defined constants in the ex-pression language.

findFile string string Given a file name relative to the userdirectory, current directory, or class-path, return the absolute file name ofthe first match, or return the name un-changed if no match is found.

filter function,arrayType(type)

arrayType(type) Extract a sub-array consisting of all ofthe elements of an array for which thegiven predicate function returns true.

filter function,arrayType(type), int

arrayType(type) Extract a sub-array with a limited sizeconsisting of all of the elements ofan array for which the given predicatefunction returns true.

freeMemory none long Return the approximate number ofbytes available for future memory al-location.

iterate function, int, type arrayType(type) Return an array that results from firstapplying the specified function to thethird argument, then applying it to theresult of that application, and repeat-ing to get an array whose length isgiven by the second argument.

map function, arrayType(type) arrayType(type) Return an array that results from ap-plying the specified function to the el-ements of the specified array.

property string string Return a system property with thespecified name from the environ-ment, or an empty string if thereis none. Some useful proper-ties are java.version, ptolemy.ptII.dir,ptolemy.ptII.dirAsURL, and user.dir.


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13. EXPRESSIONS

Table 13.16: Miscellaneous functions, part 2.Function Argument type(s) Return type Description

readFile string string Get the string text in the specified file,or throw an exception if the file can-not be found. The file can be absolute,or relative to the current working direc-tory (user.dir), the user’s home direc-tory (user.home), or the classpath. Thereadfile function is often used with theeval function.

readResource string string Get the string text in the specified re-source (which is a file found relative tothe classpath), or throw an exception ifthe file cannot be found.

totalMemory none long Return the approximate number ofbytes used by current objects plus thoseavailable for future object allocation.

yesNoQuestion string boolean Query the user for a yes-no answer andreturn a boolean. This function willopen a dialog if a GUI is available, andotherwise will use standard input andoutput.


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