(*^

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Piecewise Convolution
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by
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Kevin A. West, Brian L. Evans, and
James H. McClellan

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Digital Signal Processing Laboratory
School of Electrical Engineering
Georgia Institute of Technology
Atlanta, GA 30332-0250

EMAIL:  evans@eedsp.gatech.edu
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		This notebook contains examples of the continuous-time convolution operation and its supporting operations.
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		Remember that to expand a group of cells, just move the mouse to the downward pointing half arrow and click twice.	
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WARNING:  Under Mathematica 2.0, all graphics are generated using raw PostScript code so each graphics cell requires an extra 20 kb of memory.  Unfortunately, this notebook contains 72 graphics cells thereby requiring an extrat 1.5 Mb of memory.  Thus, the loading of this notebook may overwhelm a Mac or IBM PC.
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1.0 Introduction
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Continuous-Time Convolution
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		The continuous-time convolution operation is represented by a double asterisk ** in infix notation, as shown below:
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y(t) = x(t) ** h(t)
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The operation shown above, which takes the functions x(t) and h(t) as arguments and returns the function y(t), is defined by the integral below:
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This integral represents the effect that a linear system with impulse response h(t) would have on a signal x(t).  It has several  important properties, some of which are shown below:

		x(t) ** d(t) = x(t)
		x(t) ** h(t) = h(t) ** x(t)


Here, d(t) is the Dirac delta function (represented by Delta in Mathematica). CTPiecewiseConvolution performs the integration required, exploiting the properties shown above. 
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Discrete-Time Convolution
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		The discrete-time convolution operation is also represented by a double asterisk
** in infix notation, as shown below:
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y[n] = x[n] ** h[n]
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The operation shown above, which takes the functions x[n] and h[n] as arguments and returns the function y[n], is defined by the summation below:
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This integral represents the effect that a linear system with impulse response h[n] would have on a signal x[n].  It has several  important properties, some of which are shown below:
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x[n] ** d[n] = x[n]
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x[n] ** h[n] = h[n] ** x[n]
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Here, d[n] is the Kronecker delta function (represented by Impulse in Mathematica).  DTPiecewiseConvolution performs the integration required, exploiting the properties shown above. 
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Piecewise Functions in Mathematica
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		Often, h(t) and x(t) are defined in a piecewise manner.  A piecewise-definedF-interval function is a function which is defined separately over separate F-interval rather than as a single function defined over all time.  For example, consider the following triangular pulse:
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This function is most easily represented by the following two F-interval definitions:
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f(t) = t+1, -1 <= t < 0
f(t) = -t+1, 0 <= t < 1
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This function has two representations in the PiecewiseConvolution package:
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(a) expression notation:
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(1 - t)*CPulse[1, t] + (1 + t)*CPulse[1, 1 + t]
;[o]
(1 - t) CPulse[1, t] + (1 + t) CPulse[1, 1 + t]
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(b) list-of-intervals notation:
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{{1 + t, -1, 0}, {1 - t, 0, 1}}
;[o]
{{1 + t, -1, 0}, {1 - t, 0, 1}}
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These representations are equivalent and can be converted from one to the other.  These notations will be discussed in Data Structures (Section 3.0).  The convolution functions PiecewiseConvolution, DTPiecewiseConvolution, and CTPiecewiseConvolution automatically convert their inputs into list form before convolving them.
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2.0 Initialization
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		Before any Mathematica expressions in this notebook can be evaluated, the "Convolution.m" package and its supporting packages must be loaded.  This is accomplished by evaluating the following Mathematica expression (since it is marked as an initialization cell, it is often evaluated when the notebook is opened):
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*)
Needs[ "SignalProcessing`Support`Convolution`" ]
(*
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Support module has been loaded.
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The knowledge base of signal processing operators has\
 
  been loaded.
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Piecewise convolution rules have been loaded.
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3.0 Data Structures
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		This section discusses the data structures in the context of the continuous domain.  The data structures in the discrete domain are, however, essentially the same.  The only real difference is that specifying an impulse function in interval notation is different between the two domains.  In the continuous domain, the impulse function is the Dirac delta which is specified by its location and the area under the spike.  Hence, a special Area data tag is used in the interval format.  In the discrete domain, the impulse is the Kronecker delta function which is specified by its location and its amplitude so no special data tag is necessary.
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3.1 Finite- and Infinite-Extent F-intervals
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Finite-Extent F-intervals
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A finite-valued interval is represented as a list containing three items, called an F-interval: 
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{ f(t), t1, t2 }
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where f(t) is a particular function of t, t1 is the earliest time at which the function is non-zero, and t2 is the latest time at which the function is non-zero.  When both t1 and t2 are finite values, the interval represents a pulse which begins at time t1 and ends at time t2.
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MatrixForm[{{"Interval", "Function          "}, 
   {"{f(t), t1, t2}", "f(t), t1 <= t < t2"}, 
   {" ", "      0 otherwise "}}]
;[o]
Interval            Function          
 
{f(t), t1, t2}      f(t), t1 <= t < t2
 
                          0 otherwise
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Infinite-Extent F-intervals
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When t2 = Infinity, the interval represents a step function.  When t1 = -Infinity, the interval represents a step function, but one that is reversed in time from the conventional step function.
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MatrixForm[{{"Interval", "Expression   "}, 
   {"{f(t), t1, Infinity}", "CStep[t - t1]"}, 
   {"{f(t), -Infinity, t2}", "CStep[t2 - t]"}}]
;[o]
Interval               Expression   
 
{f(t), t1, Infinity}   CStep[t - t1]
 
{f(t), -Infinity, t2}  CStep[t2 - t]
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When t1 = -Infinity and t2 = Infinity, then f(t) is valued over all time, as in
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{ f(t), -Infinity, Infinity }
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Valid F-intervals
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An F-interval, {f, t1, t2},  is considered invalid if t1 is greater than or equal to t2.  Also, an interval which has a function f(t) = 0, while not having an invalid value, is an improper representation because such an interval represents f(t) = 0 for all t.  Such a function does not require an F-interval representation.  Therefore,
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{ f(t), t1, t2 } is valid for t1 <= t < t2
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{ 0, t1, t2 } represents 0, for any t                  
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Plotting F-intervals
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		The following example shows a function defined only on the interval [-Pi/4, Pi):
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PlotList[ {{Cos[t], - Pi/4, Pi}},
          {t, -1, 4},
          Domain -> Continuous ]
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(Note that the option Domain -> Continuous is not necessary if the value of $ConvolutionDomain is set to Continuous.)  The PlotList function will be discussed in Section 4.2.
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3.2 Delta Functions as F-intervals
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		The Dirac delta function is a special "function" which is defined by what it does rather than what it is.  The unit delta d(t), represented in Mathematica as Delta[t],  is defined as
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x(t) = d(t) ** x(t)
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for any x(t).  The unit delta function has the following properties:
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(1) 	d(t) is undefined at t = 0,
(2) 	d(t) is zero for all non-zero t,
(3)	d(t) = d/dt CStep[t], and
(4)	the integral of d(t) over all t is 1:
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A delta function can be a shifted and scaled version of the unit delta function.  The shifting affects at what time the delta function is undefined and the scaling affects the area under the function:

(1)  A d(t - t0) is undefined at t = t0, and
(2)	the integral of A d(t - t0) over all t is A:
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In Mathematica terms,
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CTPiecewiseConvolution[ A, Delta[t - t0], t ]
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{{A, DirectedInfinity[-1], DirectedInfinity[1]}}
;[o]
{{A, -Infinity, Infinity}}
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A more general example of the sifting property is
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CTPiecewiseConvolution[ Delta[t - t0], f[t], t ]
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{{f[t - t0], DirectedInfinity[-1], DirectedInfinity[1]}}
;[o]
{{f[t - t0], -Infinity, Infinity}}
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The F-interval representation of a delta function is also a list of three items.  The items are arranged similarly to those in a finite-valued interval, but with two exceptions.  Instead of a function f(t), a delta has an area A, which is represented by the tag Area.  Also, since there is only one time of interest in a delta function, the t1 and t2 values are equal to t0, the time at which a delta function occurs.
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{ Area[A], t0, t0 } represents the delta function A Delta[t - t0].
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 		This representation of a delta function can be considered a generalization of the finite-valued interval.  The Area[A] tag indicates that the function is not an ordinary function of t.  The endpoints being equal to one other is consistent with the fact that all values outside of the interval are equal to zero.  An F-interval which contains an Area tag, but different values for the endpoints in invalid.  An F-interval representing a delta function with zero area is improper because such an interval represents f(t) for all t.
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{ Area[A], t1, t2 } is invalid if t1 <> t2
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{ Area[0], t0, t0 } represents the value 0 for any t		
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3.3 List of F-intervals
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		Some functions are defined piecewise, which means that the analytic function f(t) may be different for different intervals.  To define these functions, a list of F-intervals is used.  A list of F-intervals has the following form:
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{ { e(t), a1, b1 }, { f(t), a2, b2 }, { g(t), a3, b3 }, ...  }
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where e(t), f(t), and g(t) are functions of t, and the a's and b's are the corresponding interval endpoints.  The ellipsis (...) indicates that a list of F-intervals does not have a fixed length.  It can contain as many intervals as are needed to define the piecewise function.  For example, a triangular pulse which begins at t=-1, ends at t=1, and has peak height 1 at t=0 is defined in two intervals:
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MatrixForm[{{"     = t + 1", "-1 <= t < 0"}, 
   {"f(t) = 1 - t", "0 <= t < 1 "}, 
   {"     = 0    ", "otherwise"}}]
;[o]
     = t + 1  -1 <= t < 0
 
f(t) = 1 - t  0 <= t < 1 
 
     = 0      otherwise
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Using the F-interval data structure, it would take two F-intervals to describe this function:  one for the time during which f(t) = t + 1,  and another for the time during which f(t) = 1 - t.  By placing these two intervals in a list, the function is correctly represented as a list of intervals:
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{ {-1, -2, -1}, {t, -1, 1}, {4/t, 2, Infinity} }		
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We can plot this piecewise function using PlotList:
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PlotList[ {{-1, -2, -1}, {t, -1, 2}, {4/t, 2, Infinity}},
          {t, -3, 5},
          Domain -> Continuous ]
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The Unformatted text for this cell was not generated.
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when Unformatted text is generated.
;[o]
-Graphics-
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(Again, note that the option Domain -> Continuous is not necessary if the value of $ConvolutionDomain is Continuous.)  Any piecewise function can be represented by a list of F-intervals which together define the function.  These intervals can be either finite-valued or delta functions.  Using combinations of intervals, an impulse train can be defined:
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{ {Area[1], 0, 0}, {Area[1], 1, 1}, {Area[1], 2, 2} }
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Finite-valued intervals and delta function intervals can be mixed in a list to describe a function which contains deltas and finite values over certain intervals.		
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3.4 Converting Between Expressions and Lists of F-intervals
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		Every piecewise function has two equivalent representations:  an expression and a list of F-intervals.  An expression uses Delta, CStep, and Pulse objects to define a function.  It can be readily converted to a list-of-interval representation, and vica-versa.  The conversions are based on the following corresponding representations:
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MatrixForm[{{"-- F-interval --        ", 
    "-- Expression -- "}, 
   {"{Area[a], t0, t0}       ", "a Delta[t - t0]  "}, 
   {"{f, -Infinity, Infinity}", "f                "}, 
   {"{f, t1, Infinity}       ", "f CStep[t - t1]  "}, 
   {"{f, -Infinity, t2}      ", "f CStep[t2 - t]  "}, 
   {"{f, t1, t2}             ", "f CPulse[t2 - t1]"}}]
;[o]
-- F-interval --          -- Expression -- 
 
{Area[a], t0, t0}         a Delta[t - t0]  
 
{f, -Infinity, Infinity}  f                
 
{f, t1, Infinity}         f CStep[t - t1]  
 
{f, -Infinity, t2}        f CStep[t2 - t]  
 
{f, t1, t2}               f CPulse[t2 - t1]
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Converting between expressions and lists of intervals is done using the ConvertFromList and ConvertToList operators.
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3.4.1 ConvertFromList
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		ConvertFromList takes two arguments in the following format:
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ConvertFromList[list, t]
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where list is a list of intervals (either deltas or finite intervals) , and t is the time variable. The following example Mathematica input expressions illustrate the use and results of the ConvertFromList operator.  We are assuming that the default convolution domain has been set to Continuous.
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--  Conversion of a symbolically defined delta interval to an expression:
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ConvertFromList[{{Area[a], t0, t0}}, t]
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a*Delta[t - t0]
;[o]
a Delta[t - t0]
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--  Conversion to a two-sided function (defined from -Infinity to Infinity)
     to an expression:
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ConvertFromList[{{t^2, -Infinity, Infinity}}, t]
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t^2
;[o]
 2
t
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--  Conversion to a right-sided function (defined from some t0 to Infinity)
     to an expression:
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ConvertFromList[{{Exp[t],0,Infinity}},t]
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E^t*CStep[t]
;[o]
 t
E  CStep[t]
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--  Conversion of a finite-duration interval to a pulse function expression:
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ConvertFromList[{{Sin[t],a,b}},t]
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CPulse[-a + b, -a + t]*Sin[t]
;[o]
CPulse       [-a + t] Sin[t]
       -a + b
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--  Conversion of two finite-duration intervals to an expression:
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ConvertFromList[{{t,-1,1},{1,0,1}},t]
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CPulse[1, t] + t*CPulse[2, 1 + t]
;[o]
CPulse  [t] + t CPulse  [1 + t]
       1               2
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3.4.2 ConvertToList
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		ConvertToList is the inverse operator to ConvertFromList.  ConvertToList takes two arguments in the following format:
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ConvertToList[exp, t]
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where exp is an expression which may contain Delta, CStep, or CPulse objects and where t is the time variable.  The following example Mathematica input operations illustrate the use of the ConvertToList operator.  Again, we assuming that the default convolution domain is Continuous.
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--  Conversion of a delta function expression to a list:
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ConvertToList[3 Delta[t-5], t]
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{{Area[3], 5, 5}}
;[o]
{{Area[3], 5, 5}}
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--  Conversion of a two-sided expression (defined over all time) to a list:
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ConvertToList[Exp[t], t]
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{{E^t, DirectedInfinity[-1], DirectedInfinity[1]}}
;[o]
   t
{{E , -Infinity, Infinity}}
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--  Conversion of  a right-sided function to a list:
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ConvertToList[Sin[t] CStep[t - 2 Pi], t]
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{{Sin[t], 2*Pi, DirectedInfinity[1]}}
;[o]
{{Sin[t], 2 Pi, Infinity}}
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--  Conversion of a general expression to a list:
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ConvertToList[2 t CPulse[3,t] - Delta[t-8] +
              4 CStep[-t-10], t]
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{{4, DirectedInfinity[-1], -10}, {2*t, 0, 3}, 
  {Area[-1], 8, 8}}
;[o]
{{4, -Infinity, -10}, {2 t, 0, 3}, {Area[-1], 8, 8}}
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4.0 New Functions
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		Here are the functions introduced by the convolution package:
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AutoCorrelation         ConvertFromList         ConvertToList
CTPiecewiseConvolution  DTPiecewiseConvolution  IntervalQ
PiecewiseConvolution    PlotList                SimplifyList
ValidIntervalQ
:[font = text; inactive; ]
Section 4.3 below gives help information on each new object.  Except for DTPiecewiseConvolution and CTPiecewiseConvolution, these objects work for both continuous and discrete expressions and assume that you are working in the domain the symbol to which $ConvolutionDomain evaluates (which initially is the continuous domain):
:[font = input; startGroup; ]
$ConvolutionDomain
:[font = output; inactive; output; endGroup; ]
Continuous
;[o]
Continuous
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One can reset the default domain at any time by using SetConvolutionDomain.  When you call a function, you can also override the default domain by using the Domain option:
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SimplifyList[ {{n, 1, 10}, {10, 11, 20}}, n,
              Domain -> Discrete ]
:[font = output; inactive; output; endGroup; ]
{{n, 1, 10}, {10, 11, 20}}
;[o]
{{n, 1, 10}, {10, 11, 20}}
:[font = text; inactive; ]
The three convolution objects are discussed in Sections 5 and 6.
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4.1 SimplifyList
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		A list of intervals has certain conditions which must be met in order for it to be considered in simplest form.  A list may be valid, but not in simplest form.  Take. for example, the following list:
:[font = input; center; ]
{ {t, 1, 2}, {1, 0, 3} }
:[font = text; inactive; ]
Both of the intervals are valid intervals, but they overlap.  If this overlap is interpreted as the superposition of  the intervals, then the list has the following equivalent representation:
:[font = input; center; ]
 { {1, 0, 1}, {1 + t, 1, 2}, {1, 2, 3} }
:[font = text; inactive; ]
Notice the conditions that a list must meet to be considered in simplest form:
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			1.  No invalid intervals
			2.  No overlap of intervals
			3.  The ordering of the intervals must occur from low time to high time
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The SimplifyList operation takes one argument, a list of intervals, and returns an equivalent list which is in simplest form.  The following example Mathematica input operations illustrate the use and results of the SimplifyList operator:
:[font = text; inactive; ]
---  Simplification of a list with overlapping intervals:
:[font = input; startGroup; ]
SimplifyList[{{t, 1, 2}, {1, 0, 3}}]
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{{1, 0, 1}, {1 + t, 1, 2}, {1, 2, 3}}
;[o]
{{1, 0, 1}, {1 + t, 1, 2}, {1, 2, 3}}
:[font = text; inactive; ]
---  Simplification of a list with one interval split by a delta function:
:[font = input; startGroup; ]
SimplifyList[{{t^2, -1, 1}, {Area[1], 0, 0}}, t]
:[font = output; inactive; output; endGroup; ]
{{t^2, -1, 0}, {Area[1], 0, 0}, {t^2, 0, 1}}
;[o]
   2                             2
{{t , -1, 0}, {Area[1], 0, 0}, {t , 0, 1}}
:[font = text; inactive; ]
---  Omission of  invalid intervals:
:[font = input; startGroup; ]
SimplifyList[ {{Area[1], 0, 0}, {Area[0], 1, 1},
               {1, 2, 3}, {t, 10, 9}}, t ]
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{{Area[1], 0, 0}, {1, 2, 3}}
;[o]
{{Area[1], 0, 0}, {1, 2, 3}}
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--- Reordering of intervals according to which occurs first in time:
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SimplifyList[{{t, 2, 4}, {Sin[t], 0, 2}}, t]
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{{Sin[t], 0, 2}, {t, 2, 4}}
;[o]
{{Sin[t], 0, 2}, {t, 2, 4}}
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4.2 PlotList
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The PlotList operation is similar to the Mathematica operation Plot, except that the functional representation is given as a list of F-intervals rather than as an expression.  The PlotList operation uses the following format:
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PlotList[ list, {t,t1,t2} ]
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where list is a list of intervals, t is the variable against which the list is plotted, and t1 and t2 define the range in which the list is plotted.  
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PlotList[ {{1, 0, 5}, {Area[2], 6, 6}},
          {t, -1, 8},
		  Domain -> Continuous ]
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4.3 Help Information
:[font = text; inactive; ]
		Almost every function below has the following options:
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Options[PiecewiseConvolution]
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{Domain :> $ConvolutionDomain}
;[o]
{Domain :> $ConvolutionDomain}
:[font = text; inactive; ]
This form of default options sets the default domain for the convolution objects to the value of $ConvolutionDomain, which is initially Continuous.  The global setting of $ConvolutionDomain can be reset by using SetConvolutionDomain.
:[font = subsubsection; inactive; startGroup; Cclosed; ]
AutoCorrelation    (c.f. Section 6.3)
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AutoCorrelation[e, v] computes the autocorrelation of
   expression e (in the form of a list of F-intervals,
   an F-interval, or a function) with respect to v. The
   variable v is treated as either discrete and
   continuous according to the value of the Domain
   option. The default domain is the value of
   $ConvolutionDomain.
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ConvertFromList  (c.f. Section 3.4)
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ConvertFromList[l, t] converts a list of F-intervals in
   the form {f, t1, t2} to a signal processing
   expression. The new expression will contain either
   Delta, CStep, and CPulse functions (continuous-time)
   or Impulse, Step, and Pulse functions
   (discrete-time) according to the value of the Domain
   option. The default domain is the value of
   $ConvolutionDomain.
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ConvertToList      (c.f. Section 3.4)
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ConvertToList[e, t] converts the expression e to a list
   of F-intervals in the form {fun, left, right}. The
   domain of the new expression is established
   according to the value of the Domain option. The
   default domain is the value of $ConvolutionDomain.
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CTPiecewiseConvolution  (c.f. Section 5.0)
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CTPiecewiseConvolution[f, g, t] carries out the
   one-dimensional,  continuous-time piecewise
   convolution of f and g with respect to t. See
   PiecewiseConvolution for valid representation of f
   and g. Using F-interval notation {function,
   left_endpoint, right_endpoint}, a Dirac delta is
   meant when function = Area[<area>] (where <area>
   equals the area of under the impulse function) and
   left_endpoint = right_endpoint (which is the
   location of the impulse).
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DTPiecewiseConvolution  (c.f. Section 6.0)
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DTPiecewiseConvolution[f, g, n] carries out the
   one-dimensional,  discrete-time piecewise
   convolution of f and g with respect to n. See
   PiecewiseConvolution for the valid representations
   of f and g. Using F-interval notation {function,
   left_endpoint, right_endpoint}, an impulse
   (Kronecker delta) function is meant when function =
   <value> (where <value> equals the strength of the
   impulse function) and left_endpoint = right_endpoint
   (which is the location of the impulse).
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IntervalQ
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IntervalQ[x] returns True if the argument x is an
   F-interval of the form {f, t1, t2} where none of f,
   t1, or t2 are lists and False otherwise.
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PiecewiseConvolution  (c.f.  Sections 5.0 and 6.0)
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PiecewiseConvolution[f, g, v] convolves the piecewise
   functions f and g with respect to the variable t. In
   this context, a function is represented in a
   piecewise fashion: (1) as F-intervals of the form
   {fun, left, right}, (2) as a list of F-intervals, or
   (3) as an expression. An F-interval has the form
   {function, left_endpoint, right_endpoint}. The
   F-interval notation represents a finite-extent
   function or sequence when the endpoints do not equal
   infinity. The result of convolution is always
   returned as a list of F-intervals. The variable v is
   treated as either discrete and continuous according
   to the value of the Domain option. The default
   domain is the value of $ConvolutionDomain. See also
   CTPiecewiseConvolution and DTPiecewiseConvolution.
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PlotList  (c.f. Section 4.2)
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PlotList[l, {v, vstart, vend}] plots the piecewise
   function l vs. v. The variable v is treated as
   either discrete and continuous according to the
   value of the Domain option. The default domain is
   the value of $ConvolutionDomain.
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SetConvolutionDomain
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SetConvolutionDomain[domain] establishes the default
   convolution domain as either Continuous or Discrete.
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SimplifyList  (c.f. Section 4.1)
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SimplifyList[interval_list, v] takes a list of
   F-intervals each having the form {f, t1, t2} and
   returns a simplified sorted list of F-intervals. The
   interval_list is treated as either a continuous-time
   or a discrete-time function according the value of
   the Domain option. The variable v is treated as
   either discrete and continuous according to the
   value of the Domain option. The default domain is
   the value of $ConvolutionDomain.
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ValidIntervalQ
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ValidIntervalQ[i] takes an F-interval i and returns
   True if it is a meaningful representation of an
   interval in either the domain specified by the
   Domain option or the default domain
   $ConvolutionDomain if no Domain option is present.
   Otherwise, it returns False.
:[font = section; inactive; startGroup; Cclosed; ]
5.0 Continuous Convolution
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5.1 Convolution of Expressions
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		The PiecewiseConvolution function can accept two expressions as the f(t) and g(t) arguments.  PiecewiseConvolution will return the value f(t) ** g(t) expressed as a list of F-intervals.  The following examples of Mathematica dialogue illustrate the use and results of the PiecewiseConvolution operator when performing the convolution of two expressions.
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Shifting and scaling effect of the Delta function
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PiecewiseConvolution[Delta[t - t0], f[t], t]
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{{f[t - t0], DirectedInfinity[-1], 
   DirectedInfinity[1]}}
;[o]
{{f[t - t0], -Infinity, Infinity}}
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PiecewiseConvolution[4 Delta[t - 3], t CStep[t], t]
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{{4*(-3 + t), 3, DirectedInfinity[1]}}
;[o]
{{4 (-3 + t), 3, Infinity}}
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PiecewiseConvolution[3 Delta[t + 2],
                     CStep[t] + 6 Delta[t - 1], t]
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{{3, -2, -1}, {Area[18], -1, -1}, 
  {3, -1, DirectedInfinity[1]}}
;[o]
{{3, -2, -1}, {Area[18], -1, -1}, {3, -1, Infinity}}
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Unit step function as an integrator
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PiecewiseConvolution[CStep[t], CStep[t], t]
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Integrate::divg: Integral does not converge.
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{{t, 0, DirectedInfinity[1]}}
;[o]
{{t, 0, Infinity}}
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Plot[CStep[t], {t, -1, 5}]
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PiecewiseConvolution[CStep[t], (4 Cos[t] + t) CStep[t], t]
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Limit::nlm: Could not find definite limit.
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Limit::nlm: Could not find definite limit.
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General::stop: 
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5.2 Convolution of F-intervals
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		The PiecewiseConvolution function can accept two intervals as the f(t) and g(t) arguments.  PiecewiseConvolution will return the value f(t)**g(t) expressed as a list of intervals.  Each list will have a maximum of three F-intervals.  This is because there are three ways the functions f(u) and g(t - u) can overlap when plotted against u, provided that f(u) and g(t - u) are functions that can be represented by a single F-interval.
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		Consider the convolution integral:
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Simple Example
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For the following example illustrations, let x(t) be the function corresponding to the interval {2, 0, 1}, and let h(t) be the function corresponding to the interval {1, 0, 2}.
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x(t) = 2    0 <= t < 1
 
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MatrixForm[{{"h(t) = 1", "0 <= t < 2"}, 
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h(t) = 1    0 <= t < 2
 
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If t is small enough such that the non-zero parts of x(u) and h(t-u) do not overlap at all, then their product will be zero everywhere.  The following graph illustrates x(u) and h(t-u) plotted together for t = -1/2:
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As t  increases, the functions will begin to overlap.  The first case of overlap encountered is when neither function begins and ends before the other; that is, the time interval of neither function is contained in the time interval of the other.  The following graph illustrates x(u) and h(t-u) plotted together for t = 1/2:
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The second case of overlap is when one function begins and ends before the other.  This means that the time interval of one function is contained in the other.  The following graph illustrates x(u) and h(t-u) plotted together for t = 3/2:
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The third case of overlap is when t becomes so large that the conditions of the second case no longer hold.  This case is just like the first case of overlap: neither function begins and ends before the other.  The following graph illustrates x(u) and h(t-u) plotted together for t = 5/2:
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Finally, t becomes so large that, once again, the functions x(u) and h(t-u) no longer overlap.  The product is zero everywhere.  The following graph illustrates x(u) and h(t-u) plotted together for t = 7/2:
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In general, the convolution integral can be evaluated in these different cases simply by changing the endpoints of integration.  Suppose that you have two functions f (t) and g(t) such that

			f(t) can be non-zero over the interval [t1, t2)
			g(t) can be non-zero over the interval [t3, t4)
			(t2-t1) >= (t4-t3)

Then, if  t is in the interval (-Infinity, t1+t3) or [t2+t4,Infinity), the value of f(t)*g(t) will evaluate to zero.  If t falls in one of the following intervals, then f(t)*g(t) will evaluate to the definite integral of f(t)g(t-u) in the interval [a,b], shown below:

			Interval					a								b

			[t1+t3, t1+t4]			t1								t-t3
			[t1+t4, t2+t3]			t-t4					t-t3
			[t2+t3, t2+t4]			t-t4					t2
			
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Animated Example
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		The following animation example illustrates the convolution process by "sliding" one function against another and showing the resulting function as it is generated in time.  We will convolve a with b to get c.  Please evaluate the cells containing the definitions for a, b, c, and cc.
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a = ConvertFromList[{{2,0,2},{1,2,3}},t]
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CPulse[1, -2 + t] + 2*CPulse[2, t]
;[o]
CPulse  [-2 + t] + 2 CPulse  [t]
       1                    2
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b = ConvertFromList[{{2 t,0,2}},t]
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2*t*CPulse[2, t]
;[o]
2 t CPulse  [t]
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c = PiecewiseConvolution[a,b,t]
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{{2*t^2, 0, 2}, {4 + 4*t - t^2, 2, 3}, 
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;[o]
     2                     2
{{2 t , 0, 2}, {4 + 4 t - t , 2, 3}, 
 
                  2
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cc = ConvertFromList[c,t]
(*
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  (4 + 4*t - t^2)*CPulse[1, -2 + t] + 
  2*t^2*CPulse[2, t]
;[o]
-((-5 + t) (-1 + t) CPulse  [-4 + t]) + 
                           1
 
                  2
  (-5 + 10 t - 2 t ) CPulse  [-3 + t] + 
                            1
 
              2                        2
  (4 + 4 t - t ) CPulse  [-2 + t] + 2 t  CPulse  [t]
                        1                       2
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-->  Code to Generate the Animation
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Do[Plot[{a,b},{t,-4,6},PlotRange->{0,8}],{i,5}];
Do[aa = a /. t->-t; Plot[{aa,b},{t,-4,6},PlotRange->{0,8}],{i,5}];
Do[aa = a /. t->i-t; Plot[{aa,b,cc CStep[i-t]},
{t,-4,6},PlotRange->{0,8}],{i,0,5,.2}]
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5.3 Convolution of Lists of F-intervals
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		The PiecewiseConvolution function can accept an expression, an interval, or a list of intervals for either of its first two arguments.  If these arguments are lists, then the Convolution operator automatically simplifies the result using the SimplifyList operator.  This result will be a list of intervals.  There may be large number of  intervals in the result, depending, of course, on the complexity of the arguments.  
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Example:  Convolving Two Functions in List Form
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		The following example takes two lists and computes their convolution.  Notice that the resulting list contains eleven intervals.
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a = {{1,1,2},{1,4,Infinity}}
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{{1, 1, 2}, {1, 4, DirectedInfinity[1]}}
;[o]
{{1, 1, 2}, {1, 4, Infinity}}
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b = {{1+t,0,1},{1,1,5},{5-t,5,6},{4-t,6,7}}
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{{1 + t, 0, 1}, {1, 1, 5}, {5 - t, 5, 6}, {4 - t, 6, 7}}
;[o]
{{1 + t, 0, 1}, {1, 1, 5}, {5 - t, 5, 6}, {4 - t, 6, 7}}
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c = PiecewiseConvolution[a,b,t]
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{{(-1 + t^2)/2, 1, 2}, {-(1 - 4*t + t^2)/2, 2, 3}, 
  {1, 3, 4}, {5 - 3*t + t^2/2, 4, 5}, {-5/2 + t, 5, 6}, 
  {-(29 - 12*t + t^2)/2, 6, 7}, {10 - t, 7, 8}, 
  {10 - 5*t + t^2/2, 8, 9}, {-35 + 9*t - t^2/2, 9, 10}, 
  {-25 + 8*t - t^2/2, 10, 11}, 
  {5/2, 11, DirectedInfinity[1]}}
;[o]
        2                       2
  -1 + t           -(1 - 4 t + t )
{{-------, 1, 2}, {---------------, 2, 3}, {1, 3, 4}, 
     2                    2
 
              2
             t             5
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             2             2
 
                  2
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           2
 
               2                       2
              t                       t
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              2                       2
 
                2
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Example:  Convolving an Expression with a Function in List Form
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		The next example shows the convolution of an expression with a list.  The list is a finite pulse train, and the expression is a decaying exponential.  Notice the long list of adjacent intervals.
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list= {{10,0,.1},{10,1,1.1},{10,2,2.1},{10,3,3.1},{10,4,4.1}}
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{{10, 0, 0.1}, {10, 1, 1.1}, {10, 2, 2.1}, {10, 3, 3.1}, 
  {10, 4, 4.1}}
;[o]
{{10, 0, 0.1}, {10, 1, 1.1}, {10, 2, 2.1}, {10, 3, 3.1}, 
 
  {10, 4, 4.1}}
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expr = Exp[-t] CStep[t]
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CStep[t]/E^t
;[o]
CStep[t]
--------
    t
   E
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conv = PiecewiseConvolution[list,expr,t]
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{{10 - 10/E^t, 0, 0.1}, {1.051709180756477/E^t, 0.1, 1}, 
  {(1.051709180756477 - 10*E + 10*E^t)/E^t, 1, 1.1}, 
  {(31.09336942022081 - 10*E)/E^t, 1.1, 2}, 
  {(31.09336942022081 - 10*E - 10*E^2 + 10*E^t)/E^t, 2, 
   2.1}, {(112.7550685458973 - 10*E - 10*E^2)/E^t, 2.1, 
   3}, {(112.7550685458973 - 10*E - 10*E^2 - 10*E^3 + 
      10*E^t)/E^t, 3, 3.1}, 
  {(334.7345813603137 - 10*E - 10*E^2 - 10*E^3)/E^t, 3.1, 
   4}, {(334.7345813603137 - 10*E - 10*E^2 - 10*E^3 - 
      10*E^4 + 10*E^t)/E^t, 4, 4.1}, 
  {(938.137457333933 - 10*E - 10*E^2 - 10*E^3 - 10*E^4)/
    E^t, 4.1, DirectedInfinity[1]}}
;[o]
       10            1.05171
{{10 - --, 0, 0.1}, {-------, 0.1, 1}, 
        t               t
       E               E
 
                        t
   1.05171 - 10 E + 10 E
  {----------------------, 1, 1.1}, 
              t
             E
 
   31.0934 - 10 E
  {--------------, 1.1, 2}, 
          t
         E
 
                        2       t
   31.0934 - 10 E - 10 E  + 10 E
  {------------------------------, 2, 2.1}, 
                  t
                 E
 
                        2
   112.755 - 10 E - 10 E
  {----------------------, 2.1, 3}, 
              t
             E
 
                        2       3       t
   112.755 - 10 E - 10 E  - 10 E  + 10 E
  {--------------------------------------, 3, 3.1}, 
                      t
                     E
 
                        2       3
   334.735 - 10 E - 10 E  - 10 E
  {------------------------------, 3.1, 4}, 
                  t
                 E
 
                        2       3       4       t
   334.735 - 10 E - 10 E  - 10 E  - 10 E  + 10 E
  {----------------------------------------------, 4, 
                          t
                         E
 
                               2       3       4
          938.137 - 10 E - 10 E  - 10 E  - 10 E
   4.1}, {--------------------------------------, 4.1, 
                             t
                            E
 
   Infinity}}
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      {8.75, 0.01429752301988499}, 
      {8.95833333333333, 0.01160867859977}, 
      {9.16666666666667, 0.00942550808593432}, 
      {9.375, 0.007652912595906784}, 
      {9.58333333333333, 0.006213677890530722}, 
      {10., 0.00409630892841604}}]}}, 
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5.4 Autocorrelation
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		The Autocorrelation operator, a unary operator, is defined as the convolution of a function with itself reversed in time.  It  takes two arguments.  The first argument is either an expression, interval, or list of intervals, and the second argument is the variable of  convolution.  Notice that the result of the AutoCorrelation object is always symmetric in time; that is, if f(t) is the result returned by the AutoCorrelation operator, then f(t) = f(-t).
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Example:  Autocorrelation of a Trapezoidal Function
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x = {{t,0,1},{1,1,2},{3-t,2,3}}
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{{t, 0, 1}, {1, 1, 2}, {3 - t, 2, 3}}
;[o]
{{t, 0, 1}, {1, 1, 2}, {3 - t, 2, 3}}
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autox = AutoCorrelation[x,t]
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{{(3 + t)^3/6, -3, -2}, 
  {-(-11 - 3*t + 3*t^2 + t^3)/6, -2, -1}, 
  {-(-5 + 3*t^2 + t^3)/3, -1, 0}, 
  {(5 - 3*t^2 + t^3)/3, 0, 1}, 
  {(11 - 3*t - 3*t^2 + t^3)/6, 1, 2}, 
  {-(-3 + t)^3/6, 2, 3}}
;[o]
         3                             2    3
  (3 + t)             -(-11 - 3 t + 3 t  + t )
{{--------, -3, -2}, {------------------------, -2, -1}, 
     6                           6
 
             2    3                   2    3
   -(-5 + 3 t  + t )           5 - 3 t  + t
  {-----------------, -1, 0}, {-------------, 0, 1}, 
           3                         3
 
                 2    3                   3
   11 - 3 t - 3 t  + t           -(-3 + t)
  {--------------------, 1, 2}, {----------, 2, 3}}
            6                        6
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Example:  Autocorrelation of a Sinusoid
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sin = {{Sin[t],0,16 Pi}}
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{{Sin[t], 0, 16*Pi}}
;[o]
{{Sin[t], 0, 16 Pi}}
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autosin = AutoCorrelation[sin,t]
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{{(32*Pi*Cos[t] + 2*t*Cos[t] + Sin[-t] - Sin[32*Pi + t])/
    4, -16*Pi, 0}, {(32*Pi*Cos[t] - 2*t*Cos[t] - 
      Sin[32*Pi - t] + Sin[t])/4, 0, 16*Pi}}
;[o]
  32 Pi Cos[t] + 2 t Cos[t] + Sin[-t] - Sin[32 Pi + t]
{{----------------------------------------------------, 
                           4
 
   -16 Pi, 0}, {
 
    32 Pi Cos[t] - 2 t Cos[t] - Sin[32 Pi - t] + Sin[t]
    ---------------------------------------------------, 
                             4
 
   0, 16 Pi}}
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6.0 Discrete Convolution
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		The discrete convolution routine DTPiecewiseConvolution can be invoked by passing the option Domain -> Discrete to the PiecewiseConvolution object.  If the default convolution domain $ConvolutionDomain is set to Discrete, then no option has to be specified to PiecewiseConvolution (see Section 4.0 for more information). 
:[font = text; inactive; ]
		Section 5.0 discusses the implementation of continuous piecewise convolution, which is very similar to the implementation in the discrete domain.  Continuous convolution ultimately boils down to performing symbolic integration, which is built in to Mathematica.  Discrete convolution eventually reduces to computing symbolic summations, which are carried out by the "GosperSum.m" package.
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		The rest of this section presents three examples of discrete convolution.
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Pulse Convolved with a Finite-Duration Ramp
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		In this example, we will convolve a pulse of length 10
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The code that carries out the convolution is:
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pandfdr = DTPiecewiseConvolution[{1, 0, 10}, {n, 0, 5}, n]
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{{(n*(1 + n))/2, 0, 4}, {15, 5, 10}, 
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Convolution of Two Functions of Infinite Extent
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		In this example, two functions defined over the interval n in [0, Infinity) are convolved.  Here, DTPiecewiseConvolution convolves two functions given as algebraic expressions:
:[font = input; initialization; startGroup; ]
*)
longconv = DTPiecewiseConvolution[Step[n], Exp[-n] Step[n], n]
(*
:[font = output; inactive; output; endGroup; ]
{{(-1 + E^(1 + n))/((-1 + E)*E^n), 0, DirectedInfinity[1]}}
;[o]
        1 + n
  -1 + E
{{-----------, 0, Infinity}}
            n
  (-1 + E) E
:[font = help; inactive; ]
DTPiecewiseConvolution automatically converts the two functions from algebraic forms into the list-of-interval forms and returns the result in the
list-of-interval form.  The result can be plotted:
:[font = input; startGroup; ]
PlotList[ longconv, {n, -1, 10}, Domain -> Discrete ]
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-Graphics-
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The result can be converted back into an algebraic expression:
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ConvertFromList[ longconv, n, Domain -> Discrete ]
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((-1 + E^(1 + n))*Step[n])/((-1 + E)*E^n)
;[o]
       1 + n
(-1 + E     ) Step[n]
---------------------
               n
     (-1 + E) E
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A More Complicated Example
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		In this example, we consider the inner workings of discrete-time convolution when f[n] has three adjacent F-intervals and g[n] has two.  We use the following two discrete-time signals:
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MatrixForm[{{"     = n,     ", "1 <= n <= 4  "}, 
   {"f[n] = 7 - n, ", "5 <= n <= 13 "}, 
   {"     = n - 25,", "14 <= n <= 32"}}]
;[o]
     = n,       1 <= n <= 4  
 
f[n] = 7 - n,   5 <= n <= 13 
 
     = n - 25,  14 <= n <= 32
:[font = text; inactive; ]

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MatrixForm[{{"g[n] = 2,", "10 <= n <= 43"}, 
   {"     = -2,", "44 <= n <= 90"}}]
;[o]
g[n] = 2,      10 <= n <= 43
 
     = -2,     44 <= n <= 90
:[font = text; inactive; ]

:[font = subsubsection; inactive; startGroup; Cclosed; ]
Putting f[n] and g[n] in list-of-interval form
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In the list-of-interval form, f[n] becomes
:[font = input; initialization; startGroup; ]
*)
f1 = {n, 1, 4}; f2 = {7 - n, 5, 13}; f3 = {n - 25, 14, 32};
				fdisclist = {f1, f2, f3}
(*
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{{n, 1, 4}, {7 - n, 5, 13}, {-25 + n, 14, 32}}
;[o]
{{n, 1, 4}, {7 - n, 5, 13}, {-25 + n, 14, 32}}
:[font = text; inactive; ]
and g[n] becomes
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*)
g1 = {2, 10, 43}; g2 = {-2, 44, 90}; gdisclist = {g1, g2}
(*
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{{2, 10, 43}, {-2, 44, 90}}
;[o]
{{2, 10, 43}, {-2, 44, 90}}
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Performing the convolution
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		The convolution of f[n] and g[n] will produce at most 18 F-intervals.  Each F-interval in f will be convolved with each F-interval in g which means that 6 F-intervals will be convolved.  When one F-interval is convolved with another F-interval, three F-intervals are produced.
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FconvG = DTPiecewiseConvolution[fdisclist, gdisclist, n];
:[font = text; inactive; ]
After removing overlapping F-intervals, the answer to this convolution problem actually contains eleven separate F-intervals:
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ColumnForm[ FconvG ]
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{(-10 + n)*(-9 + n), 11, 14}
{-246 + 33*n - n^2, 15, 23}
{1042 - 69*n + n^2, 24, 42}
{-92, 43, 44}
{-3876 + 174*n - 2*n^2, 45, 48}
{2*(-59 + n)*(-42 + n), 49, 57}
{-9180 + 274*n - 2*n^2, 58, 76}
{92, 77, 91}
{8282 - 181*n + n^2, 92, 95}
{-9388 + 195*n - n^2, 96, 104}
{(-123 + n)*(-108 + n), 105, 122}
;[o]
{(-10 + n) (-9 + n), 11, 14}
                2
{-246 + 33 n - n , 15, 23}
                2
{1042 - 69 n + n , 24, 42}
{-92, 43, 44}
                    2
{-3876 + 174 n - 2 n , 45, 48}
{2 (-59 + n) (-42 + n), 49, 57}
                    2
{-9180 + 274 n - 2 n , 58, 76}
{92, 77, 91}
                 2
{8282 - 181 n + n , 92, 95}
                  2
{-9388 + 195 n - n , 96, 104}
{(-123 + n) (-108 + n), 105, 122}
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Plotting the result
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PlotList[ FconvG, {n, -5, 140}, Domain -> Discrete ]
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grestore
% End of Graphics
MathPictureEnd
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The Unformatted text for this cell was not generated.
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when Unformatted text is generated.
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Autocorrelation
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		This example is the autocorrelation of a pulse function:
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AutoCorrelation[ Pulse[10, n], n, Domain -> Discrete ]
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{{10 + n, -9, 0}, {10 - n, 1, 10}}
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{{10 + n, -9, 0}, {10 - n, 1, 10}}
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In the convolution package, autocorrelation is implemented as the convolution of the function with a time-reversed version of the same function; in this case, the autocorrelation of a length 10 pulse beginning at n = 0 (written below in interval form) is
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DTPiecewiseConvolution[ {1, 0, 9}, {1, -9, 0}, n ]
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{{10 + n, -9, 0}, {10 - n, 1, 10}}
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{{10 + n, -9, 0}, {10 - n, 1, 10}}
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The autocorrelation in this case is a triangular function:
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PlotList[ %3, n, Domain -> Discrete ]
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PlotList[{{10 + n, -9, 0}, {10 - n, 1, 10}}, n, 
  Domain -> Discrete]
;[o]
PlotList[{{10 + n, -9, 0}, {10 - n, 1, 10}}, n, 
 
  Domain -> Discrete]
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This work was supported in part by the Joint Services Electronics Program
contract #DAAL-03-90-C-0004.  
^*)