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Steps, Impulses, and Convolution of Functions
Working Instructional Material Notebook
Copyright © 1998 by Fred Garber
Steps and Impulses
First, we load a package that defines the unit step function and the delta fumction:
In[3]:= << CalculusDiracDelta
The "unitstep" just defined as "UnitStep[]", denoted by u(t) has a step of size unity at t=0. The default value of u(t) at t=0 is unity.
In[4]:= UnitStep @ 0 D Out[4]= 1
Lets take a look at it:
In[110]:= Plot @ UnitStep @ t D , 8 t, - 1, 10 < , PlotRange -> 8 0, 1.5 <D
Out[110]= Ö Graphics Ö
The delta function, or impulse function, just defined is usually defined as the limit of a sequence of functions that integrate to unity, but have increasing height and decreasing width.
In[5]:= Integrate @ DiracDelta @ t D , 8 t, - Infinity, Infinity <D Out[5]= 1
Remember the defining equation for the impulse function? We had (^) Ÿ-^ • •^ dH t - a L f H t L ‚ t = f H a L Here it is in the language of Mathematica :
In[6]:= Integrate @ DiracDelta @ t - a D f @ t D , 8 t, - Infinity, Infinity <D Out[6]= f@aD
Cool, huh? So how about when the impuls is on outside of the limits of integration?
In[7]:= Integrate @ DiracDelta @ t - 5 D f @ t D , 8 t, - 23, 4 <D Out[7]= 0
How about one of the homework problems:
In[8]:= Integrate A DiracDelta @ t - 3 D t
ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ t , 8 t, - Infinity, Infinity <E
Out[8]= ÄÄÄÄÄÄÄÄ^103
Now, lets do a convolution of an impulse and a unit setp and define the result at y(t)
In[43]:= y @ t_ D : = Integrate @ DiracDelta @ t - tD UnitStep @tD , 8 t , - Infinity, Infinity <D
In[111]:= Lets look at the result
— Get::noopen : Cannot open Classic.nb.
In[64]:= Plot @8 f @ t D , h @ t D< , 8 t, - 1, 5 < , PlotRange -> 8 0, 3.5 <D
Out[64]= Ö Graphics Ö
In[65]:= y @ t_ D : = Integrate @ f @ t - tD h @tD , 8 t , - Infinity, Infinity <D
In[67]:= Plot @ y @ t D , 8 t, - 1, 5 < , PlotRange -> 8 0, 1.5 <D
Out[67]= Ö Graphics Ö
ü Example 2: Exponential * Rectangular pulse How about an exponential and a rectangular pulse. Here we go, with 3 e - t u(t) *2[u(t-1) - u(t-2)],
In[73]:= f @ t_ D : = 2 H UnitStep @ t - 1 D - UnitStep @ t - 2 DL h @ t_ D : = 3 Exp @- t D UnitStep @ t D
In[75]:= Plot @8 f @ t D , h @ t D< , 8 t, - 1, 5 < , PlotRange -> 8 0, 3.5 <D
Out[75]= Ö Graphics Ö
In[76]:= y @ t_ D : = Integrate @ f @ t - tD h @tD , 8 t , - Infinity, Infinity <D
In[81]:= Plot @ y @ t D , 8 t, - 1, 8 < , PlotRange -> 8 0, 5 <D
Out[81]= Ö Graphics Ö
ü Example 3: Boxy thing * Same Boxy thing How about this one, a convolution with itself:
ü Example 4: Big Box * Little Box How about this one, notice that f starts before t=0.
In[91]:= f @ t_ D : = H UnitStep @ t + 1 D - UnitStep @ t - 1 DL h @ t_ D : = 2 H UnitStep @ t D - UnitStep @ t - 4 DL
In[93]:= Plot @8 f @ t D , h @ t D< , 8 t, - 2, 5 < , PlotRange -> 8 0, 2.5 <D
Out[93]= Ö Graphics Ö
Now the convolution:
In[94]:= y @ t_ D : = Integrate @ f @ t - tD h @tD , 8 t , - Infinity, Infinity <D
In[95]:= Plot @ y @ t D , 8 t, - 2, 6 < , PlotRange -> 8 0, 5 <D
Out[95]= Ö Graphics Ö
ü Example 5: Box * Cosine We sort-of did this in class.
In[103]:= f @ t_ D : = Cos @ 2 p 5 t D UnitStep @ t D h @ t_ D : = 2 H UnitStep @ t D - UnitStep @ t - 1 DL
In[105]:= Plot @8 f @ t D , h @ t D< , 8 t, - 1, 2 < , PlotRange -> 8 - 2, 2.5 <D
Out[105]= Ö Graphics Ö
Now the convolution:
