Chirp Z Transform of a Moving-Average Filter

Start with the Z transform of a moving average filter:

In[1]:=

x=ConstantArray[1/5,5]

Out[1]=



1

5

,

1

5

,

1

5

,

1

5

,

1

5



Obtain the Z transform of the filter:

In[2]:=

X[z_]=ListZTransform[x,z]

Out[2]=

1

5

+

1

5

4

z

+

1

5

3

z

+

1

5

2

z

+

1

5z

Compute the chirp Z transform:

In[3]:=

DiscreteChirpZTransform[x]

Out[3]=

{1.,4.29438×

-17

10

,7.08541×

-17

10

,7.08541×

-17

10

,4.29438×

-17

10

}

Check that the result is equal to the discrete Fourier transform:

In[4]:=

Fourier[x,FourierParameters{1,-1}]

Out[4]=

{1.,4.29438×

-17

10

,7.08541×

-17

10

,7.08541×

-17

10

,4.29438×

-17

10

}

Compute the magnitude of the chirp Z transform and the complex plane contour on a spiral path defined by

a

k

w

for

k

from

0

to

n-1

:

In[5]:=

a=0.889919+0.646564I;w=0.886924+0.367376I;mag=Abs@DiscreteChirpZTransform[x,31,w,a];

Visualize the spiral path

a

k

w

:

In[6]:=

path=N[{Re[#],Im[#]}]&/@Tablea

k

w

,{k,0,30};ListLinePlot[path,AspectRatio1]

Out[6]=

Show the magnitude and path of the chirp Z transform:

In[7]:=

ShowPlot3DAbs[X[x+y]],{x,-1,1},{y,-1,1},

,ListPointPlot3DFlatten/@Thread[{path,mag}],

,Graphics3D[{Blue,Line[Flatten/@Thread[{path,mag}]]}]

Out[7]=

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