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8/2/2019 Mathematical Transforms
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PRODUCTS SOLUTIONS PURCHASE SUPPORT COMPAN OUR SITES SEARCH
EA CH MA HEMA ICA 8 DOC MEN A ION
Documentation Signals and S ste ms
5 Mathematical Transforms
S S . W
- ( -Infinity Infinity , -Infinity 0 , 0 Infinity , ),
L Z , ,
A S S , F . T
5.1 Transforms of Continuous SignalsThe Laplace Transform
T L . T
T . S S
(T .)
T . Y Ma hema ica
The L aplace transform and i ts inverse.
T L Ma hema ica LaplaceTransform . I , . O
F , .
In[1]:= Needs["SignalProcessing`"]
H L .
In[2]:= LaplaceTransform[Exp[-Abs[t]], t, s]
Out[2]=
A Ma hema ica . N ,LaplaceTransform
HI I DOC MEN A ION FO AN OB OLE E P OD C .
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H - . T ,
In[3]:= LaplaceTransform[Exp[a t1] *DiracDelta[t1 - t2] UnitStep[{t1, t2 ],{t1, t2 , {s1, s2]
Out[3]=
L , .
H .
In[4]:= InverseLaplaceTransform[s/(1 + s^2), s, t]
Out[4]=
T .
In[5]:= InverseLaplaceTransform[Exp[-a s] s / ( 1 + s^2 ),s, t]
Out[5]=
Options for LaplaceTransform .
A LaplaceTransform . T TransformDirection
.
A - UnitStep .
In[6]:= LaplaceTransform[Exp[-Abs[t]], t, s,TransformDirection -> LeftSided]
Out[6]=
T Justification . I All , Automatic , None , All
H .
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T
N , L . T
The Fourier transform and its inverse.
H F .
In[16]:= FourierTransform[Cos[t] Exp[-t] UnitStep[t],t, w]
Out[16]=
H .
In[17]:= SignalPlot[%, {w, -2, 2 ]
Out[17]=
A LaplaceTransform , , .FourierTransform LaplaceTran
H TransformDirection . T - (-Infinity 0 ).
In[18]:= FourierTransform[Exp[t^2],t, w,TransformDirection -> LeftSided]
Out[18]=
A , .
In[19]:= FourierTransform[ContinuousPulse[{1, 1, 1 ,{t1 + 1/2, t2 + 1/2, t3 + 1/2 ],{t1, t2, t3 ,{w1, w2, w3]
Out 1 9 =
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T . I ,
T F " " ,
In[20]:= InverseFourierTransform[1/2 (ContinuousPulse[a, w + a/2 - w0] +ContinuousPulse[a, w + a/2 + w0]),w, t]
Out[20]=
H . H ,
In[21]:= InverseFourierTransform[Tan[w],w, t,SeriesTerms -> 8]
Out[21]=
5.2 Transforms of Discrete Signals
The Z TransformT Z L . I
T ( ). T
. A , S S
The Z transform and its inverse.
T Ma hema ica. L
T Z .
In[22]:= ZTransform[((1/2)^n + (-1/3)^n) DiscreteStep[n],n, z]
Out[22]=
W , , .
In[23]:= PoleZeroPlot[%]
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Out[23]=
Options for ZTransform .
T Z L . TTransformDirection - , -
, StandardFormula .SimplifyOutput
T TransformPairs , , x[n] y[n]
In[24]:= Normal[ZTransform[y[n] == x[n] - (1/4) y[n - 2],n, z,TransformPairs -> {y[n] :> Y[z], x[n] :> X[z]
]]
Out[24]=
T Y[z] X[z] .
In[25]:= (Y[z]/.First[Solve[%, Y[z]]])/X[z]
Out[25]=
T .
In[26]:= InverseZTransform[%, z, n]
Out[26]=
T .
In[27]:= DiscreteSignalPlot[%, {n, -5, 15 ]
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Out[27]=
W , Z . I
H Z . N .
In[28]:= ZTransform[a^n1 b^n2 DiscreteStep[{n1, n2 ],{n1, n2 , {z1, z2]
Out[28]=
H , . N
In[29]:= InverseZTransform[ZTransformData[Normal[%],RegionOfConvergence[{0, Abs[b] ,{Abs[a], Infinity],TransformVariables[{z1, z2 ]],{z1, z2 , {n1, n2
]
Out[29]=
Options for inverse Z transform.
S Z , SeriesTerms . T
, Infinity .
H Z SeriesTerms . T .Expand
In[30]:= Expand[InverseZTransform[BesselJ[1, z],z, n,SeriesTerms -> 8]]
Out[30]=
T StandardFormula Z ,
The Discrete-Time Fourier Transform
T - F Z , Z
B Z , S S
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The discrete-time Fourier transform and its inverse.
T - F Z .
A , - F , .
In[31]:= DiscreteTimeFourierTransform[a^n DiscreteStep[n - 4],n, w]
Out[31]=
T - - F .
In[32]:= DiscreteTimeFourierTransform[(1/6) DiscreteImpulse[n1 - 1, n2 - 1] +(1/6) DiscreteImpulse[n1 + 1, n2 - 1] +(1/6) DiscreteImpulse[n1 - 1, n2 + 1] +(1/6) DiscreteImpulse[n1 + 1, n2 + 1] +(1/3) DiscreteImpulse[n1, n2],{n1, n2 , {w1, w2]
Out[32]=
T , . N ,
In[33]:= SignalPlot3D[Abs[First[%]],{w1, -Pi, Pi , {w2, -Pi, Pi]
Out[33]=
DiscreteTimeFourierTransform ZTransform , . S ,
H . N TransformPairs
In[34]:= DiscreteTimeFourierTransform[x[n + 3],n, w, TransformPairs -> {x[n] :> X[w]]
Out[34]=
T TransformPairs .
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In[35]:= InverseDiscreteTimeFourierTransform[%,w, n,TransformPairs -> {X[w] :> x[n]]
Out[35]=
The Discrete Fourier Transform
G , F - F
T
U - Ma hema ica Fourier F , S S DiscreteFourierTransform
The discrete Fourier transform and its inverse.
B F S S , N
H DiscreteFourierTransform . H , .
In[36]:= DiscreteFourierTransform[Sin[n], 10, n, k]
Out[36]=
A .
In[37]:= InverseDiscreteFourierTransform[Sin[k], 10, k, n]
Out[37]=
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Options for DiscreteFourierTransform and InverseDiscreteFourierTransform .
T DiscreteFourierTransform Z Z . T ,
W Justification Automatic , . T
In[38]:= DiscreteFourierTransform[DigitalFIRFilter[{h[0], h[1], h[2], h[3], h[4] ,n],5, n, w,Justification -> Automatic]
Out[38]=
Special syntax for transforming a numeric vector.
F , F
T ,
.
H .
In[39]:= DiscreteFourierTransform[Table[2^(-n), {n, 0, 10 ]//N]
Out[39]=
T .
In[40]:= InverseDiscreteFourierTransform[{1, 1, 1, 1, 0, 0, 0, 0 //N]
Out[40]=
5.3 Information from Transforms
E . I ,
Stabilit
F L Z ,
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Determining stability from a transform object.
T SignalStability
.
H Z .
In[41]:= ZTransform[(1/5)^n Exp[n]/20 DiscreteStep[n],n, z
]
Out[41]=
F , .
In[42]:= SignalStability[%]
Out[42]=
T .
In[43]:= SignalStability[ZTransform[
-a^n1 b^n2 DiscreteStep[{-n1 - 1, n2 ],{n1, n2 , {z1, z2]]
Out[43]=
Assumptions
S . S S
Assumptions made b y transforms during a compu tation.
D , TransformAssumptions $Line .
, . O ,
H L .
In[44]:= LaplaceTransform[
ExpIntegralEi[n t] UnitStep[t],t, s]
Out[44]=
T .
In[45]:= TransformAssumptions[-1]
Out[45]=
T . N , , %% ; , -2 Out[-2] %%
In[46]:= TransformAssumptions[%%]
Out[46]=
Transform Object Parts
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T , . T
.
Functions for extracting parts of transform objects.
A Ma hema ica ,Normal .
H L .
In[47]:= trans = LaplaceTransform[Exp[t] UnitStep[t],t, s]
Out[47]=
T .
In[48]:= TransformFunction[trans]
Out[48]=
L Z . I
. T Z , , ,
T . T ,
In[49]:= RegionOfConvergence[trans]
Out[49]=
N . T
In[50]:= TransformVariables[trans]
Out[50]=
Data objects resulting from forward transforms.
B , Part
5.4 Solving Differential and Recurrence Equations
M . F ,
T Ma hema ica DSolve RSolve . H ,
- .
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The transform-based equation solvers.
T DSolve RSolve . T
I , ZRSolve . LaplaceDSolve C[1] , C[2] , ,
H - , .
In[51]:= ZRSolve[{y[n-2] + 1/2 y[n-1] + 1/4 y[n] == 0,y[1] == 1 ,y[n], n]
Out[51]=
H - . I .
In[52]:= LaplaceDSolve[{y''[t] + 3/2 y'[t] + 1/2 y[t] ==Exp[a t] UnitStep[t],y[0] == 4, y'[0] == 0 ,y[t], t]
Out[52]=
Options for the solvin g functions.
T ZRSolve . T TransformDirection
T Justification . W None , ; Automatic ,
H Justification . M All .
In[53]:= ZRSolve[{y[n-2] + 1/2 y[n-1] + 1/4 y[n] == 0,y[0] == 1, y[1] == 0 ,
y[n], n, Justification -> Automatic]
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Newsletter
Out[53]=
2 01 2 A bo ut Wo lf ra m Wo lfra m B lo g Wo lf ra m| Al ph a Term s P ri va cy S it e M ap Co nt act