Mathematical Transforms

8/2/2019 Mathematical Transforms

1/15

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 .


2/15

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 .


3/15


4/15


5/15

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 =


6/15

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[%]


7/15

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 ]


8/15

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


9/15

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 .


10/15

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]=


11/15

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 ,


12/15

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


13/15

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 ,

- .


14/15

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]


15/15

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

Mathematical Transforms

Documents

Transcript of Mathematical Transforms