Convolution Table - Michael I. Miller PDF

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Michael I. Miller...

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TABLE 2.1 Convolution Table No.

x1 ( t )

x2 ( t )

x1 ( t )∗x2 ( t )=x2 ( t )∗x 1 ( t )

1

x (t )

δ (t−T )

x (t −T )

2

e u (t )

λt

u(t)

3

u(t)

u(t)

4

e λ t u (t )

e λ t u (t )

5

e u (t )

e u (t )

6

t e u (t)

λt

e u (t )

1

λt

1−e λt u (t ) −λ t u (t ) λ1 t

λ2 t

e −e u(t) λ1 −λ2

2

λt

λ1 ≠λ2

λt

t e u (t)

1 2 λt t e u (t ) 2

λt

N −k N ! e λt N !t u (t ) − ∑ k+1 (N −k )! u(t) λ N+1 k =0 λ N

λt

N

e u (t )

M

t u (t )

7

t u (t )

8

t u( t )

9

λ 1t

t e u (t )

e u (t )

t λ t λ t e λ −e +(λ1−λ2 ) t e u(t) (λ1 −λ2 )2

10

t M e λt u (t )

t N e λt u (t )

M! N ! t M +N +1 e λt u (t ) (N + M + 1)!

M !N! t M +N +1 u(t) ( M + N + 1)!

N

2

λ2 t

M

∑ k=0 11

t M e λ t u( t ) 1

t N e λ t u (t )

1

1

(−1)k M ! (N +k )! t M −k e λ t u(t) k ! ( M −k )! (λ1 −λ2 ) N +k +1 1

2

N

+∑ k=0

λ2 t

(−1)k N ! (M +k )! t N −k e u(t) k ! (N −k )! (λ2 −λ1 )M+k +1

cos (θ −ϕ ) e λt −e−αt cos (β t +θ −ϕ ) 12

e

−αt

cos (βt +θ ) u (t )

√ (α+λ)2 +β2

λt

e u (t )

u (t )

ϕ =atan 2(−β , α+λ) 13

e u (t )

e u (−t )

e λ t u(t)−e λ t 1u (−t ) λ2 −λ1

14

t e λ u (−t )

t e λ u (−t )

e −e u(−t) λ2 −λ1

λ1 t

1

1

λ2 t

λ1 t

2

{

2

ℜ { λ2 }>ℜ { λ1 }

λ2 t

tan −1 (−β/(α+λ))

if α+λ> 0

tan −1 (−β/(α+λ))−π

if α+λ< 0

Note ( assuming β> 0 ): atan2 (−β , α+λ)=

Version 1.3

TABLE 3.1 Convolution Sums No.

x1 [ n]]

x2 [ n]]

x1 [ n]]∗ x2 [ n ] = x 2 [ n ]∗ x 1 [ n ]

1

x [ n]

δ[ n−k ]

x [ n−k ]

2

γn u[ n ]

u[ n]

1−γn +1 u [ n] 1−γ

3

u[ n]

u[ n]

(n+1) u[n]

4

γ1n u[ n]

γ2n u[ n]

γ 1n+1−γn+1 2 γ1−γ 2 u [n ]

5

u[ n]

n u[ n]

n(n+1) u [n] 2

γ1≠γ 2

6

γ u[n ]

n u[ n]

[

7

n u[ n]

n u[ n]

1 n(n−1 )(n+ 1 ) u [ n ] 6

8

γn u[ n ]

γn u[ n ]

9

n γn1u[ n]

γ2n u[ n]

n

n

10

|γ 1|

11

γ1 u[ n]

γ2n u[ n]

cos(β n+θ) u [n ]

n

n

γ2 u[−(n+1 )]

]

γ(γ n−1)+ n(1−γ) u[ n] (1−γ)2

(n+1)γ n u [ n ] γ1 γ 2 n n n γ −γ γ2−γ1 + 1 γ2 2 n γ1 u[ n] 2 (γ1−γ 2)

[

]

1 n+1 n+1 [| γ | cos [β(n+ 1)+θ−ϕ]−γ 2 cos(θ−ϕ)] u [n] R 1 1/ 2 2 2 R=[| γ1| +γ 2−2|γ 1| γ 2 cosβ] (|γ 1|sin β) ϕ=tan −1 (|γ1|cosβ−γ2 )

[

γ2 γ1 n n γ 2−γ 1 γ 1 u[ n]+ γ2−γ 1 γ2 u[−( n+1 )]

B.7-4 Sums n

∑

k=m ∞

∑

k=m n

r n+1−r m r≠1, r−1 rm r...