EI331 Signals and Systems Lecture 5 Bo Jiang John Hopcroft Center - - PowerPoint PPT Presentation

EI331 Signals and Systems

Lecture 5 Bo Jiang

John Hopcroft Center for Computer Science Shanghai Jiao Tong University

March 12, 2019

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Contents

• 1. Linearity
• 2. DT Linear Time-invariant Systems

2.1 Impulse Response 2.2 Convolution 2.3 Properties of Convolution

• 3. CT Linear Time-invariant Systems

3.1 Impulse Response 3.2 Convolution 3.3 Properties of Convolution

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Linearity

System is linear if it has superposition property T(a1x1 + a2x2) = a1T(x1) + a2T(x2)

• r, equivalently, if it is additive and homogeneous,
• 1. additivity

T(x1 + x2) = T(x1) + T(x2)

• 2. homogeneity

T(ax) = aT(x)

• Example. y(t) = tx(t) is linear (y = ax for a(t) = t)
• Example. y(t) = x2(t) is nonlinear (y = x2)
• Example. y(t) = sin(x(t)) is nonlinear (y = sin x)
• Example. y(t) = x(sin t) is linear (y = x ◦ sin)

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Linearity

Why care about linear systems?

• 1. accurate models for many systems

◮ resistor, capacity, Newton’s law, etc

• 2. mathematical tractability, many powerful tools
• 3. linearization of nonlinear systems

◮ “small signal” perturbation around “operating point” y(t) = f(x(t)) = ⇒ ∆y(t) ≈ f ′(x0(t))∆x(t) where ∆y(t) = y(t) − f(x0(t)), ∆x(t) = x(t) − x0(t) ◮ provides insights for behavior of nonlinear system

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Linearity

General superposition property T

• k

akxk

• =
• k

akT(xk)

• 1. finitely many terms: by induction
• 2. infinitely many terms: need continuity property, i.e.

T

• lim

k→∞ xk

• = lim

k→∞ T(xk)

in some sense. Will (implicitly) assume continuity in this course; can check for concrete systems.

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Linearity

Zero-in zero-out property For linear system T(0) = 0 where 0 is zero signal, i.e. x(t) = 0, ∀t or x[n] = 0, ∀n T(0) called zero-input response of system

• Proof. Use homogeneity.
• Example. y(t) = 2x(t) + 1 is nonlinear (!) T(0) = 1

T is incrementally linear if ˜ T(x) = T(x) − T(0) is linear

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Interconnections of Linear Systems

Basic system interconnections preserve linearity T2 ◦ T1 x T1 T2 y T1 + T2 x T1 T2 + y (I − T1 ◦ T2)−1 ◦ T1 x + T1 T2 y provided I − T1 ◦ T2 is invertible

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Contents

• 1. Linearity
• 2. DT Linear Time-invariant Systems

2.1 Impulse Response 2.2 Convolution 2.3 Properties of Convolution

• 3. CT Linear Time-invariant Systems

3.1 Impulse Response 3.2 Convolution 3.3 Properties of Convolution

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Representation of DT Signals by Impulses

Recall DT unit impulse δ[n] =

• 1,

n = 0 0, n = 0 CZ = {x[n] ∈ C : n ∈ Z}, space of doubly infinite sequences of complex numbers

• 1. linear space analogous to Cn
• 2. {τkδ : k ∈ Z} is a basis of CZ

x =

∞

• k=−∞

x[k]τkδ,

• r

x[n] =

∞

• k=−∞

x[k]δ[n − k], ∀n ∈ Z Above called sifting property of DT unit impulse

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DT Linear Systems

Response of linear system y = T(x) = T

• ∞
• k=−∞

x[k]τkδ

• =

∞

• k=−∞

x[k]T(τkδ) =

∞

• k=−∞

x[k]hk where hk = T(τkδ) is system response to shifted impulse τkδ. In general, hk for different k unrelated to each other. Recall in linear algebra, x =

n

• k=1

xkek = ⇒ Ax =

n

• k=1

xkAek =

n

• k=1

xkfk where fk = Aek is image of ek under linear transformation A. Just going from finite dimensions to infinite dimensions!

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DT Linear Time-invariant (LTI) Systems

Unit impulse (sample) response h = T(δ) time invariance = ⇒ hk = T(τkδ) = τk(T(δ)) = τkh Response of LTI system – Convolution sum y =

∞

• k=−∞

x[k]τkh,

• r

y[n] =

∞

• k=−∞

x[k]h[n − k], ∀n ∈ Z LTI system is fully characterized by unit impulse response! Conversely, given h, system T(x)

∞

• k=−∞

x[k]τkh is LTI

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DT LTI Systems

n δ[n]

1

n h[n] 0 1 2 1 n x[n]

0.5 2

1 n

0.5 2.5

1

2

3 2 n x[0]δ[n]

0.5

n x[0]h[n] 0 1 2 0.5 n x[1]δ[n − 1]

2

1 n x[1]h[n − 1] 1 2 3 2

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Impulse Responses of Simple LTI Systems

Identity h[n] = δ[n] Scaler multiplication h[n] = Kδ[n] Time shift h[n] = δa[n] δ[n − a] Accumulator h[n] =

n

• k=−∞

δ[k] = u[n] First difference h[n] = δ[n] − δ[n − 1]

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Convolution of Sequences

(x1 ∗ x2)[n] =

∞

• k=−∞

x1[k]x2[n − k], ∀n ∈ Z Not always defined for arbitrary x1 and x2

• Example. For x1[n] = u[n] = x2[−n], sum divergent for all n.

Sufficient conditions for absolute convergence

• 1. Either x1 or x2 has finite support supp x = {n : x[n] = 0}, i.e.

x1[n] or x2[n] nonzero only for finitely many n.

• 2. x1, x2 both right-sided (or left-sided), i.e. xi[n] = 0 for n ≤ ni

(or n ≥ ni), ∀i = ⇒ x1 ∗ x2 also right-sided (or left-sided)

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Convolution of Sequences

Sufficient conditions for absolute convergence (cont’d)

• 3. One of x1 and x2 has finite ℓ1 norm and the other finite ℓp

norm for 1 ≤ p ≤ ∞ , where ℓp norm xp       

• ∞
• n=−∞

|x[n]|p 1/p , if 1 ≤ p < ∞ sup

n∈Z

|x[n]|, if p = ∞. If x11 < ∞, then x1 ∗ x2p ≤ x11 · x2p.

• 4. x1p < ∞ and x2q < ∞ for 1 ≤ p, q ≤ ∞ and

p−1 + q−1 = 1. In this case, x1 ∗ x2∞ ≤ x1p · x2q.

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Calculation of Convolution

• 1. Plot both x1 and x2 as functions of k, i.e. x1[k], x2[k]
• 2. Reverse x2 to obtain Rx2, i.e. x2[−k]
• 3. Given n, shift Rx2 by n to obtain τn(Rx2), i.e. x2[n − k]
• 4. Multiply x1 and τn(Rx2) component-wise to obtain

gn = x1 · τn(Rx2), i.e. gn[k] = x1[k]x2[n − k]

• 5. Sum gn over k to obtain (x1 ∗ x2)[n], i.e.

(x1 ∗ x2)[n] =

∞

• k=−∞

gn[k]

• 6. Repeat 1-5 for each n

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Convolution

• Example. Let x[n] = αnu[n] and h[n] = u[n] with α ∈ (0, 1).

For n < 0, (x ∗ h)[n] = 0 For n ≥ 0, (x∗h)[n] =

n

• k=0

αk = 1 − αn+1 1 − α Thus (x ∗ h)[n] = 1 − αn+1 1 − α

• u[n]

k x[k] = αku[k] k h[k] = u[k] k h[−k] k n (< 0) h[n − k] k n (> 0) h[n − k]

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Convolution

• Example. Let x[n] = αnu[n] and h[n] = u[n] with α ∈ (0, 1).

(x ∗ h)[n] =

∞

• k=−∞

x[k]h[n − k] =

∞

• k=−∞

αku[k]u[n − k] =

• 0≤k≤n

αk = u[n]

n

• k=0

αk = 1 − αn+1 1 − α

• u[n]

Also true for α > 1 n x[n] = αnu[n] n h[n] = u[n] n

1 1−α

(x ∗ h)[n]

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Convolution

• Example. Let

x[n] =

• 1,

0 ≤ n ≤ 4 0,

• therwise

h[n] =

• αn,

0 ≤ n ≤ 6 0,

• therwise

Five cases

• 1. n < 0
• 2. 0 ≤ n ≤ 4
• 3. 4 < n ≤ 6
• 4. 6 < n ≤ 10
• 5. n > 10

k 4 x[k] k

n − 6 n 4

h[n − k] k

n − 6 n 4

h[n − k] k

n − 6 n 4

h[n − k] k

0 n − 6 n 4

h[n − k] k

n − 6 n 4

h[n − k]

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Identity Element

Recall sifting property of δ x[n] =

∞

• k=−∞

x[k]δ[n − k], ∀n ∈ Z δ identity element for convolution x = x ∗ δ = δ ∗ x x = x ∗ δ x δ x x = δ ∗ x δ x x

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Commutative Law

x1 ∗ x2 = x2 ∗ x1

• Proof. Change variables by m = n − k,

(x1∗x2)[n] =

∞

• k=−∞

x1[k]x2[n−k] =

∞

• m=−∞

x1[n−m]x2[m] = (x2∗x1)[n] y = x ∗ h x h y y = h ∗ x h x y

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Commutative Law

• Example. Let

x[n] =

• 1,

0 ≤ n ≤ 4 0,

• therwise

h[n] =

• αn,

0 ≤ n ≤ 6 0,

• therwise

Five cases

• 1. n < 0
• 2. 0 ≤ n ≤ 4
• 3. 4 < n ≤ 6
• 4. 6 < n ≤ 10
• 5. n > 10

k

6

h[k] k

n − 4 n 6

x[n − k] k

n − 4 n 6

x[n − k] k

0 n − 4 n 6

x[n − k] k

n − 4 n 6

x[n − k] k

n − 4 n 6

x[n − k]

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Distributive Law

x1 ∗ (x2 + x3) = x1 ∗ x2 + x1 ∗ x3

• Proof. Provided x1 ∗ x2 and x1 ∗ x3 well-defined,

∞

• k=−∞

x1[k](x2[n−k]+x3[n−k]) =

∞

• k=−∞

x1[k]x2[n−k]+

∞

• k=−∞

x1[k]x3[n−k] h = h1 + h2 x h1 h2 + y h

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Bilinearity

Scalar multiplication x1 ∗ (Kx2) = K(x1 ∗ x2) (x1, x2) → x1 ∗ x2 bilinear, i.e. linear in both x1 and x2

• i

aix1i

• ∗
• j

bjx2j

• =
• i
• j

aibj(x1i ∗ x2j)

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Associative Law

(x1 ∗ x2) ∗ x3 = x1 ∗ (x2 ∗ x3) “Proof”. Under appropriate conditions ((x1 ∗ x2) ∗ x3)[n] =

∞

• m=−∞

(x1 ∗ x2)[n − m]x3[m] =

∞

• m=−∞
• ∞
• k=−∞

x1[k]x2[n − m − k]

• x3[m]

=

∞

• k=−∞

x1[k]

∞

• m=−∞

x2[n − k − m]x3[m] =

∞

• k=−∞

x1[k](x2 ∗ x3)[n − k]) = (x1 ∗ (x2 ∗ x3))[n] When associative law holds, write x1 ∗ x2 ∗ x3 without ambiguity

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Associative Law

x1 ∗ (x2 ∗ x3) = (x1 ∗ x2) ∗ x3 h = h1 ∗ h2 x h1 h2 y h commutative h = h2 ∗ h1 x h2 h1 y h Order of processing usually not important for LTI systems

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Associative Law

• Example. x1[n] = 1, x2[n] = u[n], x3[n] = δ[n] − δ[n − 1].
• 1. x2 ∗ x3 = x3 ∗ x2 = δ, so x1 ∗ (x2 ∗ x3) = x1 ∗ (x3 ∗ x2) = 1
• 2. x1 ∗ x2 and (x1 ∗ x2) ∗ x3 undefined!
• 3. x1 ∗ x3 = 0, so (x1 ∗ x3) ∗ x2 = 0

Sufficient conditions for associative law

• 1. At least two of x1, x2 and x3 have finite supports
• 2. x1, x2, x3 all right-sided (or left-sided), =

⇒ x1 ∗ x2 ∗ x3 also right-sided (or left-sided)

• 3. One signal (e.g. x3) has finite ℓp norm for 1 ≤ p ≤ ∞,

and others finite ℓ1 norm, x1 ∗ x2 ∗ x3p ≤ x11 · x21 · x3p

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Time-shift Property

(τax1) ∗ (τbx2) = τa+b(x1 ∗ x2)

• Proof. Note τax = x ∗ δa and δa+b = δa ∗ δb. Thus

(τax1) ∗ (τbx2) = (x1 ∗ δa) ∗ (x2 ∗ δb) = (x1 ∗ x2) ∗ (δa ∗ δb) = (x1 ∗ x2) ∗ δa+b = τa+b(x1 ∗ x2) Helpful in programming Example. x[−2] = −1 x[−1] = 0 x[0] = 0.5 x[1] = 1 x[2] = −0.5 h[−1] = −0.5 h[0] = 1 h[1] = 0.5

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Time-shift Property

4 2 2 4 6 8

x[n]

4 2 2 4 6 8

x[n-2]

4 2 2 4 6 8

h[n]

4 2 2 4 6 8

h[n-1]

4 2 2 4 6 8

y[n]

4 2 2 4 6 8

y[n-3]

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Calculation using Properties

• Example. Let

x[n] =

• 1,

0 ≤ n ≤ 4 0,

• therwise

h[n] =

• αn,

0 ≤ n ≤ 6 0,

• therwise
• Let h1[n] = αnu[n].

(u ∗ h1)[n] = 1 − αn+1 1 − α

• u[n]
• x = u − τ5u, h = h1 − α7τ7h1.
• x ∗ h = (I − τ5 − α7τ7 + α7τ12)(u ∗ h1)
• x = (δ − δ5) ∗ u, h = (δ − α7δ7) ∗ h1.
• x ∗ h = (δ − δ5 − α7δ7 + α7δ12) ∗ (u ∗ h1)

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Contents

• 1. Linearity
• 2. DT Linear Time-invariant Systems

2.1 Impulse Response 2.2 Convolution 2.3 Properties of Convolution

• 3. CT Linear Time-invariant Systems

3.1 Impulse Response 3.2 Convolution 3.3 Properties of Convolution

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Representation of CT Signals by Impulses

Sifting property of CT unit impulse x(t) =

• R

x(a)δ(t − a)da Interpreted as limit as ∆ → 0 of ˆ x∆(t) =

∞

• k=−∞

x(k∆)p∆(t − k∆)∆ where p∆(t) = 1 ∆[u(t) − u(t − ∆)] t x(t)

−∆ ∆ 2∆ k∆

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CT Linear Systems

Response of linear system ˆ y∆ = T(ˆ x∆) = T

• ∞
• k=−∞

x(k∆)τk∆p∆∆

• =

∞

• k=−∞

x(k∆)T(τk∆p∆)∆ =

∞

• k=−∞

x(k∆)ˆ hk∆∆ where ˆ hk∆ = T(τk∆p∆) is response to shifted pulse τk∆p∆. In the limit ∆ → 0 ,

• ˆ

x∆ → x and ˆ y∆ → y = T(x)

• for k∆ → a, have τk∆p∆ → δa, expect ˆ

hk∆ → ha = T(δa) y =

• R

x(a)hada,

• r

y(t) =

• R

x(a)ha(t)da

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CT Linear Time-invariant (LTI) Systems

Unit impulse response h = h0 = T(δ) time invariance = ⇒ ha = T(δa) = τa(T(δ)) = τah Response of LTI system – Convolution integral y(t) =

• R

x(τ)h(t − τ)dτ, ∀t ∈ R LTI system is fully characterized by unit impulse response! Conversely, given h, system T(x)(t)

• R

x(τ)h(t − τ)dτ is LTI

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Impulse Responses of Simple LTI Systems

Identity h(t) = δ(t) Scaler multiplication h(t) = Kδ(t) Time shift h(t) = δa(t) δ(t − a) Integrator h(t) = t

−∞

δ(τ)dτ = u(t)

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Convolution

(x1 ∗ x2)(t) =

• R

x1(τ)x2(t − τ)dτ, ∀t ∈ R Not always defined for arbitrary x1 and x2

• Example. For x1(t) = u(t) = x2(−t), integral divergent for all t.

Sufficient conditions for absolute convergence

• 1. Either x1 or x2 has compact support supp x = {t : x(t) = 0},

i.e. x1 or x2 vanishes outside finite interval.

• 2. x1, x2 both right-sided (or left-sided), i.e. xi(t) = 0 for t ≤ ti

(or t ≥ ti), ∀i = ⇒ x1 ∗ x2 also right-sided (or left-sided)

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Convolution

Sufficient conditions for absolute convergence (cont’d)

• 3. One of x1 and x2 has finite L1 norm and the other finite Lp

norm for 1 ≤ p ≤ ∞ , where Lp norm 1 xp     

• R

|x(t)|pdt 1/p , if 1 ≤ p < ∞ sup

t∈R

|x(t)|, if p = ∞. If x11 < ∞, then x1 ∗ x2p ≤ x11 · x2p.

• 4. x1p < ∞ and x2q < ∞ for 1 ≤ p, q ≤ ∞ and

p−1 + q−1 = 1. In this case, x1 ∗ x2∞ ≤ x1p · x2q.

1More precisely, x∞ = sup{B ≥ 0 : |x(t)| ≤ B for almost every t}

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Calculation of Convolution

• 1. Plot both x1 and x2 as functions of τ, i.e. x1(τ), x2(τ)
• 2. Reverse x2(τ) to obtain x2(−τ)
• 3. Given t, shift x2(−τ) by t to obtain x2(t − τ)
• 4. Multiply x1(τ) and x2(t − τ) pointwise to obtain

gt(τ) = x1(τ)x2(t − τ)

• 5. Integrate gt over τ to obtain (x1 ∗ x2)(t), i.e.

(x1 ∗ x2)(t) =

• R

gt(τ)dτ

• 6. Repeat 1-5 for each t

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Convolution

• Example. Let x(t) = e−atu(t) and h(t) = u(t) with a > 0.

For t < 0, (x ∗ h)(t) = 0 For t ≥ 0, (x∗h)(t) = t e−aτdτ = 1 − e−at a Thus (x ∗ h)(t) = 1 − e−at a

• u(t)

τ x(τ) = e−aτu(τ) τ h(τ) = u(τ) τ h(−τ) τ

t (< 0)

h(t − τ) τ

t (> 0)

h(t − τ)

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Convolution

• Example. Let x(t) = e−atu(t) and h(t) = u(t) with a > 0.

(x ∗ h)(t) =

• R

x(τ)h(t − τ)dτ =

• R

e−aτu(τ)u(t − τ)dτ =

• 0≤τ≤t

e−aτdτ = u(t) t e−aτdτ = 1 − e−at a

• u(t)

Also true for a < 0 t x(t) t h(t) t

1 a

(x ∗ h)(t)

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Convolution

• Example. Let

x(t) =

• 1,

0 < t < T 0,

• therwise

h(t) =

• t,

0 ≤ t ≤ 2T 0,

• therwise

Five cases

• 1. t < 0
• 2. 0 ≤ t ≤ T
• 3. T < t ≤ 2T
• 4. 2T < t ≤ 3T
• 5. t > 3T

τ

0 T

x(τ) 1 τ

t − 2T t 2T

h(t − τ) τ

t − 2T t 2T

τ

t − 2T t 2T

τ

t − 2T t 2T

τ

0 t − 2T t 2T

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Identity Element

Recall sampling property of δ x(t) =

• R

x(τ)δ(t − τ)dτ, ∀t ∈ R δ identity element for convolution x = x ∗ δ = δ ∗ x x = x ∗ δ x δ x x = δ ∗ x δ x x

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Properties of Convolution

Commutativity x1 ∗ x2 = x2 ∗ x1 Bilinearity

• i

aix1i

• ∗
• j

bjx2j

• =
• i
• j

aibj(x1i ∗ x2j) Associativity x1 ∗ x2 ∗ x3 = (x1 ∗ x2) ∗ x3 = x1 ∗ (x2 ∗ x3) Time shift (τax1) ∗ (τbx2) = τa+b(x1 ∗ x2)

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Associative Law

x1 ∗ (x2 ∗ x3) = (x1 ∗ x2) ∗ x3 h = h1 ∗ h2 x h1 h2 y h commutative h = h2 ∗ h1 x h2 h1 y h Order of processing usually not important for LTI systems

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Associative Law

• Example. x1(t) = 1, x2(t) = u(t), x3(t) = δ′(t) (later)
• 1. (x2 ∗ x3)(t) = δ(t), so x1 ∗ (x2 ∗ x3) = 1
• 2. x1 ∗ x2 and (x1 ∗ x2) ∗ x3 undefined!
• 3. x1 ∗ x3 = 0, so (x1 ∗ x3) ∗ x2 = 0

Sufficient conditions for associative law

• 1. At least two of x1, x2 and x3 have compact supports2
• 2. x1, x2, x3 all right-sided (or left-sided), =

⇒ x1 ∗ x2 ∗ x3 also right-sided (or left-sided)

• 3. One signal (say x3) has finite Lp norm for 1 ≤ p ≤ ∞

and others finite L1 norm. x1 ∗ x2 ∗ x3p ≤ x11 · x21 · x3p

2δa and its derivatives (to be defined) have support {a}.