EI331 Signals and Systems Lecture 10 Bo Jiang John Hopcroft Center - - PowerPoint PPT Presentation

EI331 Signals and Systems

Lecture 10 Bo Jiang

John Hopcroft Center for Computer Science Shanghai Jiao Tong University

March 28, 2019

1/31

Contents

• 1. Properties of CT Fourier Series

1.1 Linearity 1.2 Time and Frequency Shifting 1.3 Time Reversal and Scaling 1.4 Differentiation and Integration 1.5 Multiplication and Periodic Convolution

• 2. Convergence of Fourier Series

2.1 Mean-square Convergence

2/31

Fourier Series

Fourier series for x with period T and ω0 = 2π

T ,

x(t) =

∞

• k=−∞

ˆ x[k]ejkω0t Correspondence between periodic functions and doubly infinite sequences; time domain vs. frequency domain x

FS

← − − → ˆ x

• r

x(t)

FS

← − − → ˆ x[k]

• ˆ

x consists of expansion coefficients of x in “basis” {ejkω0t}

• ˆ

x alone does not uniquely determine x,

• also need to know basis functions, or, equivalently,

period T or fundamental frequency ω0

• same coefficients with different bases (periods) give

different functions (more later)

3/31

Linearity

If x, y have same period T, so does their linear combination ax + by, and

• ax + by = aˆ

x + bˆ y Proof. ( ax + by)[k] = ax + by, ejkω0t = ax, ejkω0t + by, ejkω0t = aˆ x[k] + bˆ y[k] Alternative proof. x(t) =

• k

ˆ x[k]ejkω0t y(t) =

• k

ˆ y[k]ejkω0t        = ⇒ ax(t) + by(t) =

• k

(aˆ x[k] + bˆ y[k])ejkω0t

4/31

Time Shifting

If x has period T and ω0 = 2π

T ,

• τt0x = E−ω0t0ˆ

x

• r

x(t − t0)

FS

← − − → e−jkω0t0ˆ x[k] where (Eaˆ x)[k] = ejakˆ x[k] Time shift ⇐ ⇒ linear phase change in frequency Proof. ( τt0x)[k] = τt0x, ejkω0t = x, τ−t0ejkω0t = x, ejkω0(t+t0) = x, ejkω0t0ejkω0t = e−jkω0t0x, ejkω0t = e−jkω0t0ˆ x[k] = (E−ω0t0ˆ x)[k]

• Example. cos(t) = 1

2ejt + 1 2e−jt, sin(t) = cos(t − π 2) = −j 2 ejt + j 2e−jt

5/31

Frequency Shifting

If x has period T and ω0 = 2π

T ,

• Emω0x = τmˆ

x

• r

ejmω0tx(t)

FS

← − − → ˆ x[k − m] where (Eax)(t) = ejatx(t) Modulation by harmonic exp. ⇐ ⇒ frequency shift Proof. ( Emω0x)[k] = Emω0x, ejkω0t = x, E−mω0ejkω0t = x, ej(k−m)ω0t = ˆ x[k − m]

• NB. Modulation by inharmonic exponential may change

fundamental frequency or even result in aperiodic signal

6/31

Frequency Shifting

• Example. x(t) = 1 + cos(ω0t) = 1 + 1

2ejω0t + 1 2e−jω0t

y(t) = ej3ω0tx(t) = ej3ω0t + 1 2ej4ω0t + 1 2ej2ω0t t x(t)

− T

2 T 2

ω

1 ω0

1 2

−ω0

1 2

t Re y(t) ω

3ω 1 4ω0

1 2

2ω0

1 2

7/31

Frequency Shifting

• Example. x(t) = 1 + cos(ω0t) = 1 + 1

2ejω0t + 1 2e−jω0t

z(t) = cos(3ω0t)x(t) = Re y(t) = 1 2ej3ω0t + 1 4ej4ω0t + 1 4ej2ω0t + 1 2e−j3ω0t + 1 4e−j4ω0t + 1 4e−j2ω0t t x(t)

− T

2 T 2

ω

1 ω0

1 2

−ω0

1 2

t z(t) ω

3ω0

1 2

−3ω0

1 2

2ω0

1 4

4ω0

1 4

−4ω0

1 4

−2ω0

1 4

8/31

Time Reversal

If x has period T,

• Rx = Rˆ

x

• r

x(−t)

FS

← − − → ˆ x[−k] Time reversal commutes with Fourier series x(t) ˆ x[k] x(−t) ˆ x[−k]

FS R R FS

Proof. ( Rx)[k] = Rx, ejkω0t = x, Rejkω0t = x, e−jkω0t = ˆ x[−k]

• Corollary. Fourier series preserves even/odd symmetry, i.e.

x even ⇐ ⇒ ˆ x even, x odd ⇐ ⇒ ˆ x odd

9/31

Time Scaling

If x has period T, then Sax has period T/a

• Sax = ˆ

x

• r

x(at)

FS

← − − → ˆ x[k] Time scaling preserves Fourier coefficients but changes fundamental frequency x(t) =

∞

• k=−∞

ˆ x[k]ejkω0t x(at) =

∞

• k=−∞

ˆ x[k]ejk(aω0)t different Fourier series

10/31

Time Scaling

x(t) =

∞

• k=−∞

ˆ x[k]ejkω0t vs. x(at) =

∞

• k=−∞

ˆ x[k]ejk(aω0)t t x(t)

− T

2 T 2

ω

1 2

2π T

− 2π

T 6π T

− 6π

T

t x( 3

2t)

− T

3 T 3

ω

1 2

3π T

− 3π

T 9π T

− 9π

T

compression in time ⇐ ⇒ expansion in frequency expansion in time ⇐ ⇒ compression in frequency

11/31

Differentiation

If x has period T, so does its derivative x′, and

• x′ = Mω0ˆ

x

• r

x′(t)

FS

← − − → jkω0ˆ x[k] where (Maˆ x)[k] = jakˆ x[k]

differentiation in time ⇐ ⇒ multiplication by jkω0 in frequency

Proof.

• x′[k] = 1

T

• T

x′(t)e−jkω0tdt = 1 T [x(t)e−jkω0t]

• T

0 + jkω0

1 T

• T

x(t)e−jkω0tdt

• = jkω0ˆ

x[k] Alternatively, differentiate term by term x(t) =

• k

ˆ x[k]ejkω0t = ⇒ x′(t) =

• k

jkω0ˆ x[k]e−jkω0t

12/31

Integration

If x has period T and Fourier series x(t) =

∞

• k=−∞

ˆ x[k]ejkω0t Integrating term by term y(t) t x(s)ds = ˆ x[0]t +

• k=0

ˆ x[k]ejkω0t − 1 jkω0 = ˆ x[0]t −

• k=0

ˆ x[k] jkω0 +

• k=0

1 jkω0 ˆ x[k]ejkω0t

• y periodic iff ˆ

x[0] = 0, i.e. x has no DC component

• if ˆ

x[0] = 0, y has period T, ˆ y[k] = 1 jkω0 ˆ x[k] for k = 0

13/31

Example: Triangle Wave

t

x(t)

− T

2 T 2

1

t

x1(t) = x′(t)

2 T

− 2

T

t

x2(t) = x′(t − T

4 )

2/T

t

x3(t) 1

ˆ x3[k] = sin(kπ/2) kπ x2 = 4 T x3 − 2 T ˆ x2[k] = 4 sin(kπ/2) kπT − 2 T δ[k] x1 = τ−T/4x2 ˆ x1[k] = 4 sin(kπ/2) kπT ejkπ/2 − 2 T δ[k] x′ = x1 ˆ x[k] = 1 jkω0 4 sin(kπ/2) kπT ejkπ/2, k = 0

14/31

Multiplication

If x and y have same period T, so does their product xy, and

• xy = ˆ

x ∗ ˆ y

• r

x(t)y(t)

FS

← − − →

∞

• m=−∞

ˆ x[m]ˆ y[k − m] multiplication in time ⇐ ⇒ convolution in frequency Proof. x(t)y(t) =

• ∞
• m=−∞

ˆ x[m]ejmω0t

∞

• ℓ=−∞

ˆ y[ℓ]ejℓω0t

• =

∞

• m=−∞

∞

• ℓ=−∞

ˆ x[m]ˆ y[ℓ]ej(m+ℓ)ω0t =

∞

• k=−∞
• ∞
• m=−∞

ˆ x[m]ˆ y[k − m]

• ejkω0t

(k = m + ℓ)

15/31

Periodic Convolution

Periodic convolution x ∗ y of x and y with same period T (x ∗ y)(t) =

• T

x(τ)y(t − τ)dτ Properties

• Commutativity

x ∗ y = y ∗ x

• Associativity

(x ∗ y) ∗ z = x ∗ (y ∗ z)

• Bilinearity
• i

aixi

• ∗
• j

bjyj

• =
• i,j

aibj(xi ∗ yj)

16/31

Periodic Convolution

Fourier coefficients satisfy

• x ∗ y = Tˆ

xˆ y

• r

(x ∗ y)(t)

FS

← − − → Tˆ x[k]ˆ y[k] convolution in time ⇐ ⇒ multiplication in frequency Proof. ( x ∗ y)[k] = 1 T

• T

(x ∗ y)(t)e−jkω0tdt = 1 T

• T
• T

x(τ)y(t − τ)dτ

• e−jkω0tdt

=

• T

x(τ) 1 T

• T

y(t − τ)e−jkω0tdt

• dτ

=

• T

x(τ)

• e−jkω0τˆ

y[k]

• dτ = Tˆ

x[k]ˆ y[k]

17/31

Conjugation and Symmetry

If x has period T, so does its complex conjugate ¯ x = x∗, and

• x∗ = Rˆ

x∗

• r

x∗(t)

FS

← − − → (ˆ x[−k])∗ Proof.

• x∗[k] = x∗, ejkω0t = x, e−jkω0t = ˆ

x[−k]

• Corollary. If x is real, ˆ

x is conjugate symmetric i.e. ˆ x[−k] = ˆ x[k]

• Corollary. If x is real and even, ˆ

x is also real and even ˆ x[k] = ˆ x[−k] = ˆ x[k]

• Corollary. If x is real and odd, ˆ

x is purely imaginary and odd −ˆ x[k] = ˆ x[−k] = ˆ x[k]

18/31

Contents

• 1. Properties of CT Fourier Series

1.1 Linearity 1.2 Time and Frequency Shifting 1.3 Time Reversal and Scaling 1.4 Differentiation and Integration 1.5 Multiplication and Periodic Convolution

• 2. Convergence of Fourier Series

2.1 Mean-square Convergence

19/31

Vector Space

Vector space V over field F (R or C) has two operations

• addition: x, y ∈ V =

⇒ x + y ∈ V

• scalar multiplication: λ ∈ F, x ∈ V =

⇒ λx ∈ V satisfying following axioms

• 1. commutativity of addition: x + y = y + x, ∀x, y ∈ V
• 2. associativity of addition: (x + y) + z = x + (y + z), ∀x, y, z ∈ V
• 3. identity element of addition: ∃0 ∈ V s.t. x + 0 = x, ∀x ∈ V
• 4. inverse elements of addition: ∀x ∈ V, ∃y ∈ V s.t. x + y = 0
• 5. identity element of scalar multiplication: 1x = x, ∀x ∈ V
• 6. compatibility of scalar and field multiplications

λ1(λ2x) = (λ1λ2)x, ∀λ1, λ2 ∈ F, x ∈ V

• 7. distributivity: λ(x + y) = λx + λy, ∀λ ∈ F, x, y ∈ V
• 8. distributivity: (λ1 + λ2)x = λ1x + λ2x, ∀λ1, λ2 ∈ F, x ∈ V

20/31

Vector Space

• Example. Rn, Cn

(x1, . . . , xn)T + (y1, . . . , yn)T = (x1 + y1, . . . , xn + yn)T λ(x1, . . . , xn)T = (λx1, . . . , λxn)T

• Example. CT real-/complex-valued signals RR, CR

(x + y)(t) = x(t) + y(t) (λx)(t) = λx(t)

• Example. DT real-/complex-valued signals RZ, CZ

(x + y)[n] = x[n] + y[n] (λx)[n] = λx[n]

21/31

Normed Vector Space

Norm on vector space V over field F · : V → R+ satisfies

• Positive definiteness

x = 0 ⇐ ⇒ x = 0

• Absolute homogeneity

λx = |λ| · x, ∀λ ∈ F, x ∈ V

• Triangle inequality

x + y ≤ x + y Vector space with norm called normed vector space

22/31

Normed Vector Space

• Example. Rn, Cn

xp =       

• n
• k=1

|xk|p 1/p , 1 ≤ p < ∞ max1≤k≤n |xk|, p = ∞

• Example. CT real-/complex-valued signals RR, CR

xp =

• |x(t)|pdt

1/p , 1 ≤ p < ∞ supt |x(t)|, p = ∞

• Example. DT real-/complex-valued signals RZ, CZ

xp =       

• ∞
• k=−∞

|x[k]|p 1/p , 1 ≤ p < ∞ supk∈Z |x[k]|, p = ∞

23/31

Inner Product Space

Inner product on vector space V over field F (R or C) ·, · : V × V → F satisfying

• Conjugate symmetry

x, y = y, x

• Linearity in first argument

ax + by, z = ax, z + bx, z

• Positive definiteness

x, x ≥ 0 x, x = 0 ⇐ ⇒ x = 0 Vector space with inner product called inner product space

24/31

Inner Product Space

• Example. Rn, Cn

x, y =

n

• k=1

xk¯ yk

• Example. CT real-/complex-valued signals RR, CR

x, y =

• R

x(t)y(t)dt

• Example. DT real-/complex-valued signals RZ, CZ

x, y =

∞

• k=−∞

x[k]y[k] In all cases x2 =

• x, x

Will write · instead of · 2 for inner product spaces

25/31

Cauchy-Schwarz Inequality

|x, y| ≤ x · y

• Proof. If y = 0, then y = 0 and x, y = x · y. Now

assume y = 0. For any λ ∈ C, 0 ≤ x − λy, x − λy = x2 − λx, y − ¯ λx, y + |λ|2y2 Let λ = x, y/y2, 0 ≤ x2−|x, y|2 y2 −|x, y|2 y2 +|x, y|2 y2 = ⇒ |x, y| ≤ x·y Integral form

• R

x(t)y(t)dt

• ≤
• R

|x(t)|2dt ·

• R

|y(t)|2dt

26/31

Orthonormal System in Inner Product Space

Two vectors x, y ∈ V are orthogonal if x, y = 0 Vector x ∈ V is unit vector if x = 1 or x, x = 1 {xk} is orthogonal system if xk, xm = 0 for m = k {ek} is orthonormal system if ek, em = δkm = δ[k − m]

• Example. Let V be set of periodic functions with period T.

Inner product x, y = 1 T

• T

x(t)y(t)dt Let ek(t) = ejk 2π

T t. Recall {ek : k ∈ Z} is orthonormal system

ek, em = δ[k − m]

27/31

Orthonormal Expansion

Given orthonormal system {ek}, suppose x =

• k

ckek Find coefficients by taking inner product with em x, em =

• k

ckek, em =

• k

ckek, em =

• k

ckδ[k − m] = cm Generalized Fourier series x =

• k

x, ekek This is how we do Fourier series expansion

28/31

Best Mean-square Approximation

Mean-square error in approximation of x by element in span{e1, . . . , en}, i.e. by element of form y = n

k=1 akek

x − y2 = x −

n

• k=1

akek, x −

n

• k=1

akek = x2 −

n

• k=1

akek, x −

n

• k=1

¯ akx, ek +

n

• k=1

|ak|2 = x2 − 2

n

• k=1

Re (¯ akx, ek) +

n

• k=1

|ak|2 ≥ x2 − 2

n

• k=1

|ak| · |x, ek, | +

n

• k=1

|ak|2 (use |Re z| ≤ |z|) = x2 −

n

• k=1

|x, ek|2 +

n

• k=1

(|ak| − |x, ek, |)2

29/31

Best Mean-square Approximation

Mean-square error in approximation x − y2 (1) ≥ x2 −

n

• k=1

|x, ek|2 +

n

• k=1

(|ak| − |x, ek, |)2

(2)

≥ x2 −

n

• k=1

|x, ek|2 Minimum mean-square error achieved iff ak = x, ek

• equality in (1) requires arg ak = argx, ek
• equality in (2) requires |ak| = |x, ek|

Fourier partial sum SN(x) = N

k=−N ˆ

x[k]ejkω0t is best mean-square approximation x − SN(x)2 = min

ak,−N≤k≤N x − N

• k=−N

akejkω0t2

30/31

Bessel’s Inequality

Minimum mean-square error x −

n

• k=1

x, ekek2 = x2 −

n

• k=1

|x, ek|2 Bessel’s inequality

• k

|x, ek|2 ≤ x2 For Fourier coefficients of signals with finite 2-norm

∞

• k=−∞

|ˆ x[k]|2 ≤ x2

• Corollary. limk→∞ ˆ

x[k] = 0

31/31

Parseval’s Identity

For Fourier coefficients of signals with finite 2-norm

∞

• k=−∞

|ˆ x[k]|2 = x2 = 1 T

• T

|x(t)|2dt Interpretation: Energy conservation

• |ˆ

x[k]|2 is average power of k-th harmonic component

• x2 is average power of x
• total power is sum of powers of all components
• Theorem. Fourier series converges in mean-square

lim

N→∞ x − SN(x) = 0

• NB. Convergence in mean-square does not imply pointwise

convergence