fund. of digital communications chapter 2: signals and systems · graz university of technology...

GRAZ UNIVERSITY OF TECHNOLOGY

Signal Processing and Speech Communications Lab

Fund. of Digital CommunicationsChapter 2: Signals and Systems

Klaus Witrisal

[email protected]

Signal Processing and Speech Communication Laboratory

www.spsc.tugraz.at

Graz University of Technology

October 6, 2016

Fund. of Digital CommunicationsChapter 2: Signals and Systems – p. 1/54



Outline

� 2-1 Signal Spaces

� 2-2 Linear Operators, Linear Systems, and a LittleLinear Algebra

� 2-3 Frequency Domain Representation of Signals

� 2-4 Matrix diagonalizations

� 2-5 Bandpass signals




2-1 Signal Spaces

� References: (Figures taken from these books.)

� Barry, Lee, Messerschmitt: “DigitalCommunications”, 3rd Ed., Kluwer AcademicPublishers, 2004

� J. G. Proakis and M. Salehi, “CommunicationSystem Engineering,” 2nd Ed., Prentice Hall, 2002

� M. Vetterli, et al., “Foundations of SignalProcessing,” Cambridge, 2014




2-1 Signal Spaces

� Idea: represent signals as vectors (in linear vectorspaces)

� allows for geometric interpretations

� linear (vector-) algebra can be used for signalprocessing algoithms

� applies for continuous-time and discrete-time signals




2-1 Signal Spaces (cont’d)

� Def: linear vector space

� set of vectors X , and scalars (in R or C) for whichthe following operations are defined:◮ vector addition and scalar multiplication

� and the following properties must hold:◮ additive identity (zero vector), additive inverse,

multiplicative identity◮ associative, commutative, and distributive laws

results are vectors in vector space

� ⇒ linearity follows in this case:

x,y ∈ X ; a, b ∈ R(or C)→ ax+ by ∈ X





� Elementary operations (in a 2D linear space)

a) sum of two vectors

b) multiplication of a vector by a scalar

[Barry 2004]





Properties of the elementary operations in a vector space:

� Given vectors x,y, z ∈ X and scalars a, b ∈ R(or C):

a) Commutativity: x+ y = y + x

b) Associativity: (x+ y) + z = x+ (y + z) and

abx = a(bx)

c) Distributivity: a(x+ y) = ax+ ay and

(a+ b)x = ax+ bx

d) Additive identity: There exists an element 0 ∈ X s.t.

x+ 0 = 0+ x = x for every x ∈ X

e) Additive inverse: There exists a unique element −x ∈ X s.t.

x+ (−x) = (−x) + x = 0 for every x ∈ X

f) Multiplicative identity: For every x ∈ X , 1x = x





� Subspaces in 3D Euclidean space X = R3

a) line (1D) ax ∈ X1 X1 ⊂ X = R3

b) plane (2D) ax+ by ∈ X2 X2 ⊂ X = R3

[Barry 2004]





� In digital communications:

� mapping of information onto sets of M waveforms

� signal space of M waveforms

S = span{s1(t), s2(t), ..., sM (t)}

� set of all signals, that can be represented as linearcombinations of these M waveforms

⇒ a subspace of continuous-time signals

� elements of this subspace; s(t) ∈ S

s(t) =α1s1(t) + α2s2(t) + · · · + αMsM (t) =

M∑

i=1

αisi(t) ∈ S





� E.g.: Signal space spanned by two signals s1(t), s2(t)

a) (basis) signals s1(t), s2(t)

b) examples of signals in S = span{s1(t), s2(t)}

[Barry 2004]Fund. of Digital CommunicationsChapter 2: Signals and Systems – p. 10/54




� inner product (scalar product) −→ inner product space

� needed f. geom. interpretations of:distance, angle, length (of/between vectors)

� E.g.: inner product on X = CN

〈x,y〉 =

N∑

i=1

xiy∗i = yHx

x = [x1, x2, ..., xN ]T

y = [y1, y2, ..., yN ]T

yH = [y∗1, y∗2, ..., y

∗N ]





Properties of the inner product:

� Def.: The inner product maps two vectors to a scalar,i.e. a = 〈x,y〉, where x,y ∈ X , a ∈ C(or R)

� The following holds:

Distributivity: 〈x+ y, z〉 = 〈x, z〉+ 〈y, z〉

Linearity in the first argument: 〈ax,y〉 = a〈x,y〉

(Conjugate linearity in the second: 〈x, ay〉 = a∗〈x,y〉)

Hermitean symmetry: 〈x,y〉∗ = 〈y,x〉

Positive definiteness: 〈x,x〉 > 0 for x 6= 0





� Def: (induced) norm (squared length, “energy”)

‖x‖2 = 〈x,x〉 =∑N

i=1 |xi|2 = xHx

� Def: angle (for real-valued vectors)

−1 ≤ 〈x,y〉‖x‖·‖y‖ = cos(θ) ≤ 1

� distance: ‖x− y‖





� Standard inner product spaces:

� DT signals (square-summable sequ.; X = ℓ2(Z))

〈x[n], y[n]〉 =

∞∑

n=−∞

x[n]y∗[n]

‖x[n]‖2 =

∞∑

n=−∞

|x[n]|2 <∞, ‖y[n]‖2 <∞

� CT signals (square-integrable functions; X = L2(R))

〈x(t), y(t)〉 =

∫ ∞

−∞

x(t)y∗(t)dt

‖x(t)‖2 =

∫ ∞

−∞

|x(t)|2dt <∞, ‖y(t)‖2 <∞





� Projection onto a subspace

� Def: subspace S ⊂ X

� for x ∈ X : projection of x onto S is unique elementx ∈ S, that is “closest” to x

‖x− x‖ = miny∈S‖x− y‖

x = argminy∈S‖x− y‖

� Projection Theorem:

� projection error x− x must be orthogonal to thesubspace S





� Projection of x onto sub-space S

� Projection of signal r(t)onto s1(t), s2(t)





� In digital communications: (review)

� mapping of information onto sets of M waveforms

� signal space of M waveforms

S = span{s1(t), s2(t), ..., sM (t)}

� set of all signals s(t) ∈ S, that can be represented aslinear combinations of these M waveforms, i.e.

s(t) =

M∑

i=1

αisi(t)





� Orthonormal basis (of a signal space)

� minimum set of N orthonormal functions (N ≤M )that can be used to represent the elements s(t) ∈ S(as linear combinations):

s(t) =

N∑

i=1

siψi(t)

� orthonormal (basis) functions {ψi(t)|i = 1, . . . , N}:

∫ ∞

−∞

ψi(t)ψ∗k(t)dt = δ[i−k] =

{1 i = k

0 i 6= k→ ‖ψi(t)‖ = 1





� Different bases for the signals s1(t), s2(t)





� Geometric representation of six waveforms inthree different bases

� geometric relations remain the same!





� arbitrary signals⇒ projection onto S, given anorthonormal basis for S

r(t) = r(t) + e(t)

r(t) =

N∑

i=1

riψi(t)

e(t) = r(t)− r(t) . . . projection error

rj = 〈r(t), ψj(t)〉 . . . coefficients

=∫∞−∞ r(t)ψ∗

j (t)dt





� Gram-Schmidt orthogonalization

� find the N ≤M orthonormal basis functions

1. ψ1(t) =s1(t)

‖s1(t)‖, S1 = span{ψ1(t)}

projection of s2(t) onto S1

2. ψ2(t) =s2(t)−s2(t)

‖s2(t)−s2(t)‖, s2(t) = c21ψ1(t)

k. ψk(t) =sk(t)−sk(t)

‖sk(t)−sk(t)‖, sk(t) =

∑k−1i=1 ckiψi(t)

no basis function, if sk(t)− sk(t) = 0





� Representation of M waveforms {sk(t)} through anorthonormal basis⇒ vectors

sk = [sk,1, sk,2, ..., sk,N ]T

where sk,i =

∫ ∞

−∞

sk(t)ψ∗i (t)dt = 〈sk(t), ψi(t)〉

� operations on signals are equivalent to operations onvectors (preservation of inner products; unitarity)

〈sj(t), sk(t)〉 = 〈sj , sk〉 = sHk sj

‖sk(t)‖2 = ‖sk‖

2 = sHk sk

‖sj(t)− sk(t)‖2 = ‖sj − sk‖

2




2-2 Linear Operators, LinearSystems, and a Little Linear

Algebra� Hilbert spaces: inner product spaces with properties

such as

� completeness: space contains all convergencepoints of sequences

� separability : space contains a countable basis

⇒ standard spaces CN , ℓ2(Z),L2(R) are all separableHilbert spaces




2-2 Linear Operators, ...

� Linear operator:

� generalizes finite-dimensional matrices

� used to perform operations with vector in-/outputs

� Def. of a linear operator (from H0 to H1):

� a function A : H0 → H1

� for all vectors x,y ∈ H0 and α ∈ C (or R),

� the following properties hold:(i) additivity: A(x+ y) = Ax+Ay

(ii) scalability: A(αx) = α(Ax)

H0 . . . domain; H1 . . . codomain

� Def.: linear operator on H0: a function A : H0 → H0





� a linear operator from CN to CM is the same as anM ×N matrix

� thus, concepts from linear algebra can be borrowed:

� e.g. the range of a lin. operator A : H0 → H1 is asubspace of H1:

R(A) = {Ax ∈ H1|x ∈ H0}

� e.g. the null space of a lin. operator A : H0 → H1 is asubspace of H0 that A maps to 0:

N (A) = {x ∈ H0|Ax = 0}





� Def.: inverse

� an operator A : H0 → H1 is invertible if a linearoperator B : H1 → H0 exists s.t.:

(a) BAx = x for every x ∈ H0 and(b) ABy = y for every y ∈ H1

� then B = A−1 is the inverse of A

� if only (a) holds, then B is the left inverse of A

� if only (b) holds, then B is the right inverse of A





� Unitary operators preserve the geometry of vectors(lengths and angles):

� Def.: A linear operator A : H0 → H1 that(i) is invertible(ii) preserves inner products

〈Ax,Ay〉H1= 〈x,y〉H0

for every for every x,y ∈ H0

� Prop. (ii) implies preservation of norms:

‖Ax‖2 = ‖x‖2 (Parseval theorem)

� Props. (i) and (ii) imply A−1 = AH

◮ (AH : H1 → H0 is the adjoint of A : H0 → H1)





� Orthogonal projection via pseudoinverse:

� projection onto R(A); least-squares approximation

� given A : H0 → H1; R(A) is a subspace of H1

� for any vector y ∈ H1, how to find best approxim.y ∈ R(A) of y?◮ find unique solution for y = Ax that minimizes

‖y − y‖2

◮ projection theorem yields

〈y −Ax, ai〉 = 0 ∀i; ai . . . i-th column of A

AH(y −Ax) = 0

AHAx = AHy . . . normal equations





� Orthogonal projection via pseudoinverse (cont’d):

� normal equations yield

x = (AHA)−1AHy = By

B . . . pseudoinverse of A

BA = I . . . i.e. B is a left inverse of A

� furthermore:

y = Ax = A(AHA)−1AHy = Py

P = AB . . . orthogonal projection onto R(A)




2-2 Linear Operators, LinearSystems, ...

� Eigenvectors and eigenvalues

� Def.: an eigenvector of an operator A : H → H is anon-zero vector v ∈ H s.t.

Av = λv

for some λ ∈ C

λ . . . eigenvalue

(λ,v) . . . eigenpair

� for H ⊆ ℓ2(Z), v is called eigensequence

� for H ⊆ L2(R), v is called eigensignal





� continuous-time (CT), linear systems

x yT

� T : L2(R)→ L2(R) is a linear operator on L2(R)

� i.e. y = Tx; x, y are CT functions x(t), y(t)

� in particular, we are interested in linear time-invariant(LTI) systems

� Def. (LTI System):

y = Hx⇒ y′ = Hx′, where x′(t) = x(t−τ) and y′(t) = y(t−τ)





� Eigenfunctions of LTI systems (complex exponentials)

v(t) = ej2πft

f . . . frequency in Hz (cycles per second)

� system response becomes

(Hv)(t) = λfv(t) = H(f)ej2πft

λf . . . eigenvalue for v(t) at frequency f

H(f) = λf . . . frequency response of the LTI system

� this motivates frequency domain signal representations




2-3 Frequency DomainRepresentation of Signals

� Why? – develop intuition for many processing steps indigital communications:

� sinusoidal signals are eigensignals of linear systems

� understanding the system response of LTI systems

� frequency occupation of communications signals

� separation of RF signals in frequency domain





� Fourier series (complex-valued form)

� x(t) is periodic with period T , i.e. x(t) = x(t+ T )

� functions with one period in (e.g.) L2([−12T,

12T ))

x(t) =

∞∑

k=−∞

ckejk(2π/T )t, ω0 =

2π

T, f0 =

1

Tfundamental freq.

� x(t) is represented in an orthogonal basis of complex

exponentials ejk(2π/T )t

� computation of the Fourier coefficients

ck =⟨x(t), ejk(2π/T )t

⟩=

1

T

∫ 12T

− 12T

x(t)e−jk(2π/T )tdt





� Fourier series properties

� given: Fourier series pair of a periodic signal

x(t) = x(t+T )F←→ {ck}; ck ∈ C,∀k ∈ Z; (i.e. {ck} ∈ ℓ

2(Z))

(the Fourier series reconstruction converges to x(t))

� Hermitean symmetry: for x(t) ∈ R: ck = c∗−k

� Parseval theorem (signal power)

P =1

T

∫ T

0

|x(t)|2dt =

∞∑

k=−∞

|ck|2

(normalized) power of a spectral component is |ck|2





� (LTI) system response in frequency domain

x(t) y(t)H(f)

x(t) =

∞∑

k=−∞

ckejk(2πf0)t → y(t) =

∞∑

k=−∞

ckH(kf0)ejk(2πf0)t

H(f) . . . frequency response of the LTI system

x(t) = x(t+ T )F←→ {ck}

y(t) = y(t+ T )F←→ {dk = ckH(kf0)}

� Exploits linearity: αx(t) + βy(t)F←→ {αck + βdk}





� E.g. periodic pulse train represented as Fourier series

� passed through first-order lowpass filter

-20 -15 -10 -5 0 5 10 15 20-0.1

0

0.1

0.2

0.3

index k (frequency k/T0)

coeffic

ient c

Koeffizienten der Fourier Reihe

-4 -2 0 2 4 6 8

0

0.5

1

time t

Zeitsignal

-20 -15 -10 -5 0 5 10 15 20-0.1

0

0.1

0.2

0.3

index k (frequency k f0)

coeffic

ients

ck, d

k

Fourier series coefficients

-4 -2 0 2 4 6 8-0.5

0

0.5

1

1.5

time t

sig

nals

x(t

), y

(t)

truncated reconstruction of the time-domain signals

ck ↔ x(t)

dk ↔ y(t)

x(t)

y(t)





� nonperiodic signals:

� Fourier transform

X(f) =

∫ ∞

−∞

x(t)e−j2πftdt

� inverse Fourier transform

x(t) =

∫ ∞

−∞

X(f)ej2πftdf

� Fourier transform pair: (if both exist ∀t,∀f ∈ R)

x(t)F←→ X(f); x(t) ∈ L2(R), X(f) ∈ L2(R)





� Fourier transform properties

� given: Fourier transform pair

x(t)F←→ X(f)

� Hermitean symmetry: for x(t) ∈ R: X(f) = X∗(−f)

� Parseval theorem (signal energy)

E =

∫ ∞

−∞

|x(t)|2dt =

∫ ∞

−∞

|X(f)|2df

|X(f)|2 . . . energy density at frequency f

� Note: E <∞ for x(t) ∈ L2(R)





� Properties of the Fourier transform (cont’d)

linearity ax(t) + by(t) ↔ aX(f) + bY (f)

time scaling x(at) ↔ 1|a|

X(

fa

)

convolution x(t) ∗ y(t) ↔ X(f)Y (f)

x(t)y(t) ↔ X(f) ∗ Y (f)

time shift x(t− τ) ↔ X(f)e−j2πfτ

frequency shift ej2πf0tx(t) ↔ X(f − f0)

real part Re{x(t)} = 12

(

x(t) + x∗(t))

↔ 12

(

X(f) +X∗(−f))

dmx(t)dtm

↔ (j2πf)mX(f)(

−jt2π

)mx(t) ↔

dmX(f)dfm

∫ t

−∞ x(τ)dτ ↔ 1j2πf

X(f) + 12δ(f)

∫∞−∞ x(τ)dτ

x(t) cos(2πf0t) ↔ 12(X(f − f0) +X(f + f0))

X(t) ↔ x(−f)





� Fourier transform pairs

ej2πf0t ↔ δ(f − f0)

δ(t− T ) ↔ e−j2πfT

cos(2πf0t) ↔ 12[δ(f − f0) + δ(f + f0)]

sin(2πf0t) ↔ 12j

[δ(f − f0)− δ(f + f0)]

sinc(Ft) ↔ 1Frect (f, F/2)

rect(t, T/2) ↔ T sinc (fT )

e−αtu(t) ↔ 1j2πf+α

; Re {α} > 0

u(t) ↔δ(f)2

+ 1j2πf

12

(

δ(t) + jπt

)

↔ u(f)∑+∞

k=−∞ δ(t− kT ) ↔ 1T

∑+∞m=−∞ δ

(

f − 1Tm)

1jt

↔ −πsgn(f)





� (LTI) system response in frequency domain

x(t) y(t)H(f)

x(t)F←→ X(f) → y(t)

F←→ Y (f) = X(f)H(f)

H(f) . . . frequency response of the LTI system

� System response in time domain:

� make use of convolution property :

y(t) = x(t) ∗ h(t)F←→ Y (f) = X(f)H(f)

h(t) . . . impulse response of the LTI system (see below)





� convolution integral (Faltungsintegral)

x(t) ∗ h(t) =

∫ ∞

−∞

x(λ)h(t− λ) dλ

=

∫ ∞

−∞

h(λ)x(t− λ) dλ

� convolution property of Fourier transform

y(t) = x(t) ∗ h(t)F←→ Y (f) = X(f)H(f)





� Dirac pulse, δ-pulse, Dirac distribution

� defined by integral (sampling, sifting property –Ausblendeeigenschaft):

∫ ∞

−∞

x(t)δ(t) dt = x(0)

∫ ∞

−∞

δ(t) dt = 1

� Fourier transform:

δ(t)F←→ 1

δ(t− T )F←→ e−j2πfT





� impulse response (of an LTI system):

x(t) = δ(t)

→ y(t) = δ(t) ∗ h(t)

=

∫ ∞

−∞

δ(λ)h(t− λ) dλ = h(t)

� time shifting property:

y(t) = δ(t− T ) ∗ x(t)

=

∫ ∞

−∞

δ(λ− T )x(t− λ) dλ = x(t− T )




2-4 Matrix factorizations

� LTI operator H on CN (circular convolution)

H = F−1HF

F . . . DFT matrix

F−1 . . . inverse DFT F−1 = 1NFH

H . . . frequency response (diagonal matrix)

� arbitrary lin. operator A on CN (full rank!)

A = VΛV−1 . . . spectral theorem; EV decomposition

V . . . Matrix of (N independent) eigenvectors

Λ . . . eigenvalues (diagonal matrix)




2-4 Matrix factorizations(cont’d)

� arbitrary lin. operator A : CN → CM

� singular value decomposition (SVD)

A = UΣVH

V . . . (Unitary) matrix of N right singular vectors

Λ . . . Singular values (diagonal matrix; sorted)

U . . . (Unitary) matrix of M left singular vectors

� for A having rank r, there are r non-zero singularvalues




2-5 Bandpass signals

� Definition:

� A bandpass (or narrowband) signal x(t) has anf -domain representation X(f) that is nonzero in a(usually small) neighborhood of some (usually high)frequency f0,

� i.e.: X(f) = 0 for |f − f0| ≥ W , where W ≪ fo

f0 . . . center frequency

W . . . bandwidth




2-5 Bandpass signals




2-5 Bandpass signals (cont’d)

� Monochromatic signal

� has bandwidth W = 0

x(t) = A cos(2πf0t+ θ)

� represented by phasor

X = Aejθ

� Bandpass signal

� time-varying phasor

xl(t) = V (t)ejθ(t)

V (t) envelope

θ(t) phase

xl(t) lowpass representation of x(t)





Relation between x(t) and xl(t):

� x(t) . . . bandpass signal

x(t) = Re{xl(t)e

j2πf0t}

� z(t) . . . analytic signalcorresponding to x(t); pre-envelope of x(t)

z(t) = xl(t)ej2πf0t

� xl(t) . . . lowpass representa-tion of x(t)

xl(t) = xc(t) + jxs(t)Fund. of Digital CommunicationsChapter 2: Signals and Systems – p. 52/54




� Generation and demodulation of bandpass signals (inpractice)

x(t) = xc(t) cos(2πf0t)− xs(t) sin(2πf0t)





� Transmission of bandpass signals through LTI systems

� given: x(t), h(t), y(t) . . . input (bandpass), linearsystem, output

� we find:

Yl(f) =1

2Xl(f)Hl(f)

yl(t) =1

2xl(t) ∗ hl(t)

� system response can be computed using lowpassequivalent representations of x(t), h(t), y(t)


fund. of digital communications chapter 2: signals and systems · graz university of technology...

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