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TSKS04 Digital CommunicationContinuation Course

Lecture 1

Repetition of Stochastic Processes, PSD of Linearly Modulated Signal

Emil Björnson

Department of Electrical Engineering (ISY)Division of Communication Systems

2016-01-18 TSKS04 Digital Communication Continuation Course - Lecture 1 2

TSKS04 Digital Communication - Formalities

Information: www.commsys.isy.liu.se/TSKS04Lecturer & examiner: Emil Björnson, [email protected] & labs: Salil Kashyap, [email protected]: Laboratory exercises (1 hp):

Solved in groups of 4-5 studentsSolve on your own – request help when needed

Written exam (5 hp):5 problems, 5 points eachGrade 3 (C): 12 pointsGrade 4 (B): 16 pointsGrade 5 (A): 20 pointsFinal grade is exam grade

Course Content

Topics Lecture§ Baseband representation, PSD of stochastic signals 1-2§ Estimation and hypothesis testing

§ Estimation of channels 3

§ Estimation for synchronization 4-5§ Estimation for equalization/decoding 6-8

§ Error control coding: Convolutional codes 9-11


Continue where TSKS01 ended and give an in-depth description of several important estimation, coding, and

decoding issues that arise in realistic digital communication systems, and theoretically founded solutions to these issues.


Repetition: Stochastic Process

Sample space: Ω𝜔$

𝜔%

𝜔&

𝑋$(𝜔$,𝑡)

𝑋%(𝜔%, 𝑡)

𝑋&(𝜔&, 𝑡)

𝑡

𝑡

𝑡

Stochastic time-continuous signal

Key propertiesMean: 𝑚 𝑡 = 𝐸 𝑋(𝑡)

ACF: 𝑟0 𝑡$,𝑡% = 𝐸 𝑋 𝑡$ 𝑋∗(𝑡%)


Examples of Stochastic Processes

Example 1: Finite number of realizations:

𝑋(𝑡) = sin (𝑡 + 𝜙), 𝜙 ∈ {0, 𝜋/2, 𝜋, 3𝜋/2}.

Example 2: Infinite number of realizations:

𝑋 𝑡 = ∑ 𝐴A𝑔(𝑡 − 𝑘)A , 𝑔 𝑡 = E cos (𝜋𝑡), |𝑡| < 1/2 0, elsewhere

Each 𝐴𝑘 is independent and 𝑁(0,1)

One realization:


Strict-Sense Stationarity

Stationarity is statistical invariance to a shift of the time origin.

Definition:Consider time instants 𝑡̅ = (𝑡$,… , 𝑡S) and shifted time instances 𝑢U = 𝑡̅ + Δ = (𝑡$ + Δ,… , 𝑡S + Δ). The process 𝑋(𝑡) is said to be strict-sense stationary (SSS) if

𝐹0(X̅) �̅� = 𝐹0(Z[) �̅�

holds for all 𝑁 and all choices of 𝑡̅ and Δ.

Equivalence:

𝐹0(X̅) �̅� = 𝐹0(Z[) �̅� ⇔ 𝑓0(X̅) �̅� = 𝑓0(Z[) �̅�

Wide-Sense Stationarity

Definition: A stochastic process 𝑋(𝑡) is said to be wide-sense stationary (WSS) if

§ Mean satisfies 𝑚0 𝑡 = 𝑚0 𝑡 + Δ for all Δ.§ Auto-correlation function (ACF) satisfies

𝑟0 𝑡$,𝑡% = 𝑟0 𝑡$ + Δ, 𝑡% + Δ for all Δ.

Interpretation:Constant mean, ACF only depends on time difference 𝜏 = 𝑡$− 𝑡%

Notation: Mean 𝑚0

ACF 𝑟0 𝜏



Gaussian Processes

Gaussian distribution:𝑋U = (𝑋$,… , 𝑋S) is jointly Gaussian, denoted as 𝑁(𝑚[, Λ0U), if

𝑓0U �̅� =1

2𝜋 Sdet (Λ0U)𝑒b

$% c̅bd[ ef[

gh c̅bd[ i

Definition: A stochastic process is called Gaussian if 𝑋(𝑡)̅ =𝑋(𝑡$),… ,𝑋(𝑡S) is jointly Gaussian for any 𝑡̅ = (𝑡$,… , 𝑡S)

Theorem: A Gaussian process that is wide sense stationary is also stationary in the strict sense.


Energy Spectrum and Power-Spectral Density

Consider a deterministic signal 𝜙(𝑡).

Fourier transform: Φ 𝑓 = ℱ 𝜙 𝑡 = ∫ 𝜙(𝑡)𝑒bm%noX𝑑𝑡qbq

Energy spectrum: 𝐸r(𝑓) = |Φ 𝑓 |%

Consider a WSS stochastic process signal 𝑥(𝑡).Power-spectral density (PSD) is Fourier transform of the ACF

𝑅0 𝑓 = ℱ 𝑟0(𝜏) = t 𝑟0(𝜏)𝑒bm%nou𝑑𝜏q

bq

Today’s topic:PSD of Linearly Modulated SignalRecall from TSKS01:

§ 𝑁 basis functions 𝜙$ 𝑡 , … , 𝜙S(𝑡) time-limited to 0 ≤ 𝑡 < 𝑇§ 𝑀 possible data symbols:

�̅�$,… , �̅�z where �̅�{ =𝑠{,$⋮𝑠{,S

Digital modulation:

𝑠{ 𝑡 =}𝑠{,m

S

m~$𝜙m(𝑡), 𝑖 = 1, … , 𝑀

Assumptions:§ Symbols intervals are non-overlapping§ No channel filtering


What if the channel filters the channel?

Selected stochastically

Dispersive Channels

§ Realistic channels are dispersive → spreads out signals§ Examples: Multi-path propagation in wireless (reflections, echos)§ Non-overlapping intervals at transmitter → Overlapping at receiver


What if the channel filters the channel?

This is essentially what the course is about!

• How to estimate the channel behavior?• How to treat consequently symbols jointly in decoding• How to utilize this for advanced encoding

Model of Linearly Modulated Signal

Pulse-amplitude modulated signal:

𝑆 𝑡 = } }𝑆{ 𝑛 𝜙{(𝑡 − 𝑛𝑇 −Ψ)S

{~$

q

�~bq


𝑛th data symbol(stochastic as before)𝑛 ∈ … , −1,0, +1, … = ℤ

𝑖th basis function(not necessarily time-limited)

Time delayRandom delay

Periodic signal• Naturally a non-WSS process• Use artificial delay Ψ, uniform on [0, 𝑇) to make it WSS

Autocorrelation Function of 𝑆(𝑡) (1/2)


Rearrange

Linearity

Indep.

Autocorrelation Function of 𝑆(𝑡) (2/2)

Notation: 𝑟��,�� 𝑛 −𝑚 = 𝐸 𝑆{ 𝑛 𝑆�∗[𝑚]


Uniformdist.

Change variable

= 𝑢= 𝑘

Simplify

Power-Spectral Density of 𝑆(𝑡) (1/2)

Recall:


Change variable

Rearrange

= 𝑣

Power-Spectral Density of 𝑆(𝑡) (2/2)

General PSD expression:

Matrix form:


Fourier transformΦ�∗ 𝑓

Fourier transformΦ{ 𝑓

Cross spectrum

Special Cases of

§ Real-valued signal 𝑆(𝑡): 𝑅�(𝑓) is also real § Imaginary part of 𝑅��,�� 𝑓𝑇 Φ{ 𝑓 Φ�

∗(𝑓) and 𝑅��,�� 𝑓𝑇 Φ� 𝑓 Φ{∗(𝑓) cancel

§ Consecutive symbols are independent§ Cross correlation:

§ Cross spectrum:

§ Special basis functions§ If Φ{ 𝑓 Φ�

∗(𝑓)is purely imaginary:


Real-valued (if indep. over basis functions)

Sine-Shaped Basis Functions (1/4) Same frequency

§ Sine and cosine with frequency 𝑓$ (2𝑓$𝑇 is positive integer)

§ Fourier transforms:


Product of cosine/sine and box function


§ Product of Fourier transforms:

§ Purely imaginary → PSD simplifies:

§ Energy spectra:




Essentially the same energy spectra!


PSD with this basis (and independent symbols):

𝑅� 𝑓 =𝜎�h%

2 +𝑚�h2 }𝛿(𝑓𝑇 −𝑚)

dsinc (𝑓 + 𝑓$ 𝑇) + −1 %oh�sinc (𝑓 − 𝑓$ 𝑇)

%

+𝜎��%

2 +𝑚��2 }𝛿(𝑓𝑇 −𝑚)

dsinc (𝑓 + 𝑓$ 𝑇) − −1 %oh�sinc (𝑓 − 𝑓$ 𝑇)

%


When does 𝛿-terms vanish?• All vanish if 𝑚�h = 𝑚�� = 0

• All but ±𝑓$ if 2𝑓$𝑇 is even (due to sinc = 0)

Examples: One-Dimensional Constellations

On-Off Keying (OOK)§ Mean value: 𝐸��/2§ Variance: 𝐸��/2

Binary Phase-Shift Keying (BPSK)§ Mean value: 0§ Variance: 𝐸��


0 2𝐸��

− 𝐸�� + 𝐸��

2𝑓$𝑇 = even integer

Examples: Two-Dimensional Constellations

𝑀-ary PSK:§ Component mean value: 0§ Component variance: 𝐸��/2

16-QAM:§ Component mean value: 0§ Component variance: 𝐸��/2


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