Lecture Slides

Digital Communication Lecture 1

Amirpasha Shirazinia

Department of Engineering Sciences

Uppsala University

January 19, 2015

Amirpasha Shirazinia (UU) Digital Communication Lecture 1 1 / 15

Todays Lecture

Course outline

Aims of the course

Introduction to digital communication


Course Outline

Part I January-March 12 + 1 Lectures 10 Tutorials 3 Hand-in assignments

Part II March-June 18 Lectures 11 Tutorials 1 Recap lecture of part I

Bonus question

Written exam


Teachers

Amirpasha Shirazinia (Part I) Email: [email protected] Office: A72411 Tel: 018 - 471 7003

Mikael Sternad (Part II) Email: [email protected] Office: A72132 Tel: 018 - 471 3078

Feel free to interrupt us!


Course Material

Main textbooks Digital Communications by Bernard Sklar (BS) ISBN 0-13-084788-7 Wireless Communications by Andrea Goldsmith (AG) ISBN 978-0-521-83716-3

Additional materials News Lecture notes, exercises, old exams


Language

The course will be taught in English since

The main textbooks are in English

International master students, exchange students, ...


Teaching style

Based on

Lecture slides: Definition, summary, important results, ...

Lecture notes (written on the board): derivations, comments onslides, ...

Some teaching notes during tutorials.

Lecture slides and lecture notes can be downloaded at the course page onstudentportalen.


Pre-requisite Courses

Signals and systems, or

Signal processing, or

Digital signal processing, or

Signal theory.


Outline Part I

Introduction (lec. 1)

Formatting: Sampling, quantization & baseband transmission (lec. 2)

Receiver structure (lec. 3-4)

Digital modulation schemes (lec. 5-7)

Channel coding (lec. 8-11)

Channel equalization (lec. 12)

Repetition and summary (lec. 13 in period 2)


Assessment

Hand-in assignments 3 assignments Work in groups of 2-3 Do not miss the deadlines Feel free to ask for help , Please read Rules Requirements on Studentportalen

Exam Covers all material Focuses on parts I and II equally


Aims Part I

We aim to teach/learn How information is transferred from source to receiver through a digitalcommunication system

How to assess the quality of the received information Some important modulation schemes The most important features of error correcting codes


Introduction

In this course, we are focusing on how the information is communicated digitally, how good the information is received.

Why communication? Humans need ...

Why digital representation/signaling?


Wired/Wireless Communication

Wired communication: ADSL: copper twisted wire Optical fiber: (almost) attenuation-free communication with highbit-rate

Wireless communication: Radio signal: e.g., satellite, mobile system, radar Infrared signals: TV remote control


Digital Communication: General Scheme

s(t)

r(t)

bit

Receiver (RX)

Transmitter (TX)

Format Encode Modulate

Channel

DemodulateDecodeFormat

Analog

info

c

c

b

b

Coded bits WaveformsInformation bits

Figure : A typical diagram of a digital communication system.


Random Signals

Random (Stochastic) processess

Signal energy and power

Cross/Auto-correlation

Power spectral density





Uppsala University

January 20, 2015


Bonus Question

Please hand in your answers in 3 minutes.

Answer:


Todays Lecture

Sampling

Quantization

Baseband modulation (coding)


Introduction

s(t)

r(t)

bit

Receiver (RX)

Transmitter (TX)


Channel


Analog

info

c

c

b

b


Formatting: Transforms the (analog) source information into digitalsymbols by sampling, quantization and baseband coding.

Applications: Analog-to-digital convertors, etc.


Sampling

Definition: is a discretizing process of a continuous signal in time.

How many samples are needed?

Sampling theorem: If the highest frequency in the signal S(t) is fmax,and the signal is sampled evenly at a rate of fs =

1

Ts> 2fmax, then

S(t) can be exactly recovered from its samples, i.e., Sis.

Nyquist frequency: 2fmax.


Sampling Methods

Impulse sampling by using a sequence of impulses.

Aliasing: happens when sampling rate is not high enough, i.e.,fs < 2fmax.

Other types of sampling methods: natural sampling, sample and hold,...


Quantization

Quantization is a compression method (a lossy source codingtechnique)

Definition: A mapping from a value in a continuous set into a digit ina discrete set.

In a general view, it can be divided into two types: Uniform quantization Non-uniform quantization


Quantization: Uniform and Non-uniform


Baseband Modulation/Coding

Digits are just abstractions a way to describe the messageinformation

Baseband modulation/coding: In practice, we present the binarydigits with electrical pulses

Two most common ways: Pulse Amplitude Modulation (PAM) Pulse Code Modulation (PCM)





Uppsala University

January 21, 2015


Bonus Question


Answer:


Todays Lecture

Mathematical Foundations of baseband demodulation/detection


Introduction

s(t)

r(t)

bit

Receiver (RX)

Transmitter (TX)


Channel


Analog

info

c

c

b

b


Baseband modulation: create pulses representing binary digits.

Baseband demodulation: The received waveforms are again pulses,but corrupted, due to thermal noise, interference, etc.

Goal of Baseband demodulation: is to recover a baseband pulse withthe best possible signal-to-noise ratio (SNR).


Noise

By noise, we normally mean thermal noise due to thermal motion ofelectrons.

Noise Statistics: We normally model the thermal noise by a randomprocess distributed according to Gaussian (normal) distribution.

n(t) N (0, 2) f(n) = 12pi2

en2

22 .

Spectral characteristics of thermal noise: Gn(f) =N02= 2, i.e., the

psd is flat at all frequencies white noise. Noise is additive additive white Gaussian noise (AWGN).


AWGN

4 3 2 1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

2 = 0.25

2 = 1

Figure : PDF of a Gaussian RV


Vector Representation of Signals

Geometric representation is useful in performance analysis of(baseband and bandpass) detectors.

Define an Ndimensional orthogonal space as a space characterizedby a set of N linearly independent functions {j(t)}Nj=1, called basisfunction.

Important condition:

T0

j(t)k(t)dt =

{Kj j = k0 otherwise

(1)

j, k = 1, . . . , N , 0 t T . Any arbitrary finite set of waveforms {si(t)}Mi=1 (pulse or sinusoid)with duration T , can be written as a linear combination of Northogonal waveforms {j(t)}Nj=1, where N M .


Vector Representation: Example 3.1 in BS

Orthogonal representation of waveforms:


Signal-to-noise ratio SNR)

In digital comm, SNR is expressed as Eb/N0. Waterfalling curves of error probability vs Eb/N0 are the mostcommon plots in research.

Figure : An example of water falling curves using different modulations.





Uppsala University

January 26, 2015


Bonus Question


Answer:


Todays Lecture

Detection Analysis of matched filter Analysis of error probability


Introduction

Receiver structure for baseband modulation

s(t)

n(t)

r(t)Matched filter

h(t)

z(t)Detector

0 or 1?s(t)

Matched filter: Aims to maximize SNR at the output of the filter. We characterize the matched filter.

Detection: Decision-making process of selecting the digital meaningof the waveform s(t).

We find the error probability (false detection) for some basebandwaveforms (signaling).


Matched Filter (MF)

Let the noisy (AWGN channel) observation r(t) = s(t) + n(t) befiltered using the impulse response h(t)

Hence, z(t) = r(t) h(t), 0 t T Our goal: is to determine h(t) that maximizes output SNR, i.e.,(S/N), at time t = T

s(t)

n(t)

r(t)Matched filter

h(t)z(t)

... MFs impulse response: h(t) = s(T t)


Error Probability

Now, we want to find the error probability for a binary waveform(signaling) si(t) (i = {1, 2}), i.e., probability of sending 1 butreceiving 0 (and vice versa).

si(t)

n(t)

r(t)

We have r(t) = si(t) + n(t), where n(t) is AWGN, s1(t) = a1 ands2(t) = a2.


Error Probability

Conditional probability of z(t) given that waveforms s1(t) and st(t)are transmitted over AWGN.

8 6 4 2 0 2 4 6 80

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

f (z |s1)f (z |s2)

a2 a1


Error Probability using a MF

We determine the threshold for detection and then error probabilityPB .

... Error probability becomes

PB = Q

Eb(1 )N0

,

where = 1E

b

T

0s1(t)s2(t)dt =

1

Eb

s1 s2 (why? Hint: use the vectorspace representation.)


Example 3.2 (BS)





Uppsala University

January 9, 2015


Bonus Question


Answer:


Todays Lecture

Passband Digital Modulation

s(t)

r(t)

bit

Receiver (RX)

Transmitter (TX)


Channel


Analog

info

c

c

b

b



Principles

Modulation: is to encode an information bit stream into a carriersignal, which is then transmitted over a communication channel.

Demodulation: is the process of extracting the information bit streamfrom the received signal.

Corruption: of the transmitted signal by the channel can lead to biterrors in the demodulation process.

Goal: to send bits at a high data rate while minimizing theprobability of error.


Modulation Types

Modulated carrier signals encode information in Amplitude Amplitude Shift Keying (ASK) Frequency Frequency Shift Keying (FSK) Phase Phase Shift Keying (PSK) Combined amplitude and phase Amplitude Modulation


Modulation Types

Modulated carrier signals encode information in Amplitude Amplitude Shift Keying (ASK) Frequency Frequency Shift Keying (FSK) Phase Phase Shift Keying (PSK) Combined amplitude and phase Amplitude Modulation

Representation:

s(t) = A(t) cos

(2pi(fc + f(t)

)t+ (t) +

)

. . .

= {u(t)ej2pifct

}, u(t) = sI(t) + jsQ(t) = (si1 + jsi2)g(t)


Amplitude and Phase Modulation

The information bit stream is encoded in the amplitude of the transmittedsignal.

Define: Symbol interval Ts and number of possible sequences (symbols): M

K = log2M bits are encoded into the amplitude and/or phase of the

transmitted signal s(t)

Amplitude/phase modulator

pi2

Shaping Filter g(t)

Shaping Filter g(t)

s(t)

InPhase branch

Quadrature Branch

i1

i2s

s i1s g(t)

s g(t)i2

csin(2 f t+ )

cos(2 f t+ )cpi 0

cos(2 f t+ )cpi 0

pi 0

Figure 5.10: Amplitude/Phase Modulator.Amirpasha Shirazinia (UU) Digital Communication Lecture 5 6 / 10

Amplitude and Phase Demodulation

Amplitude/phase demodulator (coherent demodulator if = 0)

x(t)=s (t)+n(t)i

1

m^=m

Find i: x Zi

i

T

T

InPhase branch

pi/2

g(Tt)

g(Tt)

cos (2 f t+ )

sin (2 f t+ )cpi

x =s +ni1 1

2x =s +ni2 2

Quadrature branch

s

s

pi c

Normally carrier phase recovery, i.e., , and synchronization, i.e., Ts, ischallenging in wireless communication.


Amplitude Modulation Technique: M-PAM

Pulse amplitude modulation (PAM) is the simplest (one-dimensional)form of amplitude modulation.

Representation: si(t) = {u(t)ej2pifct

}, where u(t) = Aig(t),

0 t Ts. In M-PAM, Ai = (2i 1M)d, i = 1, 2, . . . ,M . M-PAM Constellation:

M=4, K=2

00 01 11 10

M=8, K=3

000 001 011 010 110 111 101 100

2d

2d

Figure 5.12: Gray Encoding for MPAM.


Phase Modulation Technique: M-PSK

For M-PSK all the information is encoded in the phase of thetransmitted signal

Representation: si(t) = {u(t)ej2pifct

}, where

u(t) = Ag(t)e2pi(i1)/M , 0 t Ts.

M=4, K=2

0011

01

10

M=8, K=3

000

001

011

110

100

010

110

101

si1

si2

si1

si2

Figure 5.15: Gray Encoding for MPSK.


Amplitude and Phase Modulation Technique: M-QAM

For M-QAM, information is encoded in the phase and amplitude ofthe transmitted signal

More degrees of freedom Higher spectral efficiency. Representation: si(t) =

{u(t)ej2pifct

}, where u(t) = Aie

jig(t),0 t Ts.

4QAM 16QAM

Figure 5.18: 4QAM and 16QAM Constellations.





Uppsala University

February 6, 2015


Bonus Question


Answer:


Todays Lecture

Frequency modulation

Performance analysis of modulation techniques


Frequency Modulation

For frequency modulation: the information bits are encoded into thefrequency of the transmitted signal.

si(t) = Ag(t) cos

(2pi(fc + ifc)t+ i

), i = 1, . . . ,M, 0 t < Ts

Frequency separation needs to be considered to ensure orthogonalbasis functions:

f = mini,j

|fi fj| 12Ts

if i = j

f 1Ts

if i 6= j


Frequency Modulation Technique: M-FSK

In M-FSK the modulated signal is given by

si(t) = A cos

(2pi(fc + ifc

)t+ i

)

where i = (2i 1M) for i = 1, 2, . . . ,M .


Frequency Demodulators

Let the received signal be

r(t) = A cos(2pifit+ ) + n(t)

Coherent: We have perfect information about ideal!

Non-coherent: We do not have perfect knowledge of Practical.


Other Types of Modulation & Applications

Differential amplitude/frequency modulation

Addaptive Modulation

2G: Minimum Gaussian Shift Keying

3G: QPSK, 16-QAM

4G: Orthogonal Frequency Division Modulation (OFDM)


Performance Analysis of Digital Modulation Techniques

Channel: AWGN, i.e., r(t) = s(t) + n(t) where n(t) N (0, 2), 2 = N0/2 s(t) = {u(t)ej2pifct} Baseband signal u(t) has a bandwidth of B s(t) has a bandwidth of2B

Performance criterion: Probability of bit error Pb or symbol error Ps.

We define Eb: energy per bit Es: energy per symbol Tb: bit time Ts: symbol time SNR per bit b = Eb/N0 SNR per symbol s = Es/N0


Performance Analysis of BPSK and QPSK

BPSK (or binary PAM):

Pb = Q(

2Eb/N0

)= Q(

2b)

QPSK:

Pb = Q(

2Eb/N0

)= Q(

2b)

Ps = 1 (1 Pb)2





Uppsala University

February 10, 2015


Bonus Question


Answer:


Todays Lecture

Some notes on modulation

Performance analysis (SER and BER) for higher-order modulations


SER for M-PAM, -PSK, -QAM, -FSK in AWGN

Coherent M-PSK: P(MPSK)s 2Q(2s sin(pi/M)), s = (log2M)b

M-PAM: P(MPAM)s =

2(M1)M

Q(

6sM21

), s =

EsN0, Es =

1M

Mi=1A

2i

Rectangular M-QAM: P(MQAM)s 2(

M1)M

Q(

3sM1 )

Non-rectangular M-QAM: P(MQAM)s 4Q(

3sM1 )

Coherent M-FSK: P(MFSK)s Mm=1(1)m+1(M1m ) 1m+1 exp

(msm+1

)


Summary of SER and BER Formulas

Modulation Ps(s) Pb(b)

BFSK: Pb = Q(b

)BPSK: Pb = Q

(2b

)QPSK,4QAM: Ps 2Q

(s

)Pb Q

(2b

)MPAM: Ps 2(M1)M Q

(6s

M21

)Pb 2(M1)M log

2MQ

(6b log2 M(M21)

)

MPSK: Ps 2Q(2s sin(pi/M)

)Pb 2log

2MQ

(2b log2M sin(pi/M)

)Rectangular MQAM: Ps 2(

M1)M

Q

(3sM1

)Pb 2(

M1)

M log2MQ

(3b log2 M(M1)

)

Nonrectangular MQAM: Ps 4Q(

3sM1

)Pb 4log

2MQ

(3b log2 M(M1)

)

Table 6.1: Approximate Symbol and Bit Error Probabilities for Coherent Modulations


Comparison of Modulation Techniques

5 0 5 10 151016

1014

1012

1010

108

106

104

102

100

SNR per bit (b)

Bit e

rror r

ate

QPSK (or 4QAM)BFSKBPSK8PAM8PSK8QAM


Spectral Efficiency





Uppsala University

February 16, 2015


Todays Lecture

Fundamentals of channel coding


Channel Coding: Encoding & Decoding

s(t)

r(t)

bit

Receiver (RX)

Transmitter (TX)


Channel


Analog

info

c

c

b

b



Why Channel Coding?

Pros? Protect information bits Decrease BER coding gain

How? By adding redundancy (more bits) to information bits

Cons? Rate penalty (reduce data rate) Bandwidth expansion Delay


Channel Coding Techniques (Covered in the Course)

Linear Binary Codes: GF (2) block codes Cyclic codes

Non-binary codes: Reed Solomon Codes: GF (m)

Convolutional codes


Some Definitions

Hamming distance of two codewords: Modulo-2 addition of thecodewords

Code Weight: Number of 1s in a codeword

Minimum distance: Hamming distance between a codeword and theall-zero codeword.


Block Codes: Fundamentals

A code (n, k) contains a codeword of n bits from k information bits(n > k).

For an information sequence U and codeword C, encoding isperformed via a Generator matrix G {0, 1}kn such that C = UG.

We normally use systematic generator matrix G = [Ik|P].


Block Codes: Fundamentals

A code (n, k) contains a codeword of n bits from k information bits(n > k).

For an information sequence U and codeword C, encoding isperformed via a Generator matrix G {0, 1}kn such that C = UG.

We normally use systematic generator matrix G = [Ik|P]. Sometimes, it is easier to work with parity check matrixH = [Ink|P] {0, 1}(nk)n.

Syndrome check: S = CH = 0, otherwise the received codeword iscorrupt by noise.

minimum distance of linear binary block codes (n, k):dmin n k + 1.


Cyclic Codes: Fundamentals

Cyclic codes: If C = (c0c1 . . . cn1) is a codeword, thenC

i = (cici+1 . . . cni1) is a codeword.

Cyclic codes are generated via a generator polynomial:g(X) = g0 + g1X + . . .+ gnkX

nk. (g0, gn1 6= 0). Less complexity.

Systematic cyclic codes ...





Uppsala University

February 19, 2015


Bonus Question


Answer:


Todays Lecture

Channel Decoding: Techniques & Design


Channel Decoding

Hard-decision decoding (HDD): Each coded bit is demodulated aseither 0 or 1.

Soft-decision decoding (SDD): The distance between received bitfrom constellation points (in modulation) is also considered.


Hard Decision Decoding (HDD)

HDD typically uses minimum distance decoding.

It can be shown that for transmission over AWGN channels,maximum likelihood decoding (MLD) coincides with minimumdistance decoding.


Hard Decision Decoding (HDD)

HDD typically uses minimum distance decoding.

It can be shown that for transmission over AWGN channels,maximum likelihood decoding (MLD) coincides with minimumdistance decoding.

An (n, k) linear binary block code with minimum hamming distancedmin, using HHD, can

detect at most dmin 1 errors, correct at most 1

2(dmin 1 errors,

correct 2n 2k error patterns (all combinations of errors).


Probability of Error using HDD over AWGN Channel

Probability of error Pe: the probability that a transmitted codeword isdecoded in error.

Upper-bound

Pe n

j=t+1

(n

j

)pj(1 p)nj,

where p is the probability of symbol error for a modulation type.

Lower-bound (tight at high SNR)

Pe dminj=t+1

(dminj

)pj(1 p)nj.

Bit error probability Pb 1kPe


Soft Decision Decoding (SDD)

SDD typically uses correlation metric.

Using BPSK modulation, we obtain

Pe 2k

i=2

Q(

2Rcbwi)

2k

i=2

Q(

2Rcbdmin)

where wi is the hamming weight of the ith codeword.

SDD performs 2 dB better than HDD in coding gain.





Uppsala University

February 23, 2015


Bonus Question


Answer:


Todays Lecture

Convolutional Codes: Encoding & Decoding


Convolutional Codes

Codes with memory.

A Convolutional coded symbol is generated by passing informationbits through K linear shift register (memory unit + shift).

Number of shift registers: K (constraint length)

Representation: (k, n,K)

+ + +

1 2 . . . k 1 2 . . . k 1 2 . . . k. . .

k bits

1 2 . . . nTo modulator

Stage 1 Stage 2 Stage K

lengthn codeword

Figure 8.6: Convolutional Encoder.


Encoder Characterization

Encoder characterization: input transition phase output.

Three ways to characterize convolutional encoder: Tree diagram State diagram Trellis diagram (more popular!)


Trellis diagram

n = 3, k = 1, K = 3

+ +

Stage 2Stage 1 Stage 3

+

Encoder Output

S1

S S2 3

31C C

2C

S=S S2 3

t0

t1

t2

t3 t4000 000 000 000

S =01

1S =1

011 011111 111 111

111

010 010 010

001001

101 101 101

110 110

100 100

3 t000

011

111

010

001

101

110

100

5

00

01

10

11


Maximum likelihood Decoding (MLD)

Maximum likelihood decoding: check p(R|C) > p(R|C),C in oneof the two ways




1 Hard-decision: At branch i of the trellis diagram, check the metric

Bi =

nj=1

log p(Rij |Cij)

and check

iBi for all paths and choose the maximum. Equivalently,

find the minimum distance codeword by comparing the distance of allcodewords with the received codeword.





Bi =

nj=1

log p(Rij |Cij)

and check



2 Soft-decision: At branch i of the trellis diagram, check the metric

i =n

j=1

Rij(2Cij 1)

and check

ii for all paths and choose the maximum.





Bi =

nj=1

log p(Rij |Cij)

and check



2 Soft-decision: At branch i of the trellis diagram, check the metric

i =n

j=1

Rij(2Cij 1)

and check

ii for all paths and choose the maximum.

Complexity of MLD: A trellis diagram with information bits k andconstraint length K has 2K1 states, and 2k incoming/outgoingpaths to/from states. exponential complexity.


Hard Decision Viterbi Decoding

Instead of calculating all branch metrics, only calculate some of them:

Add the branch metric to the path metric for the old state. Compare the sums for paths arriving at the new state (there are only

two such paths). Select the path with the smallest value (Hamming distance). This path

corresponds to the one with fewest errors (survivor path).


Hard decision Viterbi Decoding: Example

1/2 convolutional code with received codeword: [111011000110],


Viterbi Decoding: Example (Conted)





Uppsala University

February 27, 2015


Bonus Question


Answer:


Todays Lecture

Analysis of Convolutional Codes


Trellis Diagram

n = 3, k = 1, K = 3

+ +

Stage 2Stage 1 Stage 3

+

Encoder Output

S1

S S2 3

31C C

2C

S=S S2 3

t0

t1

t2

t3 t4000 000 000 000

S =01

1S =1

011 011111 111 111

111

010 010 010

001001

101 101 101

110 110

100 100

3 t000

011

111

010

001

101

110

100

5

00

01

10

11


State Diagram

The right one is a modified version of the left one by splitting theself-loop of the all-zero state.

011

000


State Diagram

The right one is a modified version of the left one by splitting theself-loop of the all-zero state.

011

000

Transfer function (for the state diagram) is defined asT (D) = Xe/Xa =

d=df

adDd.


Extended State Diagram

The state diagram can be extended to yield information on codedistance properties. How?

Split the state a (all-zero state) into initial and final states. Label each branch by DdN lJ

path weight d denotes the Hamming weight of the n coded bits onthat branch,

data weight l = 0 when the info. bit is 0 (solid line), and l = 1 wheninfo. bit is 1 (dashed line).

Each transition on a branch represents J .


Transfer Function for Extended State Diagram

Transfer function is defined asT (D,N, J) = Xe/Xa =

d=df

m

lD

dJmN l,

In the previous extended state diagram,

T (D,N, J) =J3ND6

1 JND2L(1 + J) = J3ND6 + J4N2D8 + . . .

The minimum free distance df denotes The minimum weight of all paths (in Trellis or state diagram) that

diverge from and remerge with the all-zero state, or The lowest power of the transfer function T (D,N, J) in D.

Using long convolutional codes, we set J = 1, and the exponent of Ncan be written in terms of d. In a compact wayT (D,N) =

d adN

f(d)Dd.


Decoding Error Probability for Convolutional Codes

Assume, without loss of generality, all-zero codeword is transmitted.

An error event happens when an erroneous path is selected at thedecoder.

In general, decoding error probability Pe is upper-bounded as

Pe

d=df

adP2(d)

ad: the number of paths with the Hamming distance of d (known fromtransfer function)

P2(d): pairwise error probability of a path with Hamming distance of d(pairdepends on modulation type, hard or soft decision decoding)


Decoding Error Probability for Convolutional Codes

(Contd)

Using soft-decision decoding (Euclidean distance measure) and BPSKmodulation

P2(d) = Q

(2Ecd

N0

)= Q(

2bRcd) exp(bRcd)

Using hard-decision decoding (Hamming distance measure)

P2(d) =

dj=(d+1)/2

(d

j

)pj(1 p)dj

where p is bit error for binary symmetric channel.


BER for convolutional codes

BER is obtained by multiplying the error event probability by thenumber of data bit errors associated with each error event.

Pb

d=df

f(d)adP2(d),

where f(d) is the exponent of N in the transfer function T (D,N), or the

number of data bit errors corresponding to the erroneous path with the

Hamming distance of d.


Interleaving

Convolutional codes are suitable for memoryless channels withrandom error events.

But, some errors have bursty nature Statistical dependence among successive error events due the channel

memory: errors in multi-path fading channels.

Interleaving makes the channel looks like as a memoryless channel atthe decoder.

Interleaving is achieved by spreading the coded symbols in differentpositions before transmission.

A reverse operation, Deinterleaving, is used at the decoder.


Interleaving: Example

Consider a code with t = 1 ability of correction and 3 coded bits.

A burst error of length 3 cannot be corrected. A burst error of length 3 can not be corrected.

Let us use a block interleaver 3X3

A1 A2 A3 B1 B2 B3 C1 C2 C3

2 errors

Let us use a block interleaver 3 3

ECE 6640 13

Let us use a block interleaver 3X3

A1 A2 A3 B1 B2 B3 C1 C2 C3

Interleaver

A1 B1 C1 A2 B2 C2 A3 B3 C3

A1 B1 C1 A2 B2 C2 A3 B3 C3

Deinterleaver

A1 A2 A3 B1 B2 B3 C1 C2 C3

1 errors 1 errors 1 errors

Digital Communications I: Modulation and Coding Course, Period 3 2006, Sorour Falahati, Lecture 13





Uppsala University

March 9, 2015


Bonus Question


Answer:


Todays Lecture

Equalization


Intersymbol interference (ISI)

What is ISI? ISI is a type of distortion of a signal in which a symbol interferes with

subsequent symbols.

Causes & Consequences of ISI? Cause: Limited channel bandwidth increase in delay spread Consequence: modulation symbol time is on the same order as channel

delay spread Symbol Interference!

transmitted signal

received signal


Equalizer

Channel equalization is a way to combat against ISI, specially for highdata-rate wireless communications (e.g., 4G-LTE).

Equalizers are typically implemented at the receiver and beforedemodulation


Equalizer

Channel equalization is a way to combat against ISI, specially for highdata-rate wireless communications (e.g., 4G-LTE).

Equalizers are typically implemented at the receiver and beforedemodulation

Categories: Analog equalizer Digital equalizer (more common)

1 Linear equalizer: ZF, MMSE2 Non-linear equalizer: ML (using Viterbi)


Linear Equalizers

We have

Heq(z) =Li=0

wizi

The design goal is to optimally find wi

Zero-forcing (ZF)Heq(Z) = 1/H(Z)

Minimum mean-square error (MMSE) equalizer minimizes

E[(dk dk)2], k = 0, . . . , L


Non-linear & Other Equalizers

Maximum-likelihood sequence estimator (MLSE): is a type ofnon-linear equalizer based on

{dk}Lk=0 = argmax p([d0d1 . . . dL]

[r0r1 . . . rL])

Decision feedback equalizer (DFE), adaptive equalizers, ...


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