Equalization. Fig. Digital communication system using an adaptive equaliser at the receiver.

Equalization

Equalization

Fig. Digital communication system using an adaptive equaliser at the receiver.

Equalization

Equalization compensates for or mitigates inter-symbol interference (ISI) created by multipaths in time dispersive channels (frequency selective fading channels).

Equalizer must be “adaptive”, since channels are time varying.

Zero forcing equalizer

Design from frequency domain viewpoint.


∴ must compensate for the channel distortion.

⇒ Inverse channel filter completely eliminates ISI cau⇒sed by the channel ⇒ Zero Forcing equaliser, ⇒ ZF.



Fig. Pulses having a raised cosine spectrum



Example:

A two-path channel with impulse response

The transfer function is

The inverse channel filter has the transfer function


Since DSP is generally adopted for automatic equalizers it is convenient to use discrete time (sampled) represe⇒

ntation of signal.

Received signal

For simplicity, assume say


Denote a T-time delay element by Z− 1, then


The transfer function of the inverse channel filter is

This can be realized by a circuit known as the linear transversal filter.



The exact ZF equalizer is of infinite length but usually implemented by a truncated (finite) length approximation.

For , a 2-tap version of the ZF equalizer has coefficients

Modeling of ISI channels

Complex envelope of any modulated signal can be expressed as

where ha(t) is the amplitude shaping pulse.


In general, ASK, PSK, and QAM are included, but most FSK waveforms are not.

Received complex envelope is

where is channel impulse response.

Maximum likelihood receiver has impulse response

matched to f(t)


Output:

where nb(t) is output noise and

Least Mean Square Equalizers

Fig. A basic equaliser during training


Minimization of the mean square error (MSE), ⇒ MMSE.

Equalizer input

h(t): impulse response of tandem combination of transmit filter, channel and receiver filter.

In the absence of noise and ISI

The error due to noise and ISI at t=kT is given by The error is


The MSE is

In order to minimize , we require

……



The optimum tap coefficients are obtained as W = R−1 P.

But this is solved on the knowledge of xk's, which are the transmitted pilot data.

A given sequence of xk's called a test signal, reference signal or training signal is transmitted prior to the information signal, (periodically).

By detecting the training sequence, the adaptive algorithm in the receiver is able to compute and update the optimum wnk‘s -- until the next training sequence is sent.


Example: Determine the tap coefficients of a 2-tap MMSE for:

Now, given that


Mean Square Error (MSE) for optimum weights Let:

Mean Square Error (MSE) for optimum weights Now, the optimum weight vector was obtained as

Substituting this into the MSE formula above, we have

Mean Square Error (MSE) for optimum weights

Now, apply 3 matrix algebra rules:

For any square matrix

For any matrix product

For any square matrix

Mean Square Error (MSE) for optimum weights

For the example

MSE for zero forcing equalizers Recall for ZF equalizer

Assuming the same channel and noise as for the MMSE equalizer

for MMSE

MSE for zero forcing equalizers The ZF equalizer is an inverse filter; it amplifies noise ⇒

at frequencies where the channel transfer function has high attenuation.

The LMS algorithm tends to find optimum tap coefficients compromising between the effects of ISI and noise power increase, while the ZF equalizer design does not take noise into account.

Diversity Techniques

Mitigates fading effects by using multiple received signals which experienced different fading conditions.

Space diversity: With multiple antennas. Polarization diversity: Using differently polarized

waves. Frequency diversity: With multiple frequencies. Time diversity: By transmission of the same signal in

different times. Angle diversity: Using directive antenna aimed at

different directions.

Signal combining methods. Maximal Ratio combining.

Diversity Techniques

Equal gain combining. Selection (switching) combining.

Space diversity is classified into micro-diversity and macro-diversity.

Micro-diversity: Antennas are spaced closely to the order of a wavelength. Effective for fast fading where signal fades in a distance of the order of a wavelength.

Macro (site) diversity: Antennas are spaced wide enough to cope with the topographical conditions ( eg: buildings, roads, terrain). Effective for shadowing, where signal fades due to the topographical obstructions.

PDF of SNR for diversity systems Consider an M-branch space diversity system.

Signal received at each branch has Rayleigh distribution.

All branch signals are independent of one another.

Assume the same mean signal and noise power the s⇒ame mean SNR for all branches.

Instantaneous

PDF of SNR for diversity systems

Probability that takes values less than some threshold x is,

Selection Diversity

Selection Diversity

Branch selection unit selects the branch that has the largest SNR.

Events in which the selector output SNR, , is less than some value, x,is exactly the set of events in which each is simultaneously below x.

Since independent fading is assumed in each of the M branches,

Selection Diversity

Maximal Ratio Combining


is complex envelope of signal in the k-th branch.

The complex equivalent low-pass signal u(t) containing the information is common to all branches.

Assume u(t) normalized to unit mean square envelope such that


Assume time variation of gk (t) is much slower than that of u(t) .

Let nk(t) be the complex envelope of the additive Gaussian noise in the k-th receiver (branch).

⇒ usually all k N are equal.


Now define SNR of k-th branch as

Now,

Where are the complex combining weight factors. These factors are changed from instant to instant as the

branch signals change over the short term fading.


These factors are changed from instant to instant as the branch signals change over the short term fading.

How should be chosen to achieve maximum combiner output SNR at each instant?

Assuming nk(t)’s are mutually independent (uncorrelated), we have


Instantaneous output SNR, ,


Apply the Schwarz Inequality for complex valued numbers.

The equality holds if for all k, where K is an arbitrary complex constant.

Let


with equality holding if and only if , for each k.

Optimum weight for each branch has magnitude proportional to the signal magnitude and inversely proportional to the branch noise power level, and has a phase, canceling out the signal (channel ) phase.

This phase alignment allows coherent addition of branch signals “co-phasing”.⇒


each has a chi-square distribution.

is distributed as chi-square with 2M degrees of freedom.

Average SNR, , is simply the sum of the individual

for each branch, which is Γ,

Convolutional Codes

Department of Electrical Engineering

Wang Jin

Overview

Background Definition Speciality An Example State Diagram Code Trellis Transfer Function Summary Assignment

Background

Convolutional code is a kind of code using in digital communication systems

Using in additive white Gaussian noise channel

To improve the performance of radio and satellite communication systems

Include two parts: encoding and decoding

Block codes Vs Convolutional Codes Block codes take k input bits and produce n

output bits, where k and n are large There is no data dependency between blocks Useful for data communications

Convolution codes take a small number of input bits and produce a small number of output bits each time period Data passes through convolutional codes in a

continuous stream Useful for low-latency communication

Definition

A type of error-correction code in which each k-bit information symbol (each k-bit string) to

be encoded is transformed into an n-bit symbol, where n>k

the transformation is a function of the last M information symbols, where M is the constraint length of the code

Speciality

k bits are input, n bits are output k and n are very small (usually k=1~3, n=2~6).

Frequently, we will see that k=1 Output depends not only on current set of k input

bits, but also on past input The “constraint length” M is defined as the

number of shifts, over which a single message it can influence the encoder output

Frequently, we will see that k=1

An Example

A simple rate k/n= 1/2 convolutional code encoder (M=3)

The box represents one element of a serial register

+

+

Code digits

Binary information

digits

Input Output

An Example (cont’d) The content of the shift registers is shifted from left

to right Plus sign represents modulo-2 (XOR) addition Output by encoder are multiplexed into serial binary

digits For every binary digit enters the encoder, two code

digits are output A generator sequence specifies the connections of a

modulo-2 (XOR) adder to the encoder shift register. In this example, there are two generator sequences,

g1=[1 1 1] and g2=[1 0 1]

An Example (cont’d)

t=0

t=1

t=2

+

+

Code digits

Binary information

digits

Input Output

x2 x1 x0

x3 x2 x1

x4 x3 x2

x5 x4 x3t=3

When t=3, the content of the

initial state (x2, x1,

x0 ) is missing.

To Determine the Output Codeword There are essentially two ways

State diagram approach Transform-domain approach

Only concentrate on state diagram approach Contents of shift registers make up “state” of code:

Most recent input is most significant bit of state Oldest input is least significant bit of state (this convention is sometimes reverse)

Arcs connecting states represent allowable transitions Arcs are labeled with output bits transmitted during

transition

To Determine the Output Code Word ---State Diagram Rate k/n=1/2 convolutional code encoder

(M=3)

State is defined by the most (M-1) message bits moves into the encoder

D0 D1 D2

+

+

Code digits

Binary information

digits

State(recent M-1 digits)

State Diagram (cont’d)

There are four states [00], [01], [10], [11] corresponding to the (M-1) bits

Generally, assuming the encoder starts in the all-zero [00] state


Easiest way to determine the state diagram is to first determine the state table as shown below

Input (x3)

Start state (x2, x1)

Final state (x3, x2)

Output

0 0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 1 0 0 1 1 1 0 1 1 0 0 0 0 1 0 0 1 1 0 1 1 0 1 1 0 1 0 1 1 0 1 0 1 1 1 1 1 1 1 0


1/01 means (for example), that the input binary digit to the encoder was 1 and the corresponding codeword output is 01

10

00

11

01

1/10

1/01

0/11

0/01

0/00

1/11

1/00

0/10

Trellis Representation of Convolutional Code

State diagram is “unfolded” a function of time Time indicated by movement towards right

Code Trellis It is simply another way of drawing the

state diagram Code trellis for rate k/n=1/2 ,M=3

convolutional code shown below00

11

10

01 01

00

11

10

0/11

0/01 1/01

0/001/11

1/00

0/10

1/10

Start state Final state

Encoding Example Using Trellis Diagram Trellis diagram, similar to state diagram, also

shows the evolution in time of the state encoder

Consider the r=1/2, M=3 convolutional code

Encoding Example Using Trellis Diagram

00

01

10

11

01

10

11

000/00

0/11

1/11

1/00

0/10

1/010/01

1/10

State

Input data 0 1 0 0 1

Output 00 11 10 11 11

Distance Structure of a Convolutional code The Hamming distance between any two distinct

code sequences is the number of bits in which they differ:

The minimum free Hamming distance of a convolutional code is the smallest Hamming distance separating any two distinct code sequences:

The Transfer Function This is also known as the generating function

or the complete path enumerator. Consider the r=1/2 , M=3 convolutional code

example and redraw the state diagram.

a0 ba1c

d

JND2

JN

JD JD2

JDJND

JND

The Transfer Function (Con’d) State “a” has been split into an initial state “a0”and a

final state “a1” We are interested in the number of paths that

diverge from the all aero path at state “a” at some point in time and remerges with the all-zero path.

Each branch transition is labeled with a term , where are all integers such that: -----corresponds to the length of the branch -----Hamming weigh of the input zero for a “0” input and one

for a “1” input -----Hamming weight of the encoder output for that branch

The Transfer Function (Con’d) Assuming a unity input, we can write the set of

equations

By solving these equations,

From the transfer function, there is one path at a Hamming distance of 5 from the all-zero path. This path is of length 3 branches and corresponds to a difference of one input information bit from the all zero path. Other terms can be interpreted similarly. The minimum distance is thus 5.

Search for Good Codes

We would like convolutional codes with large free distance Must avoid “catastrophic codes”

Generators for best convolutional codes are generally found via computer search Search is constrained to codes with regular

structure Search is simplified because any permutation of

identical generators is equivalent Search is simplified because of linearity

Best Rate ½ Codes

M Generators

(in Octal)

dfree

3 5 7 5

4 15 17 6

5 23 35 7

6 53 75 8

7 133 171 10

8 247 371 10

9 561 753 12

Best Rate 1/3 Codes

M Generators

(in Octal)

dfree

3 5 7 7 8

4 13 15 17 10

5 25 33 37 12

6 47 53 75 13

7 133 145 171 15

8 225 331 367 16

9 557 663 711 18

Best Rate 2/3 Codes

M Generators

(in Octal)

dfree

2 17 16 15 4

3 27 75 72 6

4 236 155 337 7

Summary

What is convolutional code The transformation of a convolutional code We can represent convolutional codes as

generators, block diagrams, state diagrams and trellis diagrams

Convolutional codes are useful for real-time applications because they can be continuously encoded and decoded

Assignment

Question: Construct the state table and state diagram for the encoder below.

+

+

Code digits

Binary information

digits

Input (k=1) Output (n=3)

+

THANK YOU

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Equalization. Fig. Digital communication system using an adaptive equaliser at the receiver.

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