Convolution Paper

8/3/2019 Convolution Paper

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A Teaching Demo Application of Convolutional

Coding Techniques for Wireless Communications

Ali alhan, Celal eken and smail Ertrk

Computer Education Department, University of Kocaeli, Trkiye{ali.calhan, cceken, erturk }@kocaeli.edu.tr

Abstract- This paper introduces a demo application toolkit,implemented using MATLAB GUI, for teaching convolutional

coding techniques employed in wireless communications. In this

study presented, Viterbi decoding algorithm, Log-MAP and

SOVA Turbo decoding algorithms are performed using

MATLAB software. With the toolkit developed, an example image

transfer application has been carried out for comparative

performance analysis of these techniques. Using the toolkit,

performances of the convolutional coding techniques are easily

compared for various wireless channel models and through the

user friendly and visualized interface provided, learning process

of these techniques is satisfactorily achieved as well.I. INTRODUCTION

The demo application developed allows the user to be able

to compare the convolutional coding techniques, i.e., Viterbi

decoding algorithm that is the basis for convolutional coding

techniques and Turbo decoding with Log-MAP and SOVA

algorithms, widely used in wireless communications [1]. In

order to evaluate their performances comparatively, we performed several image transfer scenarios on AWGN

(Additive White Gaussian Noise), Rayleigh and Riceanchannels. All these techniques, their simulations and the demo

application toolkit are implemented using MATLAB software.

The MATLAB GUI demo tool provides several pop-up menus

and radio buttons for selecting parameters such as theconvolutional coding technique, its decoding algorithm, SNR

value, channel model, and etc. Any selected image file to be

transferred can utilise all of the available decoding algorithms

for varying SNR values. Consequently, obtained BER results

as a function of these specific selected algorithms and

parameters are denoted on GUI interface under each

transmitted image. A synopsis of BER results for different

predefined SNR values varying between 0-4 dB is also

presented on the GUI interface.There are two types of forward error correction techniques

commonly used, namely block coding and convolutional

coding [2, 3]. Block codes are based rigorously on finite field

arithmetic and abstract algebra, and as a result, convolutional

codes are commonly preferred in wireless communication.

Viterbi algorithm is the basis decoding algorithm for

convolutional codes while Turbo code is currently pointed out

as the best of them in use [4]. Turbo code enables information

transmission across the channel with arbitrary low

(approaching zero) bit error rate [5]. It is a parallel

concatenation of two component convolutional codes separated

by a random interleaver. It has been shown that a Turbo code

can achieve nearest performance to the Shannon limit [2, 5]. In

this work developed, an easy comprehension of these

convolutional coding techniques and their employment in

wireless communications is accomplished.

There are some restrictions and disadvantages of various

transmission channels in wireless medium between receiver

and transmitter where transmitted signals arrive at receiver

with different power and time delay due to the reflection,diffraction and scattering effects. In addition, the BER (Bit

Error Rate) value of the wireless medium is relatively high.

These drawbacks sometimes introduce destructive effects on

the wireless data transmission performance. As a result,

employing error control techniques is necessary in theseapplications.

During digital data transmission and storage operations,

performance criterion is commonly determined by BER which

is simply: the Number of Erroneous Bits/the Number of Total

Bits. Noise in transmission medium disturbs the signal and

causes data corruptions. Relation between signal and noise isdescribed with SNR (signal-to-noise ratio). Generally, SNR is

explained with Signal Power/Noise Power and is inverselyproportional with BER, meaning that the higher the SNR is the

less the BER and the better communication quality are.

The teaching demo application presented aims at easy

evaluation of different convolutional coding techniques interms of BER results as a function of varying SNR values and

also at displaying the distractive effects of different parameters

on the transmitted digital pictures over wireless medium.

The remainder of the paper is organized as follows. In the

next section, overall properties of convolutional codes together

with related decoding algorithms are described, followed byfundamentals of Viterbi decoding algorithms and Turbo

decoding algorithms for a better perception of the tool

developed. Section 3 introduces general information aboutwidely-used wireless channel models. Then, the summary of

the teaching demo application is provided in Section 4. Anexample image file transfer scenario with error control

mechanisms over a wireless medium modelled using the toolkit

and comparative performance evaluations of both Viterbi

decoding algorithms and Turbo decoding algorithms are

presented as well. The last section gives our summary with

final remarks.

978-1-4244-4740-4/09/$25.00 2009 IEEE


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II. CONVOLUTIONAL CODES

Convolutional encoding is accomplished using several linearshift registers and occurred to add redundant bits to data flow.

An input sequence x and contents of shift registers perform

modulo-two addition after information sequence is sent to shift

registers, so that an output sequence c is obtained. The ratio

R=k/n is called the code rate for a convolutional code. k is

the number of parallel input bits and n is the number ofparallel decoded output bits. m is the symbolized number of

shift registers shown in Fig. 1.

Shift registers store the state information of convolutional

encoder, and constraint length K relates the number of bits

upon which the output depends. A convolutional code can

become very complicated with various code rates and

constraint lengths. A simple convolutional code is shown in

Fig. 1, where D points to shift registers and points to X-OR logic operations. The trellis diagram representation shows

all possible information and encoded sequences for the

convolutional encoder and provides solution to the

convolutional codes. Fig. 2 presents the tree diagram of the

encoder given in Fig. 1.

D Dx

c1

c2

Fig. 1.Convolutional encoder with k=1, n=2, R=1/2, m=2 and K=3.

Fig. 2. Trellis diagram representation of the encoder in Fig. 1.

A. Viterbi Decoding AlgorithmViterbi decoding algorithm is usually applied to a

convolutional encoder and uses the maximum likelihood

decoding technique [6]. Noisy channels cause bit errors at

receiver. Viterbi algorithm estimates actual bit sequence using

trellis diagram. Its decoding algorithm is used in two different

forms. This difference results from the receiving form of the

bits in the receiver. Decoded information is obtained with

either hard decision or soft decision. It is explained with 1 on

hard decision operation while soft decision decoding uses multi

bit quantization [6]. Hard decision and soft decision decodingrefer to the type of quantization used on the received bits [2].

Hard decision decoding uses 1 bit quantization on the received

channel values while soft decision decoding uses multi bit

quantization on the received channel values.

For the ideal soft decision decoding (i.e., with infinite bit

quantization) the received channel values are directly used in

the Viterbi decoding process. The receiver does not assign azero or a one (single bit quantization) to each received bit, but

uses multi bit or infinite bit quantized values [7]. Ideally, the

received sequence r is infinite bit quantized and used directly

in the soft decision Viterbi decoder. The soft decision Viterbi algorithm is similar to its hard

decision algorithm except in that squared Euclidian distance is

used in the metric instead of Hamming distance. Generally,

with soft decision decoding, approximately 2 dB of coding

gain over hard decision can be obtained in Gaussian channels

[7].

B. Turbo Decoding AlgorithmsA Turbo code encoder is principally built using two identical

recursive systematic convolutional (RSC) codes with parallel

concatenation shown in Fig. 3. R is typically 1/2 for an RSC

encoder. These two component encoders are separated by aninterleaver. The interleaver changes the input sequence with a

certain rule. Only one of the systematic outputs from the two

component encoders is used as the systematic output from the

other component encoder is just a permuted version of the

chosen systematic output [5].

c1

c2Interleaver

RSC

Encoder 1

RSC

Encoder 2

u

c3

Fig. 3. Basic Turbo Encoder.

The first RSC encoder outputs are the systematic c1 and

recursive convolutional c2 sequences while the second RSC

encoder discards its systematic sequence and only output is the

recursive convolutional c3 sequence. In Fig. 3, R=1/3

convolutional encoder is shown and in order to obtain R=1/2

convolutional encoder, Turbo encoders outputs are punctured.

The Viterbi algorithm produces the ML (Maximum

Likelihood) output sequence for convolutional codes. It provides optimal sequence estimation for one stage

convolutional codes. For concatenated (multistage)

convolutional codes, there are two main drawbacks compared

to the conventional Viterbi decoders. First, the inner Viterbi

decoder produces bursts of bit errors which degrade the

performance of the outer Viterbi decoders. Second, the inner

Viterbi decoder produces hard decision outputs which prohibit

the outer Viterbi decoders from deriving the benefits of soft

decisions. Both of these drawbacks can be reduced and the

performance of the overall concatenated decoder can be

The right side of the symbol

1/11 denotes the bits which are

input to the encoder.

The left side of the symbol 1/11

denotes the state of the shift

registers.


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significantly improved if the Viterbi decoders are able to

produce reliability (soft-output) values. The reliability valuesare passed on to subsequent Viterbi decoders as a priori

information to improve decoding performance. This modified

Viterbi decoder is referred to as the soft-output Viterbi

algorithm (SOVA) decoder. Fig. 4 shows a concatenated

SOVA decoder [6, 8].

Fig. 4. A concatenated SOVA decoder (y, u and L represent the

received channel values, the hard decision output values and the associatedreliability values).

The operation of the SOVA algorithm is implemented by

using Viterbi algorithm twice. Classical Viterbi algorithm is

operated in forward direction of SOVA algorithm. It isnecessary to store path metrics while iterative decoding is

being implemented. In backward direction of iterative

decoding, survivor paths are not stored. Backward direction

operation and calculation of soft decision outputs are

concurrently implemented at the end of forward and backward

operation of SOVA, and soft outputs must be calculated as

logarithmic likelihood ratio [9].

( )( )( )( )( )

ru

ruu

i

i

i0Pr

1Prlog (1)

where r shows the received bit sequence. Metric calculationsfor obtaining soft outputs with SOVA algorithm are,

( )

( )( ){ }21121

1min

1

1

k

ibi

b

i

k

ifkk

ii

SMuBMSM

vM

++

=

(2)

{ }12,...,2,1,0, 21 mkk and ( )11

k

if SM , path metrics from

forward calculation, ( )1ib

i uBMi , branch metrics between

states, 2k

ibSM , path metrics from backward calculation.

Consequently, soft outputs are calculated as

( ) ( ) ( )01 iii MMu = (3)

At the heart of this iterative decoding algorithm is a soft-

input/soft-output module that is capable of computing the a

posteriori likelihood of the constituent codes. The original

MAP algorithm is unsuitable for practical implementations due

to the required multiplication and exponential operations [10].By formulating this algorithm in the logarithmic domain, the

multiplications become addition and the exponentials disappear

[11, 12]. The additions, however, become the soft combining

operations. Log-MAP algorithm is similar to SOVA decodingalgorithm except for that the backward and forward direction

operations are implemented together.

Decoder 1 Interleaver Decoder 2

De-Interleaver

rc1

rc2 rc3

Fig. 5. Turbo decoder scheme.

In Fig. 5, a general Turbo decoder structure is shown for

both SOVA and Log-MAP decoding algorithms. Because of

iterative decoding algorithm, in the MAP algorithm APPs (A

Posteriori Probability) are calculated for each data bits, unlikein the Viterbi algorithm. However, there are similarities in the

implementation of the both algorithms. When the decoded bit-

error probability PB is small, there is a very little performance

difference between the MAP and Viterbi algorithms. However,at low values of bit-energy to noise-power spectral density,

Eb/No, and high values of PB, the MAP algorithm canoutperform decoding with a soft-output Viterbi algorithm

called SOVA by 0.5 dB or more [6, 13]. For Turbo codes, this

can be very important since the first decoding iterations can

yield poor error performance. The implementation of the MAP

algorithm proceeds somewhat like performing a Viterbi

algorithm in two directions over a block of code bits. Once this

bidirectional computation yields state and branch metrics for

the block, the APPs and the MAP can be obtained for each data

bit represented within the block.

WIRELESS CHANNEL

In Wireless Communication, data flows through a noisy

environment. Various channel models are available for

modelling the noisy medium. This section is referred AWGN,

Rayleigh and Ricean wireless channel models.

The most important and simplest practical statement of a

mobile radio channel model in wireless communications is theadditive white Gaussian noise (AWGN) channel. When the

signals are transmitted over such channels, the signals arriving

at the modulator is perturbed only by the addition of some

noise and by some fixed, multiplicative path loss including

shadowing [14]. This channel is used in a mobile system where

the wireless nodes are not in motion. The noise is white thathas a constant power spectral density and Gaussian that has a

normal distribution. Addition to this, there is no variation in the

path loss over the signal bandwidth. All components of noise

are independent and real Gaussian processes. The SNR is the

most important parameter in calculating the performance in theAWGN channel.

Mobile radio performance will not be as good as the AWGN

channel because of fading. The detailed characteristics of the

propagation environment result of fading influence, where a

multiplicative and time-variant process included. Fading varies


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with time and so the SNR at the demodulator input varies with

time.In practice, a transmitter and receiver are surrounded by

objects which reflect and scatter the transmitted energy,

causing waves to arrive at the receiver via different paths. This

is multipath propagation. If the signal between transmitter and

receiver is blocked, this is called non-line-of-sight (NLOS)

propagation.

Consequently, the magnitude of a complex Gaussian randomvariable is a Rayleigh distributed random variable. The

Rayleigh distribution is approximation to measured fading

amplitude statistics for mobile fading channels in NLOS

situations [14].At Ricean channel model, the received signal is associated

with a random multipath component in the LOS situation,

whose amplitude is described by the Rayleigh distribution with

a constant power to cause a coherent LOS component. The

power of this component will usually be greater than the total

multipath power before it needs to be considered as affectingthe Rayleigh distribution significantly [14].

IV. GUI EDITOR

For the implemented demo application tool, MATLAB GUI

editor is designed as shown in Fig. 6. In the GUI, an image file

is chosen from the popup menu that entitled Original Picture

under Decoding Parameters and then shown on the axis that

is located on left side of the editor. After that any decoding

algorithm is marked using radio buttons located top-left side of

the GUI. Convolutional coding/decoding parameters (i.e.,

transmitting channel, SNR value, code rate, constraint length,

and iteration number for Turbo coding) are selected from

popup menus. When Transmit button is clicked with these

settings, the received image obtained is shown with the BER

result at the top-right side of the editor. These operationsexplained are relevant for only one decoding algorithm. In

order to compare BER results of all decoding algorithms for aspecific SNR value, we must use Comparison Manner popup

menu and then click on the Compare button. Besides, the

demo application enables comparison of an image file

transferred in terms of BER results obtained for different

decoding algorithms varying the SNR values through its

Comparison Manner popup menu.In order to examine BER performance results of an image

transferred with a certain convolutional coding technique

according to varying SNR values, Comparison Manner

popup menu is set to According to SNR option and

Compare button is pressed in the GUI editor. An examplescenario with its visibly comparable results illustrating thisapplication is shown in Fig. 6, Viterbi soft decision decoding

algorithm is utilized for transmitting an image on AWGN

channel in this scenario. The image to be transferred is formed

in 1024 frames, each consisting of 512 bits.

Fig. 6. BER results for different SNRs with Viterbi

soft decision decoding algorithm.

In Fig. 6, the image transmission is performed for SNR

values of 0, 1, 2, 3 and 4 dB. Obtained results show that,Viterbi soft decision decoding algorithm achieves better BER

performance with increasing SNR values. The relation between

BER and SNR is indicated with a graphic on the right side of

the GUI editor. The BER performance results of the example

scenario for all decoding algorithms are demonstrated in Fig. 7,the BER results of the image transferred are 2x10-4, 5.7x10-6,

and 0 for Viterbi, SOVA, and Log-MAP algorithms

respectively, as shown on the center bottom of the GUI editor.

In addition to these example applications and their results, it

should also be mentioned that according to the computational

complexity, Log-MAP algorithm is simpler than SOVA

algorithm. For example, considering 5 iterations, Log-MAP

decoding algorithm has 20 times and SOVA algorithm has 2times more complexity than Viterbi algorithm.

Fig. 7. BER performances according to the decoding algorithms.


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V. CONCLUSIONS

In this study, it is intended to teach convolutional codingtechniques with different decoding strategies, and in this

context, a MATLAB GUI editor application has been

implemented. In order to demonstrate and teach comparative

performance evaluation of Viterbi decoding algorithms and

Turbo decoding algorithms for various propagation channel

models, an example image transfer application is implementedthrough the proposed demo program.

Using this demo application tool, students can easily and

visually drive the conclusions that according to the simulation

results obtained under the same networking conditions, for

example, Turbo decoding algorithms provide better BER

performance when compared to Viterbi decoding algorithms,

especially for increasing SNR values (e.g., when SNR value is

4 dB and AWGN channel model, approximately 100 times

better BER results are obtained using Turbo decoding

algorithms).

It should be also noted that a fairly high computational

complexity is required for Turbo code decoding and this can be

understood with long simulation periods. The SOVAcomponent decoder is approximately two times as complex as

the Viterbi decoder. Turbo code decoding involves two SOVA

component decoders; thus, one iteration of a Turbo code

decoding requires complexity as much as four times that of the

Viterbi decoder. Therefore, the complexity of the Turbo code

decoder dramatically grows as the number of decoding

iterations is increased. The demo application presents the

delaying effect of computational complexity while producing

the BER results and showing the transferred image as well.

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Convolution Paper