tut/tlt/mr tlt-5406/1 tlt-5406 digital transmission lecture … · 2008-02-05 · tut/tlt/mr...

TUT/TLT/MR TLT-5406/1

TLT-5406 Digital Transmission Lecture Notes, Spring 2008

Markku Renfors Department of Communications Engineering

Tampere University of Technology Contents Introduction 2 Brief Introduction to Information Theory 5

Information theory, Lossless source coding; Channel capacity Transmission Channels 36 Baseband Digital Transmission 46

Baseband transmission techniques and line coding principles; Nyquist pulse shaping principles; Eye diagram.

Modulation in Digital Transmission 87 Linear digital modulation, QAM, PSK; BER calculation and performance evaluation; Bit mapping principles; VSB, CAP, D-PSK

FSK-Type Modulation Methods 141 FSK, MSK, CPFSK, GMSK

Basics of Detection and Estimation Theory 158 ML and MAP principles in BSC and AWGN cases, ML sequence detection, Viterbi algorithm

Signal Space Concepts 199 Maximum Likelihood Detection for Continuous-Time Channels 208

Optimum receiver principles, Sufficient statistics, Correlation receiver, Matched filter receiver, Sampled matched filter

Channel Equalizer Structures 232 Zero forcing and MSE principles; LE, FSE, DFE, MLSD/Viterbi

Adaptive Channel Equalization 258 General concepts, LMS algorithm in LE and DFE

Synchronization in Digital Receivers 284 Partial Response Signalling 320 Scrambling Techniques 329 Error Control Coding 334

Basics of error control coding, Coding gain, Block codes, Reed-Solomon codes

Convolutional Codes 368 Convolutional codes, Concatenated coding, Interleaving

Trellis Coding, Coded Modulation 388


INTRODUCTION A digital transmission system may or may not include conversions between analog and digital signals (sampling, A/D- and D/A-conversion)

The transmitter end of the transmission chain converts a digital bit-stream into an analog waveform which is sent to the physical channel, which is in practise analog. The receiving end converts the received analog waveform back to digital format.

The transmission chain includes:

Source coding/decoding: Reducing the bit-rate of the information signal by reducing the redundancy; Compression.

Channel coding/decoding: Error control coding, compensating the effect of bit errors that inevitably take place in a practical transmission channel.

In any 'sensible' channel, it is possible to get arbitrarily low bit-error-rate by increasing redundancy in the transmitted signal and using error control coding.

One of the central results of information theory is that source coding and channel coding can, in principle, be carried out independently of each other.

Modulation/demodulation: converting a digital signal into analog waveform.

Channel, that distorts the transmitted signal and adds noise and interference to it.


Block Diagram of a Digital Transmission System

Sampling andquantization

Channelcoding

Sourcecoding Modulation

Channel

Analoginput

Digitalinput

NoiseInterferences

Synchroni-zation

ReconstructionD/A-conversion

Sourcedecoding

Channeldecoding

Channel equal.Detection

Demodu-lation

Analogoutput

Digitaloutput

In the following, the digital transmission system is considered to include the chain between (and including) channel coding and decoding.

With this definition, the transmission chain can be designed and analyzed independently of the nature of the transmitted signal.


The Parameters of a Digital Transmission System The properties of a transmission chain are characterized by the following parameters: • Transmission rate, bit rate (bits/s, bps) • Bit error rate or block error rate (also packet error rate or frame

error rate) • Delay

o at link level due to propagation and signal processing o also due to buffering and possible re-transmissions o delay variations are also significant

In the system design, the primary target is to minimize the used bandwidth and/or the transmitted signal power, while achieving the desired data rate and error probability characteristics.

Some practical issues • When designing wireless transmission systems, the optimi-

zation has often lead to have following kind of characteristics: o Worst case bite error rate before decoding: 0.01 … 0.1 o Worst case bite error rate after decoding: 0.0001 … 0.001 o Worst case block error rate 5 … 10 %

Erroneous blocks are handled by re-transmission techniques.

• The effects due to the RF stages of the transmitter (TX) and receiver (RX) are often combined with the channel model o The effects due to TX and RX filtering can be combined

with the frequency selective channel model. o The noise generated by the receiver front end is combined

with the channel noise. o However, while the physical transmission channel can be

considered to be completely linear, the nonlinearity of practical TX power amplifiers is an important practical concern.


BRIEF INTRODUCTION TO INFORMATION THEORY In this section we study the concepts of information and entropy. With this theory, it is possible to calculate the largest possible information transmission rate through a given channel. This is called the channel capacity. Even though it is usually not possible to achieve the channel capacity in a practical system, it is a good reference point when evaluating the system performance. In fact, the Shannon-Hartley law is an important fundamental law of nature in the field of communication theory, and it is quite useful also in practical engineering work. Source: Lee&Messerschmitt, Chapter 4.

Source coding part: • B. Sklar, Digital Communications, 2nd Ed., Prentice-Hall

2001, Section 13.7 • J.G. Proakis, Digital Communications, 4th Ed., McGraw-

Hill 2001. Section 3.3


Definition of Information Here the message signal is modeled as a random process. We begin by considering observations of a random variable:

• Each observation gives a certain amount of information.

• Rare observations give more information than usual ones.

Example: The statement The sun rose this morning gives

very little information (high probability).

The statement San Francisco was destroyed this morning by an earthquake gives a lot of information (low probability).

Definition Observing a random variable X that takes its values from the set { } ,, , 21 KX aaa …=Ω , the (self) information of observation ma is

( )( )mXm apah 2log)( −=


Definition of Information - Interpretation • It is easy to see that ∞≤≤ )(0 mah • For a rare event the probability ( )mX ap is small and the

information is large. • For a usual event the probability ( ) 1≈mX ap and the

information is small. • Why logarithm?

In case of two independent random variables X and Y , { } ,, , 21 NY bbb …=Ω , the information of the joint event

nm ba , becomes

( )( ) ( )( ) ( )( ))()(

loglog,log),( 22,2

nm

nYmXnmYXnmbhah

bpapbapbah

+=

−−=−=

In case of independent events, the information is additive, which is intuitively sensible.

• Using base 2 logarithm => the unit of information is bit. • (base 10 logarithm => the unit is dit) • (base e (natural) logarithm => the unit is nat or Hartley)


Entropy The average information of a random variable is

This is called the entropy. Entropy has the following interpretations:

• Average information obtained from an observation. • Average uncertainty about X before the observation.

Example Binary random variable X , { } ( )0,1 , 1X Xp qΩ = =

The entropy is ( ) ( )qqqqXH −−−−= 1log)1(log)( 22

The maximum entropy is 1 and it is obtained when 21=q .

The entropy becomes zero for 0=q or 1=q .

( )( )[ ] ( ) ( )( )xpxpxpEXH Xx

XXX

22 loglog)( ∑Ω∈

−=−=


Example: The Entropy of an English Text Memoryless model of English text:

Source Symbol ai

Probability ip

Space 0.186 A 0.064 B 0.013 C 0.022 D 0.032 E 0.103 F 0.021 G 0.015 H 0.047 I 0.058 J 0.001 K 0.005 L 0.032 M 0.020 N 0.057 O 0.063 P 0.015 Q 0.001 R 0.048 S 0.051 T 0.080 U 0.023 V 0.008 W 0.017 X 0.001 Y 0.016 Z 0.001

Entropy 2( ) log 4.03 bitsi i

iH X p p= − =∑


Source Coding Theorem Let us consider a discrete-time and discrete-amplitude source that creates independent observations of a random variable X at rate r samples per second. The rate of the source is defined as )(XrHR = .

It is worth noting that it is often very difficult to construct codes that provide a rate arbitrarily close to the rate R. But often it is easy to get ‘rather close’ to the rate R.

Such a source can be coded using a source coder into a bit-stream, the rate of which is less than ε+R , for any 0>ε .


An Example of Source Coding Let us consider again the case of binary random variable. (1) 2

1=q

Now the entropy is ( ) 1H X = , so 1 bit per sample is needed. In this case it is best to send the samples as they are.

(2) q=0.1 47.0)9.0(log9.0)1.0(log1.0)( 22 =−−=XH

On the average, less than one bit per sample is sufficient. Now it is possible to construct codes where the average number of bits per sample is in the range 0.47 ... 1. One very simple but rather efficient code is obtained by coding two consecutive source bits as follows:

samples code word 0,0 0 0,1 10 1,0 110 1,1 111

It is also easy to decode a bit-stream constructed in this way since no code word is a prefix of another. In this code, 0.645 bits per sample are used on the average.


More Examples (1) When throwing a coin, if the result is always head (it

might be an unfair coin :-) the entropy becomes: 0)1log(10log0)( 2 =−−=XH (2) It is easy to show that the entropy satisfies KXH 2log)( ≤

where K is the number of possible outcomes (sample values, symbols, …). K2log bits is, of course, always enough. It can also be shown that KXH 2log)( = only if all possible outcomes are equally probable.


About Source Coding In the following we consider briefly binary coding methods that aim at minimizing the number of used bits as close to the source entropy as possible. These methods are lossless in the sense that the decoder is able to reproduce exactly the original source bit stream.

Many coding principles applied in this context utilize unequal code wordlengths (i.e., different number of bits for different symbols or symbol blocks). Naturally, the shortest word-lengths are used for the most common symbols/symbol blocks.

An important requirement for a code is that any coded bit-stream can be uniquely decoded. One common principle to achieve this goal is the prefix property: none of the code-words appears as the beginning of another codeword. This guarantees that, when scanning the bit-stream, a recognized codeword (belonging to the code) determines also the boundary between consecutive code-words.

The compression methods discussed below (entropy coding) are widely used, e.g., for compressing files in computer systems. They are also one ingredient in most of the current speech, audio, and video compression systems. However, in these applications much better compression ratio can be achieved by using lossy coding methods that remove some of the original source information, with small/negligible effect on the perceived quality.


a

b

c

d

e

f 0.1

0.1

0.1

0.1

0.2

0.4 0.4

0.1

0.1

0.2

0.2

0.2

0.20.2

0.40.2

0.40.6

0.6

0.4

0.4

0.2

0.2

0.41

1

1

1

1

0

0

0

0

0

1.0

Inputalphabet

Codesymbols

abcdef

1 101100

00011

01

1

0

Huffman Code In Huffman codes, variable code wordlength is utilized together with the prefix property. Coding algorithm (from Proakis) and example (from Sklar’s book):

Sort in decreasingorder of probability

Merge the twoleast probable

Number of elements = 2?

Assign 0 and 1 tothe two codewords

Is any element theresult of mergerof two elements

Append the codewordwith 0 and 1

Stop

no

yes

yes

no


Huffman Code (continued) As an example, in symbol-by-symbol compression of English text, about 43 % compression rate has been achieved. However, this doesn’t take into account the fact that certain combinations of letters (the, in, on, ...) appear in the text quite often. For Huffman code, it can be shown that the average number of bits to code a source symbol, L(X), satisfies 1)()()( +≤≤ XHXLXH When coding n symbols at a time, we obtain correspondingly nXHXLXH /1)()()( +≤≤ . So using the Huffman code blockwise, with sufficiently long block length, it is possible to get arbitrarily close to entropy limit. This is not a practical way, but this development basically constitutes one proof of the source coding theorem. One fundamental limitation of Huffman codes is that the source symbol statistics have to be known (or estimated).


Run-Length Codes Many sources produce long sequences (”runs”) of the same source symbol (e.g., think about the operation principle of telefax). In such cases, it is more efficient to send, instead of the symbol sequence, one of the repeated symbols and information about the length of the sequence.


Lempel-Ziv Codes Principle

• The method uses a codebook consisting of a number of source symbol sequences.

• The source symbol stream is scanned one symbol at a time. This is continued until the beginning of the uncoded symbol sequence is not in the codebook anymore.

This sequence can be represented as the concatenation of one of the words in the codebook and one additional symbol. This new symbol sequence will be added to the codebook.

• The same process is repeated starting from the beginning of the uncoded source symbol stream.


Lempel-Ziv Codes (continued) Example (from Proakis): Let us assume that we want to parse and encode the following sequence:

0100001100001010000010100000110000010100001001001 Parsing the sequence by the rules explained before results in the following phrases: 0, 1, 00, 001, 10, 000, 101, 0000, 01, 010, 00001, 100, 0001, 0100, 0010,

01001, … It is seen that all the phrases are different and each phrase is a previous phrase concatenated with a new source output. The number of phrases is 16. This means that for each phrase we need 4 bits, plus an extra bit to represent the new source output. The above sequence is encoded by

0000 0, 0000 1, 0001 0, 0011 1, 0010 0, 0011 0, 0101 1, 0110 0, 0001 1, 1001 0, 1000 1, 0101 0, 0110 1, 1010 0, 0100 0, 1110 1, …

Dictionary Location

DictionaryContents Codeword

1 0001 0 0000 0 2 0010 1 0000 1 3 0011 00 0001 0 4 0100 001 0011 1 5 0101 10 0010 0 6 0110 000 0011 0 7 0111 101 0101 1 8 1000 0000 0110 0 9 1001 01 0001 1

10 1010 010 1001 0 11 1011 00001 1000 1 12 1100 100 0101 0 13 1101 0001 0110 1 14 1110 0010 1010 0 15 1111 0010 0100 0 16 1110 1


Lempel-Ziv Codes (continued) In this example, one can hardly talk about compression. The efficiency of the method is realized only when coding considerably longer source symbol sequences. When compressing English text, about 55 % compression rate has been achieved. One key parameter of the method is the size of the codebook, which could be, for example, 4096, corresponding to 12-bit sequences. At some point, the codebook becomes full, and the ”oldest” (according to a proper criterion) code-word is removed from the codebook to make room for new ones. Lempel-Ziv Codes are commonly used, e.g., in file packing routines, like Zip.


The Capacity of a Discrete-Time Channel (1) Discrete-Valued Input and Output In the following, we consider the channel capacity, starting from the case of discrete-time channel with discrete-valued input and output, and moving stepwise towards the case of continuous-time channel with continuous-valued input and output. The input is represented by the random process { }kX . The output is represented by the random process { }kY . The channel is assumed to be memoryless, i.e., kY depends only on kX but not on any other input symbol (neither earlier nor later ones). Such a channel is completely determined by the conditional probabilities ( ), ,X YY Xp y x x y∈Ω ∈Ω

Example: Binary symmetric channel (BSC)

{ }1,0=Ω=Ω YX

The conditional probabilities between input and output (transition probabilities) are shown by such a graph.

p is the bit-error probability

This is the simplest possible channel. Yet it is quite useful as a model of many practical channels, and it is used often in the continuation.

x=0

x=1

y=0

y=1

p y x pY X* ( )=1-*

1-p

p


The Capacity of a Discrete-Time Channel (1) Discrete-Valued Input and Output (continued) If the input symbols are independent, the information per symbol at the input is )(XH .

The question is: How much of this information gets to the destination through the channel? The answer will be seen after a few intermediate steps.

(1) Uncertainty about X when the output symbol is yY = :

2 2( ) log ( ) ( ) log ( )

XX Y X Y X Y

xH X y E p X y p x y p x y

∈Ω⎡ ⎤= − = −⎣ ⎦ ∑

(2) Conditional entropy: Average uncertainty about X when the output has been observed:

2( ) ( ) ( ) ( ) ( ) log ( )

Y Y xY Y X Y X Y

y y xH X Y H X y p y p y p x y p x y

∈Ω ∈Ω ∈Ω= = −∑ ∑ ∑

(3) Average mutual information: Information about X obtained by observing Y:

)()()()(),(

XYHYHYXHXHYXI

−=

−=

(The proof of latter form is an exercise problem.)

The mutual information, i.e., the information transmitted over the channel depends on the input probability distribution (that depends on the source coder) and on the transition probabilities (that depend on the channel).


The Capacity of a Discrete-Time Channel (1) Discrete-Valued Input and Output (continued) (4) It is sensible to choose the input probability distribution in such a way that the mutual information is maximized. The channel capacity is defined as the maximum average mutual information over the input (X) probability distribution:

),(max)(

YXICxp

sX

=

The unit here is bits/symbol.

(5) The channel capacity can also be expressed in units of bits/second (bps) as follows: ssCC = where s is the symbol rate (symbols/second).


Example of Discrete-Time Channel Capacity Let us consider a binary symmetric channel, where the input symbol probabilities are q and q−1 .

The mutual information is

)1(log)1(log)(),( 22 ppppYHYXI −−++= .

(You can check this by writing down and simplifying the conditional entropy expression.)

This is maximized by choosing q=0.5, which gives H(Y)=1 (In this case, (0) (1) 0.5X Xp p= = and, due to symmetric channel, also (0) (1) 0.5Y Yp p= = .)

The resulting channel capacity per symbol is:

)1(log)1(log1 22 ppppCs −−++=

channel)binary free-(error 11or 0=

other)each oft independen areoutput and(input 0 21

=⇒=

=⇒=

s

sCpp

Cp


Channel Capacity Theorem Let us consider a case with

• Source rate )(XrHR = bps • Channel capacity sC sC= • CR <

If the source signal is a bit-stream and if the source bit-rate is lower than the channel capacity, then the channel coding can be designed to provide arbitrarily low bit error rate (BER). In practice, very low error rates may not be possible in case of noisy channels because the processing delay and implementation complexity might become very high.

Then there is a combination of source coding and channel coding providing error-free/distortion-free transmission.


The Capacity of a Discrete-Time Channel (2) Discrete-Valued Input, Continuous-Valued Output

Example: A channel with additive noise. • Input: Discrete-valued random process X • Output: NXY += where N is a continuous-valued random

process modeling the noise. (Notice that here the channel attenuation has been scaled away.) The noise is usually assumed to be Gaussian.

The entropy of a continuous-valued random variable Y is defined as:

[ ] dyyfyfyfEYH YYYY

)(log)()(log)( 22 ∫Ω

−=−=

The following important property is used in the continuation (proof as an exercise):

In this case the conditional entropy is calculated as:

∑ ∫Ω∈ Ω

=X Yx

XYXYX dyxyfxyfxpXYH )(log)()()( 2

Mutual information and channel capacity are defined according to the earlier models based on the definitions of entropy and conditional entropy. The channel capacity depends naturally on the channel alphabet ΩX and the noise level. Examples later.

If the variance ( 2σ ) of a random variable N is known and bounded, there is an upper bound for the entropy:

( )222

1 2log)(0 σπeNH ≤≤

The upper bound is achieved if and only if N is Gaussian distributed.


The Capacity of a Discrete-Time Channel (3) Continuous-Valued Input and Output We consider here the case where the channel alphabet is not limited in any way, i.e., the channel input is assumed to be continuous-valued. This is the most general case when considering the channel capacity (when carrying out the maximization with respect to the probability distribution of X). Discrete channel alphabet is a special case of this. The conditional entropy is now defined as:

dxdyxyfxyfxfXYHX Y

XYXYX∫ ∫Ω Ω

= )(log)()()( 2

In other respects, the derivations are similar to the earlier ones.


The Capacity of a Discrete-Time Channel (3) Continuous-Valued Input and Output Gaussian case Consider a channel with additive Gaussian noise:

NXY += where N is a zero-mean Gaussian noise process that is statistically independent of X with variance 2

Nσ . The variance of X is 2

Xσ . The mutual information is )()()()(),( NHYHXYHYHYXI −=−= Based on earlier results it is easy to see that

( )( )

2122

2 2122

( ) log 2

( ) log 2 ( )

N

X N

H N e

H Y e

π σ

π σ σ

=

≤ +

Equality applies here when X is Gaussian, in which case also Y is Gaussian (as the sum of two Gaussian processes). Clearly, the mutual information is maximised with this choice. Thus, the channel capacity becomes:

( ) ( )2

2 2 21 1 12 2 22 2 2 2log 2 ( ) log 2 log 1 X

s X N NN

C e e σπ σ σ π σσ

⎛ ⎞= + − = +⎜ ⎟

⎝ ⎠


Discrete Channel Alphabet = Constellation In the continuation we use commonly both real- and complex-valued discrete channel alphabets, referred to as constellations. Here are some examples:

2-AM

4-AM

8-AM

16-AM

4-PSK 8-PSK

8-AMPM 32-AMPM

16-QAM 64-QAM


Capacity of AWGN Channel and Discrete Constellations The following plots show the maximum information transmission rates for some common real and complex constellations as functions of the signal-to-noise ratio (SNR) in case of an AWGN (Additive White Gaussian Noise) channel. Notice that these are not the channel capacities in the strict sense, because the mutual information is not maximized with respect to the input distribution. Instead, it is assumed that all input symbols are equally probable, which is the common starting point in digital transmission systems. The figures show also the true channel capacity for the case of continuous-valued input and output. We can see that the continuous-valued channel capacity gives an upper bound for information transmission rates of the discrete-valued cases. By increasing the size of the discrete-valued constellation, it is possible to get arbitrarily close to the channel capacity. In other words, there is no essential loss in channel capacity when using discrete channel alphabets, if the size of the constellation is sufficiently large.


Capacity of AWGN Channel and Discrete Constellations (cont’d) The upmost plot shows the continuous-valued channel capacity. Reliable information transmission is possible below this curve.

Real Constellations (from Lee&Messerschmitt):

Complex Constellations:


The Capacity of a Continuous-Time Channel Most physical channels are continuous-time channels by nature. On the other hand, looking at the digital transmission chain, the part of the channel that is essential from the channel capacity point of view has a discrete-time input signal, in addition to having discrete-valued channel alphabet.

Now the question arises, how much do we loose in channel capacity in discretizing the channel in this way.

Let us consider one important (but idealized) example: Tightly band-limited baseband channel. It has the transfer function

⎩⎨⎧

>≤

=WfWf

fBfor 0for 1

)(

Here W is the bandwidth in units of Hz. The channel output signal is [ ] )(~)(~)()()()( tNtXtbtNtXtY +=∗+= where )(tX is the input signal )(tN is white Gaussian noise )(tb is the channel impulse response )(~ tX is the filtered, bandlimited input signal )(~ tN is the noise signal filtered to bandwidth W.

f-W W


The Capacity of a Continuous-Time Channel (continued) According to the sampling theorem, )(tY , )(~ tX , and )(~ tN can be represented completely in discrete-time domain by using WT 2

1= -spaced samples. The samples define discrete-time random processes )(~ kTN , )(~ kTX , and )(nTY , which define the discrete-time channel capacity. This must be the same as the continuous-time channel capacity.

With the earlier assumptions, we get the capacity of the continuous-time channel as follows:

2

2 2log (1 )X

N

C W σσ

= +

This is known as the Shannon-Hartley law.

This example demonstrates that it is possible to convert a continuous-time channel into a discrete-time channel without loosing anything in the capacity.

Earlier we have already seen that it is possible to use discrete-valued without loosing significantly in the capacity.

It is now evident that digital transmission techniques utilizing discrete-time channel inputs and outputs and discrete-valued channel alphabet are able to fully utilize the capacity of any continuous-time continuous-valued channel.


The Capacity of a Voice-Band Telephone Channel Assume that the bandwidth is W = 3.3 kHz and SNR= 40 dB, the capacity of a telephone channel can be calculated as

2

2 22log (1 ) 3300log (1 10000)

43.9 kbps

X

N

C W σσ

= + = +

=

With different values of the SNR, the channel capacity behaves as follows:

SNR/dB C/kbps 20 22.0 30 32.9 40 43.9 50 54.8 60 65.8

The date rate of 56 kbps is commonly achieved with voice-band modems with good telephone connections. This is actually quite close to the channel capacity, since the SNR cannot be assumed to be more than 60 dB even in the best connections.


The Capacity of Frequency-Selective and Fading Channels We have seen how to calculate the capacity of an AWGN channel. This basic result can be applied also to various other cases by splitting the channel in time and/or frequency domain to smaller parts that have stationary AWGN characteristics. In case of stationary frequency-selective channel, the frequency band can be divided into smaller (non-overlapping) parts and the overall capacity is the sum of the capacities of the sub-channels. If the sub-channel bandwidth is considerably smaller than the coherence bandwidth, the sub-channels have AWGN characteristics, and the Shannon-Hartley law can be used for determining the capacities of the sub-channels. In multicarrier systems (OFDM, DMT) this idea is also used as a modulation technique to overcome the channel equalization problem in heavily frequency selective channels. Furthermore, in point-to-point links with sufficiently fast feedback, the constellations and transmission powers used in each sub-channel can be optimized based on the measured sub-channel SNRs. In case of temporal variations, i.e., a fading channel, the instantaneous capacity can be calculated for the channel (or the sub-channels). Within the coherence time, the instantaneous channel capacity is practically constant, and the capacity over a longer time interval can be obtained by properly averaging the instantaneous capacities.


The Capacity in Case of Interferences In many applications, various interference sources limit the channel capacity instead of the thermal noise. The interference sources include other base-station/mobile station signals at the same frequency (in case of a cellular network), multi-access interference of the same cell in case of CDMA based cellular systems, narrowband RF interference in case of xDSL systems, spurious frequencies generated due to non-idealities of transmitter/receiver hardware, etc. In most practical cases, the desired signal, thermal noise, and the different interference sources are statistically independent of each other. In the capacity calculations, the sum of thermal noise power and the interference powers should be used instead of the noise power. The capacity of each frequency increment can be calculated from the signal-to-noise+interference value and the overall capacity is obtained by summing/integrating over the increments.


TRANSMISSION CHANNELS In this part we have a very brief look at different types of physical transmission channels.

In the following (and in communication theoretic literature in general), the term channel is assumed to include

- the physical transmission medium - antennas/other transducers used to connect to the

physical medium - high-frequency/radio frequency (RF) parts of the

transmitter and receiver.

In other words, the channel models to be discussed include here all parts of the transmission chain between the modulator and demodulator. Rather simple channel models are used (and are in most cases sufficient) when considering, e.g, the modulation and detection methods. Source: Lee&Messerschmitt, Chapter 5.


Transmission Media The commonly used transmission channels in digital transmission include:

• (Shielded/unshielded) twisted pair cable (STP/UTP), used as access cables in the telephone network and LANs.

- Plain Old Telephone Service (POTS): 3 kHz/up to 56 kbps (as composite channel, including also PCM-links, exchange equipment, etc.)

- ISDN: 144 kbps - ADSL: up to ~8 Mbps (in downlink) at several km distances - VDSL: up to ~50 Mbps at few hundred meter distances - LANs: up to ~1 Gbps (Gigabit Ethernet)

• Coaxial cables with up to ~1 GHz bandwidth - Cable TV network (CATV): hundreds of analog/digital TV

channels, with up to 40 Mbps data rate in each 8MHz BW. - Cable modems: up to ~1 Mbps date rate also in uplink. - Earlier in LANs (Ethernet).

• Optical cables - Core of the broadband fixed/mobile telecommunication

networks, CATV networks, and LANs. Terabps data rates possible in a single fiber.

- Possibly coming also to the access network (Fibre to the Home, FTTH)

• Radio Waves - Satellite communications. - Digital audio and TV broadcasting systems (DAB, DVB-T) - Microwave links. - Fixed access technologies. - Mobile communication systems (GSM, 3G/WCDMA, …) with

increasing data rates (e.g., 128 kbps everywhere, 2 Mbps in urban areas/indoors); considerably higher in the next systems

- WLANS: up to ~54 Mbps, mostly indoors; growing … - Low-power RF (e.g., Bluetooth): ~1 Mbps at few meter

distances. • Magnetic & optical storage


Twisted Pair Cable Transmission The twisted pair channel can be modeled as a linear system with more or less exponentially decaying impulse response (dispersive channel).

• The attenuation increases heavily with frequency, limiting the useful bandwidth to ~30 MHz in VDSL environment and ~1 MHz in ADSL environment.

• Also termination impedances affect the frequency response due to reflections.

• Branches (taps, e.g., having two telephone lines to different rooms of a household) may cause echoes, resulting in notches in the frequency response (like in multipath channels).

There are also various interference sources: • Leakage of signals (Ingress, RFI=radio frequency

interference) from radio communication systems • Impulsive and other types of noise signals from various

electrical appliances. • Crosstalk from other signals transmitted in other lines of the

cable bundle. The near-end crosstalk (NEXT) comes from a transmitter at the same end of the cable as the receiver under consideration. For example, in VDSL systems, frequency domain dublexing (FDD) is used to avoid NEXT problems (time domain dublexing, TDD, could be used as well). Far-end crosstalk (FEXT) comes from a transmitter in the other end of the cable in different pair, and has to be tolerated always.

RX

TX

TX

TX

RX

RXFEXT

NEXT


Coaxial Cable Transmission

The CATV network has much wider bandwidth, up to ~1 GHz. Reflections (echos) from branching points or imperfect terminations and the main imperfection in the linear system model.

RF ingress may have some significance, but in many respects, coaxial cables are rather ideal transmission media.


Radio Channel

The main impairments of the radio channel are due to multipath propagation, where the signal is received through a number of propagation paths with different delays.

Multipath effects may be caused by inhomogenity of the atmosphere, etc, but the main reasons are reflections from natural (hills, cliffs) and man-made (buildings) obstacles.

Sufficiently long delay differences are observed as echoes in the received signal.

In case of a two-path channel, the received signal can be written as )()()( τ−+= tAxtxty where τ is the delay difference of the two rays and A is the (complex) gain of delayed path. In the model, it can usually be assumed that the delay of the line-of-sight (LOS) path is zero and its gain is unity.

Directpath

Reflection


Radio Channel: Frequency Selectivity

The channel frequency response is in this case H f AeC

j( ) = + −1 ωτ

The corresponding amplitude response is in the range ( ) ... ( )1 1− +A A . If the value of A is close to unity, the frequency response has deep notches, periodically at angular frequencies

…,5,3,1τττ .

Channel delay spread is a measure of the length of the impulse response, i.e., the delay difference of the shortest and longest significant paths. The inverse of the delay spread is the coherence bandwidth, which is a measure of frequency domain variability of the channel. Depending on the channel bandwidth in relation to the channel coherence bandwidth, the channel model may be frequency non-selective, mildly frequency selective, or heavily frequency selective.


Radio Channel: Fading

Another aspect of the radio channel is fading, i.e., the variation of the channel gain (or actually, the frequency response) with time. This may also be due to atmospheric effects, but the main reason for strong fading is the movement of transmitter or receiver, causing the multipath channel model to change.

It is important to note that a movement of half wavelength (=15 cm at 1 GHz carrier frequency) may cause the propagation conditions to change greatly.

The following figure shows a typical behaviour of a mobile channel gain and phase with time. It has been observed experimentally, and also derived theoretically that the probability distribution of the magnitude of the channel gain follows Rayleigh distribution (I and Q values follow 2-D Gaussian distribution) .


Radio Channel: Fading (Continued)

The relative motion of transmitter and receiver with respect to each other causes a Doppler shift. The Doppler shift achieves the highest value when the movement straight against each other, and the smallest negative value when the movement is straight away from each other. In a mobile channel, there are, with some probability, rays from both directions, and also all the intermediate directions/Doppler shift values. A typical Doppler spectrum is shown in the following figure. This would be the received signal spectrum if the transmitted signal is an unmodulated carrier. The following figure shows a model for a two-path fading channel, which includes the delays and complex gains of the two paths, as well as the ‘modulation’ by the Doppler spectrum.

DELAYτ1

DELAYτ2

ΣOUTPUT

A1

A2u t( )

r t1( )

r t2( )

S (f)R

fc -v

λ f +c vλfc

f


Radio Channel: Example of Fading Example If the vehicle velocity is 100 km/h and the RF carrier frequency is 1 GHz, the wavewlength is

8

93 10 0.3 m10

cf

λ ⋅= = = ,

the velocity

27.8 m/sv =

and the maximum Doppler shift

27.8 m/s 92.6 Hz0.3 mD

vfλ

= = = .

The bandwidth of the received carrier is then about 185 Hz, since the Doppler shift can effect in both directions, depending on whether the reflected beam arrives from front or back. The time it takes for the vehicle to travel half a wavelength is

0.15 m 5.4 ms

27.8 m/st = = .

Thus we expect a significant fade about every 5.4 ms, i.e., at a rate of 185 Hz, which is equal to the bandwidth of the Doppler shift.


Power and Bandwidth Limitations The main constraints in digital transmission system design are imposed on the signal bandwidth and transmission power. Bandwidth is limited by

• Regulation (especially in case of radio communications)

• Bandwidth of the medium. Transmission power is limited by

• Regulations • Keeping the interferences to other users of the same

medium at a reasonable level. • Keeping the power consumption at a reasonable level

(especially in handheld equipment and in satellite communications.)


BASEBAND DIGITAL TRANSMISSION

Bits and symbols The idea of digital transmission is to transmit bit sequences or more generally multilevel symbol sequences using PAM-modulation (pulse amplitude modulation). Multilevel symbols are obtained when, e.g., 4 bit blocks are combined into symbols. In this case, the number of bit combinations is 2 164 = . In general, B bits can be represented as 2B different symbols, which in baseband transmission are usually coded as 2B equally-spaced signal levels. In practise, the number of levels or bit combinations depends on the application and channel requirements. The main requirement is that the levels can be reliably separated from each other after a noisy channel. Combining several bits into one symbol, the symbol rate (baud rate) is reduced. This affects the transmitted signal bandwidth. Example:

Bit sequence: 0 1 0 0 1 1 0 0... Rb bits/s Symbol sequence: -3A 9A ... Rb / 4 symb/s

Binary signal 16-level signal: T is the symbol period or symbol interval.

T


Pulse Waveforms

Digital PAM-signal is transmitted to the continuous-time channel as the following waveform ∑ −=

kk kTtpatx )()(

Here p(t) is a basic pulse shape whose amplitude is scaled by the transmitted symbols ka . It is important that adjacent pulses do not interfere with each other in the reception. It is assumed that the received continuos-time waveform is observed at time instants kT± for determining the transmitting symbol values. The ideal condition is:

,...2, when 0

0 when 1)(

⎩⎨⎧

±±==

=TTt

ttp

This is possible to implement in two different ways: (1) Using short pulses not overlapping in time domain

⇒ The bandwidth is not the smallest possible, but easy to implement.

(2) Using pulses overlapping in time domain, but forcing

the condition to be satisfied anyway ⇒ Signal bandwidth can be minimized, more

complicated.


Spectrum of Baseband PAM-Signal The spectrum of the transmitted digital baseband signal can be obtained from the Fourier Transform of the equation of the previous slide,

)()(1)( 22 fTjax eGfP

TfG π=

Here P(f) is the Fourier Transform of the pulse shape and G ea

j fT( )2π is the power spectral density function of the transmitted discrete-time symbol sequence.

Here, we combine both continuous-time and discrete-time signals. Therefore, it is not trivial to prove the previous result, but it is intuitively quite clear.

The transmitted signal spectrum should be matched with the channel properties.

In baseband systems, e.g., in cables, the attenuation is not constant within the used frequency band. It increases in high frequencies. So, the signal power should be concentrated on low frequencies, where the cable attenuation is smallest. This reduces also crosstalk and radio frequency interferences.

Naturally, in AC-coupled systems, there should be no DC-component in the transmitted signal. (Or at least the removal of the possible DC-component should not distort the waveform.)


Line Coding vs. Nyquist Pulse Shaping

Two different approaches to shape spectrum:

(1) Line coding • The basic pulse shape is a square pulse. ⇒ The spectrum is a sinc-type wide spectrum.

• DC-component can be removed by constructing the signal properly.

• In general, the symbol sequence is generated to have some correlations between consecutive symbols in order to shape the transmitted spectrum.

• Used mostly for binary source signal.

(2) Nyquist-pulse shaping • It is assumed that transmitted symbols are

uncorrelated. ⇒ The transmitted spectrum has the shape of the

Fourier transform of the used pulse shape. • The pulse shape is optimised so that the needed

bandwidth is small. ⇒ Adjacent pulses are overlapped in time domain.

The methods can also be combined, but in the following discussions, the focus is usually on one of these approaches. Note on terminology: In some areas, like xDSL modem literature, the term ”line coding” is used in a wider sense to cover all signal processing techniques related to the modem.


Goals of Line Coding

• Spectrum management and shaping: Keeping the spectrum reasonably narrow.

• To remove the drifting of the DC-component, “baseline

wander”, in AC-coupled systems.

• To avoid synchronization problems when the transmitted symbol sequence contains long sequences with constant level (e.g., 000000… or 1111111….).

• System monitoring during the normal operation is

possible by using suitable line codes: If sequences not belonging to the used code are received repeatedly, it can be determined that something is wrong in the transmission link.

Example: Effects of lowpass filtering (RC-filter with exponential step response) and AC-coupling (baseline wander) in case of a binary signal:


Redundancy Using an L-element channel symbol set (=alphabet) and symbol rate fb, the channel data rate (assuming no errors) is

bpslog2 LfR b= .

Let the source/user bit rate be B. Then if B=R, there is no redundancy in the code and the information bits must be mapped deterministically to the channel symbols. Usually in line coding, B<R and the difference of these values corresponds to the redundancy of the code. When there is some redundancy, it is possible to create correlations to the symbol sequence and the power spectrum of the signal can be shaped.


Running Digital Sum

Let Ak denote the symbol sequence using the line code under consideration. Then the transmitted signal can be expressed as

∑∞

−∞=−=

kk kTtgAtx )()(

where g(t) is the unit square pulse.

The baseline wander effects depend heavily on a certain characteristic of the code, the running digital sum, RDS. It is defined as

∑−∞=

=k

mmk AS

In the following, the maximum (absolute) RDS value of the code is of great interest.

The RDS value of a good code is expected to be bounded to a small value.

Example: Baseline wander effects with a code with small maximum RDS and large RDS. Upper figures with DC-coupling, lower ones with RC high-pass filtering.

t

t

t

t


Classification of Line Coding Methods

There are many (at least tens of) different line coding methods, often based on ad-hoc principles.

In the following, we consider mostly the case of binary data.

Codes can be classified, e.g., by the used signal levels, as follows: unipolar: +a, 0 polar (antipodal): +a, -a bipolar (pseudoternary): +a, 0, -a. Examples of line codes


Binary Antipodal Codes During one symbol interval, one of the following pulse waveforms or its inverse is transmitted, depending on whether the transmitted bit is 1 or 0. (Notice that there are two alternative symbol mappings.)

t t t-T/2 T/2 -T/2 T/2 -T/2

T/2

RZ NRZ BIPHASE

When using RZ (=return-to-zero) or NRZ (=non-return-to-zero) waveforms in an AC-coupled channel, the average number of positive and negative pulses should be made to be equal, by some means. In Biphase codes (=Manchester codes), the DC-level of the basic pulse is zero, and consequently, the DC-level of the transmitted symbol sequence is also zero. o There is a zero crossing in the middle of symbol time

interval that makes the synchronization more easy o Required bandwidth is higher, about twice compared to

NRZ-line code - As a theoretical model, biphase signalling can be

constructed by using NRZ-pulse and (a) using double symbol rate and sending a complement bit after each bit or (b) modulating by using square wave, which has double frequency. Both of these ideas show ways to get the spectrum of the signal from the NRZ spectrum.

o This code is simple, and good enough if the bandwidth of the system can be tolerated to be somewhat higher than the minimum.


AMI-Line Code

Alternate Mark Inversion or AMI line code is an example of bipolar line code. Its coding rule is 0 => 0 1 => +/- alternatingly Example: Incoming 0 1 1 1 1 1 1 0 0 0 0 AMI-coded + - + - + - 0 0 0 0 A long 1-bit sequence is seen as square wave AMI-code removes the long sequences of 1-bits, but it has the following problem: the coder may produce a long sequence of 0-bits which makes synchronization more difficult. This is due to the linearity of AMI-code. In more advanced codes, this problem can be avoided.


AMI-Line Code (continued)

AMI-decoding can be done easily using the following simple memoryless function:

Received Decoded + 1 0 0 - 1

However, using a more advanced decoding method, like the Viterbi algorithm, better system performance can be achieved.

It is easy to see that the RDS value for an AMI code is always 0 or 1.

The power spectrum of an AMI code sequence can be shown to be

21 cos( ) 2 (1 )

1 (1 2 ) 2(1 2 )cosj

ATG e p p

p p Tω ω

ω−= −

+ − − −

where p is the probability of bit 1. The overall spectrum depends on the used pulse shape, as explained on page 48.


Improved Line Codes Based on AMI

The starting point is AMI, which is modified. Synchronization information is added to the long sequences of 0-bits.

Basic idea: If an AMI-coded block includes just 0-bits, it is replaced by a three level sequence which

- Includes one or more + and/or – symbols, i.e., information for synchronization.

- Is not a valid sequence in the AMI-code and can thus be recognized in the decoder and changed back to the 0-sequence.

The HDB3 code is one example of this idea, which is widely used in PCM systems.

Drawbacks of the idea: - The implementation is somewhat mode complicated. - Monitoring the system performance becomes more

difficult - RDS grows.

In spite of these drawbacks, these methods are used commonly.


HDB (High-Density Bipolar) Codes

We use the following notation: 0 a transmitted 0-symbol B is a valid AMI + or - symbol V a + or – symbol violating the AMI-principle, i.e., it

has the same polarity as the most recent +/- symbol

In HDBk-code, when scanning the bit stream, a 0-sequence of length k+1 is replaced by either of the following two sequences:

B00...0V or 00...0V.

The choice is made in such a way that there is always an odd number of B-symbols between two consecutive V-symbols. This means that the polarity of the V-symbols is alternating and the RDS remains small. Notice that the polarity of the AMI sequence may be changed after such a replacement sequence.

For this code, − ≤ ≤1 1RDS .

HDB3-code is used in the PCM-transmission systems.

Example: HDB3 Input: 0 1 1 0 0 0 0 1 0 1 0 0 0 0 1 AMI: + - 0 0 0 0 + 0 - 0 0 0 0 + HDB3: + - 0 0 0 - + 0 - + 0 0 + - HDB-notation: B B 0 0 0 V B 0 B B 0 0 V B


Block Codes

Here, we consider the incoming signal blocks of k bits, which are coded into the blocks of n symbols. The alphabet size is L. The basic requirement is:

nk L≤2 A natural approach would be to choose (if possible) the code words to have zero RDS values over the whole codeword, and try to minimize also the RDS values within the code words.


kBnT Codes

The previously discussed pseudoternary codes transmit 1 bit per symbol. In principle, 3-level codes are able to transmit 58.13log2 = bits/symbol. So there is some room for improvement.

kBnT is a rather wide class of block codes. If we choose the largest possible n for each k, we obtain the following table: .

k n code ’efficiency’ 1 1 1B1T 63% = AMI 3 2 3B2T 95% 4 3 4B3T 84% 6 4 6B4T 95% 7 5 7B5T 89%

Better efficiency also reduces the sensitivity to noise, because the bandwidth, and consequently noise power, can be reduced.

On the other hand, the redundancy is reduced with increasing efficiency, and the goals of line coding become more difficult to achieve.

As a suitable compromise, the 4B3T code has received a lot of attention.

In these efficient codes, a major problem is that it is not possible to find enough code word with zero RDS. One solution is to use a number of alternative modes (typically 2...4) for each bit combination and choose the one that minimizes the RDS.


Example: Bimode 4B3T Code

Mode A is chosen if 13 −≤≤− RDS .

Mode B is chosen if 20 ≤≤ RDS .

In this code, 34 ≤≤− RDS . Improving the efficiency has thus increased the range of RDS significantly, and increased the sensitivity to baseline wander ISI.

The decoding can be done easily by using a slicer and a table to find the bit combination based on the received 3-level block. However, better performace can be achieved using, e.g., the Viterbi algorithm to be discussed later on.

Ternary Output Block Binary Input Block Mode A Mode B

Block Digital Sum

0000 +0- +0- 0 0001 -+0 -+0 0 0010 0-+ 0-+ 0 0011 +-0 +-0 0 0100 ++0 --0 ±2 0101 0++ 0-- ±2 0110 +0+ -0- ±2 0111 +++ --- ±3 1000 ++- --+ ±1 1001 -++ +-- ±1 1010 +-+ -+- ±1 1011 +00 -00 ±1 1100 0+0 0-0 ±1 1101 00+ 00- ±1 1110 0+- 0+- 0 1111 -0+ -0+ 0


Binary Block Codes

These codes are suitable for cases where the channel alphabet is binary (such as optical transmission/storage) or when there are other reasons to avoid more than two signal levels in the signaling. Zero Disparity Codes A coded n-bit block includes n/2 1-bits and n/2 0-bits. So at the end of each block, RDS=0. Within a block,

.2/2/ nRDSn ≤≤− Code properties with different block lengths:

n N Log2N k Efficiency 2 2 1 1 50% 4 6 2.58 2 50% 6 20 4.32 4 67% 8 70 6.13 6 75%

10 252 7.97 7 70%


Variable Rate Codes

In some cases it is possible to use codes where the transmission rate is not constant, but a variable number of channel symbols are transmitted for each input bit. The goal is, of course, to minimize the average symbol rate. A very simple but useful principle is bit-stuffing, where sequences of 0 bits exceeding certain length are broken by adding an extra 1-bit. Example: (0, 2) runlength limited binary code Input: 1 0 0 0 1 1 0 0 1 Coded: 1 0 0 1 0 1 1 0 0 1 1


Properties of Some Line Codes

83050E/65

Digital Transmission System Based on Baseband Pulse Shaping

CODERTRANSMIT

FILTERg t( )

CHANNELb t( )

DECODERSAMPLER

RECEIVEFILTER

f t( )

TIMINGRECOVERY

R t( )Q t( )

N t( )

QkAkBn

Bn AkS t( )

SIGNAL

NOISETRANSMITTER

BITS

ESTIMATEDSYMBOLS SLICER OR

DECISIONDEVICE

SYMBOLS

BITS

RECEIVER Channel is modeled as a linear time invariant transfer function with additive Gaussian noise. Coder maps the incoming bit sequence to a symbol sequence according to the chosen constellation/alphabet. The symbol sequence is here assumed to be uncorrelated, i.e., the consequtive symbols are independent of each other. Consequently, the spectrum of the symbol sequence, as a discrete-time signal is white. Furthermore, the symbols of the alphabet are assumed to be used with equal probability. Notice that the system between coder and sampler is basically a cascade of three continuous-time filters.

83050E/66

Transmitter Blocks The transmitter filter forms a continuous time signal from the symbol sequence, mA . The impulse response of the filter is )(tg . In the following this is called also as the transmitted pulse shape. The channel waveform is

∑∞

−∞=−=

mm mTtgAtS )()(

Here T is the symbol interval and 1/T is the symbol rate. So, the waveform is formed from pulses scaled by symbol values. The pulses may overlap in time domain. Example

t

+2+3

+1

-10 T 3T

2T t0 T 4T

TRANSMITFILTER

1

T

g t( )

t

SYMBOLS S t( )

Ak

83050E/67

Channel The received waveform is

∑

∫ ∑

∫

∞

−∞=

∞+∞

∞

−∞=

∞+∞−

+−=

+−−=

+−=+∗=

mm

mm

tNmTthA

tNdmTtgAb

tNdtSbtNS(t) b(t)tR

)()(

)()()(

)()()()( )(

- τττ

τττ

where )(th is the received pulse shape

τττ dtgbtgtbth )()()()()( −=∗= ∫∞

∞−

Example If we consider a strictly bandlimited channel

⎩⎨⎧

≥<

=WfWf

fB 0 1

)(

then a square pulse waveform of the previous page is not very good, because it would be (more or less) distorted in the channel.

f-W W

83050E/68

Receiver Blocks In general, the receiver design is more critical than the transmitter, because the channel attenuates and distorts the signal and it is important to recover the signal as much as possible, in order to minimize bit error rate. Receiver filter

1. Filters out the adjacent channels and out-of-band noise & interferences.

2. Effects on the pulse shape. 3. As equalizer compensates the linear distortion of the

channel, e.g., by using inverse transfer function. The transfer function of the channel is usually unknown, so adaptive methods are important.

Synchronization (timing recovery) defines the right symbol timings for different blocks and the correct sampling moments. The transmitted message includes often components which make the synchronization more easy. (However, this is not always necessary.) Here, it is assumed that synchronization has been done in some way. The synchronization concepts are considered in more detail later in this material. In the sampling block, samples are taken from the continuous time signal. In the ideal case, the samples are taken at time instants that correspond to the transmitted symbol and when intersymbol interference is at minimum.

83050E/69

Receiver Blocks (continued) In the decision device, an estimate of the transmitted symbol sequence kA is formed. In the simplest case, so called slicer, this is based on decision threshold levels.

Example: Consider the alphabet {-1, 0, 1}: The decision is made by using the following figure (Note: Here the receiver amplification is adjusted so that the overall gain of the link is 1.)

More advanced methods for making the decisions are needed to achieve optimal performance in case of non-ideal channels. The discussion of these methods, as well as optimal choice of the receiver filter will constitute a major part of this course. The decoder generates a bit-sequence from the detected symbols, which in the ideal case is a delayed version of the transmitted bit-sequence.

OUT

IN-0.5

0.5

1

-1

83050E/70

Example: Simple PAM System This example illustrates a simple, non-optimized PAM system, which works well if the needed bit rate is a lot smaller than the capacity of the channel. However, the bit rate should be much lower than what is possible with more optimal pulse shaping.

A binary alphabet {-a, a} is used and a is chosen such that transmission power satisfies the (regulatory or practical) limitations. If the symbols are equally probable, DC is zero.

The transmit filter is a simple 1st-order RC lowpass filter. In case of ideal channel, no channel noise, and wideband receiver filter, the received signal could look like this:

CODER

HARDWARE

CODED SIGNALBIT SIGNAL TRANSMIT FILTER

83050E/71

Example: Simple PAM System (cont’d) Here the channel is modeled as 2nd-order lowpass with 3 dB bandwidth equal to the symbol rate. Then the received signal looks as follows. Eye diagram is almost closed, in the system becomes very sensitive to noise and timing errors.

Better design of the different filtering stages in the transmission link can be expected to give significantly better waveform in the receiver, even with considerably reduced bandwidth.

83050E/72

Integrate and Dump -Principle Integrate and Dump –Principle is a widely used term for simple PAM system with square pulse shaping: A square pulse of length T is used as the impulse response for both the transmit and receive filters. (In this case the transmit and receive filters are a matched filter pair, which is optimal in case of ideal channel, as will be discussed later). As a convolution of two square pulses, the overall pulse shape (assuming ideal channel) is a triangle pulse of length 2T. In practical implementation, the receiver filter can be implemented by integrating the received signal over the symbol interval, keeping the resulting value as the sampled symbol value, and resetting the integrator for new sampling interval.

g t( ) f t( ) p t g t f t( )= ( ) ( )∗

0 0 0T T T The name is commonly used, e.g., in the spread-spectrum context, for similar basic operation, within a more advanced system concept.

83050E/73

f-W W

About Pulse Shapes in Digital Transmission Earlier, we have seen that the power spectrum of the transmitted signal comes directly as the Fourier-transform of the used pulse waveform. Usually, the channel is bandlimited,

⎩⎨⎧

≥<

=WfWf

fB01

)(

We could use a pulse whose spectrum is rectangular,

⎩⎨⎧

≥<

=WfWfWfG

02

1)(

The corresponding pulse shape would be

( )WtWt

Wttg 2sinc2

)2sin()( ==π

π

-3 -2 -1 0 1 2 3-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time in symbol intervals

It has zero crossings at each multiple of T= W2/1 , except at t=0. The pulses are over-lapping, but the requirements of p. 47 are fulfilled anyway.

This is not a practical solution, as will be discussed later, but it illustrates the principle.

83050E/74

Intersymbol Interference Let’s consider two adjacent symbols, a0=1 and a1=2. Corresponding pulses and their effects on the overall waveform are shown in the following figure.

-3 -2 -1 0 1 2 3

-0.5

0

0.5

1

1.5

2

2.5

Time in symbol intervals If we consider the earlier presented ideal bandlimited channel, this is also the received pulse waveform.

In the receiver, samples are taken at time instants mTt = . Now the adjacent symbols do not interfere with each other, and if we consider a noise free situation, the original symbol values can be obtained.

We say that intersymbol interference (ISI) is zero. Normally, this is the goal.

It should be noted that is only possible when the synchronization is done perfectly. Even a slight timing error could cause severe ISI.

83050E/75

About the Sinc Pulse The bandlimited sinc-pulse waveform is not practical because

• It is not realizable.

The pulse is too long in time domain, because the oscillations around the main lobe die out slowly.

The long tails of the pulse make also synchronization more difficult.

83050E/76

Requirements for Pulse Shape In time domain the requirement is: Intersymbol interference is zero. Mathematically, this can be written as

⎩⎨⎧

±±===

…,2,1 when0)(1)0(

mmTpp

-3 -2 -1 0 1 2 3

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time in symbol intervals Otherwise the pulse shape can be arbitrary.

83050E/77

Nyquist Criterion in Frequency Domain It follows from the previous criteria:

∑∑∞

−∞=

∞

−∞==−=−

kktkTtkTpkTttp )()()()()( δδδ

and through Fourier transform we obtain:

∑∞

−∞==⎟

⎠⎞

⎜⎝⎛ −

m TmfP

T1)1

Discussion: This result is closely related to the sampling theorem. The time domain criterion says that the sampled continuous-time pulse should be a discrete time impulse.

The left side of the frequency domain criterion is the spectrum of the sampled discrete time signal, and the right side is the Fourier-trasform of an impulse.

The latter form is the famous Nyquist Criterion. This is a criterion in frequency domain for preventing ISI.

It follows that the smallest theoretical bandwidth to obtain 0 ISI is TW 2/1= . This would obtained by using the ideal bandlimited pulse, i.e., the sinc-pulse.

In the frequency domain, the transition band is always symmetrical with respect to 1/2T.

Some (theoretical) spectra satisfying the Nyquist criterion:

ff f

P f ( ) P f ( ) P f ( )

1T

12T

1T

12T

0 1T

12T

1T

12T

0 1T

12T

1T

12T

0

83050E/78

Nyquist Criterion in Frequency Domain (cont’d) The pulses/filters with zero ISI are called Nyquist pulses/filters. In practice, the pulse shaping filter frequency response

)( fP has a symmetrical transition band with respect to Tf 2/1= .

The total signal bandwidth (to stopband edge) is

.21)1(T

α+ .

Here α is the roll-off factor, or excess bandwidth, and it is usually between 0.1…1 or 10-100%. So the excess bandwidth defines the difference between the actual and the minimum theoretical bandwidths.

1

0

δs

| |P f ( )

α2T

α2T

fp 12T

fs

f

83050E/79

Pulse Shaping in Baseband PAM-System Intersymbol interference is a very important factor in the receiver sampling. The pulse waveform at the sampling depends on the transmitter filter, receiver filter, and the transfer function of the channel.

The requirement is that the cascade of the transfer functions )()()()( fFfBfGfP =

satisfies the Nyquist criterion. The individual filter responses are not important in this respect.

This is the so-called zero-forcing criterion. It forces ISI to zero. When we have a noisy channel, it is not necessary an optimal solution, as we will see later.

• The channel transfer function is usually fixed, or it cannot be effected.

• Transmitter and receiver filters are designed together. Here, the following solutions are possible:

1. Pulse shaping in the transmitter (or receiver), the receiver (transmitter) approximates an ideal low pass filter whose bandwidth is ( ) /1 2+ α T .

2. Matched filter pair is theoretically an optimal solution in case of ideal channel. The impulse responses are mirror images, amplitude responses are the same.

3. Transmitter filter is designed as in the previous case. The receiver filter (channel equalizer) tries adaptively to minimize ISI using certain criterion.

83050E/80

Designing the Pulse Shaping Filter In general, the filter design criteria are:

1. Transmitted signal spec-trum must satisfy certain criteria in the frequency domain, usually defined in the system specifications. Example frequency mask:

f f− c

dB

0

-50

-80-26 -15 0 15 26 MHz

2. ISI should be minimized, or more generally, the overall error (including ISI and various distortions) at the decision device should be minimized to minimize the BER.

3. Out-of-band signals (adjacent channels, interferences, channel noise) should be attenuated sufficiently in the receiver.

The transmitter/receiver architecture & implementation aspects (1 and 3) are more or less (but not completely in practical implementations) separated from the communication theoretic aspect (2), and different filter stages are used to satisfy the different criteria. In the implementation, discrete time transversal filters or digital FIR filters are often used for final pulse shaping. Sampling rate conversion and digital multirate signal processing is often applied in this context. In the literature, the raised cosine pulses/filters are the standard solution.

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Raised Cosine Pulses

There are analytical expressions both for a (single-stage) raised cosine Nyquist filter and the square root raised cosine filter. The latter type of filter can be used both as transmit and receive filters, resulting in 0 ISI with ideal channel.

Raised cosine filter is constructed in frequency domain to have ideal passband and stopband response. The transition bands are formed using half a cycle of sine-wave, and the transition bandwidth is controlled by the roll-off parameter α.

0 (1 ) / 2

1( ) 1 sin (1 ) / 2 (1 ) / 22 20 f > (1 ) / 2

T f T

T TP f f T f TT

T

απ α αα

α

≤ ≤ −⎧⎪ ⎡ ⎤⎪ ⎛ ⎞⎛ ⎞= − − − ≤ ≤ +⎨ ⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦⎪⎪ +⎩

The impulse response can be shown to be:

( ) ( )( )2

sin / cos /( )

/ 1 2 /

t T t Tp t

t T t T

π αππ α

=−

i

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About Pulse Shaping Filter Design The formulas for raised cosine filters give the continuous time impulse response, and by sampling it with N times the symbol rate, symmetric (linear phase) FIR filters are obtained for N times oversampled input signal (i.e, for sampling rate of N/T). In practise, the filter has to be truncated to a finite length symmetrically around the origin, and delayed to get a causal impulse response. The finite length window causes the filters to have finite stopband attenuation. The truncation effects can be reduced by using a suitable window function, like Hanning window. Special filter optimization techniques can be used for designing filters with minimum order and satisfying the different constraints, like the Nyquist criterion and stopband attenuation requirements. One important property in this context is also the peak envelope value of the transmitted signal, which is a very important parameter from the transmitter power amplifier design point of view. Experience has shown that raised cosine filters are practically optimal in this respect, and some filter optimization methods may result in poor performance in this respect. Especially, nonlinear phase transmit filters (nonsymmetric FIR filters) seem to result in increased peak envelope values.

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Eye Diagram An eye diagram consists of many synchronized, overlaid traces of small sections (a few symbols) of a signal. It is assumed that symbols are random and independent, so all the possible symbol combinations are expected to have occurred.

Eye diagram can be measured by oscilloscope or by computer simulations. They are used for both checking the system operation and evaluation system performance in research and development work. Intersymbol interference can easily be seen in the eye diagram The eye diagram depends on the received pulse shape and the used constellation.

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Properties of Eye Diagram

a

b

c

The wider the vertical opening, the greater the noisy immunity.

ISI will reduce the vertical opening.

The ideal sampling instant is at the point of maximum vertical eye opening.

The smaller the horizontal opening, the greater the sensitivity to errors in timing phase

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Eyediagram (cont’d) The effect of excess bandwidth (Raised cosine pulses, 2 level PAM): 25% 100%

We notice that by increasing the excess bandwidth, the horizontal opening becomes wider.

In the extreme case of 0 excess bandwidth, the horizontal opening becomes zero, and the system becomes extremely sensitive to timing errors. This is one indication of the fact that sinc pulses cannot be used in digital transmission. The other bad property is that the peak envelope value of the transmitted signal grows heavily as the excess bandwidth is reduced.

4-PAM (25% excess bandwidth, Raised cosine):

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Equivalent Discrete-Time System Model If we look at the chain between coder output and sampler output, it can be modelled as a single, linear discrete-time system block (filter). The impulse response of this filter is [ ] kTtk tftbtgkTpkpp =∗∗=== )()()()()( i.e., the sampled version of the continuous-time impulse response. In many cases we can use such a simplified discrete-time system model in simulations. Of course, also the possible discrete-time/digital filter blocks in the transmitter and receiver can be included in the model. It should be noted that the channel noise is filtered by the receiver filter, which changes the noise characteristics. The noise source in this model is not necessarily white even if the channel noise is. The discrite-time noise source in the model and the actual channel noise N(t) are related through [ ] kTtk tftNU =∗= )()( .

DISCRETE-TIMEEQUIVALENT

CHANNELpk

CODER DECODER

SYMBOLSBITS BITS

Uk

QkAkBk Bk

tut/tlt/mr tlt-5406/1 tlt-5406 digital transmission lecture … · 2008-02-05 · tut/tlt/mr...

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