ECE 6640Digital Communications

Dr. Bradley J. BazuinAssistant Professor

Department of Electrical and Computer EngineeringCollege of Engineering and Applied Sciences

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Chapter 9

9. Modulation and Coding Trade-Offs.1. Goals of the Communications System Designer. 2. Error Probability Plane. 3. Nyquist Minimum Bandwidth. 4. Shannon-Hartley Capacity Theorem. 5. Bandwidth Efficiency Plane. 6. Modulation and Coding Trade-Offs. 7. Defining, Designing, and Evaluating Systems. 8. Bandwidth-Efficient Modulations. 9. Modulation and Coding for Bandlimited Channels. 10. Trellis-Coded Modulation.

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Sklar’s Communications System

Notes and figures are based on or taken from materials in the course textbook: Bernard Sklar, Digital Communications, Fundamentals and Applications,

Prentice Hall PTR, Second Edition, 2001.

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System Level Tradeoffs

• The Nyquist theoretical minimum bandwidth requirements• The Shannon-Hartley capacity theorem

– The Shannon limit

• Government regulatory involvement– frequency allocation, bandwidth limitations

• Technology limitations– physically realizable components using current technology

• Other system requirements– For satellite: orbits and energy limitations

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Error Probability Plane

• Error probability performance curves– define acceptable BER– determine required Eb/No

• We would prefer equivalent bandwidth performance curves– allows system level tradeoffs– trade-off Eb/No for modulation type at fixed BER– trade off BER vs modulation type at fixed Eb/No– show range of expected BER as Eb/No varies

BER vs Eb/No Curves

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Nyquist Minimum Bandwidth

• Nyquist showed that the theoretical minimum bandwidth needed for baseband transmission of Rs symbols per second without ISI is Rs/2 Hz. – A theoretical minimum constraint on bandwidth required. – Referred to as 2 symbols/sec/Hz– Typical systems and filters are 10%-40% wider– More likely 1.8 to ¼ symbols/s/Hz.

• Rs in terms of M symbol modulation

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sRkR M

RkRRs

2log

Example 9.1: Digital Schemes

• Orthogonal Signaling– expect improvement in BER as k or M increases

• Non-orthogonal signaling– expect a decrease in BER as k or M increases

a) Does error-performance improve or degrade with increasing M, for M-ary signaling?

b) The choices available almost always involve a tread-off. If error performance improves, what price must we pay?

c) If error-performance degrades, what benefit is exhibited?

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Example 9.1

• Expected trade-offs• M-FSK

– as M increases, the required transmission bandwidth increases for minimum frequency spacing.

– to maintain a constant bit rate, the symbol transmission rate decreases with increasing M

• M-PSK– while there is degradation as M increases, the symbol transmission

rate may be decreased as M increases– M-PSK systems plot equal-bandwidth curves, as the bit

transmission rate increases.

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Shannon-Hartley Capacity Theorem

• The capacity relation in AWGN can be stated as

– where S is the signal power, N the noise power, and W the bandwidth

– the value is defined in bits per second

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NSWC 1log2

Shannon-Hartley Capacity Theorem

• The normalized channel bandwidth vs. SNR may also be plotted

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NS

WC 1log2

1

2 1log

NS

CW

S-H Equivalent Equations

• Rearranging and defining the noise power and signal power

• For

• Letting C = Rb

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NS

WC 1log2

WR

NS

NE bb

11

0

WR

NE bbW

C

0

12

WC

NEbW

C

0

12

12

0

WC

b

CW

NE

WR

NE

WC bb

02 1log

WC

NE

WC b

02 1log

Shannon Capacity Theorem

• There is a limiting case as C/W 0– let

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WC

NEx b

0

WC

NE

CW

EN

NE b

b

b

02

0

0

1log1

WC

NE

WC b

02 1log

xxN

Eb 1log11 20

eNEx

NE bxb

x 20

12

00

log1loglim1

xb xNE 1

20

1log1

dBeN

Eb 6.1693.0log

1

20

Shannon Limit

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• As C/W 0 or W/C∞• In practice, it is not possible to

reach the bound.• Provides an improvement bound

for encoding and decoding.• For example: raw BPSK requires

approximately 9.6 dB Eb/No to achieve a BER of 10-5 which suggests that up to an 11.2 dB improvement is possible.

– Turbu Codes can achieve ~ 10 dB.

dBeN

Eb 59.1693.0log

1

20

Entropy

• To compute communication capacity, a metric for the message content of a system is also important.

• Entropy is defined as the average amount of information per source output.

• It is expressed by:

– where pi is the probability of the ith output and the sum of all pi is 1.

• For a binary system, entropy can be expressed as:

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n

iii ppH

12log

ppppH 1log1log 22

Entropy for a Binary System

• The entropy is based on the probability, p, of an event.

• This can also be looked at as the randomness of successive events or how correlated individual events are.

• Note that maximum entropy is achieved when the probability is 50%

– A sample provides no information about a succeeding sample.

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Example 9.2 English Language

• The English language is highly redundant.– The probability of the next letter in a word is not equally likely for

all possible characters.– Determine the Entropy based on the letter probabilities– p=0.10 for the letters a, e, o, t– p=0.07 for the letters h, I, n, r, s– p=0.02 for the letters c, d, f, l, m, p, u, y– p=0.01 for the letters b, g, j, k, q, v, w, x, z

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n

iii ppH

12log

bits/char17.401.0log01.0902.0log02.08

07.0log07.051.0log1.04

22

22

H

bits/char70.4

26log261log

26126 22

H

English Language Equal Probability

Equivocation

• A term used by Shannon to account for the uncertainty in a received signal. It is defined as the conditional entropy of the message X (transmitted source message), given Y (the received signal).

– based on conditional probability

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YX

YXPYXPYXH,

2 |log,|

Y X

YXPYXPYPYXH |log|| 2

Equivocation Example

• Consider a binary sequence, X, where the bits are equally likely. Assume that the channel produces on error in a received sequence of 100 bit (Pb=0.01).

• Interpretation: the channel introduces 0.081 bit/received symbol of uncertainty.

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YX

YXPYXPYXH,

2 |log,|

bbbb PPPPYXH 22 log1log1|

01.0log01.099.0log99.0| 22 YXH

081.0| YXH

Effective Transmission Rate

• Using the equivocation computation, the effective transmission rate of the channel can be computed as

– based on the previous example, the binary system would have an effective transmission rate (in terms of bit/received symbol) of

– for a communication system with R = 1000 bits/sec,the effective transmission rate would become

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YXHXHHeff |

919.0081.01 effH

919919.01000 effeff HRR

Pb vs Eb/No Curves

• It appears that Pb approaches 0.5 as Eb/No decreases … but the Shannon limits is Eb/No=-1.6 dB.Is this a contradiction or not?

• Shannon refers to received information bits based on equivocations.

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Deriving an Effective Eb/No

• As an example, take Eb/No=-10 dB for coherent BPSK

– from this form an effective Eb/No

– Thus, he effective Eb/No is well above the Shannon limit, -1.6dB

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02 NEQP bB

33.0447.0 QPB

915.033.0log33.033.01log33.01| 22 YXH

085.0915.01 effH

dBH

NENE

eff

b

eff

b 7.0176.1085.0

1.00

0

Bandwidth-Efficieny Plane

• Using Shannon-Hartley Capacity, the “normalized” channel bandwidth versus Eb/No for different symbol schemes can be compared.– Typically performed for a defined bit-error probability and under

optimal symbol detection assumptions.– Let R=C, then

– The bounds and appropriate values for MPSK, MFSK and MQAM symbol schemes are shown on Fig. 9.6

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WR

NE

WR b

02 1log

Figure 9.6: Bandwidth-Efficiency Plane

• Factors of note:– MPSK and QAM nominally

maintain the same bandwidth will increasing the bits per symbol and required Eb/No

– MFSK uses an increasing bandwidth as the bits per symbol increases while the Eb/No is decreasing

– BPSK and QPSK have the same Eb/No but different bits per symbol

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Bit and Symbol Rate Considerations

• For MPSK

– R/W increases with M

• For MFSK

– R/W decreases with M

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ss RMRkR 2log

ss

IF RT

W 1

MR

RMWR

s

s

IF2

2 loglog

ss RMRkR 2log

ss

IF RMTMW

M

MRM

RMWR

s

s

IF

22 loglog

Bandwidth versus Power

• For a bandwidth-limited system– spectral efficiency is important– expect that signal power may be increases to offset the limitation– study the bandwidth-efficient plane– PSK allows for fixed bandwidths

• For a power-limited system– a defined transmission power limit has been established– expect that signal bandwidth may increase to offset the limit– study the bit-error probability planes– FSK allows for limited spectral power

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Digital Comm. System Engineering

• Defining, designing, and evaluating communication systems.

• Comparing MPSK and MFSK (table 9.1)

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M k R Rs min W R/W Eb/No (dB) min W R/W Eb/No (dB)bits/sec sym/sec (Hz) Pb=1e‐5 (Hz) Pb=1e‐5

2 1 9600 9600 9600 1 9.6 19200 0.5 13.44 2 9600 4800 4800 2 9.6 19200 0.5 10.68 3 9600 3200 3200 3 13.0 25600 0.375 9.1

16 4 9600 2400 2400 4 17.5 38400 0.25 8.132 5 9600 1920 1920 5 22.4 61440 0.15625 7.4

MPSK Non‐Coherent MFSK

System Example #1: Bandwidth Limited

• W = 4000 Hz, Pr/No=53 dB-Hz, R=9600 bps, PB=1e-5

• Equations needed for the computations (assuming M-PSK)

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ssbr R

NER

NE

NP

000

MNEQMP s

Esin22

0

MMPP E

B2log

MNE

NE bs

200

log

W 4000 HzPr/No 53 dB‐HzR 9600 bpsPb 1.00E‐05 BER

Pr/No 199526.23 Hz

Eb/No 20.78Eb/No 13.18 dB

M‐PSK Rs lin dB sqrt(2*Es/No) sin(pi/M) x Q(x)=Pe Pb2 9600 sym/s Es/No 20.78 13.18 6.45 1.00 6.45 1.14E‐10 1.14E‐104 4800 sym/s Es/No 41.57 16.19 9.12 0.71 6.45 1.14E‐10 5.69E‐118 3200 sym/s Es/No 62.35 17.95 11.17 0.38 4.27 1.92E‐05 6.42E‐06

16 2400 sym/s Es/No 83.14 19.20 12.89 0.20 2.52 1.19E‐02 2.97E‐03

System Example #1: Bandwidth Limited

• W = 4000 Hz, Pr/No=53 dB-Hz, R=9600 bps, PB=1e-5

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MNEQMP s

Esin22

0

MMPP E

B2log

MNE

NE bs

200

logssbr R

NER

NE

NP

000

System Example #2: Power Limited

• W = 45 kHz, Pr/No=48 dB-Hz, R=9600 bps, PB=1e-5

• Equations needed for the computations (assuming M-FSK)

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021exp

21

NEMMP s

E

12

2 1

k

k

EB MPP

MNE

NE bs

200

log

ssbr R

NER

NE

NP

000

12

2 1

k

k

EB MPP

MNE

NE bs

200

logW 45000 HzPr/No 48 dB‐HzR 9600 bpsPb 1.00E‐05 BER

Pr/No 63095.73 Hz

Eb/No 6.57Eb/No 8.18 dB

M‐FSK k Rs Ws lin dB exp(‐Es/No/2) PE Pb2 1 9600 sym/s 19200 Hz Es/No 6.57 8.18 0.04 1.87E‐02 1.87E‐024 2 4800 sym/s 19200 Hz Es/No 13.14 11.19 0.00 2.10E‐03 1.40E‐038 3 3200 sym/s 25600 Hz Es/No 19.72 12.95 0.00 1.83E‐04 1.05E‐04

16 4 2400 sym/s 38400 Hz Es/No 26.29 14.20 0.00 1.47E‐05 7.82E‐0632 5 1920 sym/s 61440 Hz Es/No 32.86 15.17 0.00 1.13E‐06 5.85E‐07

ssbr R

NER

NE

NP

000

System Example #2: Power Limited

• W = 45 kHz, Pr/No=48 dB-Hz, R=9600 bps, PB=1e-5

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021exp

21

NEMMP s

E

Coded System Example

• When the previous methods do not produce a valid implementation, encoding and decoding will be required.– Monitor the effect of code rates on symbols/sec and bandwidths

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System Example #3: Encode-Decode

• W = 4000 Hz, Pr/No=53 dB-Hz, R=9600 bps, PB=1e-9• Starting with the previous 8-PSK system, we need

additional coding gain

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ss

ccbr R

NER

NER

NE

NP

0000

MNEQMP s

Esin22

0

MMPP E

C2log

Mnk

NEM

NE

NE bcs

20

200

loglog

sc RMRknR 2log

jnc

jc

n

tjB PP

jn

jn

P

11

1

Solution is Steps

• Step 1: Compute the Es/No

• Step 2: Compute the codeword symbol error rate PE(M)

• Step 3: Compute the codeword-bit-error rate

• Step 4: Compute the decoded bit error probability

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ss

ccbr R

NER

NER

NE

NP

0000

Mnk

NEM

NE

NE bcs

20

200

loglog

MNEQMP s

Esin22

0

MMPP E

C2log

jnc

jc

n

tjB PP

jn

jn

P

11

1

Excel Computations

• An excel spreadsheet can be used for all of the examples.

• see results for Example #3

• Alternate Approach– the coding gain formula can be used.

– an encoding scheme that meets the bandwidth requirement and has 2.8 dB or more coding gain is sufficient for solving this problem.

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dBinNEdBin

NEdBinG

coded

b

uncoded

b

00

8.22.1316 dBinG

Bandwidth Efficient Modulations

• Modern communication is hungry for bandwidth, demanding an every increasing communications capacity within the fixed frequ3ency bands available,

• Additional requirements to allow for non-linear amplification put a premium on using signals that are minimally effected by AM to PM conversion, limiting the amplitude variations of the signal (desiring a constant modulus).

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QPSK and Offset QPSK

• Conventional QPSK uses consecutive bits received to determine I-Q pairs for transmission.

• Offset QPSK also uses the bits, but directs them to the I and Q ports as they arrive in time (next slide)

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QPSK versus Offset QPSK

• OQPSK makes 90 degree phase transitions

• 180 degrees phase changes may result in significant amplitude variation

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Minimum Shift Keying (MSK)

• Avoiding discontinuous phase transitions of the signal– maintain a constant amplitude– use a form of continuous-phase FSK– also a modified form of OQPSK

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TktTkxtT

dfts kk

1,

42cos 0

2,

2mod 11 kkkk ddkxx

MSK Quadrature Representation

• Expanding the cosine term cos(a+b)

– the similarity to OQPSK is based on the amplitude weighted quadrature structure of this formulation

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tfTtb

tfTtats

k

k

0

0

2sin2

sin

2cos2

cos

1cos kk xa

1cos kkk xdb

2,

2mod 11 kkkk ddkxx

Bandwidth Comparison:BPSK, QPSK & OQPSK, & MSK

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Modulation and Coding for Bandlimited Channels

• Research Areas (as of 2001 copyright):– Optimum signal constellation boundaries (choosing a closely

packed signal subset from any regular array or lattice of candidate points)

– Higher density lattice structures (adding improvement to the signal subset choice by starting with the densest possible lattice for the space)

– Trellis-coded modulation (combined modulation and coding techniques for obtaining coding gain for bandlimited channels).

• Ungerboeck Partitioning

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Evolution of Telephone Modem Standards (1)

• Telephone modems have dealt with the limited power and bandwidth problem for a considerable time.

• Progress was made at different times for both leased-lines and dial-line services.

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Evolution of Telephone Modem Standards (2)

• Home modem standards – Mostly replaced by telephony DSL or cable TV access

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Signal Constellation Boundaries

• Various QAM constellations that have been investigated.– optimal packing of points with

maximum separation– reduce maximum amplitude– optimize PE(M)

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Trellis-Coded Modulation (TCM)

• Developing combined modulation and coding schemes• Use a redundant nonbinary modulation in combination

with a finite-state machine based encoding process.– FSM could be similar to convolutional encoding– A multi-level/phase modulation scheme

• The concept, when performing MATLAB simulations of encoded bit streams using MPSK or QAM symbols, is there an optimal combination? – if you know the symbols being used, could one convolutional code

leading to an appropriate trellis decoding perform better than another?

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TCM Encoding

• Ungerboeck, G., "Channel coding with multilevel/phase signals," Information Theory, IEEE Transactions on, vol.28, no.1, pp.55,67, Jan 1982.

• Initial paper describing trellis coded, soft decision encoding and modulation technique for communications.

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