10/2/07 van Alphen & Katz 1

Digital Modulation Techniques- Overview -

Digital Modulation Basics Vocabulary/Notation Basic Phase Modulation: BPSK, QPSK, MPSK, DPSK Basic Frequency Modulation: FSK Performance Measures & Fundamental Limits

QPSK Variations (π/4 QPSK, OQPSK, MSK, DQPSK) QAM & OFDM Pulse Shaping Conclusions

10/2/07 van Alphen & Katz 2

Basic M-ary Digital CommunicationSystem

A/DA/D Collect k bits;Collect k bits;bits bits →→ symbols symbols

Modulate;Modulate;

symbols symbols →→waveformswaveforms

Demodulate;Demodulate;

waveform waveform →→symbolssymbols

bits bits ←← symbols symbolsD/AD/A

(analog)(analog)informationinformation

(analog)(analog)informationinformation

channelchannel

10/2/07 van Alphen & Katz 3

M-ary Communication System:Symbol –Level Considerations

Transmits one of M possible waveforms Each symbol

corresponds to a message mi, i = 1, 2, …, M can represents k bits of information, where M = 2k

is associated with a waveform si(t), of duration T seconds

T = Ts is called the symbol time or symbol duration

To send message mi: transmit waveform si(t), 0 < t ≤ T receiver guesses which of M possible messages was sent

10/2/07 van Alphen & Katz 4

Simplified QPSK Example

Say we use Quaternary Phase Shift Keying (QPSK) as ourmodulation.

We need M = 4 waveforms, with 4 different phase angles:

cos(2πf0t)

sin(2πf0t)

00

01

11

10

Bits-to-Waveforms

(Gray-coded)

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QPSK Receiver (Generalized)

)tcos(T

20!

)tsin(T

20!

E

00

01

11

10

s4

s1

s3

s2

Dotted lines ⇔ receiver decisionboundaries (for equally likely messages inadditive white noise)

Note that the mapping between bits andwaveforms is done using a Gray Code:Noise that pushes the received point past asingle decision boundary causes only asingle bit error.

10/2/07 van Alphen & Katz 6

QPSK Example for Digital Image

Vector Value 5 7 4 …Binary-Coded 101 111 100Tx Bit Sequence for M = 4:(re-grouping) 10 11 11 10 …

Consider a vector, v, taken from a digital color image with 8quantization levels (3 bits) for R, G and B:

[5 7 4 … 4 0 3 5 … 6 2 1 0 … 5]

red green blue

(Grouping k = log2(M) = log2(4) = 2 bits per symbol)

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Quaternary Example, Continued

Bit Sequence 10 11 11 10

-sin(2πf0t)-cos(2πf0t) -sin(2πf0t)

-cos(2πf0t)

t0 T 2T 3T 4T

Say f0 = 1000 Hz, T = 1 ms

-1

0

1

0 0.001 0.002 0.003 0.004

10 11 11 10 Note: abruptphasechangesincrease thebandwidth

10/2/07 van Alphen & Katz 8

Relating Bit and Symbol Parameters

1 symbol ⇔ k = log2(M) bits T = Ts = kTb

e.g., if M = 4, then k = 2 and T = Ts = 2Tb

Baud Rate R symbols/sec = 1/T Bit Rate Rb bits/sec = 1/Tb

E = S T (E: energy in the waveform of duration T; S:average signal power) Eb = S Tb (Eb: energy allocated per bit; Tb is time allocated per bit)

Bandwidth proportional to 1/T Decreasing T ⇔ Faster signaling ⇔ Wider Bandwidth

10/2/07 van Alphen & Katz 9

Binary Phase Shift Keying (BPSK)

M = 2 ⇒ 2 waveforms, 180° out of phase

Signal Space Diagram (1-dimensional)

Throughput:

)tcos(T

E2)t(s 01 !=

)tcos(T

E2)t(s 02 !"=

0 < t ≤ T

Rb = 1/T = 1/Tb bps

X X )tcos(T

20!

0

EE!

s1s2

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BPSK: Optimum Receiver for AWGN(Assume equally likely, equal energy signals)

r(t) x !T

0

II > 0s1 si

)tcos(T

2)t( 01 !="

CorrelationReceiver

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Quaternary Phase Shift Keying (QPSK)

M = 4 ⇒ 4 waveforms, 90° out of phase

Signal Space Diagram (2-dim.)

2 bits/symbol

Throughput:

)tcos(T

E2)t(s 04 !=

)tsin(T

E2)t(s 01 !=

Rb = 2/T bps

)tcos(T

E2)t(s 03 !"=

)tsin(T

E2)t(s 02 !"=

0 < t ≤ T

)tcos(T

20!

)tsin(T

20!

xx

x

x

E

10/2/07 van Alphen & Katz 12

QPSK Modulation

diSerialParallel

xdeven

dodd

)tcos(T2)t( 01 !="

x

-90°ΣI-Q

Transmitter

1T 2T 3T0td(t)

dodd(t) 1T 2T 3T0t

deven(t) 1T 2T 3T0t

Data streamExample

s(t)

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8-ary PSK

M = 8 ⇒ 8 waveforms, 45° out of phase

Signal Space Diagram (2-dim.)

3 bits/symbol

Throughput: Rb = 3/T bps

)tcos(T

20!

)tsin(T

20!

xx

x

x

E

xx

xx

10/2/07 van Alphen & Katz 14

MPSK: Optimum Receiver for AWGN(Assume equally likely, equal energy signals)

I-QReceiver

r(t)

x !T

0

I

θ = tan-1(Q/I) si^)tcos(

T2)t( 01 !="

x !T

0

Q

)tsin(T2)t( 02 !="

Choose siw/nearest θi

I: Projection of r(t) onto ψ1 axisQ: Projection of r(t) onto ψ2 axis )t(1!

0

r

I

ψ2(t)

Qθ

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Bit Error Probabilities for MPSKSignals in AWGN

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

-2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

M=2, 4

M=8

M=16

M=32

M=64

Eb/N0, dB

Bit Error Probabililty, MPSK

Signaling

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QPSK vs. BPSK(The “Almost Free” Lunch)

QPSK has twice the throughput as BPSK

QPSK and BPSK have

the same transmission bandwidth

the same bit error probability, PB

even though QPSK has a higher symbol errorprobability

10/2/07 van Alphen & Katz 17

Differential Phase Shift Keying (DPSK)- Motivation -

Consider a BPSK system in which the transmitter local oscillatorand the receiver local oscillator are out of phase by angle θ.

Note that some phase offset between the tx and the rcvr isalways present, and usually necessitates a PLL at the rcvr

X XX

X θ

Tx Rcvr

)tsin(T2

0! )tsin(T2

0!

)tcos(T2

0!

EE!

10/2/07 van Alphen & Katz 18

(Binary) DPSK

Concept: Differentially encode the information bits, prior totransmission; differentially decode the received bits

m2 = 0 → no change in phase of the sinusoid, relative to the previousburst

m1 = 1 → 180° change in phase of the sinusoid, relative to theprevious burst

Tx phase angle θi = θi-1 + θi where θi denotes the i-thinformation bit: Δθi = θi – θi-1 = θi

Note: Differential schemes always require the transmission ofone additional reference bit prior to transmitting the data.

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Example: Transmitted Waveformsfor BPSK and DPSK

Consider the bit stream

0 1 1 0 1BPSK

-1.5

0

1.5

0 1 2 3 4 5

t, s

s(t)

DPSK

-1.5

0

1.5

0 1 2 3 4 5 6

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DPSK

Phase information is only relative to that of the previouspulse ⇒ no need to generate phase reference at thereceiver (No PLL required).

Performance is degraded from that of BPSK in AWGN, dueto induced dependence of errors from bit-to-bit.

However, DPSK eliminates degradation due to phase offsetsbetween the transmitter and the receiver.

Note that we have only discussed Binary DPSK; QuarternaryDPSK (QDPSK) is also commonly used, with similarencoding and decoding at the tx and the rcvr.

10/2/07 van Alphen & Katz 21

M-ary Frequency Shift Keying: MFSK

As M increases

Bandwidth increases (bandwidth efficiency decreases)

Pb decreases (if the frequencies are chosen to yieldorthogonality)

⇒ power efficiency increases

Constant envelope signaling⇒ No performance degradation from the use of

non-linear amplifiers

10/2/07 van Alphen & Katz 22

Binary Frequency Shift Keying (BFSK)

M = 2 ⇒ 2 waveforms, at 2 different frequencies

Signal Space Diagram (for the special case oforthogonal signaling)

Throughput:

)tf2cos(T

E2)t(s 01 !=

)t)ff(2cos(T

E2)t(s 02 !+"=

0 < t ≤ T

Rb = 1/T = 1/Tb bps

X )tf2cos(T

20!

0 E

E

s1

s2XΔf = 1/T, NC Rcvr

1/(2T), Coh. Rcvr

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BFSK Optimum (Coherent) Receiver

))(2cos(2

0 tffT

!+"

r(t)

x !T

0

si^)2cos(2

0tfT!

x !T

0

Chooselargest

10/2/07 van Alphen & Katz 24

PSD: NRZ Baseband Signaling, BPSK

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

fTs

G(f)/E

10/2/07 van Alphen & Katz 25

PSD: NRZ Signaling, BPSK(dB Scale, with Bandwidth Definitions)

-22

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

0 2 4 6 8 10 12 14 16

fTs

G(f

)/E

, dB

W3dB

WB,16 dB

WN-N

10/2/07 van Alphen & Katz 26

Communications Link: the Channel

Channels are characterized by their capacity

Capacity: An inherent limit on the rate at whichinformation can be sent “error free”

Capacity increases with bandwidth (W) and signal-to-noise ratio (SNR or S/N)

Info

Bandwidth,SNR

C = W log(1 + S/N)

10/2/07 van Alphen & Katz 27

Reviewing Shannon’s Theorem

Shannon-Hartley Theorem:

Communication (with arbitrarily small error probability)is . . .

Possible at rates Rb < C Impossible at rates Rb > C

Establishes absolute theoretical limit on tx rate Therefore,

!"

#$%

&+'=N

S1logWC 2

!"

#$%

&+=<N

S1logW/CW/R 2b

10/2/07 van Alphen & Katz 28

Efficiency Definitions and theBandwidth/Communications Efficiency Plane

Defn: Bandwidth Efficiency (or Normalized Throughput) =Rb/W, bps/Hz

Communications Efficiency: Eb/N0 required to Attain aParticular Bit Error Probability

Bandwidth/Communications Efficiency Plane: Plots Rb/W vs.Eb/N0 for particular modulation techniques, assuming aparticular pulse shaping. Goal: to be as near as possible tothe theoretical limit

Note: Separate plots are usually done for each required biterror probability

10/2/07 van Alphen & Katz 29

The Bandwidth/Communications EfficiencyPlane

0

2

4

6

8

10

12

14

-5 0 5 10 15 20 25 30 35

Rb/W,bps/Hz

Eb/N0, dB

ShannonBoundary: Rb = C

Rb < C Region:Modulation/CodingTechniques Exist w/P(E) 0

Rb > C Region: NoModulation/CodingTechniques w/P(E) 0

10/2/07 van Alphen & Katz 30

-8

-4

0

4

-2 2 6 10 14 18 22 26 30

Bandwidth Efficiency Plane, continuedPB = 10-5

Rb = C

Eb/N0, dB

log2(Rb/W)

Power-limitedRegion

Bandwidth-limitedRegion

M = 4M = 2

MPSK

10/2/07 van Alphen & Katz 31

Bandwidth Efficient Digital ModulationTechniques

Phase Modulation

Discrete Phase MPSK QPSK, OQPSK, π/4-QPSK DQPSK, ODQPSK, Oπ/4-QPSK

Continuous Phase MSK, GMSK

QAM

OFDM

More later on howto do these, if thebasic modulationtechniques aren’t“efficient enough”

10/2/07 van Alphen & Katz 32

QPSK Variations – Restricting Alowable PhaseTransitions

QPSK: max phase transition 180o

loses constant envelope afterfiltering

OQPSK: max phase transition 90o

preserves constantenvelope after filtering

Compromise - π/4-QPSK:max phase transition 135o, min 45o

• preserves constant envelope:• better than bandlimited QPSK• not as well as bandlimited OQPSK

• same BER in AWGN as QPSK, OQPSK

10/2/07 van Alphen & Katz 33

QPSK Variations, cont.

Discrete Phase Transition Techniques:

QPSK ππ//4-QPSK OQPSK max 180o max 135o max 90o

- reduction of phasetransitions to obtain lesssidelobe regrowth andtherefore less ISI

Why not completely eliminate abrupt phase transitions? “Continuous Phase” Transition Techniques (CPFSK):

MSK: applies sinusoidal weight to OQPSK (constant envelope)- same BER as QPSK & OQPSK (MF detector in AWGN)- same bandwidth efficiency (bps/Hz) as QPSK & OQPSK- spectrum has wider mainlobe than QPSK & OQPSK, but faster drop-off of sidelobes (proportional to ω-4 vs. ω-2)

10/2/07 van Alphen & Katz 34

QPSK Variations, cont.

QPSK π/4-QPSK OQPSK MSK - reduction of phasetransitions to obtain lesssidelobe regrowth andtherefore less ISI

MPSK GMSK, BT=.25 GMSK, BT = .3 GMSK, BT = .5 Ideal 0 dB .7 dB .3 dB .2 dB Practical1

.7 dB 1.7 dB

Eb/N0 Penalty in AWGN (vs. Ideal QPSK)

1 Simulation results, assuming non-optimal receiver

Special Case of MSK: GMSK (Gaussian Pulse Shape, time-bandwidth product BT)• further narrows the spectrum, at the cost of re-introducing ISI (increasing BER)• smaller BT product more compact spectrum, but larger ISI

10/2/07 van Alphen & Katz 35

Power Spectral Densities

f/Rb, Hz/bps

-60

-50

-40

-30

-20

-10

0

10

- 3 - 2 - 1 0 1 2 3

MSK, dB

QPSK, dB

G(f)

10/2/07 van Alphen & Katz 36

Quadrature Amplitude Modulation:Rectangular 16-ary QAM

-3A -A A 3A

3A

A

-A

-3A

ψ1(t)

ψ2(t)QAM: send one coordinate on in-phase carrier, one on quadraturecarrier

Throughput:

Rb = 4/T

10/2/07 van Alphen & Katz 37

Orthogonal Frequency Division Modulation(OFDM)

Choose N orthogonal tones to be used as sub-carriers:

Demux the user data (serial-to-parallel conversion) Put each sub-stream of data onto a different sub-carrier,

where the sub-carriers are the orthogonal tones describedabove

)t)ff(2cos(T

E2)t(s 0i !+"= i = 1, 2, …, N;

Δf = i/(T)

f1 f2 fN…

…x

xx

x

x

xx

x

x

xx

x

Example whereeach sub-carrieris modulatedwith QPSK

10/2/07 van Alphen & Katz 38

Motivation for Pulse Shaping

Band-limiting the signal in the frequency domain leads to timedomain spreading of the signal

Result: Intersymbol Interference, due to signal in one time-slot overlapping into the next time slot(s).

H(f)

f

Sin(f) Sout(f)

f f

sin(t)

t0 T

sin(t)Narrow relative to Signal

Channel Transfer Function

t0 T

10/2/07 van Alphen & Katz 39

Motivation for Pulse Shaping: ISI

Tails of signal overlap with body of succeeding signal

Destructive interference results when the pulses are ofopposite polarity

Problem cannot be solved by increasing signal power orenergy – as opposed to additive noise problems, which can!

t0 T

…

Pb

Eb/N0, dB

AWGN

Eb/N0, dB

PbISI

10/2/07 van Alphen & Katz 40

Commonly Used Pulse Shapes

Nyquist Pulse: Cancel ISI (ideally) Example: Raised Cosine, w/roll-off parameter α = 0):

Gaussian Pulse:where α = .5887/B, and B is the 3-dB bandwidth of the filter

0

0.2

0.4

0.6

0.8

1

1.2

-1.5 -1 -0.5 0 0.5 1 1.5

α = 0

α = 1

α = .5

fTs,

H(f)

22fG e)f(H !"

=

10/2/07 van Alphen & Katz 41

Power Efficiency Comparison for DigitalModulation Techniques in AWGN

Modulation PB = 10-1

PB = 10-2

PB = 10-3

PB = 10-4

PB = 10-5

PB = 10-6

MPSK, M = 2, 4

M = 8

M = 16

M = 32

M = 64

-0.9 dB

1.0 dB

4.0 dB

7.4 dB

11.2 dB

4.3 dB

7.3 dB

11.4 dB

16.0 dB

20.9 dB

6.7 dB

10.0 dB

14. 4 dB

19.1 dB

24.2 dB

8.4 dB

11.8 dB

16.2 dB

21.0 dB

26.1 dB

9.6 dB

12.9 dB

17.5 dB

22.3 dB

27.4 dB

10.5 dB

14.0 dB

18.5 dB

23.4 dB

28.5 dB

DQPSK, MSK 2.0 dB 6.8 dB 9.2 dB 10.8 dB 12.0 dB 12.9 dB

GMSK, BT = .25

BT=infinity

0.8 dB

-.1 dB

6.0 dB

5.0 dB

8.5 dB

7.5 dB

10.1 dB

9.1 dB

11.3 dB

10.3 dB

12.2 dB

11.2 dB

16-QAM

64-QAM

1.9 dB

5.0 dB

7.9 dB

11.5 dB

10.5 dB

15.0 dB

12.3 dB

16.0 dB

13.6 dB

17.8 dB

14.2 dB

18.5 dB

Required Eb/N0 for Various Bit Error Probabilities

10/2/07 van Alphen & Katz 42

Relative Complexityof Modulation Techniques [Ref: Oetting]

BPSK

QPSK

OQPSK

MSK

CPFSK

QPR

MPSK

QAMDQPSK

DPSK

CPFSK

NCFSK

OOK

(ED) (DISC.DET.)

(OPT.DET.)

HIGHLOW