Fundamentals of Digital Modulation_Ahmed

8/8/2019 Fundamentals of Digital Modulation_Ahmed

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Fundamentals ofDigital Modulation

Dr. Ahmed BassyouniResearch Professor

Electrical Engineering Dept.Boise State University


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Digital Modulation

The input is discrete signal

Time sequences of pulses or symbols

Offers many advantages

Robustness to channel impairments

Easier multiplexing of various sources of information: voice, data,video.

Can accommodate digital error-control codes

Enables encryption of the transferred signals

More secure link

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Digital Modulation Example

The modulating signal is represented as a time-sequence of symbolsor pulses.

Each symbol has mfinite states: That means each symbol carries nbitsof information where n= log2m bits/symbol.

...0 1 2 3 T

One symbol(has mstates voltage levels)

(represents n= log2mbits of information)

Modulator

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Factors that Influence Choice of DigitalModulation Techniques

A desired modulation scheme

Provides low bit-error rates at low SNRs

Power efficiency

Performs well in multipath and fading conditions

Occupies minimum RF channel bandwidth

Bandwidth efficiency

Is easy and cost-effective to implement

Depending on the demands of a particular system or

application, tradeoffs are made when selecting a digitalmodulation scheme.

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Bandwidth Efficiency of Modulation

Ability of a modulation scheme to accommodate data within a

limited bandwidth.

Bandwidth efficiency reflect how efficiently the allocatedbandwidth is utilized

bps/Hz:EfficiencyBandwidthB

RB

R: the data rate (bps)B: bandwidth occupied by the modulated RF signal

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Shannons Bound

There is a fundamental upper bound on achievable bandwidth

efficiency. Shannons theorem gives the relationship betweenthe channel bandwidth and the maximum data rate that can betransmitted over this channel considering also the noise presentin the channel.

)1(log2maxN

S

B

CB

Shannons Theorem

C: channel capacity (maximum data-rate) (bps)B: RF bandwidthS/N: signal-to-noise ratio (no unit)

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Tradeoff between BW Efficiency and Power Efficiency

There is a tradeoff between bandwidth efficiency and power

efficiency Adding error control codes

Improves the power efficiency

Reduces the requires received power for a particular biterror rate

Decreases the bandwidth efficiency Increases the bandwidth occupancy

M-ary keying modulation

Increases the bandwidth efficiency

Decreases the power efficiency

More power is requires at the receiver M-FSK keying modulation

Increase the power efficiency

Decrease the bandwidth efficiency

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Example

SNR for a wireless channel is 30dB and RF bandwidth is

200kHz. Compute the theoretical maximum data rate that canbe transmitted over this channel?

Answer:

Example 6.6

Example 6.7

MbpsxN

SBC

NS

dB

99.1)10001(log102)1(log

10

2

5

2

10

30

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Noiseless Channels and Nyquist Theorem

For a noiseless channel, Nyquist theorem gives the relationship

between the channel bandwidth and maximum data rate thatcan be transmitted over this channel

mBC 2log2

Nyquist Theorem

C: channel capacity (bps)B: RF bandwidthm: number of finite states in a symbol of transmitted signal

Example: A noiseless channel with 3kHz bandwidth can only transmitmaximum of 6Kbps if the symbols are binary symbols.

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Power Spectral Density of Digital Signals and Bandwith

What does signal bandwidth mean?

Answer is based on Power Spectral Density (PSD) of Signals

For a random signal w(t), PSD is defined as:

elsewhere

oftransformfourierthis

0

22)()(

)()(

)(lim)(

2

Tt

Ttwtw

twfW

T

fWfPw

T

TT

T

T

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Fourier Analysis

Joseph Fourier has shown that any periodic function F(f) with period T, can

be constructed by summing a (possibly infinite) number of sins and coss.

Such a decomposition is called Fourier series and the coefficients are called

the Fourier coefficients.

A line graph of the amplitudes of the Fourier series components can bedrawn as a function of frequency. Such a graph is called a spectrum or

frequency spectrum. f0 is called thefundamental frequency.

The nth term is called nth harmonic. The coefficients of the nth harmonic are

an and bn.

0

1

0

22

)sin()cos(2

)(

fT

tnbtnaa

tF

T

TnT

n

n

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Fourier Analysis

The coefficients can be obtained from the periodic function F(t)

as follows:

,...2,1,sin)(2

,...2,1,cos)(2

)(2

0

0

0

0

ntdtntFT

b

ntdtntFT

a

dttFT

a

T

Tn

T

Tn

T

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Example: A Periodic Function

Find the Fourier series of the periodic function f(x), where one

period of f(x) is defined as: f(x) = x, -p < x < p

2T

T=2

0 2

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Example: Its Fourier Approximation

-4

-3

-2

-1

0

1

2

3

4

-3 -2 -1 0 1 2 3

x

2*sin(x)

-4

-3

-2

-1

0

1

2

3

4

-3 -2 -1 0 1 2 3

x

2*(sin(x)-sin(2*x)/2)

-4

-3

-2

-1

0

1

2

3

4

-3 -2 -1 0 1 2 3

x

2*(sin(x)-(sin(2*x)/2)+(sin(3*x)/3))

-4

-3

-2

-1

0

1

2

3

4

-3 -2 -1 0 1 2 3

x

2*(sin(x)-(sin(2*x)/2)+(sin(3*x)/3)-(sin(4*x)/4))

1 harmonic 2 harmonics

3 harmonics 4 harmonics

Domain: [-, ]

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Example: Frequency Spectrum22

nn ba :Magnitude

Harmonics

For First 10 harmonics

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6 7 8 9 10 11 12

Each harmonic corresponds to a frequency that is multiple of the fundamentalfrequency

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Digital Modulation - Continues

Line Coding

Base-band signals are represented as line codes

UnipolarNRZ

BipolarRZ

ManchesterNRZ

Tb

Tb

Tb

V

0

V

-V

V

-V

1 0 1 0 1 0 1

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PSD of various line codes

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Summary of line coding schemes

Plus HDB3 and B8ZS

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Baseband binary data transmission system.

ISI arises when the channel is dispersive

Frequency limited -> time unlimited -> ISI

Time limited -> bandwidth unlimited -> bandpass channel ->time unlimited -> ISI

p(t)

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ISI Example

5T

0 t

Sequence of three pulses (1, 0, 1)

sent at a rate 1/T

sequence sent 1 0 1

sequence received 1 1(!) 1

Signal received

Threshold

4T3T2TT0-T-2T-3T

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ISI

Nyquist three criteria

Pulse amplitudes can be detected

correctly despite pulse spreading or

overlapping, if there is no ISI at the

decision-making instants

1: At sampling points, no ISI

2: At threshold, no ISI

3: Areas within symbol period iszero, then no ISI

At least 14 points in the finals

4 point for questions

10 point like the homework

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1st Nyquist Criterion: Time domain

p(t): impulse response of a transmission system (infinite length)

Equally spaced zeros,

interval Tfn

2

1

Tf

N

2

1

02t0t

t

0

1p(t)

-1

shaping function

no ISI !

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1st Nyquist Criterion: Time domain

Suppose 1/T is the sample rate

The necessary and sufficient condition for p(t) to satisfy

0,0

0,1

n

nnTp

Is that its Fourier transformP(f) satisfy

TTmfPm

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1st Nyquist Criterion: Frequency domain

2a N

f f 4 Nff

0

(limited bandwidth)

TTmfPm

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Proof

T

T

T

Tm

m

T

T

m

Tm

Tm

dffnTjfB

dffnTjTmfP

dffnTjTmfP

dffnTjfPnTp

21

21

21

21

21

21

212

212

2exp

2exp

2exp

2exp

dfftjfPtp 2exp

dffnTjfPnTp 2expAt t=T

m

TmfPfB

Fourier Transform

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Proof

n

n nfTjbfB 2exp

T

Tn nfTjfBTb

21

212exp

m

TmfPfB

nTTpbn

000

nnTbn

TfB TTmfPm

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Sample rate vs. bandwidth

W is the bandwidth of P(f)

When 1/T > 2W, no function to satisfy Nyquist condition.

P(f)

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,,0

,sinc

sin

otherwise

WfTfP

T

t

t

Tttp

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When 1/T < 2W, numbers of choices to satisfy Nyquist

condition

A typical one is the raised cosine function

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Cosine rolloff/Raised cosine filter

Slightly notation different from the book. But it is the same

20 )2(1

)cos()sin()(

Tt

Tt

Tt

Tt

rc tp

r

r

r: rolloff factor 10 r

)1()1(2

1

2

1 rfrTT

Trf

21)1(

)1(21 rfT

)2(0 fjPrc

1

0

ifrr

))1(cos(122

1 T

f

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Raised cosine shaping

W

W

P()r=0

r = 0.25

r = 0.50

r = 0.75

r = 1.00

W

0

t0

p(t)

W

2w

Tradeoff: higher r, higher bandwidth, but smoother in time.

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Modulated time domain

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Cosine rolloff filter: Bandwidth efficiency

Vestigial spectrum

data rate 1/ 2 bit/s

bandwidth (1 ) / 2 1 Hzrc

T

r T r

Hz

bit/s2

)1(

2

Hz

bit/s1

r

2nd Nyquist (r=1) r=0

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2ndNyquist Criterion

Values at the pulse edge are distortion-less

p(t) =0.5, when t= -T/2 or T/2; p(t)=0, when t=(2k-1)T/2, k0,1

-1/T f 1/T

0])/()1(Im[)(

)2/cos(])/()1(Re[)(

n

n

I

n

n

r

TnfPfP

fTTTnfPfP

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Example

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3rdNyquist Criterion

Within each symbol period, the integration of signal (area) is

proportional to the integration of the transmit signal (area)

Tw

Tw

wT

wt

wP

,0

,)2/sin(

2/)(

)(

T

T

jwtdwe

wT

wttp

/

/)2/sin(

)2/(

2

1)(

0,0

0,1)(

212

212 n

ndttpA

T

T

n

n

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Cosine rolloff filter: Eye pattern

nd Nyquist

1st Nyquist

2nd Nyquist:

1st Nyquist:

2nd Nyquist:

1st Nyquist: 2nd Nyquist:

1st Nyquist:

2nd Nyquist:

1st Nyquist:

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Gaussian Pulse Shaping Filter

Tradeoff between bandwidth and ISI

Example 6.8

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Gaussian minimum shift keying

GMSK is similar to MSK except it incorporates a premodulation Gaussian

LPF

Achieves smooth phase transitions between signal states which can

significantly reduce bandwidth requirements

There are no well-defined phase transitions to detect for bit synchronization

at the receiving end.

With smoother phase transitions, there is an increased chance in intersymbol

interference which increases the complexity of the receiver.

Used extensively in 2nd generation digital cellular and cordless telephone

apps. such as GSM

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spread-spectrum transmission

Three advantages over fixed spectrum

Spread-spectrum signals are highly resistant to noise andinterference. The process of re-collecting a spread signal spreadsout noise and interference, causing them to recede into thebackground.

Spread-spectrum signals are difficult to intercept. A Frequency-

Hop spread-spectrum signal sounds like a momentary noise burstor simply an increase in the background noise for shortFrequency-Hop codes on any narrowband receiver except aFrequency-Hop spread-spectrum receiver using the exact samechannel sequence as was used by the transmitter.

Spread-spectrum transmissions can share a frequency band withmany types of conventional transmissions with minimalinterference. The spread-spectrum signals add minimal noise tothe narrow-frequency communications, and vice versa. As aresult, bandwidth can be utilized more efficiently.

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PN Sequence Generator

Pseudorandom sequence

Randomness and noise properties Walsh, M-sequence, Gold, Kasami, Z4

Provide signal privacy

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Direct Sequence (DS)-CDMA

It phase-modulates a sine wave pseudo-randomly with a

continuous string of pseudo-noise code symbols called "chips",each of which has a much shorter duration than an informationbit. That is, each information bit is modulated by a sequence ofmuch faster chips. Therefore, the chip rate is much higher than

the information signal bit rate.

It uses a signal structure in which the sequence of chipsproduced by the transmitter is known a priori by the receiver.The receiver can then use the same PN sequence to counteract

the effect of the PN sequence on the received signal in order to

reconstruct the information signal.

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Direct Sequence Spread Spectrum

Unique code to

differentiate all users Sequence used for

spreading have low

cross-correlations

Allow many users to

occupy all thefrequency/bandwidth

allocations at thatsame time

Processing gain is the

system capacity How many users

the system can

support

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Spreading & Despreading

Spreading

Source signal is multiplied by a PN signal: 6.134, 6.135

Processing Gain:

Despreading

Spread signal is multiplied by the spreading code

Polar {1} signal representation

DataRate

ChipRate

T

TG

c

sp

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Direct Sequence Spreading

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Spreading & Despreading

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Frequency Hopping Spread Spectrum

Frequency-hopping

spread spectrum(FHSS) is a spread-spectrum method oftransmitting radiosignals by rapidly

switching a carrieramong manyfrequency channels,using apseudorandom

sequence known toboth transmitter andreceiver.

Military, bluetooth

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http://en.wikipedia.org/wiki/Spread-spectrumhttp://en.wikipedia.org/wiki/Spread-spectrumhttp://en.wikipedia.org/wiki/Carrier_wavehttp://en.wikipedia.org/wiki/Channel_%28communications%29http://en.wikipedia.org/wiki/Pseudorandomhttp://en.wikipedia.org/wiki/Transmitterhttp://en.wikipedia.org/wiki/Receiver_%28radio%29http://en.wikipedia.org/wiki/Receiver_%28radio%29http://en.wikipedia.org/wiki/Transmitterhttp://en.wikipedia.org/wiki/Pseudorandomhttp://en.wikipedia.org/wiki/Channel_%28communications%29http://en.wikipedia.org/wiki/Carrier_wavehttp://en.wikipedia.org/wiki/Spread-spectrumhttp://en.wikipedia.org/wiki/Spread-spectrumhttp://en.wikipedia.org/wiki/Spread-spectrum


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Baseline: Stationary Channel

BPSK modulation

y: the received signal

x: the transmitted signal with

amplitude a

w: white noise with power N0

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Baseline: Stationary Channel

BPSK modulation

Error probability decays exponentially with signal-noise-ratio(SNR).

y: the received signal

x: the transmitted signal with

amplitude a

w: white noise with power N0

0

22/

0

,2

1

Q(x),

)2()2/

(

2

N

a

SNRduewhere

eSNRQN

aQP

x

u

SNRe

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Flat Fading Channel

BPSK:

Conditional on h,

Averaged over h, which follows chi-square distribution

at high SNR.

Assume h is Gaussian random:

6.154, 6.155

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Irreducible Bit Error Rate due to multipath

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Simulation of Fading and Multipath

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Irreducible BER due to fading

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Irreducible BER due to fading

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BER due to fading & multipath

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Q estions?


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Questions?

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Biography


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BiographyDr. Ahmed Bassyouni has received the B.Sc. degree in Electrical Engineering, M.Sc. and

Ph.D. degrees in Adaptive controls from the Faculty of Engineering, Alexandria

University in 1982, 1987, and 1992. Post Doctoral Research on RF Transceivers with the

Caltech Institute, and Boise State University in 1995, and 2001.Dr. Bassyouni is an expert of RF and Microwave applications oriented to air defense

systems technology, and a designer of complex sensors such as phased array surveillance

radars, fire control radars, MTI, SAR, Airborne, Mortars Fires tracking radars, and

integrated UAV battle field systems. Currently he is a consultant with the Sensors and

Systems in New York. He was a Research Professor with the Department of Electrical

Engineering, Boise State University, Idaho (1997-2002); Visitor Professor with the

Department of Electrical Engineering at the University of Arkansas in (1993-1995);

Teaching Assistant (1987-1992); Instructor of Engineering Science with the Air Defense

College in Alexandria, Egypt (1981-1987).Dr. Bassyouni is a senior member, technical committee member, and co-chair conferences

participant, of the IEEE, International Society for Optical Engineering (SPIE), (American

Institute of Aeronautics and Astronautics) AIAA, (Society of Manufacturing Engineers)

SME, and (Robotic Institute) RI, the New York Academy of Sciences, and the American

Society of Engineering Education (ASEE). Also, He is an active member with the

American Society of Quality (ASQ), the American Association for the Advancement of

Science (AAAS), and the International Association of Online Engineering (IAOE). He is a

member of Tau Beta Pi (The Engineering Honor Society), Sigma Xi (The Scientific

Research Society), Phi Beta Delta (The International Honor Society). He is the author and

coauthor of seven books, and over 120 papers, trades analysis documents andpresentations in the areas of adaptive controls, robotics, RF/microwave systems, and radar

systems. He leads the radars and missile control systems projects from the requirements to

the prototype field testing, participating with innovative ideas, and advanced design

techniques. He is a Certified Reliability Engineer (C.R.E.), and a Certified System

Engineer (CSE). Dr. Bassyouni provides consulting service to the USA companies, and

organizations of defense industry. He has been awarded over 25 research grants to pursue

his work in integrated sensors.

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Fundamentals of Digital Modulation_Ahmed