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6.4 Digital Modulation

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6.4 Digital Modulation• cost effective because of advances in digital technology (VHDL, DSP, FPGA…)

• digital vs analog- digital has better noise immunity- digital can be robust to channel impairments- digital data can be multiplexed- error control: detect & correct corrupt bits- able to encrypt digital data- flexible software modulation & demodulation- digital requires complex signal conditioning- digital is “all or nothing”

6.4.1 Factors in Digital Modulation6.4.2 Bandwidth & Power Spectral Density (PSD) of Signals

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6.4.1 Factors in Digital Modulation

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6.4.1 Factors in Digital Modulation

Significant Factors •efficiency: desire low BER at low SNR• channel: multipath & fading characteristics• minimize bandwidth required• cost and ease of implementation

Performance Measures for Modulation Schemes

(1) p = power efficiency

(2)B = bandwidth efficiency

modulating signal (message) represented as pulses • n bits represented by m finite states• n = log2m bits per state

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(1) Power Efficiency, p

Ability to preserve signal fidelity at low power • increasing signal power increases noise immunity• specifics depend on modulation technique• measures trade-off between fidelity & signal power

Eb / N0 = to achieve BER <

Eb = bit energyN0 = noise power spectral density

p often expressed as ratio of Eb to N0 at receiver input to achieve specified BER

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(2) Bandwidth Efficiency, B

Ability to accommodate data in limited bandwidth• increasing data rate requires increased bandwidth• direct relationship to system capacity• measured in terms of bit rate, Rb & RF bandwidth, B

B =Rb/B 6.36

Fundamental Upper Bound on achievable Bit Rate per given Bandwidth (aka Shannon Bound)

• C = maximum channel capacity (bps)

Bmax = C/B

Bmax = log2

NS1 6.37

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typically there is a tradeoff between B & p

e.g

addition of error control codes increases p and decreases B

- increases bandwidth for given data rate- reduces required received power for specified BER

use of M-ary keying increases B and decreases p

- decreases bandwidth for given data rate- requires increased receive power for specified BER

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Additional Factors in digital modulation

cost & complexity •cost vs performance improvement•complexity vs robustness

channel impairments (Rayleigh, Ricean fading)• multipath dispersion• interference from other users or random sources

detection sensitivity to timing jitter – time varying channel

typically system is simulated & all factors are analyzed prior to selection of methods and specification of parameters

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6.4.2: Bandwidth & Power Spectral Density (PSD) of Signals

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6.4.2: Bandwidth & Power Spectral Density (PSD) of Signals

assume w(t) is a random signal (measured in volts)

wT(t) =

elsewhereTtTtw

02/2/)(

6.39

let wT(t) be a truncated version of w(t)

Pw(f) =

TfWT

T

2|)(|lim 6.38

• WT(f) is Fourier transform of wT(t)

• bar denotes ensemble average of WT(f)2

PSD of w(t) is given by:

-T/2 T/2

wT(t)

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Fourier Transform of Real Even and Odd Signals

for real signal x(t):

X(f) = dtftjtx

2exp

= dtftjtxjdtftjtx

2sin2cos

real part imaginary part

since cos(x) is an even function and sin(x) is an odd function:• X(-f) = X*(f)• Re[X(-f)] = Re[X(f)]• Im[X(-f)] = - Im[X(f)]• |X(-f)| = |X(f)|• X(-f) = -X(-f)

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s(t) = bandpass (modulated) signal

g(t) = complex envelope of baseband signal (magnitude & phase)

s(t) = Re(g(t)exp(j2fc)) 6.40

PSD of s(t) is related to PSD of g(t) by

• Ps(f) = PSD of s(t)

• Pg(f) = PSD of g(t)

6.41 Ps(f) = ¼[Pg(f-fc) + Pg(-f-fc)]

Bandpass vs Baseband Signals

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i. absolute bandwidth = range of frequencies over which PSD 0

(sin f)2/f 2

• for symbols represented as baseband pulses PSD has form of

- extends over infinite range of frequencies- absolute bandwidth =

ii. null-to-null bandwidth = bandwidth of main spectral lobe• simpler measure of bandwidth

iii. ½ power bandwidth = frequency range between -3dB points

iv. FCC definition: leaves exactly 0.5% of signal above upper band and below lower band (99% of signal power)

v. outside a specified band, PSD < given level (e.g. 45dB, 60dB )

Definitions of Bandwidth

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6.5 Line Coding

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6.5 Line Coding

digital baseband signal often use line codes to provide spectral characteristics of pulse train

i. RZ = pulse returns to 0 with every bit period• wider spectrum – easier to synchronize

ii. NRZ = pulse stays at constant level during bit period • narrower spectrum – harder to synchronize

iii. Manchester: 0-crossing guaranteed for each bit

Classification of line codes: unipolar range = 0 .. V bipolar range = -V.. +V

• some line codes have dc components not used for circuits that block dc signal (e.g. PSTN)

• manchester has no dc component

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Tb

Rb 2Rb 3Rb f

0.5Tb

PSD

0.5Rb Rb 2Rb f

0.5Tb PSD

0.5Rb R 1.5Rb f

PSD

weight ½

1 0 1 0 …V

0

(i) Unipolar NRZ

t

Tb

(ii) Bipolar RZ

1 0 1 0 …V

-Vt

1 0 1 0 …V

-V Tb

(iii) Manchester NRZ

t

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6.6 Pulse Shaping Techniques

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6.6 Pulse Shaping Techniques

Intersymbol Interference (ISI)• rectangular pulses passing through a bandlimited channel experience delay spread from MPCs

• ISI results when pulses smear – overlap with adjacent pulses

• increasing channel bandwidth alleviates this, but often isn’t practical

e.g. mobile systems use techniques that reduce bandwidth andsuppress out-of-band radiation

- requires out-of-band radiation 40dB-80dB below desired passband

Pulse Shaping: reduce spectral width (and ISI) of modulated data signal

• pulse shaping done at baseband or IF• difficult to manipulate RF spectrum

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6.6.1 Nyquist Criteria for ISI Cancellation

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6.6.1 Nyquist Criteria for ISI Cancellation

Specifies system design criteria to nullify Effects of ISI

• at receiver’s ith sampling instant (recovering symbol i), the system response for any symbol j i is zero

• system response includes transmitter, receiver, channel

Ts = symbol period n is an integer K is non-zero constant

if n = 0 heff(nTs) = Kif n 0 heff(nTs) = 0

6.42

Nyquist derived transfer function, Heff(f), which satisfied 6.42

Let heff(t) = system’s impulse response (transfer function)Mathematical Statement of Nyquist Criteria

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Two key considerations for selecting Heff(f) that satisfies 6.42

(1) heff(t) should have fast decay & small magnitude near samples at n 0

(2) assume an ideal channel: hc(t) = (t),

• requires an equalizer for FSF channels• reduces problem to designing approximate shaping filters at both transmitter & receiver to produce desired Heff(f)

• pulse shape = p(t)• channel impulse response = hc(t)

• receiver impulse response = hr(t)

heff(t) = (t) p(t) hc(t) hr(t) 6.43

System transfer function satisfying Nyquist Criteria given as

( = convolution operation)

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-6T -4T -2T 0 2T 4T 6T

10.80.60.40.2

0-0.2

xNYQ(t)

s

sTt

Tt/

)/sin(

heff(nTs) = 0

heff(t) = 6.44s

s

TtTt

/)/sin(

Consider Sinc Function:

for n > 0 heff(nTs) = 0

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Fourier Transform of sinc function yields

Heff(f) = 6.45

ss ffrect

f1

• corresponds to “brick wall” filter with absolute bandwidth = fs/2

fs = symbol rate (Hz) = 1/Ts (symbol period)

• impulse response satisfies Nyquist condition for ISI cancellation with minimal bandwidth• to eliminate ISI model total system as filter with impulse response of 6.44 including effects from

- transmitter filtering - receiver filtering- channel filtering

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Heff(f)

sT21

sT21

Frequency Response of Sinc Function

f

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practical issues:

(1) non-causal: heff(t) exists for t < 0 difficult to approximate

(2) sin t/t pulse has waveform slope = 1/t at each zero-crossing

• waveform = 0 only at exact multiples of Ts

• errors in sampling time of zero crossings (timing jitter) cause sampling to occur when adjacent symbols to overlap

- results in significant ISI- slope of 1/t2 or 1/t3 desirable - minimize ISI due to timing jitter in adjacent samples

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Nyquist also showed that if 2 conditions hold then frequency domain convolution of filter & Z(f) satisfies zero ISI condition

(1) Assume function Z(f) exists such that

• Z(f) is an arbitrary even function: Z(f) = Z(-f)

• Z(f) has zero magnitude outside passband of rectangular filter

• f0= filter bandwidth

• Ts = symbol period

(2) Assume a filter exists with transfer function = rectangular filter such that

f0 sT2

1

ISI solution based on sinc function and an even function in f

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Nyquist Criteria for zero-ISI is expressed mathematically as

6.46Heff(f) = )(0

fZffrect

6.47heff(t) = )()/sin( tzt

Tt s

Impulse response expressed as

• for | f | f0 sT2

1 Z(f)=0

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Nyquist Response Filter can eliminate ISI

• entire communications link must have Nyquist Response

Transmitter must filter Baseband Signal to • constrain modulated BW to regulated values • minimize adjacent channel interference

Receiver must filter Incoming Signal to • remove strong interference• reject noise that is not the passband

therefore the following approach is taken

(i) split RC Filter between transmitter & receiver

(ii) assume channel response is flat (use adaptive equalizers, etc)

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Assume channel distortion can be 100% neutralized by equalizer

• equalizers transfer function = inverse of channel response

• overall transfer function, Heff(f) can be approximated as

Heff(f) = HTX(f)HRX(f)HTX(f) = transmit filters transfer function

HRX(f) = receive filters transfer function

Effective end-end Heff(f) achieved using

• provides matched filter response• minimizes bandwidth & ISI

HTX(f) = )( fHeff

HRX(f) = )( fHeff

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6.6.2 Raised Cosine (RC) Rolloff Filter

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6.6.2 Raised Cosine (RC) Rolloff Filter• popular pulse shaping filter with Nyquist Response (zero-ISI)• Transfer Function HRC(f) given as

Ts = 1/2Wmin (Wmin = minimum channel bandwidth)

= 1/Rb ( Rb= bit rate)

HRC(f) = 6.48

212

cos121 sTf

1sT

f2

1 || 0 for

ss Tf

T 21 ||

21

for

sTf

21 ||

for0

is the rolloff factor that determines transition bandwidth 0 1

• = 0 RC filter = rectangular filter with minimum bandwidth

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RRC filter operating bands and response = 1 = 0.5 = 0

HRC(f)

1.0

0.5

0

sT1

sT1

sT21

sT21

sT43

sT43

pass band range

cut off band range

transition band

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HNYQ(f) 1

sT21

sT21

sT21

sT21

sT21

sT21

Frequency Response for RRC filter for specific

scaling factor , 0 ≤ ≤ 1

= 0

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RC impulse response, hRC(t) obtained from IFT of HRC(f)

hRC(t) = 6.49

22/41/cos/sin

s

ss

TtTt

tTt

large hRC(t) decays faster at zero crossings

• for t >>Ts rolloff 1/t3

- less sensitive to timing jitter - rapid rolloff allows temporal truncation with little penalty

• temporal sidelobe levels decrease in adjacent symbol slots

• however occupied bandwidth increases

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impulse response

large : • increased bandwidth • faster decay less sensitive to timing jitter• smaller temporal lobes

magnitude transfer function

HRC(f) 1.0

0.5

0 -Rs -¾Rs -½Rs 0 ½Rs ¾Rs Rs f

= 1 = 0.5 = 0

RC filter impulse response and transfer function

1/Ts

-3Ts -2Ts -Ts 0 Ts 2Ts 3Ts

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Rs = feasible symbol rate through baseband RC filter

• Ts = 1/Rs is symbol period• B = absolute filter bandwidth

Rs = 12B

6.50

½ Rs ≤ B ≤ Rs for 0 ≤ ≤ 1

Baseband

For RF systems passband bandwidth is doubled

Rs = 1B

6.51

Rs ≤ B ≤ 2Rs for 0 ≤ ≤ 1

Passband

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Root Raised Cosine (RRC)• use identical filters at transmitter & receiver

• filter transfer function = )( fHRC

• matched filter realization provides optimum performance in flat fading channel

• filter can be implemented either at baseband – before modulation passband at transmitter output

• typically – pulse shaping filters implemented using DSP at baseband

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hRC(t) (6.49) is non-causal must be truncated

• for each symbol, pulse shaping filters typically implemented for 6Ts about t = 0 point

• to reduce impact of truncation – modulator stores several symbols at a time

• group of symbols clocked out simultaneously – using lookup table

-6Ts -5Ts-4Ts -3Ts -2Ts -Ts 0 Ts 2Ts 3Ts 4Ts 5Ts 6Ts 7Ts 8Ts

hRC(t)

6Ts 6Ts

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e.g. RC Filter with input binary pulses with = 0.5

• 3 bits (symbols) stored at a time 23 = 8 possible waveforms

• if 6Ts represents time span for each symbol time span of discrete waveform = 14Ts

• optimal bit decision points are zero-crossing points of other symbols- don’t always coincide with peak values of waveform

- optimal decision points are 4Ts, 5Ts, and 6Ts

• pulse is inherently time dispersive1 0 1

0 Ts 2Ts 3Ts 4Ts 5Ts 6Ts 7Ts 8Ts 9Ts 10Ts

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e.g. pulse shaping comparison between RC and RRC• binary sequence 01100 • for binary ‘1’ multiply p(t) by ‘+’ symbol• for binary ‘0’ multiply p(t) by ‘-’ symbol

RRC waveform occupies larger dynamic range than RC

-3 -2 -1 0 1 2 3 4 t/Tb

p(t)1.5

1.0

0.5

0

-0.5

-1.0

-1.5

RRC shaping pulse p(t), α = 1.0 RC shaping pulse p(t), α = 0.5

-1 1 1 -1 -1

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RC filter

• 1st zero crossing (null-to-null bandwidth) = 11

sT

2/Ts = 48.7 kHz

rectangular pulse, 1st zero crossing (null-to-null bandwidth) = sT

2e.g. assume Ts = 41.04us (Rs = 24.35k symbols/second)

Bnull = 32.9 kHz = 0.35

Bnull = 24.35 kHz = 0.0

Bnull = 48.7 kHz = 1.0

smaller smaller Bnull, • bigger side lobes• more sensitive to timing jitter

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6.6.1.1 Practical Issues 1. Finite Impulse Response (FIR) Filters 2. Amplifiers: power efficiency vs linearity 3. Symbol Timing Recovery

1. FIR Filters (aka digital non-recursive linear phase filters)• can approximate perfect RRC filter to any accuracy for small • as order of filter increases

- sharper transition bands occur- longer processing time more propagation delay

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2. Amplifiers

For RC filters exact pulse shape must be preserved by carrier for spectral efficiency

• linear amplifiers preserve shape but are power inefficient• non-linear amplifiers are power efficient

- difficult to preserve pulse shape- small distortions at baseband can lead to large changes in transmitted pulse- can result in significant adjacent channel interference

For mobile communications – power efficiency is crucial !RC filters must use linear amplifiers with real-time feedback to improve efficiency

*active research area

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3. Symbol Timing Recovery requires sampling of symbol• optimal zero-ISI sampling points = maximum eye opening

• 3 methods:

(1) send separate timing reference as a continuous tone at nTs

• Ts = symbol period (e.g. seconds per symbol)

(2) send burst clock between message transmissions

(3) send timing information encoded into data (frequently used)e.g. 0-crossings in baseband bipolar data

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4. Symbol Timing Circuits needed for timing recovery because

• large is bandwidth inefficient

• reducing results in sensitivity at zero-crossings however, required bandwidth decreases towards minimum

• noise in received signal causes imperfect zero-crossing

Practically, accept compromise between • quick symbol timing acquisition for rapid data decoding• long averaging time to minimize jitter

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• with strings of “000…” or “111…” - estimate correct sample times until next 0-crossing- data scrambling or bit stuffing to increase frequency of 0-crossings

RRC filtered data with = 1 timing of sample point is simplified

• 0-crossings of filtered waveform occur at Ts/2 before optimum zero ISI detection points

• start timer at 0-crossings sample data Ts/2 later

Ts

start timer zero crossingssampleTs/2

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(1) Averaging over many 0-crossings can be used to • achieve accurate symbol timing on receiver with < 1• eliminate 0-mean noise

Feedback Timing Control Circuit• timing circuit has local clock, f(t) running at close to incoming Rs

• monostable creates a pulse of duration Ts/2 at each 0-crossing

• monostable & local clock are multiplied (mixed)• mixer output is integrated & filtered to produce smoothed DC voltage

- magnitude represents difference between incoming Rs & local clock

• DC voltage is used to feed VCO to adjust local clock until it matches incoming Rs

represents a compromise between • long averaging time to eliminate jitter • quick acquisition of symbol timing for rapid data decoding

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b(t) c(t) e(t) f(t) a(t) 0-crossingdetector

mono-stable VCO

increasing e(t) increases frequency of f(t)decreasing e(t) decreases frequency of f(t)

a(t)

b(t)

f(t)

c(t)

e(t)

Ts/2

T’s

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(2) square received filtered data stream

• yields signal with strong discrete frequency component at symbol timing frequency 2f0 = 1/Ts

• extract signal with narrow band filter or PLL yields symbol clock

- if = 1 works well

- for small doesn’t yield discrete spectral line at 1/Ts

X2

t

t

t

1/Ts f

f0 = 1/2Ts

2f0 = 1/Ts

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6.6.3 Gaussian Pulse Shaping Filter

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6.6.3 Gaussian Pulse Shaping Filter

non-Nyquist technique - doesn’t satisfy ISI cancellation conditions

• reduces spectral occupancy temporal spreading results in increased ISI

• trade-off is desired RF bandwidth vs irreducible error from ISI of adjacent symbols

effective with modulating techniques that use non-linear amplifiers• filter’s preserve pulse properties• non-linear amplifiers are power efficient – distort pulse shape

e.g. MSK uses non-linear amplifiers

Simple, Power Efficient technique that reduces BW at the cost of BER

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Guassian filter has narrow absolute bandwidth • not as narrow as RC filter• sharp cut-off frequency & low overshoot• smooth transfer function & no zero crossings

good design choice when • cost & power efficiency are most important • BER from ISI is less critical issue

Nyquist (RRC) filters have• zero crossings at adjacent symbol peaks• truncated transfer function• assume flat channel response (e.g. equalizer)

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Impulse response of Guassian filter realizes a transfer function that depends heavily on B3dB

• B3dB = 3dB bandwidth of baseband Gaussian shaping filter

HG(f) = exp(-2 f 2) 6.52

Gaussian LPF transfer function given by

hG(t) =

2

2

2exp t

6.54

Impulse response is given by

6.53 = dBdB BB 33

5887.02

2ln ’s relation to B3dB

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Impulse Response of baseband Gaussian filter (time dispersion)

• plotted for different values of B3dBTs

= 0.5 = 0.75 = 1.0 = 2.0

B3dB

0.5 1.1780.75 0.7851.0 0.589 2.0 0.294

t

hG(t)

2sT

2sT

23 sT

23 sT

as increases • B3dB (spectral occupancy) decreases• time dispersion increases

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-4T -3T -2T -T 0 T 2T 3T 4T

hRC(t)

1/Ts

-4T -3T -2T -T 0 T 2T 3T 4T

hRC(t)

1/Ts

-4T -3T -2T -T 0 T 2T 3T 4T

hRC(t)

1/Ts

Baseband Gaussian Filter• impulse response plotted for different B3dBTs

Baseband RC Filter• impulse response plotted for 0 1

= 0 = 0.5 = 1

t

hG(t)

2sT

2sT

2

3 sT

23 sT t

hG(t)

2sT

2sT

2

3 sT

23 sT

2sT

2sT

2

3 sT

23 sT

= 0.5 = 0.75 = 1.0 = 2.0

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6.7 Geometric Representation of Modulation Signals

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6.7 Geometric Representation of Modulation Signals

• digital modulation involves specifying a symbol si(t) from a finite set of possible symbols (waveforms)• based on mapping of information bits to symbols

let M = total possible signal states & S = modulation signal set

S = {s1(t), s2(t),…,sM(t)}

binary modulation, M = 2 m = 1 S = {s1(t), s2(t)}

m-ary modulation: m = log2M bits/symbols

quadrature modulation. M = 4 m = 2,

S = {s1(t), s2(t), s3(t), s4(t)}

e.g. S = {a45°, a 135°, a 225°, a 315°}

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basics of geometric view: • select a finite set of physically realizable waveforms (symbols) in S • assume N orthonormal symbols form the basis of S each si(t) S can be expressed as linear combination (LC) of basis

Representing modulation signals on S – requires finding a basis of S

• basis = set of signals that can represent any point in S as a linear combination of its elements

Useful to view S as vector space• general concept applied to any type of modulation• provides useful insight into performance of modulation schemes

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Let {j(t)| j = 1,2,…,N} represent a basis of S such that

si(t)= )(1

tsN

jjij

(1) For any symbol, si(t) 6.56

j i dttt ji

0)()(

(2) Basis signals are orthogonal to each other in time

6.57

(3) Each basis signal is normalized to have unit energy

1)(2 dtti

E = 6.58

a Basis signals coordinate system for S

Gram-Schmidt process systematic way to obtain basis for S

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Components of Complex Envelope for si(t)

• x-axis = in-phase component (I)

• y-axis = quadrature component (Q)

• distance between symbols indicates receivers ability to differentiate

different symbols

•dimension of S, N = number of signals in basis for S

let S = {s1(t), s2(t),…,sM(t)} N M

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Density of constellation diagram (vs sparse)

• implies spacing of symbols is closer

• increasing M/N occupied bandwidth of modulated signal decreases

• increasing N occupied bandwidth of modulated signal increases

• BER is related to distance between closest points in constellation

A dense constellation diagram has large M/N

• is more BW efficient• has higher BER less energy efficient

e.g. for BPSK N=1, M = 2 M2 M1

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Upper Bound Estimate of Ps(|si) in AWGN channel using Union Bound:

Ps(|si) = average probability of error for ith modulation symbol, si

• si was transmitted and sj was decoded, where i j

N0 = noise spectral density for arbitrary constellation

dij = euclidian distance between points i & j

Ps (|si) =

ijj

ij

N

dQ

1 026.62

Union bound provides estimate of Ps(|si)

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duux

2

exp21 2

F(x) =

Q(x) = dxxx

)2/exp(21 2

6.63

p(x)

0 xmxx0

F(x0) Q(x0)

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

100

10-1

10-2

10-3

10-4

10-5

10-6

Q(x)

x

F(x) = 1 – Q(x)

Reminder: Q function and Normal Distributions

04/22/23 64

more signal states:

• dij decreases Q(x) increases

• Ps (|si) = ij Q(x) increases

d13

d12d14

3

4 2

Q

I

Ps (|s1) =

0

14

0

13

0

12

222 NdQ

NdQ

NdQ

e.g. QPSK

04/22/23 65

Assume • all M modulations are equally likely to be transmitted• selection of symbols are independent• P(si) = probability of symbol si being generated

estimate Ps() = average probability of error in modulation as:

Ps() = Ps (|si) P(si)

M

iis sP

M 1)|(1 Ps() =

6.64

Ps () =

ijj

ijM

i Nd

QM 1 01 21