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6.4 Digital Modulation. 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 - PowerPoint PPT Presentation
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04/22/23 1
6.4 Digital Modulation
04/22/23 2
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
04/22/23 7
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)
04/22/23 12
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
04/22/23 23
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
04/22/23 24
Heff(f)
sT21
sT21
Frequency Response of Sinc Function
f
04/22/23 25
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
04/22/23 28
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
04/22/23 32
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
04/22/23 33
HNYQ(f) 1
sT21
sT21
sT21
sT21
sT21
sT21
Frequency Response for RRC filter for specific
scaling factor , 0 ≤ ≤ 1
= 0
04/22/23 34
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
04/22/23 35
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
04/22/23 36
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
04/22/23 37
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
04/22/23 38
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
04/22/23 39
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
04/22/23 40
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
04/22/23 41
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
04/22/23 42
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
04/22/23 43
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
04/22/23 44
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
04/22/23 45
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
04/22/23 46
• 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
04/22/23 47
(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
04/22/23 48
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
04/22/23 49
(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
04/22/23 50
6.6.3 Gaussian Pulse Shaping Filter
04/22/23 51
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
04/22/23 52
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)
04/22/23 53
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
04/22/23 54
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
04/22/23 55
-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
04/22/23 56
6.7 Geometric Representation of Modulation Signals
04/22/23 57
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°}
04/22/23 58
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
04/22/23 59
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
04/22/23 60
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
04/22/23 61
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
04/22/23 62
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)
04/22/23 63
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