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EE 580
Wire less Com m unic at ions Syst em s
Fal l 2004
Dig i t a l Modulat ion
Richard S. Wolff, Ph. D.
[email protected] 994 7172
509 Cobleigh Hall
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Modu la t ion and w i re less syst em s
• Analog:– used in early radio systems,
– first generation cellular• Digital:
– more efficient use of spectrum
– Better noise immunity (regeneration)
– inter-works better with other system elements and data applications,
– typical in second generation cellular
– Supports complex signal conditioning and processing techniques
• Source coding
• Encryption
• Equalization
– DSPs used to implement modulators and demodulators in software
– Software radio: alterations and improvements easy to implement!!
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Represent a t ion o f m odula t ing
s ignal
• Message is a time sequence of symbols• Each symbol can have m states
• Each symbol represents n bits of information
lbits/symbolog2 mn =
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Represent a t ion o f m odula t ing
s ignal : ex am ple
Number of
states, m
log2
m n bits/symbol
1 0 0
2 1 1
4 2 2
8 3 3
16 4 4
Transmit m symbols/sec
Results in n bits/sec
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Choos ing a m odula t ion t ec hn ique:
des i rab le feat ures
• Low bit error rate at low received signal tonoise ratio
• Performs well under multipath and fading
conditions
• Occupies minimum bandwidth
• Easy and inexpensive to implement
Numerous trade offs implied by these attributes!!
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K ey issues in selec t ing a
m odulat ion t ec hnique
• Power efficiency• Bandwidth efficiency
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Pow er e f f ic ienc y
• Power efficiency ηp
: ability of modulation technique to
preserve the fidelity of the digital message at low power
levels
• Increase in signal power can raise noise immunity (forthermal noise)
• Trade off between fidelity (low errors) and signal power
• Metric: signal energy per bit/noise power spectral density:Eb /N0 needed to achieve a a certain probability of error
(BER)
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BER versus EB/N0 for d i f feren t PSK
and FSK m odula t ion t ec hn iques
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Ex am ple : BER versus Eb/N0
Ef fec t s o f Rayleigh fad ing
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Bandw idt h ef f i c ienc y
• bandwidth efficiency : ability of modulation technique to
accommodate data within a limited bandwidth
• Increase in signal rate will decrease pulse width of digital
symbol, which increases the signal bandwidth (think of
Fourier transform: ∆T~1/ ∆f)
• Result is a trade off between symbol rate R and bandwidth
occupancy B
• Metric: ηB=R/B bps/Hz• System capacity of digital mobile system directly related to
bandwidth efficiency of the modulation technique
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Bandw idt h ef f i c ienc y : ex am ples
• Occupied bandwidth B ~ 1/R, R=symbol rate
• BPSK: 2 states (m=2), n=1bit/symbol,
η=R/B=1/1=1 bps/Hz
• DPSK: 4 states (m=4), n=2 bits/symbol η=R/B=2/1=2 bps/Hz• 8PSK: 8 states (m=8), n=3 bits/symbol
η=R/B=3/1=3 bps/Hz
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Channel c apac i t y : Shannon l im i t
• Upper bound on achievable bandwidth efficiency– Assumes non-fading channel
– Assumes additive white Gaussian noise (AWGN)
rationoisetosignal /
bandwidthRF
bpscapacity,channel
)1(log 2max
=
=
=
+==
N S
B
C
N
S
B
C
Bη
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Shannon l im i t , ex am ples
• AMPS channel, B = 30kHz. Assume SNR=20dB.
What is maximum channel capacity C?
kbpsC
N
S BC
7.199
)1001(log000,30)1(log 22
=
+=+=
Note that data rate actually used in an AMPS channel is well
below the Shannon limit!
Voice message signal = 64kbpsVoice band data signal = 48kbps
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Shannon l im i t , ex am ples
• Deep space link, B = 30kHz. Assume SNR=0.1dB.
What is maximum channel capacity C?
kbpsC N
S BC
ratio N S
ratio N
SdB
N
S dB N
S
49.30
)023.11(log000,30)1(log
023.11010
10,1.0
22
01.010 / 1.0
10 /
=
+=+=
===
==
Earth-space links are typically power limited: low SNR do to:1. Limited spacecraft transmit power
2. Large path loss
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Ot her fac t o rs af fec t ing c ho ic e of
m odulat ion t ec hnique
• In addition to power and bandwidthefficiency, must consider:
– Cost and complexity of subscriber equipment
– A modulation that is simple to detect– Performance under Ricean and Rayleigh fading
– Performance under multipath conditions
– Interference environment
– Effects of timing jitter
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Bandw idt h and pow er spec t ral
dens i ty
• Signal spread in frequency inversely proportional tosymbol rate R
• Narrow pulses (high symbol rate) lead to large spectralBW
• Define absolute bandwidth of signal as frequencyrange over which the signal has a non-zero powerspectral density PSD
rangefrequencyoverextends
/ )(sin~
shape,pulserrectangulawithsymbolsFor
22
∞
f f PSD
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PSD for rec t angula r pulses
Time
A m
p l i t u d e
Tp
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Def in i t ions o f signa l bandw idt h
• Null-to-null bandwidth: width of main spectral
lobe
• Half-power bandwidth: frequency where PSD has
dropped to half (3dB) of its peak value• Occupied bandwidth: defined by FCC as band
which leaves 0.5% of power below lower and
above upper band limit. 99% of power in occupied
bandwidth
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L ine c od ing
• Line coding is the scheme used to representsuccessive zeros and ones
• Return to zero (RZ): pulse returns to zero
within each symbol period
1 1 0 1
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Line c od ing I I• Non-return to zero (NRZ): pulse does NOT return
to zero within each symbol period
• Manchester coding: special case of NRZ
– “1” represented by a positive pulse followed bya negative pulse (within a symbol period)
– “0” represented by a negative pulse followed bya positive pulse (within a signal period
• Line codes can be– unipolar (0 or +V)
– Bipolar(-V or +V)
L i di l
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L ine c oding ex am ples
Unipolar NRZ
Bipolar NRZ
Unipolar RZ
Bipolar RZ.
Manchester code
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L ine c od ing t rade of fs
• RZ: return to zero every symbol adds transitions
– Leads to spectral broadening
– Improves timing synchronization
• NRZ codes
– More spectral efficiency– Poorer synchronization
• Manchester code
– No DC component– Simple synchronization
– More spectral spreading than NRZ
P t f U i l NRZ
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frequency is normalized with respect to the bit rate 1/ T b, and the
average power is normalized to unity
Pow er spec t ra o f Unipolar NRZ
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Pow er spec t ra o f b ipo lar NRZ
frequency is normalized with respect to the bit rate 1/ T b, and theaverage power is normalized to unity.
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frequency is normalized with respect to the bit rate 1/ T b, and the
average power is normalized to unity.
Pow er spec t ra o f unipo la r RZ
Pow er spec t ra of Bipo lar RZ
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The frequency is normalized with respect to the bit rate1/ T b, and the average power is normalized to unity.
Pow er spec t ra o f Bipo lar RZ
P f M h d
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frequency is normalized with respect to the bit rate 1/ T b, and the
average power is normalized to unity.
Pow er spec t ra o f Manc hes t e r c ode
C i f l i d i t
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Unipolar NRZ
Bipolar RZ
Manchester NRZ
Com par ison o f l ine c od ing spec t ra
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Ef fec t s of bandw idt h l im i t ing a
d ig i t a l s igna l
• Typically band pass limit the transmittedsignal
• Cutting off some of the energy in the
frequency domain leads to spreading in the
time domain
• Net effect is inter-symbol interference(ISI)!!
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Graphic a l “ proof ” o f ISI due t o
band l im i t ing
Transmit rectangular pulses in time domain
time
Fourier transform of a pulse
frequency
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Graphic a l “ proof ” o f ISI due t o
band l im i t ing I I
Pass transmitted pulse through a band pass filter
frequency
Transmitted pulse is truncated in the frequency
domain
frequency
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Graphic a l “ proof ” o f ISI due t o
band l im i t ing I I I
Fewer frequency components result in wider pulse intime domain
Intersymbol interference results where received
pulses overlap in time domain
time
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Pulse shaping
• Technique to limit the bandwidth of thereceived signal while at the same time
minimizing ISI
• Typical approaches: round off the “sharp
edges”
– Raised cosine roll-off filter– Gaussian filter
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Raised Cosine t ransfer func t ion
ss
s RC
T
f
T
T f COS f H
2
)1(
2
)1(
2)12(1
21)(
α α
α
α π
+≤≤
−
⎥⎥⎦⎤
⎢⎢⎣⎡ ⎥
⎦⎤⎢
⎣⎡ +−+=
α +==
1
1 B
T
Rs
s
RF bandwidth increases with α, but ISI goes down
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Spec t rum of Raised Cosine pu lse
Vi r t ue of pu lse shaping
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Vi r t ue o f pu lse shap ing
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Represent ing d ig i t a l s ignals in
vec t o r spac e
• Consider a set S of waveforms representing
discrete symbols
• For binary modulation, bit= symbol , S has two
symbols• For higher order modulation, S>2
symbolainedtransmittbecanthatbitsof numbern,log 2 == M n
Example: QPSK has 4 symbols (0, π /2,π,3π /2), log2(4)=2 bits/symbol
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BPSK c onst e l la t ion d iagram
•Two symbols (0, π)
•Distance between the symbols represents how different the signals are
•Large symbol separation makes it easier for receiver to detect symbol
correctly
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Noise e f fec t s on c ons t e l lat ion
As the noise increases, the symbol smears out, and can overlap
with another symbol, causing a symbol error