dept. of ee, ndhu 1 chapter three baseband demodulation/detection
TRANSCRIPT
1Dept. of EE, NDHU
Chapter Three
Baseband Demodulation/Detection
2Dept. of EE, NDHU
Error Probability Performance
• Error probability function
where is the time cross-correlation coefficient between two signals
• Antitpodal signal
– equals to -1, then
• Orthogonal signal
– equals to 0, then
))1(
()2
(00 N
EQ
N
EQP bd
B
)2
(0N
EQP b
B
)(0N
EQP b
B
3Dept. of EE, NDHU
Error Probability of Binary Signaling
• Unipolar signaling
• Detection of unipolar baseband signaling0binary for 0 0)(
1binary for 0 )(
2
1
TttS
TtAtS
TAaa
Ta
TAdttnAAEtsTzETaT
2210
2
2
011
)2/1(2
thresholdoptimal the
,0)(
}))(({)](|)([)(
)(0N
EQP b
B
4Dept. of EE, NDHU
Error Probability of Binary Signaling
• Bipolar signaling
• Detection of bipolar baseband signaling0binary for 0 )(
1binary for 0 )(
2
1
TtAtS
TtAtS
02
thresholdoptimal the 210
21
aa
aa
)2
(0N
EQP b
B
5Dept. of EE, NDHU
Bit Error Performance of Unipolar and Bipolar Signaling
6Dept. of EE, NDHU
Intersymbol Interference in the Detection Process
)()()()( fHfHfHfH rct
7Dept. of EE, NDHU
Nyquist Channels for Zero ISI
8Dept. of EE, NDHU
Pulse Shaping to Reduce ISI
• Goals and Trade-offs
– Compact signaling spectrum is to provide the higher allowable data rate
– Time pulse would become spread in time, which induces ISI
• The Raised-Cosine filter
where W is the absolute bandwidth and W0=1/2T represents the minimum Nyqu
ist bandwidth and the -6 dB bandwidth
Wffor
WfWWforWW
WWfWWffor
fH
0
2 )2
4(cos
2 1
)( 00
02
0
20
000
])(4[1
])(2cos[)2(sin2)(
tWW
tWWtWcWth
9Dept. of EE, NDHU
Raised-Cosine Filter Characteristics
10Dept. of EE, NDHU
Two Types of Error-Perfformance Degradation
11Dept. of EE, NDHU
Example 3.3 Bandwidth Requirements
(a) Find the minimum required bandwidth for the baseband transmission of
a four-level PAM pulse sequence having a data rate of R=2400 bits/s if t
he system transfer characteristic consists of a raised-cosine spectrum wit
h 100% excess bandwidth (r=1)
(b) The same 4-ary PAM sequence is modulated onto a carrier wave, so that
the baseband spectrum is shifted and centered at frequency f0. Find the
minimum required DSB bandwidth for transmitting the modulated PAM
sequence
12Dept. of EE, NDHU
Nyquist Pulse
13Dept. of EE, NDHU
Square-root Nyquist Pulse and Raised-cosine Pulse
14Dept. of EE, NDHU
Equalization
• Maximum-likelihood sequence estimation (MLSE)
– Make measurement of channel response and adjust the receiver to the transmission environment
– Enable the detector to make good estimates from the distorted pulse sequence (ex. Viterbi equaliza
tion)
• Equalization with filtering
– Use filter to compensate the distorted pulse
– Linear filter contains only feedforward elements (ex. transversal equalizers)
– Non-linear filter contains both feedforward and feedback elements (ex. decision feedback equalize
rs)
– Preset or adaptive filter design
– Filter’s resolution and update rate
15Dept. of EE, NDHU
Receiving / Equalizing Filter
• The overall transfer function
• System design goal
then Ht(f) and Hr(f) each have frequency transfer functions that are the square r
oot of the raised cosine.
• Equalizing filter sometimes not only compensates the channel effect but compen
sates the ISI brought by the transmitter and receiver (ex. Gaussian filter)
)()()()()( fHfHfHfHfH erctRC
)()()(
)(
1
)(
1)( )(
fHfHfH
efHfH
fH
rtRC
fj
cce
c
16Dept. of EE, NDHU
Eye Pattern
• Eye pattern is a filtering effect
17Dept. of EE, NDHU
Distorted Pulse Response
18Dept. of EE, NDHU
Transversal Equalizer
• A training sequence (like PN sequence) is needed to estimate the channel freque
ncy response
• A transversal filter is the most popular form of an easily adjustable equal
izing filter consisting of a delay line with T-second tapes
• The main contribution is from a central tap of a transversal filter
• In practice, a finite-length transversal filter is realized to approximate the ideal fi
lter (infinite-length transversal filter)
• Consider there are (2N+1) taps with weights c-N, c-N+1, …,cN, the equalizer output
samples {z(k)}
NNnNNkcnkxkzN
Nnn , 2,2 , )()(
19Dept. of EE, NDHU
Transversal Filter
20Dept. of EE, NDHU
Zero-Forcing Solution
• Relationship among {z(k)}, {x(k)}, and {cn} for the transversal filter
• Disposing the top N the bottom N rows of the matrix X into a square matrix with dimension
of 2N+1 and transform Z vector into a vector of 2N+1
• Rewrite the relationship
• Select the weights {cn} so that the equalizer output is
)(0000
)1()(000
)()1()2()1()(
0)()1(
0000)(
and
)2(
)0(
)2(
0
Nx
NxNx
NxNxNxNxNx
NxNx
Nx
X
c
c
c
c
Nz
z
Nz
z
N
N
zXccXz 1
Nkfor
kforkz
,,2,1 0
0 1)(
21Dept. of EE, NDHU
Example: A Zero-Forcing Equalizer
• Consider a three-taps transversal filter, the given received data {x(k)} are 0.0, 0.2,
0.9, -0.3,0.1. Using the zero-forcing solution to find the weights {c-1, c0, c1}
– For the relationship
1
0
1
1
0
1
9.03.01.0
2.09.03.0
02.09.0
)0()1()2(
)1()0()1(
)2()1()0(
0
1
0
c
c
c
c
c
c
xxx
xxx
xxx
Xcz
3448.0
9631.0
2140.0
1
0
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c
c
22Dept. of EE, NDHU
Minimum MSE Solution
• Minimize the mean-square error (MSE) of all the ISI terms plus the
noise power at the output of the equalizer
• MSE is defined as the expected value of the squared difference between
the desired data symbol and the estimated data symbol
• MSE solution
• Minimum MSE solution is superior to zero-forcing solution
• Minimum MSE is more robust in the presence of noise and large ISI
xzxxxxxz
TT
RRccRR
XcXzX
1
23Dept. of EE, NDHU
Decision Feedback Equalizer
• Limitation of a linear equalizer is that it performs poor on channel
having spectral nulls
• Decision feedback equalizer (DFE) is a non-linear equalizer and uses
previous detector decisions to eliminate the ISI on pulse
• Basic idea is that if the values of the symbols previously detected are
known, then the ISI contributed by these symbols can be cancelled out
• Forward filter and feedback filter are used in the DFE
• The advantage of DFE is that the feedback filter operates on noiseless
quantized levels, and thus its output is free of channel noise
24Dept. of EE, NDHU
Decision Feedback Equalizer
25Dept. of EE, NDHU
Preset and Adaptive Equalization
• The equalizer weights remain fixed during transmission of data, then the
equalization is called preset equalization
• Preset equalization sets the tap weights according to some average
knowledge of the channel (Ex. Voice-grade telephone)
• Adaptive equalization can be implemented to perform tap-weight
adjustments periodically or continually
• Periodic adjustments are accomplished by periodically transmitting a
preamble sequence
• Continually adjustment are performed by the decision directed procedure
26Dept. of EE, NDHU
Preset and Adaptive Equalization
• Disadvantages of preset equalization
– Require an initial training period
– A time-varying channel can degrade system performance
• If the probability of error exceeds one percent (rule of thumb), decision-directed adaptive
equalizer might not converge
• Common solution to the adaptive equalization
– Initialize the equalizer with a preamble to provide good channel-error performance
– Then switch to the decision-directed mode
– Blind equalization algorithm can be used to form initial channel estimates without a
preamble