unit iv signal processing in wireless communications ref.: 1.“wireless communications”, molisch...
TRANSCRIPT
UNIT IV
SIGNAL PROCESSING IN WIRELESS COMMUNICATIONSRef.: 1.“Wireless Communications”, Molisch 2. "Wireless Communications", Rappaport 3. "Wireless Communications", Andrea Goldsmith
Diversity
- Principle of Diversity
- Macrodiversity- Microdiversity- Signal Combining Techniques- Transmit Diversity
Channel Impairments
1) ACI/CCI → system generated interference [ACI - Adjacent Channel Inteference, CCI - Co-channel Interference]
2) Shadowing → large-scale path loss from LOS obstructions
3) Multipath Fading → rapid small-scale signal variations
4) Doppler Spread → due to motion of mobile unit
Note:All can lead to significant distortion or attenuation of Rx signalDegrade Bit Error Rate (BER) of digitally modulated signal
Techniques used to improve Rx signal quality
• Three techniques are used to improve Rx signal quality and lower BER:
1) Equalization2) Diversity3) Channel Coding
- They can be used independently or combined
Diversity Techniques
Principle of Diversity- Primary goal is to reduce depth & duration of small-scale fades- To ensure that the same information reaches the receiver on statistically indeendent channels.
Types of DiversitySpatial or antenna diversity → most common•Use multiple Rx antennas in mobile or base station•Even small antenna separation ( λ ) changes phase of signal ∝→ constructive /destructive nature is changedOther diversity types → polarization, frequency, & time diversity
Diversity arrangementsLet’s have a look at fading again
Illustration of interference pattern from aboveReceived power [log scale]
Movement
Position
A B
A B
Transmitter
Reflector
Diversity arrangementsThe diversity principle
The principle of diversity is to transmit the same information onM statistically independent channels.
By doing this, we increase the chance that the information willbe received properly.
The example given on the previous slide is one such arrangement:antenna diversity.
Diversity arrangementsGeneral improvement trend
Bit error rate (4PSK)100
10-1
10-2
10-3
10-4
10-5
10-6
0 2 4 6
10 dB
10 dB
8 10 12 14
Eb/N0 [dB]
10 x
10M x
16 18 20
Rayleigh fadingNo diversity
Rayleigh fadingMth order diversity
No fading
Microscopic diversity
- Most widely used
- Combat small-scale fading (fading created by interference effects)
- Use multiple antennas separated in space
- At a mobile, signals are independent if separation > λ / 2
- But it is not practical to have a mobile with multiple antennas separated by λ / 2 (7.5cm apart at 2 GHz)
- Can have multiple receiving antennas at base stations, but must be separated on the order of ten wavelengths (1 to 5 meters).
Microscopic diversity
- Since reflections occur near receiver, independent signals spread out a lot before they reach the base station.
- a typical antenna configuration for 120 degree sectoring.
- For each sector, a transmit antenna is in the center, with two diversity receiving antennas on each side.
- If one radio path undergoes a deep fade, another independent path may have a strong signal.
- By having more than one path selection can be made, instantaneous and average SNRs at the receiver may be improved
Microscopic diversity Techniques
- Spatial Diversity (several antenna elemenst separated in space)
- Temporal Diversity (repetition of the transmit signal at different times)
- Frequency Diversity (transmission of the signal on different frequencies)
- Angular Diversity (multiple antennas with different antenna patterns)
- Polarization Diversity (multiple antennas receiving different polarizations)
Diversity arrangementsSome techniques
Spatial (antenna) diversity
We will focus on thisone today!TX Signal combiner
Frequency diversity
TX
D D D
Signal combiner
Temporal diversity
CodingInter- De-inter-leaving leaving De-coding
(We also have angular and polarization diversity)
Spatial (antenna) diversityFading correlation on antennas
Isotropicuncorrelatedscattering.
Macroscopic diversity
- Combat large-scale fading (fading created by shadowing effects)
- Frequency diversity/Polarization Diversity/Spatial Diversity/ Temporal Diversity are not suitable here.
- If there is a hill in between Tx and Rx antennas on either the BS or MS does not help.
Macroscopic diversity (contd.)
- To solve the problem use a separate BS
- Large distance between BS1 and BS2 gives rise to macrodiversity.
- Use on-frequency repeaters (receive the signal and retransmit the amplified version). It is simpler as synchronization is not necessary but delay and dispersion are larger.
- Simulcast (same signal transmitted simultaneously from different BSs.)
- Simulcast widely used for broadcast applications like digital TV.
- Disadvantage of simulcast is the large amount of signaling information that has to be carried on landlines, synchronization information and transmit data have to be transported on landlines to BSs.
- Select path with best SNR or combine multiple paths
→ improve overall SNR performance
Selection diversity - 'Best' signal copy is selected and processed (demodulated and decoded) and all other copies are discarded
Combining Diversity - All signal copies are combined and combined signal decoded
Note: Combining diversity leads to better performance but Rx complexity higher than Selection Diversity.
Gain of Multiple Antennas - Diversity Gain and Beamforming Gain.
Signal Combining
Spatial (antenna) diversitySelection diversity
RSSI = receivedsignal strengthindicator
Spatial (antenna) diversitySelection diversity, cont.
- Selection criteria (Power/BER) of all diversity branches monitored to select the best.
- Alternately to reduce hardware cost and spectral inefficiency, switched diversity done.
Switched Diversity - Active branch monitored if signal strength falls below threshold Rx switches to a different antenna
Demerits - Works well if sufficient signal quality in one of the branches
- If all branches signal strength < threshold then repeated switching Free Parameters of Switched Diversity - switching threshold (neither too low nor too high), hysteresis time (not too long or short)
Disadvantages of Selection Diversity
- Selection Diversity wastes signal energy by discarding M-1 copies of Rxd signal
- Combining Diversity - All branches are considered
- Combining Diversity Types - Maximal Ratio Combining, Equal Gain Combining
Disadvantages of Selection Diversity
Spatial (antenna) diversityMaximum ratio combining
Spatial (antenna) diversity
Spatial (antenna) diversityPerformance comparison
Cumulative distribution of
SNR
MRC
Comparison ofSNR distribution
for different numberof antennas M andtwo different diversitytechniques.
RSSI selection
[Fig. 13.9]
Copyright: Prentice-Hall
-
Spatial (Antenna) Diversity
• Spatial or Antenna Diversity – M independent branches– Variable gain & phase at each branch → G θ∠– Each branch has same average SNR:
– Instantaneous
– the pdf of
0
bESNRN
iSNR i
0 0
1Pr ( ) 1
i
i i i ip d e d e
i
-
Spatial (Antenna) Diversity
– The probability that all M independent diversity branches Rx signal which are simultaneously less than some specific SNR threshold γ
– The pdf of :– Average SNR improvement offered by selection diversity
/1
/
Pr ,... (1 ) ( )
Pr 1 ( ) 1 (1 )
MM M
Mi M
e P
P e
1( ) ( ) 1
M
M M
d Mp P e e
d
1
0 0
1
( ) 1 ,
1
Mx xM
M
k
p d Mx e e dx x
k
Spatial (antenna) diversityPerformance comparison
Cumulative distribution of
SNR
MRC
Comparison ofSNR distribution for different numberof antennas M f
RSSI selection
[Fig. 13.9]
Copyright: Prentice-Hall
Space diversity types/methods:
1) Selection diversity
2) Feedback diversity
3) Maximal radio combining
4) Equal gain diversity
Selection diversity Technique
Selection Diversity → simple & cheap– Rx selects branch with highest instantaneous SNR
• new selection made at a time that is the reciprocal of the fading rate
• this will cause the system to stay with the current signal until it is likely the signal has faded
– SNR improvement :• is new avg. SNR
• Γ : avg. SNR in each branch
Selection Diversity Technique (Contd.):
Selection Diversity Technique:Ref: Rappaport (Wireless Communications)
Scanning/Feedback Diversity
Scanning/Feedback Diversity– scan each antenna until a signal is found that is above
predetermined threshold– if signal drops below threshold → rescan– only one Rx is required (since only receiving one signal
at a time), so less costly → still need multiple antennas
Maximal Ratio Combiner Diversity
– signal amplitudes are weighted according to each SNR– summed in-phase– most complex of all types– a complicated mechanism, but modern DSP makes this
more practical → especially in the base station Rx where battery power to perform computations is not an issue
Maximal Ratio Combiner Diversity
The resulting signal envelop applied to detector:
Total noise power:
SNR applied to detector:
1
M
M i ii
r G r
2
1
M
T ii
N N G
2
2M
MT
r
N
Maximal Ratio Combiner Diversity
The voltage signals from each of the M diversity branches are co-phased to provide coherent voltage addition and are individually weighted to provide optimal SNR
( is maximized when )Mr NrG ii /
The SNR out of the diversity combiner is the sum of the SNRs in each branch.
Maximal Ratio Combiner Diversity
The probability that less than some specific SNR threshold γ
gives optimal SNR improvement :Γi: avg. SNR of each individual branchΓi = Γ if the avg. SNR is the same for each branch
1 1
M M
M i ii i
M
Maximal Ratio Combiner Diversity
Equal Gain Combining Diversity
• Combine multiple signals into one
• G = 1, but the phase is adjusted for each received signal.
• The signal from each branch are co-phased vectors add in-phase.
• Better performance than selection diversity
Transmit Diversity
• Multiple antennas installed at just one link (usually at BS)
•Uplink transmission from MS to BS - multiple antennas act as Rx diversity branches
•For downlink diversity branches originate at Txr.
- Transmit Diversity with channel-state information
- Transmit Diversity without channel-state information
Time Diversity
• Time Diversity → transmit repeatedly the information at different time spacings
• Time spacing > coherence time (coherence time is the time over which a fading signal can be considered to have similar characteristics)
• So signals can be considered independent
• Main disadvantage is that BW efficiency is significantly worsened – signal is transmitted more than once BW must ↑ to obtain the same Rd (data rate)Note: If data stream repeated twice then either BW doubles for the same Rd or Rd is reduced by ½ for the same BW
Time Diversity - RAKE Receiver
• Powerful form of time diversity available in spread spectrum (DS) systems → CDMA• Signal is transmitted only once• Propagation delays in the MRC provide multiple copies of Tx signals delayed in time• If time delay between multiple signals > chip period of spreading sequence (Tc) → multipath signals can be considered uncorrelated (independent)• In a basic system, these delayed signals only appear as noise, since they are delayed by more than a chip duration and ignored.• Multiplying by the chip code results in noise because of the time shift.• But this can be used to our advantage by shifting the chip sequence to receive that delayed signal separately from the other signals.
Time Diversity - RAKE Receiver
• attempts to collect the time-shifted versions of the original signal by providing a separate correlation receiver for each of the multipath signals.
• Each correlation receiver may be adjusted in time delay, so that a microprocessor controller can cause different correlation receivers to search in different time windows for significant multipath.
• The range of time delays that a particular correlator can search is called a search window.
Time Diversity - RAKE Receiver
The RAKE Rx is a time diversity Rx that collects time-shifted versions of the original Tx signal
Time Diversity - RAKE Receiver
The RAKE Rx is a time diversity Rx that collects time-shifted versions of the original Tx signal
M branches or “fingers” of correlation Rx’s
Separately detect the M strongest signals
Weighted sum computed from M branches
faded signal → low weightstrong signal → high weightovercomes fading of a signal in a single branch
SNR statistics for diversity receivers
Nr.
1
1
BER of diversity receivers
1
Computation via moment-generating function
Spatial (antenna) diversityPerformance comparison, cont.
MRC
Comparison of2ASK/2PSK BER
for different numberof antennas M andtwo different diversitytechniques.
RSSI selection
Copyright: Prentice-Hall
Spatial (antenna) diversityErrors due to signal distortion
Comparison of2ASK/2PSK BERfor different numberof antennas M andtwo different diversitytechniques.
Copyright: Prentice-Hall
Optimum combining in flat-fadingchannel
• Most systems interference limited
• OC reduces not only fading but also interference
• Each antenna can eliminate one interferer or give onediversity degree for fading reduction:
(“zero-forcing”).
• MMSE or decision-feedback gives even better results
• Computation of weights for combiningK
R 1h R 2I ErrTwopt d
k1
Performance of Optimum Combining
• Define channel matrix H:
Hkm is transfer function fork-th user to m-th diversityantenna 2 interferers, optimum combining
• Error of BPSK, QPSK for onechannel constellation bounded
asH −1
-15x10
-110
M=1-210BER ≤exp[−h R h]static d
• average behavior:
BER≤[1+SNR
ni d
10
10
−(M−K)]
-3
M=5 M=3
-4
-10 -5 0 5
Γ / dΒ
From Winters 1984,
M=2
10 15 20
Review of Channel coding & Speech Coding Techniques
Contents
• Overview
• Block codes
• Convolution codes
• Trellis-coded modulation
• Turbo codes and LDPC codes
• Fading channel and interleaving
OVERVIEW
Basic types of codes
Channel codes are used to add protection against errors in the channel.
It can be seen as a way of increasing the distance between transmittedalternatives, so that a receiver has a better chance of detecting thecorrect one in a noisy channel.
We can classify channel codes in two principal groups:
BLOCK CODES
Encodes data inblocks of k, usingcode words of
length n.
CONVOLUTION CODES
Encodes data in a stream,without breaking it into
blocks, creating codesequences.
Information and redundancy (1)
EXAMPLE
Is the English language protected by a code, allowing us to correcttransmission errors?
When receiving the following sentence with errors marked by ´-´:
“D- n-t w-rr- -b--t ---r d-ff-cult--s -n M-th-m-t-cs.- c-n -ss-r- --- m-n- -r- st-ll gr--t-r.”
it can still be “decoded” properly.
What does it say, and who is quoted?
There is something more than information in the original sentencethat allows us to decode it properly, redundancy.
Redundancy is available in almost all “natural” data, such as text, music,images, etc.
Information and redundancy (2)
Electronic circuits do not have the power of the human brain andneeds more structured redundancy to be able to decode “noisy”messages.
”Pure information”without
redundancy
Original source data Sourcewith redundancy coding
E.g. a speechcoder
Channel ”Pure information” withcoding structured redundancy.
The structured redundancy addedin the channel coding is often calledparity or check sum.
Illustration of code words
Assume that we have a block code, which consists of k informationbits per n bit code word (n > k).
Since there are only 2k different information sequences, there can beonly 2k different code words.
2n differentbinary sequencesof length n.
Only 2k are validcode words inour code.
Illustration of decoding
Received word
Distances
Two common ones:
Hamming distance Measures the number of bitsbeing different between two
binary words.
Euclidean distance Same measure we have usedfor signal constellations.
Used for binarychannels with
random bit errors.
Used for AWGNchannels.
Coding gain
When applying channel codes we decrease the Eb/N0 required toobtain some specified performance (BER).
BER
BERspecGcode
Eb/N0 [dB]
BLOCK CODES
Channel codingLinear block codes
Channel codingSome definitions
min
0 0 0G G
i≠ ji +x j )
x+x = 1ij + 0 = 1 1 1 0
G Gw(x)
d( i j i j
Channel codingEncoding example
For a specific (n,k) = (7,4) code we encode the informationsequence 1 0 1 1 as
1 0 0 0 1 Systematic bits
0 1 0 0
10
0 0 1 0
0
=
1
0 0 0 1 11
1 1 0 1 0 1
parity bits.
1 0 1 1
1
0 1 1 1 0
Generator matrix
Channel codingEncoding example, cont.
Encoding all possible 4 bit information sequences gives:
Information Code word Hammingweight
0 0 0 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 1 1 1 1 40 0 1 0 0 0 1 0 0 1 1 30 0 1 1 0 0 1 1 1 0 0 30 1 0 0 0 1 0 0 1 0 1 30 1 0 1 0 1 0 1 0 1 0 30 1 1 0 0 1 1 0 1 1 0 40 1 1 1 0 1 1 1 0 0 1 41 0 0 0 1 0 0 0 1 1 0 31 0 0 1 1 0 0 1 0 0 1 31 0 1 0 1 0 1 0 1 0 1 41 0 1 1 1 0 1 1 0 1 0 41 1 0 0 1 1 0 0 0 1 1 41 1 0 1 1 1 0 1 1 0 0 41 1 1 0 1 1 1 0 0 0 0 31 1 1 1 1 1 1 1 1 1 1 7
This is a (7,4) Hamming code, capable of correcting one bit error.
Channel codingError correction capability
td
=min −1
2
t t
dmin
From Ericsson radio school
Channel codingPerformance and code length
Eb/N0
CONVOLUTION CODES
Channel codingEncoder structure
L = 3
Copyright: Ericsson
Channel codingEncoding example
Input State Output Next state
0 00 000 001 00 111 100 01 001 001 01 110 100 10 011 011 10 100 110 11 010 011 11 101 11
We usually start the encoder in the all-zero state!Copyright: Ericsson
Channel codingEncoding example, cont.
We can view the encoding process in a trellis created from the table onthe previous slide.
Copyright: Ericsson
Channel codingTermination
Copyright: Ericsson
Channel codingA Viterbi decoding example
Received sequence:
010
0001
000
0001
100
0002
001
0003
011
0005
110
0004
001
0005
2 4
4
3101
Decoded data:
63
5
75
44 101
54
8
46
55 101
24
6
42
77
5 6
68
Tail bits
0 0 1 0 0 0 0
Channel codingSurviving paths
Copyright: B. Mayr
TRELLIS-CODED MODULATION
Principle of TCM
• Goal: improve BER performance while leaving thebandwidth requirement unchanged
• “Conventional” coding introduces redundancy, andtherefore increases the requirement for bandwidth
• Therefore, TCM increases the constellation size of themodulation, while at the same time using a convolutionalcode
Trellis-coded modulation (1)
• Simple example: TCM with 8-PSK and rate 2/3 coding
Copyright: B. Mayr
Trellis-coded modulation (2)
Signal-space diagram Admissible transitions
Copyright: B. Mayr
TCM: BER computation (1)
d2 8EB
Copyright: B. Mayr
TCM: BER computation (2)
• Asymptotic coding gain of 3 dB- Euclidean distance is 8E, compared to 4E for QPSK
Copyright: B. Mayr
Set partitioning
Copyright: B. Mayr
TURBO CODES AND LDPC CODES
Turbocoders
• Generates long codewords by- encoding data with two different convolutional encoders
- for each of the encoders, data are interleaved withdifferent interleavers
Copyright: M. Valenti
Decoding of turbocodes
• Iterative decoding
• Two separate decoders (corresponding to the twoconvolutional encoders) that exchange information
• Quantity of interest is the log-likelihood ratio
log Prbi1|x
Prbi1|x
Block diagram of turbo decoder
Copyright: IEEE
Performance of turbo codes
#2#18 #6 #2
Copyright: IEEE
Principle of LDPC codes
• LDPC: low density parity check codes
• Block codes with large block length
• Defined by the parity-check matrix H, not the generatormatrix
Construction of parity-check matrix
1. Divide matrix horizontally into p submatrices
2. Put a “1” into each column of the submatrix. Make sure that there areq “1”s per row1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1
3. Let other submatrices be column permutations of first submatrix1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1
1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0
0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0
H 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0
0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0
0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1
1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0
0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0
0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0
0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0
0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
Encoding of bits
• Generator matrix has to be computed
• First step:
H PT I• Second step: generator matrix is
G I P
Decoding: Tanner graph
• Method for iterative decoding
• Represent code in a Tanner graph (bipartite graph)
Check nodes
Variable nodes
1 0 1 1Tanner graph for parity check matrix H = 0 1 1 1
Decoding: step-by-step procedure
1. Variable nodes decide what they think they are, given externalevidence only
,0 0, for all i ,0 2/2rj , for all j
2. Constraint nodes compute what they think variable nodes have to bel
2tanh1 kAij
l1i,k
tanh2
Ai j is "all the members of ensemble Ai with the exception of j"
3. Update opinion of what variable nodes have to be
li,j 2/ 2
n rj kBji l
k,j
Bj i is "all variable nodes that connect to the j th constraint node, with the exception of i. "
4. compute the pseudoposterior probabilities that a bit is 1 or 0
Lj 2/2rj i l,
5. If codeword has syndrome 0, stop iteration; otherwise goto 2
FADING CHANNELSAND INTERLEAVING
Channel codingDistribution of low-quality bits
Without interleaving With interleaving
Eb/N0 Eb/N0
bit bitFading dip gives many With interleaving the fading diplow-quality bits in spreads more evenly acrossthe same code word code words
Code words Code words
Channel codingBlock interleaver
Channel codingInterleaving - BER example
BER of a R=1/3 repetition code over a Rayleigh-fading channel,with and without interleaving. Decoding strategy: majority selection.
10 dB Div. order 2
10 dB
100x
Div. order 1
10x
Summary
• Channel coding is used to improve error performance
• For a fixed requirement, we get a coding gain thattranslates to a lower received power requirement.
• The two main types of codes are block codes andconvolution codes
• Depending on the channel, we use different metrics tomeasure the distances
• Decoding of convolution codes is efficiently done with theViterbi algorithm
• In fading channels we need interleaving in order to breakup fading dips (but causes delay)
Equalization
30
Contents
• Inter-symbol interference
• Linear equalizers
• Decision-feedback equalizers
• Maximum-likelihood sequence estimation
INTER-SYMBOL INTERFERENCE
Inter-symbol interference - Background
Transmitted symbols Received symbols
Channel withdelay spread
Modeling of channel impulse response
What we have used so far (PAM and optimal receiver):
n( t) kTc δ(t −kT) ϕk
g (t) g (T−t)
PAM
Matched filter
Including a channel impulse response h(t):n (t)
cδ(t −kT)
k
ISI-free andwhite noisewith properpulses g(t)
kTϕ
k
g (t) h(t)
PAM
Can be seen as a “new”basis pulse
(g∗h)*(T−t)
Matched filter
k
This one is nolonger ISI-free andnoise is not white
Modeling of channel impulse response
We can create a discrete time equivalent of the “new” system:
cn
k
F(z)
k
F (ϕ
1z−) k
where we can say that F(z) represent the basis pulse and channel, whileF*(z-1) represent the matched filter. (This is an abuse of signal theory!)
We can now achieve white noise quite easily, if (the not unique) F(z) ischosen wisely (F*(z-1) has a stable inverse) :
nk
ck ϕ uF(z) F k* (z−1
) 1/ F *( kz−1)
NOTE:Noisewhitening
filter
F*(z -1)/F *(z -1)=1
The discrete-time channel model
With the application of a noise-whitening filter, we arrive at a discrete-timemodel
cn
k
F(z)
k
uk
This is themodel we are
where we have ISI and white additive noise, in the form
Lgoing to use
whenu = f c +nk
The coefficients f
∑ j=0 j k− j k designingequalizers.
j represent the causal impulse response of thediscrete-time equivalent of the channel F(z), with an ISI that extendsover L symbols.
Channel estimation
LINEAR EQUALIZER
Principle
The principle of a linear equalizer is very simple: Apply a filter E(z) at thereceiver, mitigating the effect of ISI:
nkc k uk
F(z)ck
E(z)
Linearequalizer
Now we have two different strategies:
1) Design E(z) so that the ISI is totally removed
2) Design E(z) so that we minimize the meansquared-error ofε=c−ck
Zero-forcing
MSEk k
Zero-forcing equalizer
nkck u ck
F(z)
Information Channel
f
k
Noise
f f
1/ F(z)
ZFequalizer
Equalizer
f
Informationand noise
fNoise enhancement!
MSE equalizer
The MSE equalizer is designed to minimize the error variance
nkck u 2 −1σ z ck
F(z) k
σs2
s F ( )F(z)2+N
MSE
0
equalizer
InformationInformation Channel
f
Noise Equalizer
f f f
and noise
fLess noise enhancement than Z-F!
DECISION-FEEDBACKEQUALIZER
DFE - Principle
Decisiondevice
ck
F(z)
nk
E(z)
Forwardfilter
This part shapesthe signal to workwell with the
decisionfeedback.
+
-
D(z)
Feedbackfilter
This part removes ISI on“future” symbols from thecurrently detected symbol.
ck
If we makea wrongdecisionhere, we
mayincrease theISI instead
of remove it.
Zero-forcing DFE
In the design of a ZF-DFE, we want to completely remove all ISI beforethe detection.
ISI-freenk
ck
F(z) E(z) +
-
D(z)
ck
This enforces a relation between the E(z) and D(z), which is (we assumethat we make correct decisions!)
F (z)E (z)−D(z)=1
MSE-DFE
minimal MSEnk
ck
F(z) E(z) +
-
D(z)
ck
MAXIMUM-LIKELIHOODSEQUENCE ESTIMATION
Principle
“noise free signal alternative”
L
umNF=∑
j=0
fcj m− j
The squared Euclidean distance (optimal for white Gaussian noise) tothe received sequence {um} is
d 2 {u u NF = u −u2NF
L
= u −
2
fc( m},{ m }) ∑m
m m ∑m
m ∑j=0
j m− j
The MLSE decision is then the sequence of symbols {cm} minimizing this
distancec m =arg min
2L
u − fc{cm}
∑m
m ∑j=0
j m− j
The Viterbi-equalizer
Let’s use an example to describe the Viterbi-equalizer.
Discrete-time channel: ck1
F(z)2 This would caseserious noisez−
-0.9
f
enhancement inlinear equalizers.
Further, assume that our symbol alphabet is -1 and +1 (representingthe bits 0 and 1, respectively).
State-1
The fundamentaltrellis stage:
1
-0.1
1.9
-1.9
0.1
Input cm
-1+1
The Viterbi-equalizer (2)
Transmitted:1 1 -1 1 -1
The filter startsin state -1.
1−
Noise free sequence:1.9 0.1 -1.9 1.9
Noise10.9z−
-1.9
At this stage,Received noisy sequence:0.72
State -0.1
0.19 -1.70 1.09
-0.1 -0.1 -0.1
the path ending-1.06here has the best
metric!
-0.1-1
VITERBIDETECTOR
1
0.68 0.76
1.9 1.95.75
1.39 3.60
-1.9
3.32 2.86 3.78
1.9 1.9 1.91.44 13.58 2.79
13.72 2.09 11.62
-1.9 -1.9 -1.9
0.1 1.40 0.1 4.64 0.1 5.62 0.1 3.43
Detected sequence:1 1 -1 Correct!1 -1
Summary
• Linear equalizers suffer from noise enhancement.• Decision-feedback equalizers (DFEs) use decisions on data
to remove parts of the ISI, allowing the linear equalizer partto be less ”powerful” and thereby suffer less from noiseenhancement.
• Incorrect decisions can cause error-propagation in DFEs,since an incorrect decision may add ISI instead of removingit.
• Maximum-likelihood sequence estimation (MLSE) is optimalin the sense of having the lowest probability of detecting thewrong sequence.
• Brut-force MLSE is prohibitively complex.• The Viterbi-equalizer (detector) implements the MLSE with
considerably lower complexity.