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Lecture 12 1
Lecture 12
Equalization
Lecture 12 2
Intersymbol Interference
– With any practical channel the inevitable filtering effect will cause a spreading (or smearing out) of individual data symbols passing through a channel.
Lecture 12 3
• For consecutive symbols this spreading causes part of the symbol energy to overlap with neighbouring symbols causing intersymbol interference (ISI).
Lecture 12 4
ISI can significantly degrade the ability of the data detector to differentiate a current symbol from the diffused energy of the adjacent symbols.With no noise present in the channel this leads to the detection of errors known as the irreducible error rate.It will degrade the bit and symbol error rate performance in the presence of noise
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Lecture 12 5
Pulse Shape for Zero ISI
Careful choice of the overall channel characteristic makes it possible to control the intersymbol interference such that it does not degrade the bit error rate performance of the link.Achieved by ensuring the overall channel filter transfer function has what is termed a Nyquist frequency response.
Lecture 12 6
Nyquist Channel Responsetransfer function has a transition band between passband and stopband that is symmetrical about a frequency equal to 0.5 x 1/ Ts.
Lecture 12 7
For such a channel the the data signals are still smeared but the waveform passes through zero at multiples of the symbol period.
Lecture 12 8
If we sample the symbol stream at the precise point where the ISI goes through zero, spread energy from adjacent symbols will not affect the value of the current symbol at that point.This demands accurate sample point timing -a major challenge in modem / data receiver design. Inaccuracy in symbol timing is referred to as timing jitter.
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Lecture 12 9
Achieving the Nyquist Channel
Very unlikely that a communications channel will inherently exhibit a Nyquist transfer response.Modern systems use adaptive channel equalisers to flatten the channel transfer function. Adaptive equalisers use a training sequence.
Lecture 12 10
Eye Diagrams
Visual method of diagnosing problems with data systems.Generated using a conventional oscilloscope connected to the demodulated filtered symbol stream.Oscilloscope is re-triggered at every symbol period or multiple of symbol periods using a timing recovery signal.
Lecture 12 11 Lecture 12 12
Example eye diagrams for different distortions, each has a distinctive effect on the appearance of the ‘eye opening’:
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Lecture 12 13
– Example of complex ‘eye’ for M-ary signalling:
Lecture 12 14
Raised Cosine Filtering• Commonly used realisation of a Nyquist filter. The
transition band (zone between pass- and stopband) is shaped like a cosine wave.
Lecture 12 15
The sharpness of the filter is controlled by the parameter β, the filter roll-off factor. When β = 0 this conforms to the ideal brick-wall filter.
The bandwidth B occupied by a raised cosine filtered data signal is thus increased from its minimum value, Bmin = 0.5 x 1/Ts, to:
Actual bandwidth B = Bmin ( 1 + β)
Lecture 12 16
Impulse Response of Filter
– The impulse response of the raised cosine filter is a measure of its spreading effect.
– The amount of ‘ringing’ depends on the choice of β.
– The smaller the value of a (nearer to a ‘brick wall’ filter), the more pronounced the ringing.
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Lecture 12 17
Choice of Filter Roll-off β
• Benefits of small β– maximum bandwidth efficiency is achieved
• Benefits of large β– simpler filter with fewer stages hence easier
to implement– less signal overshoot– less sensitivity to symbol timing accuracy
Lecture 12 18
Symbol Timing Recovery
– Most symbol timing recovery systems obtain their information from the incoming message data using ‘zero crossing’ information in the baseband signal.
Lecture 12 19
Three kinds of systems
mT T>>
Narrowband system:Flat fading channel, single-tap channel model.
bit or symbolbit or symbol
Tm = delay spread of multipath channel
T = bit or symbol duration
System bandwidth
T W∝
Lecture 12 20
No intersymbol interference (ISI)
Adjacent symbols (bits) do not affect the decision process (in other words there is no intersymbol interference).
However: Fading (destructive interference)
is still possible
No intersymbolinterference at
decision time instant
Received replicas of same symbol overlap in multipath channel
Narrowband system:
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Lecture 12 21
Decision circuit
In the binary case, the decision circuit compares the received signal with a threshold at specific time instants (usually somewhere in the middle of each bit period):
Decision time instant
Decision circuitDecision circuitDecision threshold
Noisy and distorted symbols “Clean” symbols
Lecture 12 22
Three kinds of systems, Cont.
...m mT T T T≈ <<
Wideband system (TDM, TDMA):Selective fading channel, transversal filter channel model, good performance possible through adaptive equalization
T = bit or symbol duration
Intersymbol interference causes signal distortion
Tm = delay spread of multipath channel
Lecture 12 23
Receiver structure
The intersymbol interference of received symbols (bits) must be removed before decision making (the case is illustrated below for a binary signal, where symbol = bit):
Decision circuit
Decision circuit
Adaptiveequalizer
Adaptiveequalizer
Symbols with ISI
Symbols with ISI removed
“Clean”symbols
Decision time instantDecision threshold
Lecture 12 24
Three kinds of systems: BER performance
S/N
BER
Frequency-selective channel(no equalization)
Flat fading channel
AWGN channel
(no fading)
Frequency-selective channel(with equalization)
“BER floor”
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Lecture 12 25
Three kinds of systems, Cont.
m cT T T>> >>
Wideband system (DS-CDMA):Selective fading channel, transversal filter channel model, good performance possible through use of Rake receiver(this lecture).
Bit (or symbol)Bit (or symbol)
Tm = delay spread of multipath channel
T = bit (or symbol) duration
Tc = Chip duration ...
cT W∝
Lecture 12 26
Three basic equalization methods
1)- Linear equalization (LE):Performance is not very good when the frequency response of the frequency selective channel contains deep fades.
Zero-forcing algorithm aims to eliminate the intersymbolinterference (ISI) at decision time instants (i.e. at the centre of the bit/symbol interval).
Least-mean-square (LMS) algorithm will be investigated in greater detail in this presentation.
Recursive least-squares (RLS) algorithm offers faster convergence, but is computationally more complex than LMS (since matrix inversion is required).
Lecture 12 27
Three basic equalization methods
2)-Decision feedback equalization (DFE):Performance better than LE, due to ISI cancellation of tails of previously received symbols.
Decision feedback equalizer structure:
Feed-forward filter (FFF)
Feed-forward filter (FFF)
Feed-back filter (FBF)
Feed-back filter (FBF)
Adjustment of filter coefficients
Input Output
++
Symbol decision
Lecture 12 28
Three basic equalization methods, Cont.
3)- Maximum Likelihood Sequence Estimation using the Viterbi Algorithm (MLSE-VA):Best performance. Operation of the Viterbi algorithm can be visualized by means of a trellis diagram with m K-1 states, where m is the symbol alphabet size and K is the length of the overall channel impulse response (in samples).
State trellis diagram
Sample time instants
State
Allowed transition between states
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Lecture 12 29
Linear equalization, zero-forcing algorithm
Raised cosine
spectrum
Raised cosine
spectrum
Transmitted symbol
spectrum
Transmitted symbol
spectrum
Channel frequencyresponse
(incl. T & R filters)
Channel frequencyresponse
(incl. T & R filters)
Equalizer frequency response
Equalizer frequency response
=
( ) ( ) ( ) ( )Z f B f H f E f=
0 ffs = 1/T
( )B f
( )H f
( )E f
( )Z f
Basic idea:
Lecture 12 30
Zero-forcing equalizer
Communication channel
Equalizer
FIR filter contains 2N+1 coefficients
( )r k ( )z kTransmitted impulse sequence Input to
decision circuit
( ) ( )N
nn N
h k h k nδ=−
= −∑Channel impulse response Equalizer impulse response
( ) ( )M
mm M
c k c k mδ=−
= −∑
Coefficients of equivalent FIR filter
( )M
k m k mm M
f c h M k M−=−
= − ≤ ≤∑(in fact the equivalent FIR filter consists of 2M+1+2N coefficients, but the equalizer can only “handle” 2M+1 equations)
FIR filter contains 2M+1 coefficientsOverall
channel
Lecture 12 31
Zero-forcing equalizer
We want overall filter response to be non-zero at decision time k = 0 and zero at all other sampling times k ≠ 0 :
1, 00, 0
M
k m k mm M
kf c h
k−=−
=⎧= = ⎨ ≠⎩∑
0 1 1 2
1 0 1 2 1
1 1
2 1 2 2 1 1
2 2 1 1 0
... 0... 0
:... 1
:... 0
... 0
M M M M
M M M M
M M M M M M
M M M M M
M M M M M
h c h c h ch c h c h c
h c h c h c
h c h c h ch c h c h c
− − − + −
− − + − +
− − − + −
− − − − + −
− − − +
+ + + =+ + + =
+ + + =
+ + + =+ + + =
This leads to a set of 2M+1equations:
(k = –M)
(k = 0)
(k = M)
Lecture 12
32 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Equalization: Removing Residual ISIConsider a tapped delay line equalizer with
Search for the tap gains cN such that the output equals zero at sample intervals D except at the decision instant when it should be unity. The output is (think for instance paths c-N, cN or c0)
that is sampled at yielding
( ) ( )=−∑= − −%N
eq nn N
p t c p t nD ND
[ ]( ) ( ) ( )N N
eq n nn N n N
p kD ND c p kD nD c p D k n=− =−∑ ∑+ = − = −% %
= +kt kD ND
( ) ( 2 )N Np t c p t ND= −%
( ) ( )N Np t c p t− −
= %
0 ( ) ( )Np t c p t ND−
= −%
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Lecture 12
33 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Example of Equalization
Read the distorted pulse values into matrix from fig. (a)
and the solution is
1
0
1
1.0 0.1 0.0 00.2 1.0 0.1 10.1 0.2 1.0 0
−⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥− =⎢ ⎥ ⎢ ⎥ ⎢ ⎥
−⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦
ccc
1
0
1
0.0960.960.2
− −⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
ccc
Zero forced values
0p
1p
2p1p−
2p−
Lecture 12 34
Minimum Mean Square Error (MMSE)
The aim is to minimize:2
kJ E e=
ˆ ˆk k k k ke z b b z= − −(or depending on the source)
EqualizerEqualizerChannelChannelkz kb
ke
( )r k( )s k
+
Estimate of k:thsymbol
Input to decision circuit
( )z k ( )b k
Error
Lecture 12 35
MSE vs. equalizer coefficients
2kJ E e= =
J
1c
2c
quadratic multi-dimensional function of equalizer coefficient values
MMSE aim: find minimum value directly (Wiener solution), or use an algorithm that recursively changes the equalizer coefficients in the correct direction (towards the minimum value of J)!
Illustration of case for two real-valued equalizer coefficients (or one complex-valued coefficient)
Lecture 12 36
Wiener solution
opt =Rc p
R = correlation matrix (M x M) of received (sampled) signal values
p = vector (of length M) indicating cross-correlation between received signal values and estimate of received symbol
copt = vector (of length M) consisting of the optimal equalizer coefficient values
(We assume here that the equalizer contains M taps, not 2M+1 taps like in other parts of this presentation)
We start with the Wiener-Hopf equations in matrix form:
kb
kr
kr
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Lecture 12 37
Correlation matrix R & vector p
Before we can perform the stochastical expectation operation, we must know the stochastical properties of the transmitted signal (and of the channel if it is changing). Usually we do nothave this information => some non-stochastical algorithm like Least-mean-square (LMS) must be used.
( ) ( )*TE k k⎡ ⎤= ⎣ ⎦R r r
( ) *kE k b⎡ ⎤= ⎣ ⎦p r
( ) [ ]1 1, ,..., Tk k k Mk r r r− − +=rwhere
M samples
Lecture 12 38
Algorithms
Stochastical information (R and p) is available:
1. Direct solution of the Wiener-Hopf equations:
2. Newton’s algorithm (fast iterative algorithm)
3. Method of steepest descent (this iterative algorithm is slowbut easier to implement)
R and p are not available:
Use an algorithm that is based on the received signal sequence directly. One such algorithm is Least-Mean-Square (LMS).
opt =Rc p 1opt
−=c R p Inverting a large matrix is difficult!
Lecture 12 39
Conventional linear equalizer of LMS type
k Mr −k Mr +TT TT TT
ΣΣ
LMS algorithm for adjustment of tap coefficients
Transversal FIR filter with 2M+1 filter taps
Estimate of kth symbol after symbol decision
Complex-valued tap coefficients of equalizer filter
ke
+
kbkz
Mc− 1 Mc − 1Mc − Mc
WidrowReceived complex
signal samples
Lecture 12 40
Joint optimization of coefficients and phase
Equalizer filterEqualizer filter
Coefficient updating
Coefficient updating
Phase synchronization
Phase synchronization
ke
je φ
+
( )r k kz
kb
( )ˆ ˆexpM
k k k m k m km M
e z b c r j bφ−=−
⎛ ⎞= − = −⎜ ⎟⎝ ⎠∑
Minimize:2
kJ E e=
11
Lecture 12 41
Least-mean-square (LMS) algorithm(derived from “method of steepest descent”)
for convergence towards minimum mean square error (MMSE)
( ){ } ( ){ } { }
2
Re 1 ReRe
kn n
n
ec i c i
c∂
+ = − ∆∂ ⎡ ⎤⎣ ⎦
Real part of n:th coefficient:
( ){ } ( ){ } { }
2
Im 1 ImIm
kn n
n
ec i c i
c∂
+ = −∆∂ ⎡ ⎤⎣ ⎦
Imaginary part of n:th coefficient:
( ) ( )2
1 kei i φφ φ
φ∂
+ = − ∆∂
Phase:
( )2 2 1 1M + +equations Iteration index Step size of iteration
2k k ke e e ∗=
Lecture 12 42
LMS algorithm (cont.)
After some calculation, the recursion equations are obtained in the form
( ){ } ( ){ } ˆRe 1 Re 2 ReM
j jn n m k m k k n
m M
c i c i e c r b r eφ φ∗ −− −
=−
⎧ ⎫⎛ ⎞+ = − ∆ −⎨ ⎬⎜ ⎟⎝ ⎠⎩ ⎭
∑
( ){ } ( ){ } ˆIm 1 Im 2 ImM
j jn n m k m k k n
m Mc i c i e c r b r eφ φ∗ −
− −=−
⎧ ⎫⎛ ⎞+ = − ∆ −⎨ ⎬⎜ ⎟
⎝ ⎠⎩ ⎭∑
( ) ( ) ˆ1 2 ImM
jk m k m
m Mi i b e c rφ
φφ φ ∗−
=−
⎧ ⎫+ = − ∆ ⎨ ⎬
⎩ ⎭∑
ke
Lecture 12 43
Effect of iteration step size
Slow acquisitionSlow acquisition
∆∆smaller larger
Poor tracking performance
Poor tracking performance
Poor stabilityPoor stability
Large variation around optimum
value
Large variation around optimum
value
∆φ∆φ
Lecture 12 44
Decision feedback equalizer
TT TT TT
ΣΣ
LMS algorithm
for tap coefficient adjustment
TT TT
ΣΣ
FFFFFF
FBFFBF +
+ ke
kb
kz
k Qb −1kb −
k Mr + k Mr −
Mc− 1 Mc − 1Mc − Mc
1Qq − Qq1q?
12
Lecture 12 45
The purpose is again to minimize
Decision feedback equalizer (cont.)
1
ˆ ˆ ˆQM
k k k m k m n k n km M n
e z b c r q b b− −=− =
= − = − −∑ ∑
2 2k kJ E e e= ≈
Feedforward filter (FFF) is similar to filter in linear equalizer tap spacing smaller than symbol interval is allowed => fractionally spaced equalizer => oversampling by a factor of 2 or 4 is common
Feedback filter (FBF) is used for either reducing or cancelingsamples of previous symbols at decision time instants
Tap spacing must be equal to symbol interval
where
Lecture 12 46
The coefficients of the feedback filter (FBF) can be obtained in either of two ways:
Recursively (using the LMS algorithm) in a similar fashion as FFF coefficients
By calculation from FFF coefficients and channel coefficients (we achieve exact ISI cancellation in this way, but channel estimation is necessary):
Decision feedback equalizer (cont.)
1, 2, ,M
n m n mm M
q c h n Q−=−
= − =∑ K
Lecture 12 47 Lecture 12 48
Channel estimation circuit
TT TT TT
ΣΣ
LMS algorithm
Estimated symbols
+
Mc1Mc −1c0c
k Mb −kb
krkr
ˆm mh c=
k:th sample of received signal Estimated channel coefficients
Filter length = CIR length
13
Lecture 12 49
Channel estimation circuit (cont.)
1. Acquisition phaseUses “training sequence”Symbols are known at receiver, .
2. Tracking phaseUses estimated symbols (decision directed mode) Symbol estimates are obtained from the decision circuit (note the delay in the feedback loop!) Since the estimation circuit is adaptive, time-varying channel coefficients can be tracked to some extent.
k kb b=
Alternatively: blind estimation (no training sequence)
Lecture 12 50
Channel estimation circuit in receiver
Channel estimation
circuit
Channel estimation
circuit
Equalizer & decision
circuit
Equalizer & decision
circuit
( )h m ( )b k
( )b k
( )r k
Estimated channel coefficients
“Clean” output symbolsReceived signal samples
Symbol estimates (with errors)
Training symbols
(no errors)
Mandatory for MLSE-VA, optional for DFE
Lecture 12 51
MLSE-VA receiver structure
Matched filter
Matched filter
MLSE(VA)
MLSE(VA)
Channel estimation circuit
Channel estimation circuit
NW filter
NW filter
( )r t
( )f k
( )y k ( )b k
( )f k
MLSE-VA circuit causes delay of estimated symbol sequence before it is available for channel estimation
=> channel estimates may be out-of-date (in a fast time-varying channel)
Lecture 12 52
MLSE-VA receiver structure (cont.)
The probability of receiving sample sequence y (note: vector form) of length N, conditioned on a certain symbol sequence estimate and overall channel estimate:
( ) ( )( )
21
2 21 01
1 1 ˆ ˆˆ ˆˆ ˆ, , exp22
N N K
k k n k nN Nk nk
p p y y f bσπ σ
−
−= ==
⎧ ⎫⎪ ⎪= = − −⎨ ⎬⎪ ⎪⎩ ⎭
∑ ∑∏y b f b f
Metric to be minimized
(select best ..using VA)
Objective: find symbol
sequence that maximizes this
probability
This is allowed since noise samples are
uncorrelated due to NW (= noise whitening) filter
Length of f (k)Since we have AWGN
b
14
Lecture 12 53
MLSE-VA receiver structure (cont.)
We want to choose that symbol sequence estimate and overall channel estimate which maximizes the conditional probability.
Since product of exponentials <=> sum of exponents, the metric to be minimized is a sum expression.
If the length of the overall channel impulse response in samples (or channel coefficients) is K, in other words the time span of the channel is (K-1)T, the next step is to construct a state trellis where a state is defined as a certain combination of K-1 previous symbols causing ISI on the k:th symbol.
K-10 k
( )f k Note: this is overall CIR, including response of matched
filter and NW filter
Lecture 12 54
MLSE-VA receiver structure (cont.)
At adjacent time instants, the symbol sequences causing ISI are correlated. As an example (m=2, K=5):
1 0 0 1 0
1 0 0 1 0
0 0 1 0 0
0
1
At time k-3
At time k-2
At time k-1
:
:
Bits causing ISI not causing ISI at time instant
At time k
1
0 1 0 0 11 0 1
Bit detected at time instant16 states
Lecture 12 55
MLSE-VA receiver structure (cont.)
State trellis diagram
k-2 k-1 kk-3
1Km −
Number of states The ”best” state
sequence is estimated by
means of Viterbi algorithm (VA)
k+1
Of the transitions terminating in a certain state at a certain time instant, the VA selects the transition associated with highest accumulated probability (up to that time instant) for further processing.
Alphabet size
Lecture 12 56
Rake receiver structure and operation
Rake receiver <=> a signal processing example that illustrates some important concepts
Rake receiver is used in DS-CDMA (Direct Sequence Code Division Multiple Access) systems
Rake “fingers” synchronize to signal components that are received via a wideband multipath channel
Important task of Rake receiver is channel estimation
Output signals from Rake fingers are combined, for instance calculation of Maximum Ratio Combining (MRC)
15
Lecture 12 57
Principle of RAKE Receiver
Lecture 12 58
To start with: multipath channel
in which case the received (equivalent low-pass) signal is of the form
Suppose a signal s (t) is transmitted. A multipath channel with Mphysical paths can be presented (in equivalent low-pass signal domain) in form of its Channel Impulse Response (CIR)
( )1
0
( ) ( ) ( ) m
Mj
m mm
r t s t h t a e s tφ τ−
=
= ∗ = −∑
( )1
0
( ) m
Mj
m mm
h t a e tφ δ τ−
=
= −∑
.
Lecture 12 59
Sampled channel impulse response
Delay (τ )
( )1
0
( ) n
Nj
nn
h t a e t nφ δ τ−
=
= − ⋅ ∆∑
Sampled Channel Impulse Response (CIR)
The CIR can also be presented in sampled form using N complex-valued samples uniformly spaced at most 1/W apart, where W is the RF system bandwidth:
τ∆
CIR sampling rate = for instance sampling rate used in receiver during A/D conversion.
Uniformly spaced channel samples
1 Wτ∆ ≤
Lecture 12 60
Rake finger selection
Delay (τ )
( )1
( ) i
Lj
rake i ii
h t a e tφ δ τ=
= −∑
Channel estimation circuit of Rake receiver selects L strongest samples (components) to be processed in L Rake fingers:
In the Rake receiver example to follow, we assume L = 3.
1τ 2τ 3τ
Only one sample chosen, since adjacent samples may be correlated
Only these samples are constructively utilized in
Rake fingers
16
Lecture 12 61
Received multipath signal
Received signal consists of a sum of delayed (and weighted) replicas of transmitted signal.
All signal replicas are contained in received signal
:
Signal replicas: same signal at different delays, with different amplitudes and phases
Summation in channel <=> “smeared” end result
Blue samples indicate signal replicas processed in Rake fingersGreen samples only cause interference
Lecture 12 62
Rake receiver
Finger 1Finger 1
Finger 2Finger 2
Channel estimationChannel estimation
Received baseband multipath signal
Finger 3Finger 3
ΣΣ
Output signal
(to decision circuit)
Rake receiver Path combining
(Generic structure, assuming 3 fingers)
WeightingWeighting
Lecture 12 63
Channel estimation
Channel estimationChannel estimationA
BC
A
B
C
Amplitude, phase and delay of signal components detected in Rake fingers must be estimated.
ia iφ iτ
Each Rake finger requires delay (and often also phase) information of the signal component it is processing.
iaiτ
Maximum Ratio Combining (MRC) requires amplitude (and phase if this is utilized in Rake fingers) of components processed in Rake fingers.
( )iφ ( )iφ
Lecture 12 64
Rake finger processing
Case 1: same code in I and Q branches
Case 2: different codes in I and Q branches
- for purpose of easy demonstration only
- the real case e.g. in IS-95 and WCDMA
- no phase synchronization in Rake fingers
- phase synchronization in Rake fingers
17
Lecture 12 65
DelayDelay
Rake finger processing
Tdt⋅∫
ΣΣ
Received signal
To MRC
Tdt⋅∫( )if τ
Stored code sequenceStored code sequence
(Case 1: same code in I and Q branches)
I branch
Q branch
I/QI/Q
Output of finger: a complex signal value for each detected bit
Lecture 12 66
Correlation vs. matched filtering
Tdt⋅∫Received
code sequence
Received code sequence
Stored code sequenceStored code sequence
Basic idea of correlation:
Same result through matched filtering and sampling:
Received code sequence
Received code sequence
Matched filter
Matched filter
Sampling at t = T
Sampling at t = T
Sam
e end resu
lt (in th
eory)
Lecture 12 67
Rake finger processing
( ) ( ) ( ) ( )
( ) ( ) ( )1
i n
Nj j
i i n nnn i
r t z t v t w t
a e s t a e s t w tφ φτ τ=≠
= + +
= − + − +∑
Correlation with stored code sequence has different impact on different parts of the received signal
= desired signal component detected in i:th Rake finger
= other signal components causing interference
= other codes causing interference (+ noise ... )
( )z t
( )v t( )w t
Lecture 12 68
Rake finger processing
Illustration of correlation (in one quadrature branch) with desired signal component (i.e. correctly aligned code sequence)
Desired component
Stored sequence
After multiplication
Strong positive/negative “correlation result” after integration
“1” bit “0” bit “0” bit
18
Lecture 12 69
Rake finger processing
Illustration of correlation (in one quadrature branch) with some other signal component (i.e. non-aligned code sequence)
Other component
Stored sequence
After multiplication
Weak “correlation result” after integration
“1” bit “0” bit
Lecture 12 70
Rake finger processing
Mathematically:
( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( )
0
2
0
1 0 0
i
n
T
i
Tj
i
T TNj
n n inn i
C z t v t w t s t dt
a e s t dt
a e s t s t dt w t s t dt
φ
φ τ τ=≠
= + +⎡ ⎤⎣ ⎦
=
+ − + +
∫
∫
∑ ∫ ∫
Correlation result for bit between
Interference from same signal
Interference from other signals
Desired signal
( )0,T
Lecture 12 71
Rake finger processing
Set of codes must have both: - good autocorrelation properties (same code sequence)- good cross-correlation properties (different sequences)
( )
( ) ( ) ( ) ( )
2
0
1 0 0
i
n
Tj
i i
T TNj
n n inn i
C a e s t dt
a e s t s t dt w t s t dt
φ
φ τ τ=≠
=
+ − + +
∫
∑ ∫ ∫
Large
Small Small
Lecture 12 72
DelayDelay
Rake finger processing
Tdt⋅∫
Received signal
Tdt⋅∫
Stored I code sequenceStored I code sequence
(Case 2: different codes in I and Q branches)
I branch
Q branch
I/QI/Q
Stored Q code sequenceStored Q code sequence
iφ
To MRC for I signal
To MRC for Q signal
Required: phase synchronization
( )if τ
19
Lecture 12 73
Phase synchronization
I/QI/Q
iφ
When different codes are used in the quadrature branches (as in practical systems such as IS-95 or WCDMA), phase synchronization is necessary.
Phase synchronization is based on information within received signal (pilot signal or pilot channel).
Signal in I-branch
Pilot signalPilot signal
Signal in Q-branch
I
Q
Note: phase synchronization must
be done for each finger separately!
Lecture 12 74
Weighting
Maximum Ratio Combining (MRC) means weighting each Rake finger output with a complex number after which the weighted components are summed “on the real axis”:
3
1
i ij ji i
iZ a e a eφ φ−
=
= ⋅∑
Component is weighted
Phase is aligned
Rake finger output is complex-valued
real-valued
(Case 1: same code in I and Q branches)
Instead of phase alignment: take absolute value of finger outputs ...
Lecture 12 75
Phase alignment
The complex-valued Rake finger outputs are phase-aligned using the following simple operation:
1i ij je eφ φ− ⋅ =
Before phase alignment:
ije φ−
ije φ
1
After phase alignment:
Phasors representing complex-valued Rake
finger outputs
Lecture 12 76
Maximum Ratio Combining
The idea of MRC: strong signal components are given more weight than weak signal components.
The signal value after Maximum Ratio Combining is:
2 2 21 2 3Z a a a= + +
(Case 1: same code in I and Q branches)
20
Lecture 12 77
Maximum Ratio Combining
Output signals from the Rake fingers are already phase aligned (this is a benefit of finger-wise phase synchronization).
Consequently, I and Q outputs are fed via separate MRC circuits to the decision circuit (e.g. QPSK demodulator).
(Case 2: different codes in I and Q branches)
Quaternarydecisioncircuit
Quaternarydecisioncircuit
Finger 1Finger 1
Finger 2Finger 2
MRCΣ
MRCΣ
MRCΣ
MRCΣ
:
I
Q
I
Q
I
Q