point-to-point wireless communication (ii): isi & equalization, diversity...
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Point-to-Point Wireless Communication (II): ISI & Equalization, Diversity (Time/Space/Frequency). Shiv Kalyanaraman Google: “Shiv RPI” [email protected]. Ref: Chapter 3 in Tse/Viswanath texbook. - PowerPoint PPT PresentationTRANSCRIPT
Shivkumar KalyanaramanRensselaer Polytechnic Institute
1 : “shiv rpi”
Point-to-Point Wireless Communication (II):ISI & Equalization,
Diversity (Time/Space/Frequency)
Shiv Kalyanaraman
Google: “Shiv RPI”
Based upon slides of P. Viswanath/Tse, Sorour Falahati, Takashi Inoue, J. Andrews, Scott Baxter,& textbooks by Tse/Viswanath, A. Goldsmith, J. Andrews et al, & Bernard Sklar.
Ref: Chapter 3 in Tse/Viswanath texbook
Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Multi-dimensional Fading
Time, Frequency, Space
Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Plan First, compare 1-tap (i.e. flat) Rayleigh-fading channel vs
AWGN. i.e. y = hx + w vs y = x + w Note: all multipaths with random attenuation/phases are
aggregated into 1-tap
Next consider frequency selectivity, i.e. multi-tap, broadband channel, with multi-paths Effect: ISI Equalization techniques for ISI & complexities
Generalize! Consider diversity in time, space, frequency, and develop efficient schemes to achieve diversity gains and coding gains
Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Single-tap, Flat Fading (Rayleigh) vs AWGN
Why do we have this huge degradation in performance/reliability?
Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Rayleigh Flat Fading Channel
BPSK: Coherent detection.
Conditional on h,
Averaged over h,
at high SNR.
Looks like AWGN, but…
pe needs to be “unconditioned”
To get a much poorer scaling
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SNR
BER
Frequency-selective channel (no equalization)
Flat fading channel
AWGN channel
(no fading)
Frequency-selective channel (equalization or Rake receiver)
“BER floor”
BER vs. SNR (cont.)
01 4eP
( )eP
means a straight line in log/log scale
0( )
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Typical error event is due to: channel (h) being in deep fade!… rather than (additive) noise being large.
Conditional on h,
When the error probability is very small.
When the error probability is large:
Typical Error Event
Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Preview: Diversity Gain: Intuition Typical error (deep fade) event probability: In other words, ||h|| < ||w||/||x||
i.e. ||hx|| < ||w|| (i.e. signal x is attenuated to be of the order of noise w)
Chi-Squared pdf of
Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Recall: BPSK, QPSK and 4-PAM
BPSK uses only the I-phase.The Q-phase is wasted. QPSK delivers 2 bits per complex symbol. To deliver the same 2 bits, 4-PAM requires 4 dB more transmit power. QPSK exploits the available degrees of freedom in the channel better.
A good communication scheme exploits all the available d.o.f. in the channel.
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MQAM doesn’t change the asymptotics…
QPSK does use degrees of freedom better than equivalent 4-PAM
(Read textbook, chap 3, section 3.1)
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Frequency Selectivity: Multipath fading & ISI
Mitigation: Equalization & Challenges
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ISI Mitigation: Outline
Inter-symbol interference (ISI): review Nyquist theorem
Pulse shaping (last slide set)
1. Equalization receivers 2. Introduction to the diversity approach
Rake Receiver in CDMA OFDM: decompose a wideband multi-tap channel
into narrowband single tap channels
Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Recall: Attenuation, Dispersion Effects: ISI!
Source: Prof. Raj Jain, WUSTL
Inter-symbol interference (ISI)
Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Base Station (BS)Mobile Station (MS)
multi-path propagation
Path Delay
Po
we
r
path-2
path-2path-3
path-3
path-1
path-1
Recall: Multipaths: Power-Delay Profile
Channel Impulse Response: Channel amplitude |h| correlated at delays . Each “tap” value @ kTs Rayleigh distributed
(actually the sum of several sub-paths)
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Inter-Symbol-Interference (ISI) due to Multi-Path Fading
Transmitted signal:
Received Signals:Line-of-sight:
Reflected:
The symbols add up on the channel
Distortion!
Delays
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Multipath: Time-Dispersion => Frequency Selectivity
The impulse response of the channel is correlated in the time-domain (sum of “echoes”) Manifests as a power-delay profile, dispersion in channel autocorrelation function A()
Equivalent to “selectivity” or “deep fades” in the frequency domain Delay spread: ~ 50ns (indoor) – 1s (outdoor/cellular). Coherence Bandwidth: Bc = 500kHz (outdoor/cellular) – 20MHz (indoor) Implications: High data rate: symbol smears onto the adjacent ones (ISI).
Multipath effects
~ O(1s)
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BER vs. S/N performance: AWGN
Typical BER vs. S/N curves
S/N
BER
Frequency-selective channel (no equalization)
Flat fading channel
Gaussian channel(no fading)
In a Gaussian channel (no fading) BER <=> Q(S/N)erfc(S/N)
Shivkumar KalyanaramanRensselaer Polytechnic Institute
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BER vs. S/N performance: Flat Fading
Typical BER vs. S/N curves
S/N
BER
Frequency-selective channel (no equalization)
Flat fading channel
Gaussian channel(no fading)
Flat fading: BER BER S N z p z dzz = signal power level
Shivkumar KalyanaramanRensselaer Polytechnic Institute
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BER vs. S/N performance:
ISI/Freq. Selective Channel
Typical BER vs. S/N curves
S/N
BER
Frequency-selective channel (no equalization)
Flat fading channel
Gaussian channel(no fading)
Frequency selective fading <=> irreducible BER floor!!!
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BER vs. S/N performance:
w/ Equalization
Typical BER vs. S/N curves
S/N
BER
Flat fading channel
Gaussian channel(no fading)
Diversity (e.g. multipath diversity) <=>
Frequency-selective channel(with equalization)
improved performance
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Equalization
Frequencydown-conversion
Receiving filter
Equalizingfilter
Threshold comparison
For bandpass signals Compensation for channel induced ISI
Baseband pulse(possibly distored)
Sample (test statistic)
Baseband pulseReceived waveform
Step 1 – waveform to sample transformation Step 2 – decision making
)(tr)(Tz
im
Demodulate & Sample Detect
Shivkumar KalyanaramanRensselaer Polytechnic Institute
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What is an equalizer?
We’ve used it for music in everyday life! Eg: default settings for various types of music to emphasize bass, treble etc… Essentially we are setting up a (f-domain) filter to cancel out the channel mpath filtering effects
Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Equalization: Channel is a LTI Filter
ISI due to filtering effect of the communications channel (e.g. wireless channels) Channels behave like band-limited filters
)()()( fjcc
cefHfH
Non-constant amplitude
Amplitude distortion
Non-linear phase
Phase distortion
Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Pulse Shaping and Equalization Principles
Square-Root Raised Cosine (SRRC) filter and Equalizer
)()()()()(RC fHfHfHfHfH erctNo ISI at the sampling time
)()()()(
)()()(
SRRCRC
RC
fHfHfHfH
fHfHfH
tr
rt
Taking care of ISI caused by tr. filter
)(
1)(
fHfH
ce Taking care of ISI
caused by channel
* Equalizer: enhance weak freq., dampen strong freq. to flatten the spectrum* Since the channel Hc(f) changes with time, we need adaptive equalization, i.e. re-estimate channel & equalize
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Equalization: Channel examples Example of a (somewhat) frequency selective, slowly changing (slow fading)
channel for a user at 35 km/h
Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Equalization: Channel examples … Example of a highly frequency-selective, fast changing (fast fading) channel for a
user at 35 km/h
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Recall: Eye pattern
Eye pattern:Display on an oscilloscope which sweeps the system response to a baseband signal at the rate 1/T (T symbol duration)
time scale
ampl
itude
sca
le
Noise margin
Sensitivity to timing error
Distortiondue to ISI
Timing jitter
Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Example of eye pattern with ISI:Binary-PAM, SRRC pulse
Non-ideal channel and no noise)(7.0)()( Tttthc
Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Example of eye pattern with ISI:Binary-PAM, SRRC pulse …
AWGN (Eb/N0=20 dB) and ISI
)(7.0)()( Tttthc
Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Example of eye pattern with ISI:Binary-PAM, SRRC pulse …
AWGN (Eb/N0=10 dB) and ISI)(7.0)()( Tttthc
Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Equalizing filters … Baseband system model
Tx filter Channel
)(tn
)(tr Rx. filterDetector
kz
kTt
ka1a
2a 3aT )(
)(
fH
th
t
t
)(
)(
fH
th
r
r
)(
)(
fH
th
c
c
k
k kTta )( Equalizer
)(
)(
fH
th
e
e
)(tz
Equivalent system
)(ˆ tn
)(tzDetector
kz
kTt )(
)(
fH
th
filtered (colored) noise
)()()()( fHfHfHfH rct
1a
2a 3aT
k
k kTta )( )(tx Equalizer
)(
)(
fH
th
e
e
)()()(ˆ thtntn r
ka)(tz
Equivalent model
Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Equalizer Types
Source: Rappaport book, chap 7
Covered later in slideset
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Linear Equalizer
Equalizer
Heq(f)1
Hc(f)
Channel
Hc(f)
n(t)
• A linear equalizer effectively inverts the channel.
• The linear equalizer is usually implemented as a tapped delay line.
• On a channel with deep spectral nulls, this equalizer enhances the noise. (note: both signal and noise pass thru eq.)
poor performance on frequency-selective fading channels
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Noise Enhancement w/ Spectral Nulls
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Decision Feedback Equalizer (DFE)
=> doesn’t work well w/ low SNR. Optimal non-linear: MLSE… (complexity grows exponentially w/ delay spread)
• The DFE determines the ISI from the previously detected symbols and subtracts it from the incoming symbols.
• This equalizer does not suffer from noise enhancement because it estimates the channel rather than inverting it.
The DFE has better performance than the linear equalizer in a frequency-selective fading channel. • The DFE is subject to error propagation if decisions are
made incorrectly.
Hc(f)Forward
Filter
n(t)
x(t)
DFE
Feedback Filter
+
-
x(t)^
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Equalization by transversal filtering Transversal filter:
A weighted tap delayed line that reduces the effect of ISI by proper adjustment of the filter taps.
N
Nnn NNkNNnntxctz 2,...,2 ,..., )()(
Nc 1 Nc 1Nc Nc
)(tx
)(tz
Coeff. adjustment
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Training the Filter
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Transversal equalizing filter … Zero-forcing equalizer:
The filter taps are adjusted such that the equalizer output is forced to be zero at N sample points on each side:
Mean Square Error (MSE) equalizer: The filter taps are adjusted such that the MSE of ISI and noise power at
the equalizer output is minimized. (note: noise is whitened before filter)
Nk
kkz
,...,1
0
0
1)(
N
Nnnc
Adjust
2))((min kakTzE N
Nnnc
Adjust
Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Equalization: Summary Equalizer “equalizes” the channel response in frequency domain to remove ISI Can be difficult to design/implement, get noise enhancement (linear EQs) or error
propagation (decision feedback EQs)
Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Summary: Complexity and Adaptation Nonlinear equalizers (DFE, MLSE) have better performance but
higher complexity
Equalizer filters must be FIR Can approximate IIR Filters as FIR filters Truncate or use MMSE criterion
Channel response needed for equalization Training sequence used to learn channel
Tradeoffs in overhead, complexity, and delay
Channel tracked during data transmissionBased on bit decisionsCan’t track large channel fluctuations
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Diversity Techniques: Time, Frequency, Code, Space
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Introduction to Diversity Basic Idea
Send same bits over independent fading pathsIndependent fading paths obtained by time, space,
frequency, or polarization diversity Combine paths to mitigate fading effects
Tb
tMultiple paths unlikely to fade simultaneously
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Diversity Gain: Short Story…
AWGN case: BER vs SNR:
(any modulation scheme, only the constants differ)
Note: γ is received SNR
Rayleigh Fading w/o diversity:
Rayleigh Fading w/ diversity: (MIMO):
Note: “diversity” is a reliability theme, not a capacity/bit-rate one…For capacity: need more degrees-of-freedom (i.e. symbols/s)
& packing of bits/symbol (MQAM).
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Time Diversity
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Time Diversity Time diversity can be obtained by interleaving and coding
over symbols across different coherent time periods.
Coding alone is not sufficient!
Channel: timediversity/selectivity, but correlated acrosssuccessive symbols
(Repetition) Coding…w/o interleaving: a full codeword lost during fade
Interleaving: of sufficient depth: (> coherence time)At most 1 symbol of codeword lost
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Forward Error Correction (FEC): Eg: Reed-Solomon RS(N,K)
Data = K
FEC (N-K)
Block Size (N)
RS(N,K) >= K of Nreceived
Lossy Network
Recover K data packets!
Block: of sufficient size: (> coherence time), else need to interleave, or use with hybrid ARQ
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Hybrid ARQ/FEC ModelPackets • Sequence Numbers
• CRC or Checksum• Proactive FEC
Status Reports • ACKs• NAKs, • SACKs• Bitmaps
• Packets• Reactive FEC
Retransmissions
Timeout
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Example: GSM
The data of each user are sent over time slots of length 577 μs Time slots of the 8 users together form a frame of length 4.615 ms
Voice: 20 ms frames, rate ½ convolution coded = 456 bits/voice-frame Interleaved across 8 consecutive time slots assigned to that specific user:
0th, 8th, . . ., 448th bits are put into the first time slot, 1st, 9th, . . ., 449th bits are put into the second time slot, etc.
One time slot every 4.615 ms per user, or a delay of ~ 40 ms (ok for voice). The 8 time slots are shared between two 20 ms speech frames.
Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Time-Diversity Example: GSM
Amount of time diversity limited by delay constraint and how fast channel varies.
In GSM, delay constraint is 40ms (voice). To get full diversity of 8, needs v > 30 km/hr at fc = 900Mhz.
Recall: Tc < 5 ms = 1/(4Ds) = c/(8fcv)
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GSM contd
Walking speed of say 3 km/h => too little time diversity. GSM can go into a frequency hopping mode, Consecutive frames (each w/ time slots of 8 users) can hop
from one 200 kHz sub-channel to another.
Typical delay spread ~ 1μs => the coherence bandwidth (Bc) is 500 kHz.
The total bandwidth of 25 MHz >> Bc
=> consecutive frames can be expected to fade independently.
This provides the same effect as having time diversity.
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Repetition Code: Diversity Analysis
After interleaving over L coherence time periods,
Repetition coding: for all
This is classic vector detection in white Gaussian noise.
where and
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Repetition Coding: Matched Filtering
hx1 only spans a 1-dimensional space(similar to MPAM, w/ random channel gains instead!)
Multiply by conjugate => cancel phase!
||h||
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Repetition Coding: Fading Analysis (contd) BPSK Error probability:
Average over ||h||2 i.e. over Chi-squared distribution,
L-degrees of freedom!
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Diversity Gain: Intuition Typical error (deep fade) event probability: In other words, ||h|| < ||w||/||x||
i.e. ||hx|| < ||w|| (i.e. signal x is attenuated to be of the order of noise w)
Chi-Squared pdf of
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Key: Deep Fades Become Rarer
Note: this graph plotsreliability (i.e. BER vs SNR)
Repetition code trades off information rate (i.e. poor use of deg-of-freedom)
Deep fade ≡ Error event…
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Beyond Repetition Coding: Coding gains
Repetition coding gets full diversity, but sends only one symbol every L symbol times. i.e. trades off bit-rate for reliability (better BER)
Does not exploit fully the degrees of freedom in the channel. (analogy: PAM vs QAM)
How to do better?
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Example: Rotation code (L=2)
where d1 and d2 are the normalized distances between the codewords in the two basis directions (axes).
x1, x2 are two BPSK symbols before rotation (each, either a or –a).
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Product-Distance Criterion
product distanceChoose the rotation angle to maximize the worst-case product distance to all the other codewords:
If d1 = 0 or d2 = 0, the the diversity gain of the code is only 1.
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Rotation vs Repetition Coding
Recall repetition coding was like PAM (see matched filter slide before)Rotation code uses the degrees of freedom better!
Coding gain over the repetition code in terms of a saving in transmit power by a factor of sqrt(5) or 3.5 dB for the same product distance
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Time Diversity + Coding + Fading: The gory details!
If we plot this pe vs SNR curve vs the one for repetition code, then we can get the coding gain (for any target pe)
Note: the squared-product-distance idea will reappear as a determinant criteria in space-time codes
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Antenna Diversity
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Antenna Diversity
Receive(SIMO)
Transmit(MISO)
Both(MIMO)
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Antenna Diversity: Rx
Receive(SIMO)
Transmit(MISO)
Both(MIMO)
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Receive Diversity
Same mathematical structure as repetition coding in time diversity (!), except that there is a further power gain (aka “array gain”).
Optimal reception is via matched filtering/MRC
(a.k.a. receive beamforming).
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Array Gain vs Diversity Gain Diversity Gain: multiple independent channels between the transmitter and
receiver, and is a product of the statistical richness of those channels
Array gain does not rely on statistical diversity between the different channels and instead achieves its performance enhancement by coherently combining the actual energy received by each of the antennas. Even if the channels are completely correlated, as might happen in a line-
of-sight (LOS) system, the received SNR increases linearly with the number of receive antennas,
Eg: Correlated flat-fading:
Single Antenna SNR:
Adding all receive paths:
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Recall: Diversity Gain: Short Story…
AWGN case: BER vs SNR:
(any modulation scheme, only the constants differ)
Note: γ is received SNR
Rayleigh Fading w/o diversity:
Rayleigh Fading w/ diversity: (MIMO):
Note: “diversity” is a reliability theme, not a capacity/bit-rate one…For capacity: need more degrees-of-freedom (i.e. symbols/s)
& packing of bits/symbol (MQAM).
Shivkumar KalyanaramanRensselaer Polytechnic Institute
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Receive Diversity: Selection Combining
Recall: Bandpass vs matched filter analogy. Pick max signal, but don’t fully combine signal
power from all taps. Diminishing returns from more taps.
Source: J. Andrews et al, Fundamentals of WIMAX
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Receive Beamforming: Maximal Ratio Combining (MRC)
Weight each branch
SNR:
MRC Idea: Branches with better signal energy should be enhanced, whereas branches with lower SNR’s given lower weights
Source: J. Andrews et al, Fundamentals of WIMAX
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Recall: Maximal Ratio Combining (MRC) or “Beamforming” … is just Matched Filtering in the Spatial Domain!
Generalization of this f-domain picture, for combining multi-tap signal
Weight each branch
SNR:
Source: J. Andrews et al, Fundamentals of WIMAX
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Selection Diversity vs MRC
Source: J. Andrews et al, Fundamentals of WIMAX
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Antenna Diversity: Tx
Receive(SIMO)
Transmit(MISO)
Both(MIMO)
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Transmit Diversity
If transmitter knows the channel, send:
maximizes the received SNR by in-phase addition of signals at the receiver (transmit beamforming), i.e. closed-loop Tx diversity.
Reduce to scalar channel:
same as receive beamforming.
What happens if transmitter does not know the channel?
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Open-Loop Tx Diversity: Space-Time Coding
Alamouti : Orthogonal space-time block code (OSTBC). 2 × 1 Alamouti STBC
Rate 1 code: Data rate is neither increased nor decreased; Two symbols are sent over two time intervals. Goal: harness spatial diversity. Don’t care about ↑ rate
Alamouti Code:
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Alamouti Scheme
Over two symbol times:
Projecting onto the two columns of the H matrix yields:
•double the symbol rate of repetition coding.
•3dB loss of received SNR compared to transmit beamforming (i.e. MRC or matched filtering).
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What was that, again? Alamouti STBC
Flat fading channel. h1(t), h2(t) are the complex channel gains from antenna 1 &
antenna 2 Channel is constant over 2 symbol times,
i.e. h1(t = 0) = h1(t = T) = h1.
Like MRC, but 3dB (i.e. ½) lower power
Received Signal:
Receiver: Project on columns of H:
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Space-time Codes Note: Transmitter does NOT know the channel instantaneously (open-loop)
Using the antennas one at a time and sending the same symbol over the different antennas is like repetition coding. Repetition scheme: inefficient utilization of degrees of freedom Over the two symbol times, bits are packed into only one dimension of
the received signal space, namely along the direction [h1, h2]t. More generally, can use any time-diversity code by turning on one
antenna at a time.
Space-time codes are designed specifically for the transmit diversity scenario. Alamouti scheme spreads the information onto two dimensions - along
the orthogonal directions [h1, h2*]t and [h2,−h1* ]t.
Repetition: Alamouti:
Shivkumar KalyanaramanRensselaer Polytechnic Institute
78 : “shiv rpi”
Space-time Code Design: In Brief
A space-time code is a set of matrices
Full diversity is achieved if all pairwise differences have full rank.
Coding gain determined by the (min) determinants of
Time-diversity codes have diagonal matrices and the determinant reduces to squared product distances.
Shivkumar KalyanaramanRensselaer Polytechnic Institute
79 : “shiv rpi”
ST-Coding Design: Details Space-time code as a set of complex codewords {Xi}, where
each Xi is an L by N matrix. L: number of transmit antennas N: block length of the code.
Repetition: Alamouti:
Normalize the codewords so that the average energy per symbol time is 1, hence SNR = 1/N0.
Assume channel constant for N symbol times
Shivkumar KalyanaramanRensselaer Polytechnic Institute
80 : “shiv rpi”
ST-Coding Design: Details
Note: λl here instead of dl
in rotation code analysis
Shivkumar KalyanaramanRensselaer Polytechnic Institute
81 : “shiv rpi”
ST Coding Design: Details
If all the λ2l are strictly positive for all the codeword
differences, then the maximal diversity gain of L is achieved. Number of positive eigenvalues λ2
l equals the rank of the codeword difference matrix, this is possible only if N ≥ L.
Min-determinant over codeword pairs controls the coding gain! (det-criterion)If XA etc are diagonal, then the determinant = squared-prod-distance!For Alamouti, min-det is 4; Repetition ST-code: min-det = 16/25
=> Alamouti coding gain: factor-of-6 (or 7.8 dB!)
(Recall: determinant= product of e-values)
Shivkumar KalyanaramanRensselaer Polytechnic Institute
82 : “shiv rpi”
Space-time Code Design: Summary
A space-time code is a set of matrices
Full diversity is achieved if all pairwise differences (eg: XA – XB have full rank (i.e. all e-values positive).
Coding gain determined by the (min) determinants of
Time-diversity codes have diagonal matrices and the determinant reduces to squared product distances.
Shivkumar KalyanaramanRensselaer Polytechnic Institute
83 : “shiv rpi”
Code Design & Degrees of Freedom
Shivkumar KalyanaramanRensselaer Polytechnic Institute
84 : “shiv rpi”
Antenna Diversity: Tx+Rx = MIMO
Receive(SIMO)
Transmit(MISO)
Both(MIMO)
Shivkumar KalyanaramanRensselaer Polytechnic Institute
85 : “shiv rpi”
MIMO: w/ Repetition or Alamouti Coding
Transmit the same symbol over the two antennas in two consecutive symbol times (at each time, nothing is sent over the other antenna). ½ symbol per degree of freedom (d.f.)
MRC combining w/ repetition:
Alamouti scheme used over the 2 × 2 channel: Sends 2 symbols/2 symbol times (i.e. 1symbol/d.f), Same 4-fold diversity gain as in repetition.
But, the 2x2 MIMO channel has MORE degrees of freedom!
Shivkumar KalyanaramanRensselaer Polytechnic Institute
86 : “shiv rpi”
MIMO: degrees of freedom Degrees of freedom =
dimension of received signal space
1xL: One-dimensional 2x2: Has 2 dimensions hj: vector of channel gains
from Tx antennas. Space gives new degrees of
freedom. A “spatial multiplexing”
scheme like V-BLAST can leverage the additional d.f.
Shivkumar KalyanaramanRensselaer Polytechnic Institute
87 : “shiv rpi”
Spatial Multiplexing: V-BLAST
Transmit independent uncoded symbols over antennas and over time!
V-BLAST: poorer diversity gain than Alamouti. But exploits spatial degrees of freedom better
Space-only coding: no Tx diversity. Diversity order only 2. Coding gain possible by coding across space & time (increased
degrees of freedom) with spatial multiplexing
Shivkumar KalyanaramanRensselaer Polytechnic Institute
88 : “shiv rpi”
MIMO Receiver Issues
V-BLAST uses joint ML reception (complex)
Zero-forcing linear receiver loses one order of diversity. Interference nuller,
decorrelator Noise samples
correlated (colored).
Shivkumar KalyanaramanRensselaer Polytechnic Institute
89 : “shiv rpi”
Summary: 2x2 MIMO Schemes
Need closed-loop MIMO to be able to reap both diversity and d.f. gains
Shivkumar KalyanaramanRensselaer Polytechnic Institute
90 : “shiv rpi”
Frequency Diversity: MLSD, CDMA Rake, OFDM
Shivkumar KalyanaramanRensselaer Polytechnic Institute
91 : “shiv rpi”
Frequency Diversity
Resolution of multi-paths provides diversity. Full diversity is achieved by sending one symbol every L
symbol times. But this is inefficient (like repetition coding). Sending symbols more frequently may result in intersymbol
interference. Note: ISI is not intrinsic, but frequency-diversity is!
Challenge is how to mitigate the ISI while extracting the inherent diversity in the frequency-selective channel.
Shivkumar KalyanaramanRensselaer Polytechnic Institute
92 : “shiv rpi”
Approaches
Time-domain equalization (eg. GSM)
Direct-sequence spread spectrum (eg. IS-95 CDMA)
Orthogonal frequency-division multiplexing OFDM (eg. 802.11a, Flash-OFDM)
Shivkumar KalyanaramanRensselaer Polytechnic Institute
93 : “shiv rpi”
ISI Equalization
Suppose a sequence of uncoded symbols are transmitted.
Maximum likelihood sequence detection is performed using the Viterbi algorithm.
Can full diversity be achieved?
Shivkumar KalyanaramanRensselaer Polytechnic Institute
94 : “shiv rpi”
Reduction to Transmit Diversity (Flat-Fading)
Shivkumar KalyanaramanRensselaer Polytechnic Institute
95 : “shiv rpi”
MLSD Achieves Full Diversity
Space-time code matrix for input sequence
Difference matrix for two sequences first differing at
is full rank.
Shivkumar KalyanaramanRensselaer Polytechnic Institute
96 : “shiv rpi”
Uncoded Max Likelihood Seq. Detection (MLSD)
Tradeoff: MLSD too complex!
MLSD:
Shivkumar KalyanaramanRensselaer Polytechnic Institute
97 : “shiv rpi”
MLSD: Viterbi Algorithm A brute-force exhaustive search would require a complexity that grows
exponentially with the block length n. Key: exploit the structure of the problem and should be recursive in n so
that the problem does not have to be solved from scratch for every symbol time.
Solution: Viterbi algorithm. Key Observation: memory in the frequency-selective channel can be
captured by a finite state machine. At time m, define the state (an L dimensional vector) # states is ML, where M is the constellation size
L: # of taps (diversity order)
Shivkumar KalyanaramanRensselaer Polytechnic Institute
98 : “shiv rpi”
MLSD: Viterbi Algo (Contd)
Re-write MLSD, conditioned on states s[i], instead of input sequence x
Conditional independence =>
MLSD ≡ finding the shortest path through an n-stage trellis the cost associated with the m-th transition (or “hop”) is
Shivkumar KalyanaramanRensselaer Polytechnic Institute
99 : “shiv rpi”
MLSD/Viterbi: Trellis
Note: a trellis is a state diagram that evolves with time as well.
Shivkumar KalyanaramanRensselaer Polytechnic Institute
100 : “shiv rpi”
Viterbi: Dynamic Programming
We only consider the states that the finite state machine can be in at stage m− 1 Subset of shortest path, also a shortest path! The complexity of the Viterbi algorithm is linear in the number of stages n
Complexity is also proportional to the size of the state space, which is ML, … where M is the constellation size of each symbol
Shivkumar KalyanaramanRensselaer Polytechnic Institute
101 : “shiv rpi”
Rake Receiver for Frequency Diversity
Detour: Spread Spectrum, CDMA,
Ref: Chapter 3 & 4, Tse/Viswanath book,Chap 13, 15: A. Goldsmith book
Shivkumar KalyanaramanRensselaer Polytechnic Institute
102 : “shiv rpi”
Sender Receiver
Code A
A
Code B
B
AB
AB
CBC
A
Code A
AB
C
Time
Freq
uenc
y
BC
B
A
Base-band Spectrum Radio Spectrum
spread spectrum
What is CDMA ?
Shivkumar KalyanaramanRensselaer Polytechnic Institute
103 : “shiv rpi”
Types of CDMA
Shivkumar KalyanaramanRensselaer Polytechnic Institute
104 : “shiv rpi”
Spread Spectrum
Spread-spectrum modulation is considered “secondary” modulation after the usual primary modulation.
Shivkumar KalyanaramanRensselaer Polytechnic Institute
105 : “shiv rpi”
Direct Sequence Spread Spectrum
Bit sequence modulated by chip sequence
Spreads bandwidth by large factor (K)
Despread by multiplying by sc(t) again (sc(t)=1)
Mitigates ISI and narrowband interference
s(t) sc(t)
Tb=KTc Tc
S(f)Sc(f)
1/Tb 1/Tc
S(f)*Sc(f)
Shivkumar KalyanaramanRensselaer Polytechnic Institute
106 : “shiv rpi”
Chips & Spreading
Shivkumar KalyanaramanRensselaer Polytechnic Institute
107 : “shiv rpi”
Processing Gain / Spreading Factor
Shivkumar KalyanaramanRensselaer Polytechnic Institute
108 : “shiv rpi”
Processing Gain & Shannon
With 8K vocoders, above 32 users, SNR becomes too low.
Practical CDMA systems restrict the number of users per sector to ensure processing gain remains at usable levels
Shivkumar KalyanaramanRensselaer Polytechnic Institute
109 : “shiv rpi”
ISI and Interference Rejection
Narrowband Interference Rejection
Multipath Rejection (Two Path Model)
S(f) S(f)I(f)S(f)*Sc(f)
Info. Signal Receiver Input Despread Signal
I(f)*Sc(f)
S(f) S(f)S(f)*Sc(f)[(t)+(t-)]
Info. Signal Receiver Input Despread Signal
S’(f)
Shivkumar KalyanaramanRensselaer Polytechnic Institute
110 : “shiv rpi”
Direct Sequence (DS)
Modulation(primary modulation)
Modulation(primary modulation)
user data
Sp
rea
din
g(s
ec
on
da
ry m
od
ula
tio
n)
Sp
rea
din
g(s
ec
on
da
ry m
od
ula
tio
n)
Tx
Base-bandFrequency
Po
we
rD
en
sity
RadioFrequency
Po
we
rD
en
sity
TIME
data rate
10110100
spreading sequence(spreading code)
How to Spread Spectrum
Shivkumar KalyanaramanRensselaer Polytechnic Institute
111 : “shiv rpi”
Spreading: Time-Domain View
Shivkumar KalyanaramanRensselaer Polytechnic Institute
112 : “shiv rpi”
Spreading: Freq-Domain View
Shivkumar KalyanaramanRensselaer Polytechnic Institute
113 : “shiv rpi”
If you know the correct spreading sequence (code) ,
RadioFrequency
Po
we
rD
en
sity
received signal
spreading sequence(spreading code)
you can find the spreading timing which gives the maximum detected power, and
Accumulate for one bit duration
Accumulate for one bit duration
Demodulated data
Base-bandFrequency
gathering energy !
10110100
1011010010110100 10110100
TIME
0100101110110100 10110100
0 01
1111111100000000 00000000
Demodulation 1/2
Shivkumar KalyanaramanRensselaer Polytechnic Institute
114 : “shiv rpi”
If you don’t know the correct spreading sequence (code) •••
Base-bandFrequency
received signal
spreading sequence(spreading code)
you cannot find the spreading timing without correct spreading code, and
Accumulate for one bit duration
Accumulate for one bit duration
Demodulated data
RadioFrequency
Po
we
rD
en
sity
01010101 01010101 01010101
10101010 10101010 10101010
TIME
0100101110110100 10110100
No data can be detected
- --
1011010010110100 10110100
Demodulation 2/2
Shivkumar KalyanaramanRensselaer Polytechnic Institute
115 : “shiv rpi”
Privacy, Security
RadioFrequency
Po
we
rD
en
sity
Power density of SS-signals could be lower than the noise density.
transmitted SS-signal
••••
••
Noise
Po
we
rD
en
sity
RadioFrequency
Noise
••••
••received signal de-
modulator
de-modulator
Base-bandFrequency
Po
we
rD
en
sityWith incorrect code
(or carrier frequency),SS-signal itself cannot be detected.
They cannot perceive the existence of communication, because of signal behind the noise.
With correct code (and carrier frequency), data can be detected.
Base-bandFrequency
Po
we
rD
en
sity
Security Aspects of Spread Spectrum
Shivkumar KalyanaramanRensselaer Polytechnic Institute
116 : “shiv rpi”
Spreading: Details
Shivkumar KalyanaramanRensselaer Polytechnic Institute
117 : “shiv rpi”
Spreading: Mutually Orthogonal, Walsh Codes
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118 : “shiv rpi”
Spreading: Walsh Codes
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119 : “shiv rpi”
Walsh Codes (Contd)
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120 : “shiv rpi”
Numerical Example: Walsh Codes
-1
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121 : “shiv rpi”
Properties of Walsh Codes
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122 : “shiv rpi”
Multiplexing using Walsh Code
Code for 00
Code for 01
Code for 10
Code for 11
Data
Modulator
Code for 01
Code for 10
Code for 11
0dtT
Select maximum
value
Code for 00
0dtT
0dtT
0dtT
Demodulator
Shivkumar KalyanaramanRensselaer Polytechnic Institute
123 : “shiv rpi”
Freq.Freq.
BPFDespreader
Code B
Freq.Freq.
BPFDespreader
Code A
CDMA is a multiple spread spectrum.
Difference between each communication path is only the spreading code
Data B
Code B
BPF
Freq.Freq.
•••
Data A
Code A
BPF
Freq.Freq.
MS-A
•••
MS-B
BS
Data A
Data B
DS-CDMA System Overview (Forward link)
Shivkumar KalyanaramanRensselaer Polytechnic Institute
124 : “shiv rpi”
The IS-95 CDMA (2G) Forward Link
Shivkumar KalyanaramanRensselaer Polytechnic Institute
125 : “shiv rpi”
Forward Link(Down Link)
Synchronous Chip Timing
A
B
AA
Signal for B Station(after re-spreading)
Less Interference for A station
Synchronous CDMA Systems realized in Point to Multi-point System.e.g., Forward Link (Base Station to Mobile Station) in Mobile Phone.
Synchronous DS-CDMA
Shivkumar KalyanaramanRensselaer Polytechnic Institute
126 : “shiv rpi”
The IS-95 Reverse Link
Shivkumar KalyanaramanRensselaer Polytechnic Institute
127 : “shiv rpi”
In asynchronous CDMA system, orthogonal codes have bad cross-correlation.
Reverse Link(Up Link)
BA
Signal for B Station(after re-spreading)
Big Interference from A station
Asynchronous Chip Timing
Signals from A and B are interfering each other.
A
B
Asynchronous DS-CDMA
Shivkumar KalyanaramanRensselaer Polytechnic Institute
128 : “shiv rpi”
Cross-Correlation: PN Sequences
Cross-Correlationbetween Code A and Code B = 5/16
Self-Correlationfor each code is 16/16.
one data bit duration
Spreading Code A
1 0 11 1 1 0 0 10 1 0 1 0 0 1
one data bit duration
Spreading Code A
1 0 01 1 1 0 0 10 1 0 1 0 0 1
Spreading Code A
1 0 01 1 1 0 0 10 1 0 1 0 0 1
0 0 00 0 0 0 0 00 0 0 0 0 0 0
Spreading Code B
1 0 01 1 0 0 1 11 0 0 1 0 1 1
0 0 00 0 1 0 1 01 1 0 0 0 1 0
0
Shivkumar KalyanaramanRensselaer Polytechnic Institute
129 : “shiv rpi”
In order to minimize mutual interference in DS-CDMA , the spreading codes
with less cross-correlation should be chosen.
Synchronous DS-CDMA :Orthogonal Codes are appropriate. (Walsh code etc.)
Asynchronous DS-CDMA :• Pseudo-random Noise (PN) codes / Maximum sequence
• Gold codes
Preferable Codes
Shivkumar KalyanaramanRensselaer Polytechnic Institute
130 : “shiv rpi”
Generating PN Sequences
Shivkumar KalyanaramanRensselaer Polytechnic Institute
131 : “shiv rpi”
M-Sequences
Autocorrelation: like impulse
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132 : “shiv rpi”
Near-Far Problem: Power Control
Shivkumar KalyanaramanRensselaer Polytechnic Institute
133 : “shiv rpi”
(((
②
①
Open Loop Power Control Closed Loop Power Control
estimating path loss
calculating transmission
power
transmitmeasuring received power
transmit receive
decide transmission
power
transmit measuring received power
power control command
about 1000 times per second
①
②
Power Control (continued)
Shivkumar KalyanaramanRensselaer Polytechnic Institute
134 : “shiv rpi”
Effect of Power Control
AB
Time
De
tect
ed
Po
we
r
from MS B from MS A
closed loop power
control for MS B.
for MS A
.
Effect of Power Control• Power control is capable of compensating the fading fluctuation.
• Received power from all MS are controlled to be equal.
... Near-Far problem is mitigated by the power control.
Shivkumar KalyanaramanRensselaer Polytechnic Institute
135 : “shiv rpi”
CDMA: Issues
Shivkumar KalyanaramanRensselaer Polytechnic Institute
136 : “shiv rpi”
Key: Interference Averaging!
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137 : “shiv rpi”
Voice Activity: Low Duty Cycle
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138 : “shiv rpi”
Variable Rate Vocoders
Shivkumar KalyanaramanRensselaer Polytechnic Institute
139 : “shiv rpi”
Sector Antennas in CDMA
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140 : “shiv rpi”
Capacity Comparison
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141 : “shiv rpi”
Handoff :• Cellular system tracks mobile stations in order to maintain their communication links.
• When mobile station goes to neighbor cell, communication link switches from current cell to the neighbor cell.
Hard Handoff :• In FDMA or TDMA cellular system, new communication establishes after breaking current communication at the moment doing handoff. Communication between MS and BS breaks at the moment switching frequency or time slot.
Hard handoff : connect (new cell B) after break (old cell A)
switching
Cell B Cell A
Soft Handoff
Shivkumar KalyanaramanRensselaer Polytechnic Institute
142 : “shiv rpi”
Σ
Cell B Cell A
Soft handoff : break (old cell A) after connect (new cell B)
transmitting same signal from both BS A and BS B simultaneously to the MS
Soft Handoff :• In CDMA cellular system, communication does not break even at the moment doing handoff, because switching frequency or time slot is not required.
Soft Handoff
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143 : “shiv rpi”
Soft vs Hard Handover Hard handover: the connection to the current
cell is broken, and then the connection to the new cell is made. "break-before-make" handover.
Universal freq. reuse in CDMA "make-before-break" or "soft" handover.
Soft handovers require less power, which reduces interference and increases capacity.
Mobile can be connected to more that two BTS the handover.
"Softer" handover is a special case of soft handover where the radio links that are added and removed belong to the same node.
Shivkumar KalyanaramanRensselaer Polytechnic Institute
144 : “shiv rpi”
CDMA: Rake Receiver for Frequency Diversity
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145 : “shiv rpi”
Path Delay
Po
we
r path-1
path-2
path-3
With low time-resolution,different signal paths cannot be discriminated.
•••These signals sometimes strengthen,
and sometimes cancel out each other, depending on their phase relation.••• This is “fading”.
•••In this case, signal quality is damaged
when signals cancel out each other.In other words, signal quality is dominated
by the probability for detected power to be weaker than minimum required level.
This probability exists with less than two paths.
Time
Po
we
r
Detected Power
In non-CDMA system, “fading” damages signal quality.
Frequency-Selective Fading in non-CDMA Broadband System
Shivkumar KalyanaramanRensselaer Polytechnic Institute
146 : “shiv rpi”
Because CDMA has high time-resolution,different path delay of CDMA signals
can be discriminated.•••Therefore, energy from all paths can be summed
by adjusting their phases and path delays.••• This is a principle of RAKE receiver.
Path Delay
Po
we
r path-1
path-2
path-3
CDMAReceiver
CDMAReceiver
•••
Synchron
ization
Add
er
Path Delay
Po
we
r
CODE Awith timing of path-1
path-1
Po
we
r
path-1
path-2
path-3
Path Delay
Po
we
r
CODE Awith timing of path-2
path-2
interference from path-2 and path-3
•••
Fading in CDMA System: Rake Principle
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147 : “shiv rpi”
In CDMA system, multi-path propagation improves the signal quality by use of RAKE receiver.
Time
Po
we
r Detected Power
RAKEreceiver
Less fluctuation of detected power, because of adding all
energy .
Po
we
r
path-1
path-2
path-3
Fading in CDMA System (continued)
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148 : “shiv rpi”
Frequency Diversity via Rake Receiver (details)
Consider a simplified situation (uncoded). Each information bit is spread into two pseudorandom
sequences xA and xB (xB= -xA).
Each tap of the match filter is a finger of the Rake.
Shivkumar KalyanaramanRensselaer Polytechnic Institute
149 : “shiv rpi”
Frequency Diversity via Rake Receiver
Project y … (assuming h is known)
What the Rake actually does is take inner products of the received signal … with shifted versions of the candidate transmitted sequences. Each output is then weighted by the channel tap gain of the appropriate
delay and summed.
The signal path associated with a particular delay is sometimes called a finger of the Rake receiver.
Shivkumar KalyanaramanRensselaer Polytechnic Institute
150 : “shiv rpi”
Recall: Maximal Ratio Combining (MRC), “Beamforming” , Rake Receiving: are just Matched Filtering operations!
Generalization of this f-domain picture, for combining multi-tap signal
Weight each branch
SNR:
Source: J. Andrews et al, Fundamentals of WIMAX
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151 : “shiv rpi”
Rake Receiver: Max-Ratio-Combiner
Due to hardware limitations, the actual number of fingers used in a Rake receiver may be less than the total number of taps L in the range of the delay spread. => a tracking mechanism in which the Rake receiver
continuously searches for the strong paths (taps) to assign the limited number of fingers to.
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152 : “shiv rpi”
Rake Receiver: Summary Counter-Intuitive: Increase rate and bandwidth PN Code Autocorrelation attenuates ISI Not particularly effective for wideband signals (no spreading
gain)
Shivkumar KalyanaramanRensselaer Polytechnic Institute
153 : “shiv rpi”
ISI vs Frequency Diversity
In narrowband systems, ISI is mitigated using a complex receiver.
In asynchronous CDMA uplink, ISI is there but small compared to interference from other users.
But ISI is not intrinsic to achieve frequency diversity.
The transmitter needs to do some work too!
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154 : “shiv rpi”
Multi-Carrier Modulation and OFDM
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155 : “shiv rpi”
Frequency Diversity & Multicarrier Modulation, i.e. OFDM
Key Idea: Since we avoid ISI if Ts > Tm, just send a large number of narrowband carriers
M subcarriers each with rate R/M, also have Ts’ = Ts*M. Total data rate is unchanged.
subchannel
frequency
ma
gn
itude
carrier
channel
Figure courtesy B. Evans
Shivkumar KalyanaramanRensselaer Polytechnic Institute
156 : “shiv rpi”
Multicarrier Modulation
Breaks data into N substreams Substream modulated onto separate carriers
Substream bandwidth is B/N for B total bandwidth B/N<Bc implies flat fading on each subcarrier (no ISI)
Can overlap substreams (OFDM)
x
cos(2f0t)
x
cos(2fNt)
R bps
R/N bps
R/N bps
QAMModulator
QAMModulator
Serial To
ParallelConverter
Shivkumar KalyanaramanRensselaer Polytechnic Institute
157 : “shiv rpi”
Multicarrier vs Equalizers
Equalizers use signal processing in receiver to eliminate ISI.
Linear equalizers can completely eliminate ISI (ZF), but this may enhance noise. MMSE better tradeoff.
Equalizer design involves tradeoffs in complexity, overhead, and performance (ISI vs. noise). Number of filter taps, linear versus nonlinear, complexity and
overhead of training and tracking
Multicarrier is an alternative to equalization Divides signal bandwidth to create flat-fading subchannels.
Shivkumar KalyanaramanRensselaer Polytechnic Institute
158 : “shiv rpi”
Multicarrier: Time vs Freq. Domain Multicarrier: interesting interpretation in both
time and frequency domains.
In the time domain, the symbol duration on each subcarrier has increased to T = LTs, …
… so by letting L grow larger, it can be assured that the symbol duration exceeds the channel delay spread,
… which is a requirement for ISI-free communication.
In the frequency domain, …the sub-carriers have bandwidth B/L << Bc, … which assures “flat fading”, … the frequency domain equivalent to ISI-free
communication.
Shivkumar KalyanaramanRensselaer Polytechnic Institute
159 : “shiv rpi”
OFDM: Parallel Tx on Narrow Bands
Channel impulse response
1 Channel (serial)
Channeltransfer function(Freq selective fading)
Channels are “narrowband”(flat fading, ↓ ISI)
2 ChannelsFrequency
Frequency
8 ChannelsFrequency
FrequencyTime
Signal is “broadband”
Shivkumar KalyanaramanRensselaer Polytechnic Institute
160 : “shiv rpi”
Multicarrier & ISI
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161 : “shiv rpi”
Issues w/ Multicarrier Modulation
1. Large bandwidth penalty since the subcarriers can’t have perfectly rectangular pulse shapes and still be time-limited.
2. Very high quality (expensive) low pass filters will be required to maintain the orthogonality of the subcarriers at the receiver.
3. This scheme requires L independent RF units and demodulation paths.
OFDM overcomes these shortcomings!
Ch.2 Ch.3 Ch.4 Ch.5 Ch.6 Ch.7 Ch.8 Ch.9 Ch.10Ch.1
Conventional multicarrier techniques frequency
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OFDM OFDM uses a computational technique known as the Discrete Fourier
Transform (DFT) … which lends itself to a highly efficient implementation commonly
known as the Fast Fourier Transform (FFT). The FFT (and its inverse, the IFFT) are able to create a multitude of
orthogonal subcarriers using just a single radio.
Ch.1
Saving of bandwidth
Ch.3 Ch.5 Ch.7 Ch.9Ch.2 Ch.4 Ch.6 Ch.8 Ch.10
Orthogonal multicarrier techniques
50% bandwidth saving
frequency
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Concept of an OFDM signal
Ch.1
Ch.2 Ch.3 Ch.4 Ch.5 Ch.6 Ch.7 Ch.8 Ch.9 Ch.10
Saving of bandwidth
Ch.3 Ch.5 Ch.7 Ch.9Ch.2 Ch.4 Ch.6 Ch.8 Ch.10
Ch.1
Conventional multicarrier techniques
Orthogonal multicarrier techniques
50% bandwidth saving
frequency
frequency
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Spectrum of the modulated data symbols
Rectangular Window of duration T0
Has a sinc-spectrum with zeros at 1/ T0
Other carriers are put in these zeros
sub-carriers are orthogonal
Frequency
Magnitude
T0
Subcarrier orthogonality must be preservedCompromised by timing jitter, frequency offset, and fading.
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OFDM Symbols Group L data symbols into a block known as an OFDM symbol.
An OFDM symbol lasts for a duration of T seconds, where T = LTs. Guard period > delay spread OFDM transmissions allow ISI within an OFDM symbol, but by
including a sufficiently large guard band, it is possible to guarantee that there is no interference between subsequent OFDM symbols.
The next task is to attempt to remove the ISI within each OFDM symbol
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Circular Convolution & DFT/IDFT
Circular convolution:
Detection of X (knowing H):
(note: ISI free! Just a scaling by H)
Circular convolution allows DFT!
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Cyclic Prefix: Eliminate intra-symbol interference! In order for the IFFT/FFT to create an ISI-free channel, the channel must appear to provide a circular
convolution If a cyclic prefix is added to the transmitted signal, then this creates a signal that appears to be x[n]L, and so
y[n] = x[n] * h[n].
The first v samples of ycp interference from preceding OFDM symbol => discarded. The last v samples disperse into the subsequent OFDM symbol => discarded. This leaves exactly L samples for the desired output y, which is precisely what is required to recover the L data symbols embedded in x.
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Cyclic Prefix (Contd) These L residual samples of y will be equivalent to
By mimicking a circular convolution, a cyclic prefix that is at least as long as the channel duration (v+1)…… allows the channel output y to be decomposed into a simple multiplication of the channel frequency response H = DFT{h} and the channel frequency domain input, X = DFT{x}.
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Cyclic Prefix & Circular Convolution
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Circulant Matrix & DFT
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Recall: DFT/Fourier Methods ≡ Eigen Decomposition!
Applying transform techniques is just eigen decomposition! Discrete/Finite case (DFT/FFT):
Circulant matrix C is like convolution. Rows are circularly shifted versions of the first row
C = FΛF* where F is the (complex) fourier matrix, which happens to be both unitary and symmetric, and multiplication w/ F is rapid using the FFT.
Applying F = DFT, i.e. transform to frequency domain, i.e. “rotate” the basis to view C in the frequency basis.
Applying Λ is like applying the complex gains/phase changes to each frequency component (basis vector)
Applying F* inverts back to the time-domain. (IDFT or IFFT)
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Cyclic Prefix overhead
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Cyclic Prefix Overhead: final thoughts
OFDM overhead
= length of cyclic prefix / OFDM symbol time Cyclic prefix dictated by delay spread. OFDM symbol time limited by channel coherence
time. Equivalently, the inter-carrier spacing should be much
larger than the Doppler spread. Since most channels are underspread, the overhead
can be made small.
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OFDM Implementation
1. Break a wideband signal of bandwidth B into L narrowband signals (subcarriers) each of bandwidth B/L. The L subcarriers for a given OFDM symbol are represented by a vector X, which contains the L current symbols.
2. In order to use a single wideband radio instead of L independent narrow band radios, the subcarriers are modulated using an IFFT operation.
3. In order for the IFFT/FFT to decompose the ISI channel into orthogonal subcarriers, a cyclic prefix of length v must be appended after the IFFT operation. The resulting L + v symbols are then sent in serial through the wideband channel.
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-60 -40 -20 0 20 40 60-50
-40
-30
-20
-10
0
10
f [MHz]
pow
er s
pect
rum
mag
nitu
de [
dB] OFDM spectrum for N
FFT = 128, N
w in = 12, N
guard = 24, oversampling = 1
0 20 40 60 80 100 120 140 160 180 200-0.2
-0.1
0
0.1
0.2time domain signal (baseband)
sample nr.
imaginaryreal
OFDM Block Diagram
OFDM modulation
(IFFT)
Channel coding /
interleaving
Guard interval
I/Q I/QSymbol mapping
(modulation)
Transmitter
N symbols
OFDM demod. (FFT)
Decoding / deinter-leaving
Guard interval removal
Time sync.
I/Q I/Q
symbol de-mapping
(detection)
Channel est.
ReceiverFFT-part
time
1 OFDM symbol
Channel impulse response:
0101010010110
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OFDM in WiMAX
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OFDM in Wimax (Contd)
Pilot, Guard, DC subcarriers: overhead Data subcarriers are used to create “subchannels” Permutations & clustering in the time-frequency domain used
to leverage frequency diversity before allocating them to users.
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Example: Flash OFDM (Flarion)
Bandwidth = 1.25 Mz OFDM symbol = 128 samples = 100 s Cyclic prefix = 16 samples = 11 s delay spread 11 % overhead.
• Permutations for frequency diversity for each user (gaps filled by other users)
• Recall: like repetition coding• Efficiency gained across users•(multi-user & frequency diversity)
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Summary: OFDM vs Equalization
CMAC: complex multiply and accumulate operations per received symbol
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P/S
QAM demod
decoder
invert channel
=frequency
domainequalizer
S/P
quadrature amplitude
modulation (QAM)
encoder
N-IFFTadd
cyclic prefix
P/SD/A +
transmit filter
N-FFT S/Premove
cyclic prefix
TRANSMITTER
RECEIVER
N subchannels 2N real samples
2N real samplesN subchannels
Receive filter
+A/D
multipath channel
Summary: An OFDM Modem
Bits
00110
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OFDM: summary
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Channel Uncertainty
In fast varying channels, tap gain measurement errors may have an impact on diversity combining performance.
The impact is particularly significant in channel with many taps each containing a small fraction of the total received energy. (eg. Ultra-wideband channels)
The impact depends on the modulation scheme.
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Summary: Diversity
Fading makes wireless channels unreliable.
Diversity increases reliability and makes the channel more consistent.
Smart codes yields a coding gain in addition to the diversity gain.
This viewpoint of the adversity of fading will be challenged and enriched in later parts of the course.