Point-to-Point Wireless Communication (II):
ISI & Equalization,
Diversity (Time/Space/Frequency)
Shiv Kalyanaraman
Google: “Shiv RPI”
shivkuma@ecse.rpi.edu
Ref: Chapter 3 in Tse/Viswanath texbook
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.
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
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Multi-dimensional Fading

Time, Frequency, Space
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Rensselaer 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
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Single-tap, Flat Fading (Rayleigh) vs AWGN
Why do we have this huge degradation in performance/reliability?
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Rayleigh Flat Fading Channel
BPSK:
Coherent detection.
Looks like
AWGN, but…
Conditional on h,
pe needs to be
“unconditioned”
To get a much
poorer scaling
Averaged over h,
at high SNR.
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BER vs. SNR (cont.)
Frequency-selective channel
(equalization or Rake receiver)
BER
(  Pe )
Frequency-selective channel
(no equalization)
“BER floor”
AWGN
channel
(no fading)
Flat fading channel
SNR
Pe  1 4 0
( 0)
means a straight line in log/log scale
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Typical Error Event
Conditional on h,
When
When
the error probability is very small.
the error probability is large:
Typical error event is due to: channel (h) being in deep fade!
… rather than (additive) noise being large.
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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
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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.
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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

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Recall: Attenuation, Dispersion Effects: ISI!
Inter-symbol interference (ISI)
Shivkumar Kalyanaraman
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Source: Prof. Raj Jain, WUSTL
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Power
Recall: Multipaths: Power-Delay Profile
multi-path propagation
path-1
path-2
path-3
path-2
Path Delay
path-1
path-3
Mobile Station (MS)
Base Station (BS)
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 MultiPath 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) – 1s (outdoor/cellular).
Coherence Bandwidth: Bc = 500kHz (outdoor/cellular) – 20MHz (indoor)
Implications: High data rate: symbol smears onto the adjacent ones (ISI).
Multipath
effects
~ O(1s)
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BER vs. S/N performance: AWGN
In a Gaussian channel (no fading)
BER <=> Q(S/N)
erfc(S/N)
Typical BER vs. S/N curves
BER
Frequency-selective channel
(no equalization)
Gaussian
channel
(no fading)
Flat fading channel
S/N
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BER vs. S/N performance: Flat Fading
BER   BER  S N z  p  z  dz
Flat fading:
z
= signal power level
Typical BER vs. S/N curves
BER
Frequency-selective channel
(no equalization)
Gaussian
channel
(no fading)
Flat fading channel
S/N
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BER vs. S/N performance:
ISI/Freq. Selective Channel
Frequency selective fading <=>
irreducible BER
floor!!!
Typical BER vs. S/N curves
BER
Frequency-selective channel
(no equalization)
Gaussian
channel
(no fading)
Flat fading channel
S/N
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
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BER vs. S/N performance:
w/ Equalization
Diversity (e.g. multipath diversity) <=>
improved
performance
Typical BER vs. S/N curves
BER
Gaussian
channel
(no fading)
Frequency-selective channel
(with equalization)
Flat fading channel
S/N
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Equalization
Step 1 – waveform to sample transformation
Step 2 – decision making
Demodulate & Sample
Detect
z (T )
r (t )
Frequency
down-conversion
For bandpass signals
Received waveform
Receiving
filter
Equalizing
filter
Threshold
comparison
m̂i
Compensation for
channel induced ISI
Baseband pulse
(possibly distored)
Baseband pulse
Sample
(test statistic)
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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 Kalyanaraman
Rensselaer 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
Hc ( f )  Hc ( f ) e
j c ( f )
Non-constant amplitude
Non-linear phase
Amplitude distortion
Phase distortion
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Pulse Shaping and Equalization Principles
No ISI at the sampling time
H RC ( f )  H t ( f ) H c ( f ) H r ( f ) H e ( f )

Square-Root Raised Cosine (SRRC) filter and Equalizer
H RC ( f )  H t ( f ) H r ( f )
H r ( f )  H t ( f )  H RC ( f )  H SRRC ( f )
1
He ( f ) 
Hc ( f )
Taking care of ISI
caused by tr. filter
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
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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)
Distortion
due to ISI
amplitude scale
Noise margin
Sensitivity to
timing error
Timing jitter
timeShivkumar
scale
Kalyanaraman
Rensselaer Polytechnic Institute
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Example of eye pattern with ISI:
Binary-PAM, SRRC pulse

Non-ideal channel and no noise
hc (t )   (t )  0.7 (t  T )
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Example of eye pattern with ISI:
Binary-PAM, SRRC pulse …

AWGN (Eb/N0=20 dB) and ISI
hc (t )   (t )  0.7 (t  T )
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Example of eye pattern with ISI:
Binary-PAM, SRRC pulse …

AWGN (Eb/N0=10 dB) and ISI
hc (t )   (t )  0.7 (t  T )
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Equalizing filters …
Baseband system model

a1
 a  (t  kT ) Tx filter
ht (t )
Channel
hc (t )
Ht ( f )
Hc ( f )
k
k
Ta a
2
3

 a  (t  kT )
k
k
Ta a
2
3
he (t )
He ( f )
âk 
Rx. filter z (t ) zk
hr (t )
Detector
t  kT
Hr ( f )
n(t )
Equivalent model
a1
r (t ) Equalizer
Equivalent system
h(t )
H( f )
H ( f )  Ht ( f )Hc ( f )H r ( f )
z (t )
x(t )
Equalizer z (t )
he (t )
He ( f )
âk 
zk
t  kT
Detector
nˆ (t )
filtered (colored) noise
nˆ(t )  n(t )  hr (t )
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Equalizer Types
Covered later
in slideset
Shivkumar Kalyanaraman
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Source: Rappaport book, chap 7
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Linear Equalizer
• A linear equalizer effectively inverts the channel.
n(t)
Equalizer
1
Heq(f)
Hc(f)
Channel
Hc(f)
• 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)
DFE
n(t)
x(t)
Hc(f)
Forward
Filter
^
x(t)
+
Feedback
Filter
• 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.


=> doesn’t work well w/ low SNR.
Optimal non-linear: MLSE… (complexity grows exponentially w/ delay
spread)
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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.
z (t ) 
N
 c x(t  n )
n N
n

x(t )
c N
n   N ,..., N k  2 N ,...,2 N


c N 1

cN 1
cN

Coeff.
adjustment
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z (t )
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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:
k 0
1
z (k )  
0 k  1,..., N
Adjust
cn nN N

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)
Adjust
c 
N
n n  N

min E ( z (kT )  ak ) 2

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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 Kalyanaraman
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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 transmission
 Based on bit decisions
 Can’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 paths
 Independent fading paths obtained by time, space,
frequency, or polarization diversity
 Combine paths to mitigate fading effects
Tb
Multiple paths unlikely to fade simultaneously
t
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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.
Channel: time
diversity/selectivity,
but correlated across
successive 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
Coding alone is not sufficient!
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Shivkumar Kalyanaraman
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Forward Error Correction (FEC):
Eg: Reed-Solomon RS(N,K)
>= K of N
received
RS(N,K)
Recover K
data packets!
FEC (N-K)
Block
Size
(N)
Lossy Network
Block: of sufficient size: (> coherence time), else need
to interleave, or use with hybrid ARQ
Data = K
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Hybrid ARQ/FEC Model
• Sequence Numbers
• CRC or Checksum
• Proactive FEC
Packets
Timeout
Status Reports
• ACKs
• NAKs,
• SACKs
• Bitmaps
Retransmissions
• Packets
• Reactive FEC
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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.
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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
where
and
This is classic vector detection in white Gaussian noise.
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Repetition Coding: Matched Filtering
hx1 only spans a
1-dimensional space
(similar to MPAM, w/
random channel gains instead!)
||h||
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Multiply by conjugate => cancel
52 phase!
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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
Deep fade ≡ Error event…
Note: this graph plots
reliability (i.e. BER vs SNR)
Repetition code trades
off information rate
(i.e. poor use of deg-of-freedom)
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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)
x1, x2 are two BPSK symbols before rotation (each, either a or –a).
where d1 and d2 are the normalized distances between the codewords in the two
basis directions (axes).
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Product-Distance Criterion
If d1 = 0 or d2 = 0, the the diversity gain of the code is only 1.
product distance
Choose the rotation angle to maximize the worst-case product distance to all
the other codewords:
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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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
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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 lineof-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:
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
66
: “shiv rpi”
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 Kalyanaraman
Rensselaer Polytechnic Institute
67
: “shiv rpi”
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.
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
Source: J. Andrews et al, Fundamentals of WIMAX
68
: “shiv rpi”
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
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
Source: J. Andrews et al, Fundamentals of WIMAX
69
: “shiv rpi”
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:
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
Source: J. Andrews et al, Fundamentals of WIMAX
70
: “shiv rpi”
Selection Diversity vs MRC
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
Source: J. Andrews et al, Fundamentals of WIMAX
71
: “shiv rpi”
Antenna Diversity: Tx
Receive
(SIMO)
Transmit
(MISO)
Both
(MIMO)
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
72
: “shiv rpi”
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?
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
73
: “shiv rpi”
Open-Loop Tx Diversity: Space-Time Coding
Alamouti Code:


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
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
74
: “shiv rpi”
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).
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
75
: “shiv rpi”
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.
Received Signal:

Receiver:
Project on columns of H:
Like MRC, but 3dB (i.e. ½) lower power
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
76
: “shiv rpi”
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.
Alamouti:
Repetition:
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
77
: “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 Kalyanaraman
Rensselaer Polytechnic Institute
78
: “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 Kalyanaraman
Rensselaer Polytechnic Institute
79
: “shiv rpi”
ST-Coding Design: Details
Note: λl here instead of dl
in rotation
code
analysis
Shivkumar
Kalyanaraman
Rensselaer Polytechnic Institute
80
: “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 λ2l equals the rank of the
codeword difference matrix, this is possible only if N ≥ L.
(Recall: determinant
= product of e-values)
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!)
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
81
: “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 Kalyanaraman
Rensselaer Polytechnic Institute
82
: “shiv rpi”
Code Design & Degrees of Freedom
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
83
: “shiv rpi”
Antenna Diversity: Tx+Rx = MIMO
Receive
(SIMO)
Transmit
(MISO)
Both
(MIMO)
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
84
: “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 Kalyanaraman
Rensselaer Polytechnic Institute
85
: “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 Kalyanaraman
Rensselaer Polytechnic Institute
86
: “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 Kalyanaraman
Rensselaer Polytechnic Institute
87
: “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 Kalyanaraman
Rensselaer Polytechnic Institute
88
: “shiv rpi”
Summary: 2x2 MIMO Schemes

Need closed-loop MIMO to be able to reap both diversity and
d.f. gains
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
89
: “shiv rpi”
Frequency Diversity:
MLSD, CDMA Rake, OFDM
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
90
: “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 Kalyanaraman
Rensselaer Polytechnic Institute
91
: “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 Kalyanaraman
Rensselaer Polytechnic Institute
92
: “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 Kalyanaraman
Rensselaer Polytechnic Institute
93
: “shiv rpi”
Reduction to Transmit Diversity (Flat-Fading)
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
94
: “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 Kalyanaraman
Rensselaer Polytechnic Institute
95
: “shiv rpi”
Uncoded Max Likelihood Seq. Detection
(MLSD)
MLSD:

Tradeoff: MLSD too complex!
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
96
: “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 Kalyanaraman
Rensselaer Polytechnic Institute
97
: “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 Kalyanaraman
Rensselaer Polytechnic Institute
98
: “shiv rpi”
MLSD/Viterbi: Trellis
Note: a trellis is a state diagram that evolves with time as well.
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute

99
: “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 Kalyanaraman
Rensselaer Polytechnic Institute
100
: “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 Kalyanaraman
Rensselaer Polytechnic Institute
101
: “shiv rpi”
What is CDMA ?
spread spectrum
Radio Spectrum
Base-band Spectrum
Code B
Code A
B
B
A
Code A
A
B
A
Sender
C
A
B
A
Time
C
C
B
A
B
C
B
Receiver
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
102
: “shiv rpi”
Types of CDMA
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
103
: “shiv rpi”
Spread Spectrum

Spread-spectrum modulation is considered
“secondary” modulation after the usual primary
modulation.
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
104
: “shiv rpi”
Direct Sequence Spread Spectrum

Bit sequence modulated by chip sequence
s(t)
S(f)
sc(t)
Sc(f)
S(f)*Sc(f)
1/Tb
Tc
Tb=KTc

Spreads bandwidth by large factor (K)

Despread by multiplying by sc(t) again (sc(t)=1)

Mitigates ISI and narrowband interference
1/Tc
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
105
: “shiv rpi”
Chips & Spreading
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
106
: “shiv rpi”
Processing Gain / Spreading Factor
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
107
: “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 Kalyanaraman
Rensselaer Polytechnic Institute
108
: “shiv rpi”
ISI and Interference Rejection

Narrowband Interference Rejection
S(f)
I(f)
S(f)
S(f)*Sc(f)
Info. Signal

I(f)*Sc(f)
Despread Signal
Receiver Input
Multipath Rejection (Two Path Model)
S(f)
Info. Signal
S(f)*Sc(f)[a(t)+b(t-)]
Receiver Input
aS(f)
bS’(f)
Despread Signal
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
109
: “shiv rpi”
How to Spread Spectrum
Power
Density
Direct Sequence (DS)
user data
TIME
Modulation
(primary modulation)
Radio
Frequency
Rensselaer Polytechnic Institute
spreading sequence
(spreading code)
Spreading
(secondary modulation)
Power
Density
Base-band
Frequency
data rate
10110100
Tx
Shivkumar Kalyanaraman
110
: “shiv rpi”
Spreading: Time-Domain View
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
111
: “shiv rpi”
Spreading: Freq-Domain View
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
112
: “shiv rpi”
Demodulation 1/2
Power
Density
If you know the correct spreading sequence (code) ,
received signal
TIME
10110100 01001011 10110100
spreading sequence
(spreading code)
Radio
Frequency
gathering energy !
you can find the
spreading timing
which gives the
maximum detected
power, and
Accumulate for
one bit duration
10110100
10110100 10110100 10110100
00000000 11111111 00000000
Demodulated data
0
Base-band
Frequency
Rensselaer Polytechnic Institute
1
0
Shivkumar Kalyanaraman
113
: “shiv rpi”
Demodulation 2/2
Power
Density
If you don’t know the correct spreading sequence (code) •••
received signal
TIME
10110100 01001011 10110100
spreading sequence
(spreading code)
01010101 01010101 01010101
Radio
Frequency
you cannot find
the spreading
timing
without correct
spreading code,
and
10101010 10101010 10101010
10110100 10110100 10110100
Accumulate for
one bit duration
No data can be detected
Demodulated data
Base-band
Frequency
Rensselaer Polytechnic Institute
-
-
-
Shivkumar Kalyanaraman
114
: “shiv rpi”
Security Aspects of Spread Spectrum
Privacy, Security
Power
Density
Power
Density
Power
Density
Power density of SS-signals could be lower than the noise density.
transmitted SS-signal
received signal
Noise
Radio
Frequency
demodulator
With incorrect code
(or carrier frequency),
SS-signal itself
cannot be detected.
They cannot perceive the existence of communication,
because of signal behind the noise.
Rensselaer Polytechnic Institute
115
Base-band
Frequency
With correct code
(and carrier frequency),
data can be detected.
Power
Density
Radio
Frequency
••••••
••••••
Noise
Base-band
Frequency
Shivkumar Kalyanaraman
: “shiv rpi”
Spreading: Details
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
116
: “shiv rpi”
Spreading: Mutually Orthogonal, Walsh Codes
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
117
: “shiv rpi”
Spreading: Walsh Codes
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
118
: “shiv rpi”
Walsh Codes (Contd)
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
119
: “shiv rpi”
Numerical Example: Walsh Codes
-1
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
120
: “shiv rpi”
Properties of Walsh Codes
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
121
: “shiv rpi”
Multiplexing using Walsh Code
Modulator
Code for 00
Code for 01
Data
Code for 10
Demodulator
Code for 11
Code for 00
 dt
T
0
Code for 01
 dt
T
Select
maximum
value
0
Code for 10
 dt
T
0
Code for 11
 dt
T
0
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
122
: “shiv rpi”
DS-CDMA System Overview
(Forward link)
CDMA is a multiple spread spectrum.
BPF
BPF
Data A
BPF
MS-B
•••
BS
Freq.
Despreader
Data B
Code B
•••
Code B
Data A
Code A
Freq.
Freq.
BPF
Data B
Despreader
MS-A
Code A
Freq.
Freq.
Freq.
Freq.
Freq.
Difference between each communication path is only the spreading
code
Shivkumar
Rensselaer Polytechnic Institute
123
Kalyanaraman
: “shiv rpi”
The IS-95 CDMA (2G) Forward Link
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
124
: “shiv rpi”
Synchronous DS-CDMA
Synchronous CDMA Systems realized in Point to Multi-point System.
e.g., Forward Link (Base Station to Mobile Station) in Mobile Phone.
Forward Link
(Down Link)
Synchronous Chip Timing
Ａ
A
A
Less Interference for A station
Ｂ
Signal for B Station
(after re-spreading)
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
125
: “shiv rpi”
The IS-95 Reverse Link
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
126
: “shiv rpi”
Asynchronous DS-CDMA
Reverse Link
Asynchronous Chip
Timing
(Up Link)
A
A
B
Big Interference
from A station
B
Signal for B Station
(after re-spreading)
Signals from A and B are
interfering each other.
In asynchronous CDMA system, orthogonal codes have bad cross-correlation.
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
127
: “shiv rpi”
Cross-Correlation: PN Sequences
Spreading Code A
Spreading Code A
1 0 1 01 1 1 0 0 0 1 1 0 1 0 0 1
1 01 0 11 0 0 0 11 0 1 00 1
one data bit duration
one data bit duration
Spreading Code A
Spreading Code B
1 01 0 11 0 0 0 11 0 1 00 1
1 01 0 10 0 1 1 10 0 1 01 1
0 00 0 00 0 0 0 00 0 0 00 0
0 00 0 01 0 1 1 01 0 0 01 0
Self-Correlation
for each code is 16/16.
Cross-Correlation
between Code A and Code B = 5/16
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
128
: “shiv rpi”
Preferable Codes
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
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
129
: “shiv rpi”
Generating PN Sequences
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
130
: “shiv rpi”
M-Sequences
Autocorrelation:
like impulse
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
131
: “shiv rpi”
Near-Far Problem: Power Control
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
132
: “shiv rpi”
Power Control (continued)
Open Loop Power Control
Closed Loop Power Control
①
②
measuring
received power
decide
transmission
power
transmit
estimating path
loss
power control
command
about 1000 times
per second
transmit
calculating
transmission
power
transmit
②
①
measuring
received power
receive
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
133
: “shiv rpi”
Effect of Power Control
Effect of Power Control
• Power control is capable of compensating the fading fluctuation.
• Received power from all MS are controlled to be equal.
Detected Power
... Near-Far problem is mitigated by the power control.
Ｂ
from MS B
from MS A
Time
Ａ
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
134
: “shiv rpi”
CDMA: Issues
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
135
: “shiv rpi”
Key: Interference Averaging!
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
136
: “shiv rpi”
Voice Activity: Low Duty Cycle
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
137
: “shiv rpi”
Variable Rate Vocoders
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
138
: “shiv rpi”
Sector Antennas in CDMA
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
139
: “shiv rpi”
Capacity Comparison
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
140
: “shiv rpi”
Soft Handoff
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.
switching
Cell A
Cell B
Hard handoff : connect (new cell B) after break (old cell A)
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
141
: “shiv rpi”
Soft Handoff
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.
transmitting same signal from both BS A and
BS B simultaneously to the MS
Σ
Cell
B
Cell A
Soft handoff : break (old cell A) after connect (new cell B)
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
142
: “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 Kalyanaraman
Rensselaer Polytechnic Institute
143
: “shiv rpi”
CDMA: Rake Receiver for Frequency Diversity
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
144
: “shiv rpi”
Power
Frequency-Selective Fading in
non-CDMA Broadband System
path-1
path-2
path-3
Power
Path Delay
Detected Power
Time
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.
In non-CDMA system, “fading” damages signal quality.
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
145
: “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-2
path-3
CDMA
Receiver
Power
Path Delay
•••
CODE A
with timing of path-2
Path Delay
Power
CDMA
Receiver
Synchronization
Adder
CODE A
with timing of path-1
interference from path-2 and path-3
path-1
path-2
Path Delay
path-3
Power
path-1
path-2
path-1
•••
Power
Fading in CDMA System: Rake Principle
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
146
: “shiv rpi”
Fading in CDMA System (continued)
In CDMA system, multi-path propagation improves
the signal quality by use of RAKE receiver.
Power
path-3
path-2
Power
path-1
Detected Power
Time
RAKE
receiver
Less fluctuation of detected power,
because of adding all energy .
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
147
: “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 Kalyanaraman
Rensselaer Polytechnic Institute
148
: “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 Kalyanaraman
Rensselaer Polytechnic Institute
149
: “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:
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
Source: J. Andrews et al, Fundamentals of WIMAX
150
: “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.
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
151
: “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 Kalyanaraman
Rensselaer Polytechnic Institute
152
: “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!
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
153
: “shiv rpi”
Multi-Carrier Modulation and OFDM
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
154
: “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.

magnitude
channel
carrier
subchannel
frequency
Figure courtesy B. Evans
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
155
: “shiv rpi”
Multicarrier Modulation
R/N bps
R bps
Serial
To
Parallel
Converter
QAM
Modulator
x
cos(2pf0t)
R/N bps
QAM
Modulator
S
x
cos(2pfNt)



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)
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
156
: “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 Kalyanaraman
Rensselaer Polytechnic Institute
157
: “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 Kalyanaraman
Rensselaer Polytechnic Institute
158
: “shiv rpi”
OFDM: Parallel Tx on Narrow Bands
Channel impulse
response
Time
Frequency
1 Channel (serial)
Frequency
2 Channels
Channel
transfer function
(Freq selective fading)
Signal is
“broadband”
Frequency
8 Channels
Frequency
Channels are
“narrowband”
(flat fading, ↓ ISI)
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
159
: “shiv rpi”
Multicarrier & ISI
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
160
: “shiv rpi”
Issues w/ Multicarrier Modulation
Ch.1
Ch.2
Ch.3
Ch.4
Ch.5
Ch.6
Conventional multicarrier techniques
Ch.7
Ch.8
Ch.9
Ch.10
frequency

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!


Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
161
: “shiv rpi”
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.2 Ch.4 Ch.6
Ch.8 Ch.10
Ch.1 Ch.3 Ch.5
Ch.7 Ch.9
Saving of bandwidth
50% bandwidth saving
Orthogonal multicarrier techniques
frequency
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
162
: “shiv rpi”
Concept of an OFDM signal
Ch.1
Ch.2
Ch.3
Ch.4
Ch.5
Ch.6
Ch.7
Ch.8
Ch.9
Conventional multicarrier techniques
Ch.10
frequency
Ch.2 Ch.4 Ch.6
Ch.8 Ch.10
Ch.1 Ch.3 Ch.5
Ch.7 Ch.9
Saving of bandwidth
50% bandwidth saving
Orthogonal multicarrier techniques
frequency
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
163
: “shiv rpi”
Spectrum of the modulated data symbols




Rectangular Window of
duration T0
Has a sinc-spectrum with
zeros at 1/ T0
Magnitude
T0
Other carriers are put in
these zeros
 sub-carriers are
orthogonal
Frequency
Subcarrier orthogonality must be preserved
Compromised by timing jitter, frequency offset, and fading.
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
164
: “shiv rpi”
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
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
165
: “shiv rpi”
Circular Convolution & DFT/IDFT

Circular convolution:

Circular convolution allows DFT!

Detection of X (knowing H):
(note: ISI free! Just a scaling by H)
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
166
: “shiv rpi”
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.

Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
167
: “shiv rpi”
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}.

Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
168
: “shiv rpi”
Cyclic Prefix & Circular Convolution
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
169
: “shiv rpi”
Circulant Matrix & DFT
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
170
: “shiv rpi”
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)
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
171
: “shiv rpi”
Cyclic Prefix overhead
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
172
: “shiv rpi”
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.

Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
173
: “shiv rpi”
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.
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
174
: “shiv rpi”
OFDM Block Diagram
Transmitter
0110
Symbol
mapping
(modulation)
010101001
Receiver
Decoding /
deinterleaving
power spectrum magnitude [dB]
Channel
coding /
interleaving
OFDM
modulation
(IFFT)
I/Q
I/Q
Guard
interval
OFDM spectrum for NFFT = 128, Nw in = 12, Nguard = 24, oversampling = 1
N symbols
10
0
-10
1 OFDM symbol
-20
-30
-40
-50
symbol de-60
mapping
0.2
(detection)
Channel 0.1
impulse
Channel response:
est.
FFT-part
Guard
-40 OFDM
-20
0
20
f
[MHz]
demod.
interval
time domain signal (baseband)
(FFT)
removal
I/Q
40
I/Q
Time sync.
0
-0.1
Rensselaer Polytechnic Institute
-0.2
60
time
imaginary
real
0
20
175
40
60
80
Shivkumar Kalyanaraman
100
120
sample nr.
140
160
180
200
: “shiv rpi”
OFDM in WiMAX
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
176
: “shiv rpi”
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.
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
177
: “shiv rpi”
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)
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
178
: “shiv rpi”
Summary: OFDM vs Equalization
Shivkumar
Kalyanaraman
CMAC: complex multiply and accumulate operations per
received
symbol
Rensselaer Polytechnic Institute
179
: “shiv rpi”
Summary: An OFDM Modem
N subchannels
Bits
00110
S/P
quadrature
amplitude
modulation
(QAM)
encoder
2N real samples
add
cyclic
prefix
N-IFFT
P/S
D/A +
transmit
filter
TRANSMITTER
multipath channel
RECEIVER
N subchannels
P/S
QAM
demod
decoder
invert
channel
=
2N real samples
N-FFT
frequency
domain
equalizer
remove
S/P cyclic
prefix
Receive
filter
+
A/D
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
180
: “shiv rpi”
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
181
: “shiv rpi”
OFDM: summary
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
182
: “shiv rpi”
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.
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
183
: “shiv rpi”
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.
Shivkumar Kalyanaraman
Rensselaer Polytechnic Institute
184
: “shiv rpi”