Point-to-Point Wireless Communication (II): ISI & Equalization, Diversity (Time/Space/Frequency)

Shivkumar KalyanaramanRensselaer Polytechnic Institute

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Point-to-Point Wireless Communication (II):ISI & Equalization,

Diversity (Time/Space/Frequency)

Shiv Kalyanaraman

Google: “Shiv RPI”

[email protected]

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


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Multi-dimensional Fading

Time, Frequency, Space


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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.

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


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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.

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


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Recall: Attenuation, Dispersion Effects: ISI!

Source: Prof. Raj Jain, WUSTL

Inter-symbol interference (ISI)


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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


Flat fading channel

Gaussian channel(no fading)

In a Gaussian channel (no fading) BER <=> Q(S/N)erfc(S/N)


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BER vs. S/N performance: Flat Fading


S/N

BER


Flat fading channel


Flat fading: BER BER S N z p z dzz = signal power level


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BER vs. S/N performance:

ISI/Freq. Selective Channel


S/N

BER


Flat fading channel


Frequency selective fading <=> irreducible BER floor!!!


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BER vs. S/N performance:

w/ Equalization


S/N

BER

Flat fading channel


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


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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


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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


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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


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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


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Example of eye pattern with ISI:Binary-PAM, SRRC pulse

Non-ideal channel and no noise)(7.0)()( Tttthc


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Example of eye pattern with ISI:Binary-PAM, SRRC pulse …

AWGN (Eb/N0=20 dB) and ISI

)(7.0)()( Tttthc


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Example of eye pattern with ISI:Binary-PAM, SRRC pulse …

AWGN (Eb/N0=10 dB) and ISI)(7.0)()( Tttthc


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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


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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


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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)


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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.


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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).


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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



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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:



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Selection Diversity vs MRC



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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:


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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.


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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


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ST-Coding Design: Details

Note: λl here instead of dl

in rotation code analysis


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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)


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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.


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Code Design & Degrees of Freedom


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Antenna Diversity: Tx+Rx = MIMO

Receive(SIMO)

Transmit(MISO)

Both(MIMO)


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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!


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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.


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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


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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).


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Summary: 2x2 MIMO Schemes

Need closed-loop MIMO to be able to reap both diversity and d.f. gains


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Frequency Diversity: MLSD, CDMA Rake, OFDM


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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.


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Approaches

Time-domain equalization (eg. GSM)

Direct-sequence spread spectrum (eg. IS-95 CDMA)

Orthogonal frequency-division multiplexing OFDM (eg. 802.11a, Flash-OFDM)


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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?


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Reduction to Transmit Diversity (Flat-Fading)


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MLSD Achieves Full Diversity

Space-time code matrix for input sequence

Difference matrix for two sequences first differing at

is full rank.


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Uncoded Max Likelihood Seq. Detection (MLSD)

Tradeoff: MLSD too complex!

MLSD:


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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)


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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


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MLSD/Viterbi: Trellis

Note: a trellis is a state diagram that evolves with time as well.


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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



Rake Receiver for Frequency Diversity

Detour: Spread Spectrum, CDMA,

Ref: Chapter 3 & 4, Tse/Viswanath book,Chap 13, 15: A. Goldsmith book



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 ?



Types of CDMA



Spread Spectrum

Spread-spectrum modulation is considered “secondary” modulation after the usual primary modulation.



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)



Chips & Spreading



Processing Gain / Spreading Factor



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



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)



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



Spreading: Time-Domain View



Spreading: Freq-Domain View



If you know the correct spreading sequence (code) ,

RadioFrequency

Po

we

rD

en

sity

received signal


you can find the spreading timing which gives the maximum detected power, and

Accumulate for one bit duration


Demodulated data

Base-bandFrequency

gathering energy !

10110100

1011010010110100 10110100

TIME

0100101110110100 10110100

0 01

1111111100000000 00000000

Demodulation 1/2



If you don’t know the correct spreading sequence (code) •••

Base-bandFrequency

received signal


you cannot find the spreading timing without correct spreading code, and



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



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



Spreading: Details



Spreading: Mutually Orthogonal, Walsh Codes



Spreading: Walsh Codes



Walsh Codes (Contd)



Numerical Example: Walsh Codes

-1



Properties of Walsh Codes



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



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)



The IS-95 CDMA (2G) Forward Link



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



The IS-95 Reverse Link



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



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



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



Generating PN Sequences



M-Sequences

Autocorrelation: like impulse



Near-Far Problem: Power Control



（（（

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)



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.



CDMA: Issues



Key: Interference Averaging!



Voice Activity: Low Duty Cycle



Variable Rate Vocoders



Sector Antennas in CDMA



Capacity Comparison



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



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



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.



CDMA: Rake Receiver for Frequency Diversity



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



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



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)



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.



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.



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:




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.



Rake Receiver: Summary Counter-Intuitive: Increase rate and bandwidth PN Code Autocorrelation attenuates ISI Not particularly effective for wideband signals (no spreading

gain)



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!



Multi-Carrier Modulation and OFDM



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



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



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.



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.



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”



Multicarrier & ISI



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



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



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



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.



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



Circular Convolution & DFT/IDFT

Circular convolution:

Detection of X (knowing H):

(note: ISI free! Just a scaling by H)

Circular convolution allows DFT!



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.



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}.



Cyclic Prefix & Circular Convolution



Circulant Matrix & DFT



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)



Cyclic Prefix overhead



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.



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.



-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



OFDM in WiMAX



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.



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)



Summary: OFDM vs Equalization

CMAC: complex multiply and accumulate operations per received symbol



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





OFDM: summary



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.



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.

Documents

Point-to-Point Wireless Communication (II): ISI & Equalization, Diversity (Time/Space/Frequency)