OFDM_slides(Good Basic Material)

8/3/2019 OFDM_slides(Good Basic Material)

1/58

Spring'09 ELE 739 - OFDM 1

Orthogonal Frequency Division Multiplexing

(OFDM)


2/58


Frequency Selectivity of the Channel

System bandwidth vs. Coherence Bandwidth

Frequency Non-Selective Channel Frequency-Flat channel

Channel impulse response is a

simple impulse function.

Detecting symbols at the MF outputis optimum in the ML sense

No need for an equalizer/MLSD.

Generally low data rate.

Frequency Selective Channel Channel impulse response has a

certain width in time, max. Detecting symbols at the MF output

is NOT optimum in the ML sense An equalizer/MLSD required.

Generally high data rate.


3/58


SC vs. MC If we use a single carrier modulation scheme,

For relatively low bit rate there is no problem

Channel is flat,

No equalizer is required.

For increased data rate, system bandwidth will increase

May cause ISI

Gets worse as data rate (system BW) increases.

ISI causes severe error in the detected symbols.

An Equalizer/MLSD is required for better reception.

What if we have many independent low bit rate (system BW)transmissions in parallel?

Single Carrier Multi-Channel (Carrier)

no.chnl=1, Rk=R, Rk=R no.chnl=N, Rk=R/N, Rk=R

W R >> Bc Wk R/N


4/58


FDMA

One way of generating independent multi-channel systems is to divide

the frequency range into smaller parts subcarriers (freq. bins)

How can we seperate subchannels in freq. so that they do not interfere

FDMA: Subcarriers must be separated at least by the BW of the xmission

Waste of precious spectrum.


5/58


OFDM

Instead place the subcarriers at frequencies

Obviously,

Pulse shape is rectangular spectrum is the sinc function.

Spectra of the above pulses overlap but the sub-carrier frequencies areplaced at the spectral nulls of all other pulses.


6/58


OFDM

How can we generate these pulses,

Analog way, have N oscillators at frequencies

Many practical problems

ci,0

ci,1

ci,N-1

ci,0

ci,1

ci,N-1

~

~

~

(i+1)T


7/58


OFDM

Alternative (digital) way, use the IFFT/FFT pair

Much easier to implement on a digital platform

Overcomes the problems of the analog implementation

ci,0

ci,1

ci,N-1

ci,0

ci,1

ci,N-1

~

~

~

(i+1)T


8/58


OFDM

Consider the transmitted signal

where the normalized (rectangular) basis pulse gn(t) is

Now, w.l.o.g. consider only i= 0, and sample at instances

This is the IDFT of the transmit symbols {c0,n} IfNis a power of 2, can be realized by IFFT.

i: OFDM Symbol indexn: Subcarrier index

IFFTN

(n)N

(k)

freq. time


9/58


OFDM {sk} are time samples transmitted sequentially in time P/S

At the receiver, the procedure is reversed

Collect N samples in time S/P

FFT

Obtain the estimates cn regarding the transmitted cn.

Works fine for the AWGN channel, subcarriers are orthogonal.

~

time

freq.N datasymbols

OFDMSymbol

(frame)

Guard

Interval

time

freq.N datasymbols

DataPacketDelay

dispersionDelay

dispersion

Single Freq.

OFDM


10/58


Cyclic Prefix Delay dispersion may destroy the orthogonality of the subcarriers.

Causes Inter-Carrier Interference (ICI).

Can be prevented by adding a Cyclic Prefix (CP) to the OFDM symbol. Copy the last Ncp samples of the OFDM symbol to the beginning.

Number of samples per OFDM symbol increases from Nto N+Ncp.

IfNcp L-1 (delay dispersion of the channel (no. samples)) ICI is prevented.


11/58


CP

Define a new basis function

where W/N is the carrier spacing and Ts=N/W.

OFDM symbol duration is Ts=Ts+Tcp. -Tcp < t < 0 : cyclic prefix part.

0 < t < T s : data part.

Normally, the signal arriving from a delay-dispersive channel is the

linear convolution of the transmitted signal and the channel IR.

CP converts this linear convolution to cyclical convolution.

If max Tcp.

^

^

^


12/58


CP

At the receiver, to eliminate Linear ISI from the previous OFDM

symbol (i-1), simply discard the CP part of the signal received

corresponding to OFDM symbol i.

We end up with cyclical ISI, totally contained in the i-th symbol.

Assume that Tcp = max, also let i=0.

In the receiver, there is bank of filters matched to the basis functionswithout the CP:

After removing CP, can be implemented by the FFT operation.

OFDM symbol i OFDM symbol i+1OFDM symbol i-1

Linear ISICyclical ISICP data


13/58


OFDM

The signal at the output of the MF is the convolution of the

Transmit signal (transmit data + IFFT),

The channel impulse response, and

The receive filter:

nn: noise at the MF output. Assume quasi-static channel, h(t, )=h()

Substituting gk(t) into the inner integral:


14/58


OFDM

Moreover, since the basis functions are orthogonal during 0 < t < Ts

The OFDM system is respresented by a number of parallel and

orthogonal (non-interfering) non-dispersive (flat) fading channels,

each with its own complex attenuation H(nW/N).

Equalization is very simple:

Divide signal from each subchannel by the transfer function at that freq.

^


15/58


OFDM

A simple structure which employs FFT/IFFT

Very simple equalization (1-tap) if subcarriers are orthogonal

ICI occurs when subcarriers are not orthogonal,

Happens when max Tcp

CP does not convey data, only used to prevent ISI/ICI

Decreases useful SNR

Decreases throughput by Tcp/(Ts+Tcp)


16/58


Another Perspective

Assume that a data block composed of N symbols.

Consider two consequent data blocks.

c0,i-1

c1,i-1

cN-1,i-1

c0,i

c1,i

cN-1,i

(i-1)thdata block. ithdata block.


17/58


Another Perspective

FFT matrix

IFFT matrix:

Time-domain transmit signal:

Parallel-to-serial convert and trasmit through the channel.


18/58


Another Perspective

If we transmit both sequentially

In an ISI channel with length L taps

Last L-1 samples of the (i-1)th OFDM symbol will interfere the first L-1

samples of the ith OFDM symbol (in time).

Causing ISI between OFDM symbols.

c0,i-1 c1,i-1 cN-1,i-1

(i-1)th

OFDM symbol.

c0,i c1,i cN-1,i

ith

OFDM symbol.

L-1

ISI


19/58


Guard Period

In order to prevent ISI, place a guard period of at least L-1 samplesbetween adjacent OFDM symbols.

Place all-zeros in the guard period.

No ISI between OFDM symbols ISI only within an OFDM symbol controllable

Received signal is the convolution of the transmit signal and the

channel.

GP

L-1

ISI

No ISI

ISI


20/58


Received Signal

GP

Co

llecttheseN

samplesforFFT

discard

CollecttheseN

sam

plesforFFT

discard

GP


21/58


22/58


FFT/IFFT

Now consider the system after removing CP

Taking FFT ofyi, we obtain the estimates ofcn,i

For a one-to-one relation between cn,iand cn,i, i.e. without ICI

must be a diagonal matrix.

Q must diagonalize H.

^


23/58


Diagonalization

Let us consider the Eigendecomposition of the matrix H

where E is a unitary matrix containing the eigenvectors ofH, and

is a diagonal matrix with the eigenvalues ofH on the main diagonal.

For an arbitrary H matrix E will also be arbitrary.

To diagonalize the channel, we should use this matrix (EHmore

precisely) at the transmitter and at the receiver.

Channel estimation is performed at the receiver, transmitter does

not know the channel, hence E.

We need a feedback channel to move the E matrix to the transmitter,

Not practical in many cases.


24/58


Diagonalization

As a special case, when the H is a circulant matrix, E becomes the

FFT matrix.

For our previous example with N=8, L=3,

?


25/58


Cyclic Prefix

Instead of a Guard Period L-1 samples, repeat the last L-1 samples

ofsias the Cyclic Prefix at the beginning of and OFDM symbol.

same


26/58


Diagonalization

The eigenvectors ofHcirc form the FFT matrix, hence the

eigenvalues correspond to the samples of the transfer function of

the channel


27/58


Cyclic Prefix

FFT and IFFT (DFT/IDFT) are the pairs for

Circular convolution in the time domain, and

Product of the transfer functions in the frequency domain.

For regular packet structure, ie. with the zero guard period, we have

the linear convolution of the transmit signal and channel at the

channel output.

Product of the transfer functions will not give what we want.

Cyclic prefix enables circular convolution


28/58


Channel Estimation

We have seen the estimation of the Channel Impulse Response for

single-carrier modulation.

In OFDM, we have a number of narrowband (freq. flat) subcarriers

Each subcarrier channel can be represented by a single complex

coefficient, Hn,i

In total these coef.s give the Transfer Function of the channel. We wish to obtain the estimates of the N samples of the Transfer

Function.

There are three approaches:

Pilot OFDM symbols,

Scattered pilot symbols,

Eigenvalue-decomposition based methods.


29/58


Pilot-OFDM Symbol Based

We have a dedicated OFDM symbol containing known data.

Data on each subcarrier is known.

Appropriate for initial acquisition of the channel at the beginning of a

transmission burst.

If the known data on subcarriern for OFDM symbol index iis cn,i, then

in the LS sense, or the Transfer Function is


30/58


31/58


Pilot-OFDM Symbol Based

Produces very good estimates (the transfer function of the OFDMsymbol of concern).

Very high computational complexity.

Suitable for initial channel acquisition.

Uses all OFDM frame as pilot: no room for data.

Less than N subcarrier can be used as pilot tones since neighbouringsubcarriers are correlated (in frequency).

Time correlation can be exploited: subsequent OFDM symbols are notused as pilots.


32/58


Scattered Pilots (in time & frequency)

We could use a grid structure for the pilots, i.e.

Pilots scattered in frequency and time

What should be the spacing between the pilots? Nt= ?

Nf= ?

Sampling Theory:

Interpolate between

the pilots to estimate the

channel.


33/58


Scattered Pilots (in time & frequency)

Interpolation can be done by MMSE estimation,

Let be the LS estimates of the

pilot tones, then

where

Less complexity.


34/58


Eigenvalue Decomposion Based Methods

Length of the Channel Impulse Response < OFDM symbol length

Channel can be represented by less coefficients in the time domain.

Consider the LMMSE estimator:

, approximately Ncp

+1 eigenvalues of will havesignificant values, rest can be ignored.


35/58


Channel Shortening

Assume that the length of the communication channel is L taps,

Then a CP of length at least L-1 is required in an OFDM symbol to

eliminate ISI/ICI,

Spectral Efficiency of OFDM with Nsubcarriers is

IfL is fixed, Ndetermines the efficiency, N (gets closer to 1)

We cannot arbitrarily increase Ndue to the time-selectivity of the

channel.

Then we may decrease L ? Equalization effectively decreases L to 1.

If the system allows taps for CP, we can have an equalized

channel of length +1 taps more degrees of freedom.


36/58


Channel Shortening

Complete Equalization

Channel Shortening Equalization

Filter, wEffective

Channel, f+ +

Delay,

x[n] t[n]

z[n]

-

[n]

[n]z[n]

TIR, b

^

Forced to be these values.

Freely determined by the

cost function.{bk} are variable!


37/58


Channel Shortening (MMSE)

Design a receiver filter ofnw taps

whose output is

Toeplitz Matrix

Same expressions as we have seen before.

?


38/58


Cost Function

Now, the MMSE cost function is

We can proceed with the same derivations as complete equalization

by substituting

For complete equalization

we have


39/58


Cost Function

Expanding the cost function

Using the property that data and noise are uncorrelated E{x*}=0

This is a quadratic function ofw and b, first optimize wrt. w


40/58


41/58


Optimum TIR

To avoid the trivial solution, impose a constraint on the filters:

Unit norm constraint on w, wTw=1

Unit tap constraint on TIR, bk=1, bk: variable

Unit norm constraint on TIR, bTb=1

Unit norm constraint on TIR gives better performance:

Cost and constraint are convex: Use Lagrangian method

This is the eigenvalue

problem.

Hermitian symmetric


42/58


Optimum TIR

defines a square windows in T

Lagrangian becomes

is the minimum eigenvalue ofT and b is the corresponding

eigenvector.

Obviously, this still a function of delay . Above problem has to besolved for every possible .

T=

P k A P R i (PAPR)


43/58


Peak-to-Average Power Ratio (PAPR)

For frequency-time domain conversion we use the FFT/IFFT matrix:

Let the symbols to be transmitted be

Then the transmitted signal is ( + CP)

P k t A P R ti (PAPR)


44/58



Most practical RF power amplifiers have limited dynamic range.

They can be linear only in a limited range.

There is no problem for a constant modulus signal since theamplitude of the signal does not change.

But, even if the symbols to be transmitted are drawn from a constant

modulus constellation (such as M-PSK), the output of the IFFT

operator may have different amplitudes for every sample (in time).

Example: constantmodulus

non-constant

modulus

time IFFT freq.

P k t A P R ti (PAPR)


45/58



The problem is; with the IFFT operation, the symbols on the

subcarriers sometimes add constructively and at other times

destructively. Amplitude of the signal to be transmitted is proportional to N

Power goes with N2

Another point of view is We can consider the symbols on subcarriers as random variables

If the number of subcarriers is large, adding these symbols up will result in

a complex Gaussian distribution with a variance of unity (mean power),

due to the central limit theorem. Absolute amplitude is Rayleigh distributed.

Example: Probability that the peak power is 6 dB above the average power

4 times

I t f th A lit d Di t ib ti


46/58


Impact of the Amplitude Distribution

An amplifier that can amplify nearly up to the possible peak value of

the transmit signal is not practical

Requires expensive, inefficient class-A amplifiers.

Using a non-linear amplifier will cause distortions in the output signal.

Destroys the orthogonality of the subcarriers, causes ICI, increased BER.

Nonlinearity causes spectral regrowth, increased out-of-band emissionsinterfering systems in the neighbouring frequency bands.

We may use PAPR reduction techniques.

C di f PAPR R d ti


47/58


Coding for PAPR Reduction

In error control coding, for an (n,k) code and M-ary modulation, There are Mn possible combinations of the symbols.

But, we use a subset of these combinations of dimension Mk(#codewords)

We choose one these Mkcodewords of length n, i.e. For sending ksymbols

we use n symbols (n k) Redundancy of (n - k) symbols.

Codewords chosen wisely, so that the distance between them is maximized.

We can think of an OFDM symbol as a possible combination ofN, M-ary

symbols (MNsymbols)

Among these combinations, choose appropriate codewords so that PAPR is

guaranteed to be below a certain level. Completely eliminates PAPR problem,

Significant loss of throughput due to redundancy,

Have some coding gain (but less than a dedicated ECC code).

Phase Adjustments


48/58


Phase Adjustments

Define an ensemble of phase adjustment vectors I={n}l, l= 1,...,L,n = 1,...,N

Known both at the transmitter and receiver.

Transmitter multiplies the OFDM symbol to be transmitted Ciby each ofthese phase vectors to get

and then selects:

to get the lowest PAPR possible.

Instead of the sequence {cn}, {cn}l is transmitted together with the index l.

Receiver undos phase adjustment by using the index l.

Less overhead,

Cannot guarantee a certain level of PAPR.

^ ^^^

Correction by Multiplicative Function


49/58


Correction by Multiplicative Function

Multiply the OFDM signal by a time-dependent function whenever the

peak value is high.

Simplest example, Clipping (penalize by saturating the output) If signal attains a level sk>A0, multiply the signal to be transmitted byA0/sk, i.e.

Another example, multiply the signal by a Gaussian function centered attimes when the level exceeds the threshold (penalized by the Gauss func.)

Multiplying by a Gaussian in time is equivalent to convolving with a Gaussian infrequency spectral regrowth controlled by t

2

Causes significant ICI increased BER.

Correcting by Additive Function


50/58


Correcting by Additive Function

Instead of multiplicative, we can use an additive correction function.

Correction function acts as pseudo-noise increased BER.

No best PAPR reduction technique. There is trade-offbetween

PAPR

Redundancy/Overhead

Guaranteeing a certain PAPR level

ICI/BER

Out-of-band interference.

Inter Carrier Interference


51/58


Inter-Carrier Interference

Cyclic prefix completely eliminates Inter-Carrier Interference and Inter-Symbol Interference caused by the quasi-static frequency selective

channel

If the channel delay spread is less than the cyclic prefix

If the channel is time-varying (time-selective) and changes within an

OFDM symbol,

Orthogonality of subcarriers is destroyed, Doppler shift of one subcarrier causes ICI in many adjacent subcarriers.

Doppler shiftedSubcarrier.

ICI


52/58


ICI

Impact of time-selectivity is mostly determined by

Product of maximum Doppler frequency and duration of the OFDM symbol.

Spacing between subcarriers is inversely proportional to symbol duration

Large symbol duration, a small Doppler shift can cause considerable ICI.

Delay dispersion can also be a source of ICI if CP is shorter than the maximum excess delay.

Length of the channel maybe changing from time to time.

Tradeoff: Large excess delay requires long CP reduced spectral efficiency.

Shorter CP ICI. CP does not have to be chosen to cope with the worst case channel if the

loss due to ICI is amenable, to increase spectral efficiency.

ICI


53/58


ICI

Received signal in case of ICI occuring as a result of Doppler shift or

insufficient CP.

h[n, l] : sampled version of the time-variant channel IR h(t, )

L : maximum excess delay in units of samples L = maxN/Ts.

u[n] : unit step function.

For a time-invariant channel (h[q, l]=h[ l][q-l]) and sufficiently longCP, above expression reduces to

ICI


54/58


ICI

Optimum choice of carrier spacing and OFDM symbol length: Tradeoff between ICI and spectral efficiency (N/(N+Ncp))

Short symbol duration (Ts) (large subcarrier spacing) is good for

reducing Doppler-caused ICI,

Long symbol duration (narrow spacing) is good for satisfactory

spectral efficiency

TCP

is limited by the maximum excess delay,

CP should be around 10% of the OFDM symbol for high efficiency

Choose Ts (N) to maximize

: function of the channel

Adaptive Modulation


55/58


Adaptive Modulation

Transfer function of a frequency selective channel has peaks and

valleys.

Subchannels at peaks are good (narrowband) channels Almost no information can be transmitter through valleys

Energy over subcarriers

are the same for this example.

For satisfying a target BERe.g. 10-2, too much energy

would be required at these

subcarriers.

Power is limited!.

Adaptive Modulation


56/58


Adaptive Modulation

We see that fixed power allocation loads bad channels with low

SNR (high BER and low capacity) waste of energy.

Problem: We want to maximize the capacity of the system by wisely distributing

the energy over subchannels

Under the constraint of limited power

Solution is the waterfilling algorithm

where is the water level chosen to satify

Adaptive Modulation


57/58


Adaptive Modulation

Total power

Power allocatedto subcarrier - 1

No power is allocated

to this subcarrier

waterlevel,

Adaptive Modulation


58/58


Adaptive Modulation

We have found the power allocation for each subcarrier which

maximizes the capacity.

What modulation should we use in the subcarriers to get as close aspossible to the assigned capacity?

This means, the transmitter has to adapt the data rate according to the

SNR available for a subcarrier.

A constellation with Na points has a capacity of log2(Na) bits/channel use

A higher order modulation (64-128-QAM) can be used for a subcarrier

with high SNR, For low SNR, modulation order has to be decreased.

OFDM_slides(Good Basic Material)

Documents

Transcript of OFDM_slides(Good Basic Material)