Orthogonal Frequency Division Multiplexingwireless.phys.uoa.gr/docs/2009/ofdm.pdf · Orthogonal...

Introduction Frequency Division Multiplexing Orthogonal Frequency Division Multiplexing Signal Processing in OFDM Summary

Orthogonal Frequency Division Multiplexing

Stelios Stefanatos

Department of Physics,National & Kapodistrian University of Athens,

Athens, Greece

April 7, 2009

Stelios Stefanatos Orthogonal Frequency Division Multiplexing


Outline

1 Introduction

2 Frequency Division Multiplexing

3 Orthogonal Frequency Division Multiplexing

4 Signal Processing in OFDM

5 Summary



System Model (Complex Baseband Notation)

Information (transmitted) signal:s(t), t ∈ [0,T],S(f ) , (Fs)(f ), f ∈ [−W,W]

Energy Constraint: E{∫|s(t)|2dt} = E{

∫|S(f )|2df} ≤ P

LTI channel:c(t), t ∈ [0,Tc]C(f ) , (Fc)(f ), f ∈ [?, ?]

Received signal:y(t) , s(t) ∗ c(t) + w(t), t ∈ [0,T], (Tc << T; realistic?)Y(f ) = C(f )S(f ) + W(f ), f ∈ [−W,W]

Questions:information theoretic limitssignal structure for simple processing



Capacity of the LTI Channel

Assumption: Tx and Rx have perfect Channel State Information(CSI)

System’s mutual information (in bits/sec/Hz):

I(y(t); s(t)) = 12W

∫ W−W log2

(1 + |C(f )|2P(f )

N0

)df

P(f ) , E{|S(f )|2} ≥ 0,∫ W−W P(f ) = P

Capacity achieved by:s(t) should be a white (circularly symmetric) Gaussian processPower P should be allocated in frequency in a waterfilling manner



Waterfilling Power Allocation



Channel compensation (equalization)

Single carrier linearly modulated signal:

s(t) =∑K−1

k=0 xkgT(t − kTs), t ∈ [0,T]

xk ∈ A, Ts ≈ 1/(2W) (Nyquist pulses)

Question: How do we choose A and gT(t) so that s(t)

1 is Gaussian(-like)?2 has a non–flat power spectral density

Generic Rx waveform processing (generation of sufficientstatistics)

Received signal y(t) goes through a low pass filter gR(t)Sampling at the symbol period (Nyquist sampling)



Discrete-time signal processing

Discrete-time I/O relation:

y[k] , y(kTs) =∑Lh−1

l=0 h[l]xk−l + w[k], k = 0, . . . ,K − 1

h[k] = h(kTs), h(t) , gR(t) ∗ c(t) ∗ gT(t)K = bT/TscLh = bTh/Tsc : # of channel tapsSmaller Ts results in larger Lh (increasing ISI)!w[k] :AWGN

Matrix-vector relationship:

y = T (h)x + w

y = [y[0], y[1], . . . , y[K − 1]]T , h, x, w similarly definedT (h): K × K Toeplitz matrix with first column h (convolutionmatrix)



Discrete-time signal processing

T (h) =

h[0]h[1] h[0]

h[1]. . .

......

. . .

h[Lh − 1]. . .

h[Lh − 1]. . .

h[Lh − 1] · · · h[1] h[0]



Data Detection

Optimal decision rule:x = arg minx∈AK ||y− T (h)x||2

For {xk} i.i.d. from an M-ary constellation“Brute force” maximization complexity: O(MK)Dynamic programming (Viterbi) complexity: O(MLh−1)

Complexity unacceptable for high-rate/broadband applicationsExample: DSL channel has Lh ≈ O(100)

Low complexity channel compensation (equalization) is required

linear equalization (ZF, MMSE)non-linear equalization (DFE)

Non-optimal techniques (sub-optimal performance)

Need for a “channel-matched” signaling



Frequency Division Multiplexing

Basic idea:

Divide the channel BW into small non− overlapping bandscentered at {fn}N−1

n=0 (say N bands)

Separate original high− rate data stream into N low− ratestreams (symbol period increases from Ts to NTs)

Transmit each low-rate signal within each of the frequency bands

If frequency band is “small” each stream faces a flat fading (onetap) channel H(fn)




Multiplexed signal:

s(t) =∑N−1

n=0∑

k xk[n]gT(t − kNTs)ej2πfn(t−kNTs)

k: time indexn: frequency index

set of sub− carriers: {gT(t − kNTs)ej2πfn(t−kNTs)} generatedfrom a prototype pulse gT(t) of small bandwidth




Advantages:waterfilling can be approximated by appropriate power loadingchannel effect can be easily compensated for by simple one tapequalization

Disadvantages:high Tx complexity (need for N parallel oscillators)need for frequency guard bands (reduced bandwidth efficiency)



Orthogonal Frequency Division Multiplexing

Spectral efficiency requirement:zero guard bands (possibly allow for overlapping in frequency)desirable property: decoupled processing for each stream

Solution: Employ orthonormal sub-carriers∫gT(t − k1NTs)ej2πf1(t−k1NTs)g∗T(t − k2NTs)e−j2πf2(t−k1NTs))dt =

δ(f1 − f2)δ(k1 − k2)

the set of subcarriers {gT(t − kNTs)ej2πfn(t−kNTs)} is an(orthonormal) basis for its span

for maximum spectral efficiency we choose the span to be thespace of band- and time-limited functions (dimension of space?)



Optimal Rx structure

Rx also requires a filter bank!!!

if a frequency selective channel is present a possible Rx structurewould divide each stream with the corresponding channel gain

problem: How to choose gT(t) and {fn}N−1n=0 ?



Fourier Basis

one particularly simple choice of the orthonormal basis is basedon the rectangular pulse, i.e.,

gT(t) = 1/√

NTs, t ∈ [0,NTs], for all n

fn = n/(NTs), n = 0, . . . ,N − 1

the resulting basis is called the Fourier basis

in that case the transmitted signal (OFDM symbol) equals

s(t) = 1√NTs

∑N−1n=0 x[n]ej2πnt/(NTs), t ∈ [0,NTs]



Time-frequency plane



OFDM Spectrum

nominal BW: 2W = 1/Ts

“sinc” shape of the sub-carriers spectrum introduces out-of-bandleakagePSD tends to a perfect rectangular for large number ofsub-carriers (increasing symbol duration)



Discrete-time implementation

sampling the OFDM symbol at a rate 1/Ts results in thediscrete-time sequence

s[n] = s(nTs) = 1√NTs

∑N−1k=0 x[k]ej2πnk/N

important observation: this is the Inverse Discrete FourierTransform (IDFT) of the (symbol) sequence {x[k]}N−1

k=0

OFDM signal generator can be implemented as follows:



Discrete-time implementation

(I)DFT transformation F : CN 7→ CN requires a complexity oforder O(N2)

In 1965 an efficient implementation of DFT was proposed: FastFourier Transform (FFT)

complexity scales as O(N log2 N)

IFFT-based implementation of OFDM has allowed for systemsemploying up to 8K sub-carriers (DVB-T)

FFT also utilized at the Rx also

sample the received signal at 1/Ts

perform an FFT on the N-length sequence



FFT vs DFT



Channel effect

sampled received OFDM symbol (assuming no noise) can bewritten as:

y[n] =∑N−1

k=0 x[k]gk[n], n = 0, . . . ,N + Lh − 1

where gk[n] =∑Lh−1

l=0 h[l]ej2πk(n−l)/N

what happens for h[l] = aδ[l]?

The channel and the finite symbol duration introduce two effects:

received symbol now contained in [0,N + Lh − 1] (ISI)

received version of sub-carriers {gk[n]} no longer orthogonal



Cyclic Prefix

Cyclic prefixed (CP) signal generation:x ∈ CN : vector of symbols, input to IFFTs = FHx ∈ CN : IFFT outputscp = [s[N − Lh], s[N − Lh + 1], s[N − 1], s]T = Tcps ∈ CN+Lh−1 :cyclic prefixed symbol (input to A/D)

sampled received signal (no noise): ycp = T (h)scp

drop CP:

y = Rcpycp

= RcpT (h)Tcps= C(h)s

C(h): N × N circulant matrix



Cyclic Prefix

C(h) =

h[0] h[3] h[2] h[1]h[1] h[0] h[3] h[2]

h[1]. . . h[3]

......

. . .

h[Lh − 1]. . .

h[Lh − 1]. . .

h[Lh − 1] · · · h[1] h[0]



Cyclic Prefix

The introduction of the CP has transformed the channel’s effectfrom linear convolution to circular convolution

Proposition: A circulant matrix is diagonalized by the Fourierbasis, i.e.,

C(h) = FHD(h)F

h , Fh (DFT of h; channel’s frequency response)DSP interpretation?Other examples of diagonalization?

received time-domain signal: y = FHD(h)x

performing an FFT on y results in N parallel (decoupled)channels: y = Fy = D(h)x



Cyclic Prefix

End-to-end practical OFDM system:

OFDM system abstraction:



Cyclic Prefix

Advantages of CP:eliminates ISI“diagonalizes channel”simple equalization (complexity scales as O(NM)) (why?)simple power (rate) allocationnegligible complexity

Disadvantages of CP:spectral efficiency and power loss: Lh/(N + Lh)increase ratio: N/Lh

DSL: “channel shortening” before FFT



Channel Estimation (LTI case)

typical case for fixed wireless scenarios (DVB, WiFi)

channel is estimated at the beginning of transmission (frame) bytransmission of pilot OFDM symbols (preamble)




Least Squares (LS) channel estimation

h[k] = y[k]/xp[k], k = 0, 1, . . . ,N − 1

typically a LS channel estimate is preformed for a number ofOFDM pilot symbols (usually two) and the average value isextracted

advantage: very simple implementation (e.g., set all pilotsymbols to 1)

Minimum Mean Square Error (MMSE) channel estimationchannel’s frequency response {h[k]} is correlated, i.e.,information of the channel response at frequency k providessome information about its values in other frequencies

more complex estimation procedure



Channel Estimation (LTV case)

case of slow mobility (e.g., walking speed)

channel assumed quasi− static during each OFDM symbolduration



Channel Estimation (LTV case)

periodic pilot insertion is mandatory

channel is initially estimated at the “pilot slots” and subsequentlyinterpolated at the “data slots”



Phase Impairments

Oscillator instabilities, non-perfect carrier synch., Doppler, e.t.c.,introduce a phase error on the received signal

Mathematical model (AWGN only):

y(t) = ejθ(t)s(t) + w(t)

θ(t) = 2πfFOt + φ(t)

frequency offset: fFO (constant)

phase noise: φ(t)typical model: dφ(t)/dt = n(t) =⇒ φ(t) =

∫ t0 n(t)dt (Wiener

process)n(t): AWGN process



Phase Impairments

sample realization of φ(t)



Phase Impairments

Consider FO only

Received OFDM signal (no channel):

y(t) =∑N−1

n=0 xnej2π(fFO+n/(NTs))t + w(t), t ∈ [0,NTs]

FFT output will be (with no FO compensation)

xn = ACPExn + ICIn + noisen

FO introduces two effects:Common Phase Error ACPE (same for all symbols)Inter Carrier interference (ICI)



Phase Impairments

Demodulator output (64-QAM, SNR = 20dB, no FO)



Phase Impairments

Demodulator output (FO = 0.1% of sub-carrier spacing)



Phase Impairments

Demodulator output (FO = 0.5% of sub-carrier spacing)



Phase Impairments

OFDM much more sensitive to FO that SC!!!“improved” pulse shape may be beneficial

compensation is mandatory

two methods:estimate the CPE in every OFDM symbol with pilots (ICI stillremains)estimate the value of fFO and (digitally) “de-rotate” the receivedsignal (ideally, no FO effect)

phase noise has exactly the same effect

compensation is more difficult since it is a random process

pilot aided CPE estimation only solution



Summary

OFDM is the modulation of choice for transmitting high ratesover dispersive channels

major step for wider acceptance: use of FFT

simple channel equalization

increased sensitivity to phase impairments

not well suited for high mobility applications (due to ICI)

expected to be prevalent in future wireless standards



Wireless standards based on OFDM

point-to-point transmissionDigital Audio Broadcasting (DAB)Digital Video Broadcasting (DVB)IEEE 802.11a, ETSI HiperLAN (WiFi applications, low mobility)

multiuser transmission (OFDMA)IEEE 802.16 (WiMAX)UMTS LTE


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