OFDM and DMT: System Model

EIT 140, tom<AT>eit.lth.se

Review: LTI channel and dispersion

Time-dispersive LTI channel

g1g3 g2

���� ���� ��s

r

g0

z zz

Dispersion M = length of impulse response − 1 = order of channel filter

LTI channel performs linear convolution

r(n) = ∑k

s(k)h(n− k)

Length-N input s(n), n = 0, . . . ,N − 1 yields length-(N +M) outputr(n), n = 0, . . . ,N +M − 1

Input block is “smeared out” (dispersed) over M additional samples

Review: Dispersion can cause unacceptable ISI/ICI

High datarate −→ high symbolrate −→ severe ISI

high data rate

severe ISI!

Dispersion causes transients: severe ICI ←→ loss of orthogonality

channel input

0

0 N

0

channel output

0

0 M N N+M

0

Review: Ideas of multicarrier transmission

Use several, mutually orthogonal, longer, sinusoidal transmit waveforms

“several”: compensate for the “longer” in terms of data rate“mutually orthogonal”: avoid ICI“sinusoidal”: eigenfunctions of LTI systems, efficientimplementation (FFT)

Cyclic extension avoids ISI and ICI in time-dispersive channels

channel input

0

−M 0 N

0

channel output

0

−M 0 M N N+M

0

System design limits

TMC � τmax

TMC � min{Tcoh, latency limit, carrier-spacing limit}

CP: linear convolution −→ circular convolution

CP CPh(n)s̃(n) s(n) r̃(n) r(n)

CP-add block: s(n) =

{s̃(n), n = 0, . . . ,N − 1

s̃(N + n), n = −L, . . . ,−1

Channel: linear convolution r̃(n) = ∑k s(k)h(n− k)CP-remove block: r(n) = r̃(n), n = 0, . . . ,N − 1

For L ≥ M: CP-add + channel + CP-remove yield circular convolution

r(n) = r̃(n) = ∑k

s̃((n− k) mod N)h(k), n = 0, . . . ,N − 1

and consequently R [k ] = S̃ [k ]H [k ], k = 0, . . . ,N − 1 holds.

R [k ], S̃ [k ], H [k ] are the N-point FFTs of r (n), s̃(n), h(n), respectively.

Orthogonal Frequency Division Multiplex (OFDM) system

IDFT

(IFF

T)

DFT

(FFT

)

CPCP h(m)↑p ↓p

ej2πfcm 2 e−j2πfcm

Re{}

hL

P(m

)

OFDMOFDM RFRFmodulator demodulatormodulator demodulatorchannel

{xk}N−1

k=0{y

k}

s(n) u(m) v(m) r(n)

s (n) = IDFTN {xk}N−1k=0 n = 0, . . . ,N − 1

s (n) = s (n+N), n = −L, . . . ,−1s (m) = {s (n)}↑p , m = −Lp, . . . ,Np−1

u(m) = Re{s (m)e j2πfcm

}

v(m) = u(m) ∗ h(m)

r (m) =(v(m) 2e−j2πfcm

)∗ hLP(m)

r (n) = {r (m)}↓p , n = 0, . . . ,N−1

yk= DFTN{r (n)}N−1

n=0 k = 0, . . . ,N − 1

Receive signal components (passband)

Real-valued passband receive signal components1:

v(i)k (m) =

1√N

cos(2π

(fc+

k

Np

)m),

v(q)k (m) = − 1√

Nsin(2π

(fc+

k

Np

)m),

m = 0, . . . ,Np − 1

for k = −N/2 + 1, . . . ,N/2.

fc is a positive carrier frequency

p is the oversampling factor and sufficiently large to ensure thatfc +

kpN < 1

2 , k = −N/2 + 1, . . . ,N/2 holds for all the N

normalised frequencies

1Throughout the lecture, we assume that N is even.

Receive signal components (passband) in time-domain

v(i)−3

(m) v(q)−3

(m)

v(i)−2

(m) v(q)−2

(m)

v(i)−1

(m) v(q)−1

(m)

v(i)0

(m) v(q)0

(m)

v(i)1

(m) v(q)1

(m)

v(i)2

(m) v(q)2

(m)

v(i)3

(m) v(q)3

(m)

v(i)4

(m) v(q)4

(m)

00

00

00

00

00

00

00

00

Np−1Np−1

Np−1Np−1

Np−1Np−1

Np−1Np−1

Np−1Np−1

Np−1Np−1

Np−1Np−1

Np−1Np−1

v(i)k (m) (cosine signals, left

column) and v(q)k (m) (sine signals,

right column); parameters:N = 8, p = 32, fc = 1/32,k = −3, . . . , 4

There are 2N mutually orthogonalreal-valued length-pN discrete-timesinusoidal waveforms with aninteger number of periods

Transmit signal components (passband)

Similarly, we consider the transmit signals of length (N + L)p samples

u(i)k (m) =

1√N

cos(2π

(fc+

k

Np

)m),

u(q)k (m) = − 1√

Nsin(2π

(fc+

k

Np

)m),

m = −Lp, . . . ,Np − 1

where k = −N/2 + 1, . . . ,N/2.

Cyclic prefix of length pL (passband)

RF modulator

We write the passband transmit signal as

u(m)=1√N

N/2

∑k=−N/2+1

(x

(i)[k ]N

cos(2π

(fc+

k

Np

)m)− x

(q)[k ]N

sin(2π

(fc+

k

Np

)m)

)

= Re

1√N

N/2

∑k=−N/2+1

x [k ]N ej2π k

Npm

︸ ︷︷ ︸s (m)

ej2πfcm

,

where [k ]N denotes the modulo-N operation applied to k.

This operation corresponds to a cos/sin modulator structure

Perfect interpolation by factor p of s (n) yields s (m)

Note that s (n) = 1√N

N/2∑

k=−N/2+1x [k ]N e

j2π kN n = 1√

N

N−1∑k=0

xkej2π k

N n

Baseband transmit signal processing (OFDM modulator)

s (n) =1√N

N/2

∑k=−N/2+1

x [k ]N ej2π k

N n

=1√N

N/2

∑k=0

xkej2π k

N n +−1

∑k=−N/2+1

x (N+k︸ ︷︷ ︸,i ,→k=i−N

)ej2π k

N n

=1√N

N/2

∑k=0

xkej2π k

N n +N−1

∑i=N/2+1

x i ej2π i−NN n

︸ ︷︷ ︸e j2π i

N ne−j2π NN n=e j2π i

N n

=1√N

(N/2

∑k=0

xkej2π k

N n +N−1

∑k=N/2+1

xkej2π k

N n

)

=1√N

N−1

∑k=0

xkej2π k

N n, which is the IDFT of xk

Channel output

The channel coefficients are given by

H [k ]N=A[k ]N

+jB[k ]N=(M+1)p−1

∑m=0

h(m)e−j2π(fc+kpN )m, k = −N/2+ 1, . . . ,N/2.

The passband transmit signal u(m) yields the channel output

v(m) =1√N

N/2

∑k=−N/2+1

((A[k ]N

x(i)[k ]N− B[k ]N

x(q)[k ]N

) cos(2π

(fc+

k

pN

)m)−

(B[k ]Nx

(i)[k ]N

+ A[k ]Nx

(q)[k ]N

) sin(2π

(fc+

k

pN

)m)

)

RF demodulator

Receiver performs the cosine-sine demodulation, low pass filtering anddownsampling by factor p of the receive signal, which yields

r (n)=1√N

N2

∑k=−N

2 +1

((A[k ]N

x(i)[k ]N−B[k ]N

x(q)[k ]N

)+ j(B[k ]Nx

(i)[k ]N

+A[k ]Nx

(q)[k ]N

))ej2π k

N n,

which can be written as

r (n) =1√N

N−1

∑k=0

((Akx

(i)k − Bkx

(q)k ) + j(Bkx

(i)k + Akx

(q)k )

)ej2π k

N n

(1)We can interpret (1) as receive multiplex using the receive signalcomponents

rk (n) =1√Nej2π kn

N , n = 0, . . . ,N−1

Baseband receive signal processing (OFDM demodulator)

Finally, we pass the complex-valued receive multiplex

r (n) =1√N

N−1

∑k=0

((Akx

(i)k − Bkx

(q)k ) + j(Bkx

(i)k + Akx

(q)k )

)e+j2π k

N n

through a bank of correlators, which yields

〈r , rk 〉 =1√N

N−1

∑n=0

r (n)e−j2π kN n = (Akx

(i)k −Bkx

(q)k )+ j(Bkx

(i)k +Akx

(q)k )

where k = 0, . . . ,N − 1. The operation performed by these correlators isequivalent to the scaled N-point Discrete Fourier Transform (DFT) ofthe complex-valued baseband receive multiplex r (n)

yk=

1√N

N−1

∑n=0

r (n)e−j2π kN n.

Complex baseband representation

We now introduce a model of a fictitious system which has the samebehaviour as the real-world, implementable passband system above. Thereceived complex signal can be written as

r (n) = h(n) ∗ s (n),

where the complex channel impulse response h(n) is given by

h(n) = IDFTN {Hk} =N−1

∑k=0

Hkej2π kn

N .

We model the (real-valued) passband signals as (complex-valued)baseband signals

In essence, we omit RF modulator and RF demodulator andtransform the passband channel into an equivalent baseband channel

OFDM spectra (flat channel)

Re {DTFT (s(n))}

Im {DTFT (s(n))}

Re {DTFT (u(m))}

Im {DTFT (u(m))}

Re{

DTFT(v(m)2e−j2πfcm

)}

Im{

DTFT(v(m)2e−j2πfcm

)}

Re {DTFT (r(n))}

Im {DTFT (r(n))}

IFFT

outp

ut

chan

nel i

n/ou

tput

sin/c

osde

mod

ulat

orou

tput

FFT

inpu

t

0−fc−2fc fc f

solid blue: DTFT of Re {x}. dashed red: DTFT of Im {x}. dashed-dotted green:DTFT of x = Re {x}+ j Im {x}.

Power spectral density (PSD)

Average measure for spectral allocation of transmit power

Stationary signals: PSD(f ) = Fm (r(m))PSD is Fourier transform of transmit signal’s autocorrelationfunction r(m)

Cyclostationary signals: PSD(f ) = Fm

(1N ′ ∑

nr(m, n)

)

PSD is Fourier transform of time-averaged transmit signal’sautocorrelation function r(m, n). (r(m, n) = r(m, n+N ′))

For uncorrelated data (over both time and subchannels):

PSD(f ) = ∑k∈set of used subchannels

Pk |DTFT of kth basis function|2

where Pk is the power on the kth subchannel.

cf. psdmc.m

F·(r (·, . . .)) is the Fourier transform (DTFT) of the continuous(discrete)-time signal rwith respect to ·

Discrete Multi-Tone (DMT) spectra (flat channel)In case we have a baseband channel, we

do not need the RF modulator/demodulator

need a real-valued transmit multiplex

Re {DTFT (s(n))}

Im {DTFT (s(n))}

Re {DTFT (r(n))}

Im {DTFT (r(n))}

IFFT

outp

ut=

chan

nel i

nput

FFT

inpu

t =ch

anne

l out

put

0 f

solid blue: DTFT of Re {x}. dashed red: DTFT of Im {x}. dashed-dotted green:DTFT of x = Re {x}+ j Im {x}.

Receive signal components: baseband ↔ passband

r(i)0

(n) r(q)0

(n)

r(i)1

(n) r(q)1

(n)

r(i)2

(n) r(q)2

(n)

r(i)3

(n) r(q)3

(n)

r(i)4

(n) r(q)4

(n)

r(i)5

(n) r(q)5

(n)

r(i)6

(n) r(q)6

(n)

r(i)7

(n) r(q)7

(n)

00

00

00

00

00

00

00

00

N−1N−1

N−1N−1

N−1N−1

N−1N−1

N−1N−1

N−1N−1

N−1N−1

N−1N−1

v(i)−3

(m) v(q)−3

(m)

v(i)−2

(m) v(q)−2

(m)

v(i)−1

(m) v(q)−1

(m)

v(i)0

(m) v(q)0

(m)

v(i)1

(m) v(q)1

(m)

v(i)2

(m) v(q)2

(m)

v(i)3

(m) v(q)3

(m)

v(i)4

(m) v(q)4

(m)

00

00

00

00

00

00

00

00

Np−1Np−1

Np−1Np−1

Np−1Np−1

Np−1Np−1

Np−1Np−1

Np−1Np−1

Np−1Np−1

Np−1Np−1

Sinusoidal receive signal components

r(i)0

(n) r(q)0

(n)

r(i)1

(n) r(q)1

(n)

r(i)2

(n) r(q)2

(n)

r(i)3

(n) r(q)3

(n)

r(i)4

(n) r(q)4

(n)

r(i)5

(n) r(q)5

(n)

r(i)6

(n) r(q)6

(n)

r(i)7

(n) r(q)7

(n)

00

00

00

00

00

00

00

00

N−1N−1

N−1N−1

N−1N−1

N−1N−1

N−1N−1

N−1N−1

N−1N−1

N−1N−1 There are N mutually orthogonalreal-valued length-N discrete-timesinusoidal waveforms with aninteger number of periods (N = 8in this example).

For k ∈ {bN/2c+ 1, . . . ,N−1},the number of oscillations per Nsamples decreases with increasing k.

In fact, it is easy to verify that

cos(2πk

Nn) = cos(2π

(N − k)

Nn), k = 0, . . . ,N,

sin(2πk

Nn) = − sin(2π

(N − k)

Nn) k = 0, . . . ,N

(2)

Discrete Multi-Tone (DMT) system

()∗

DFTIDFT CP CPh(n)

DMT DMTmodulator demodulatorchannel

{xk}N/2

k=0{yk}

Nk=0

s(n) r(n)

xN−k = x∗k k = 0, . . . ,N/2

s(n) = IDFTN {xk}N−1k=0 n = 0, . . . ,N − 1

s(n) = s(n+N), n = −L, . . . ,−1

r(n) = h(n) ∗ s(n), n = 0, . . . ,N − 1

yk= DFTN{r(n)}N−1

n=0 k = 0, . . . ,N/2

DMT: Hermitian symmetry of transmit symbols

We make the following choice for the transmit symbols

x(i)k = x

(i)N−k and x

(q)k = −x(q)

N−k , (3)

which implies x(q)0 = x

(q)N/2 = 0. (3) ensures Hermitian symmetry of the

transmit symbol blocks. With (2), we obtain for n = −L, . . . ,N − 1

s(n) =1√N

N−1

∑k=0

(x

(i)k cos(2π

k

Nn)− x

(q)k sin(2π

k

Nn)

)

+ j1√N

N−1

∑k=0

(x

(i)k sin(2π

k

Nn) + x

(q)k cos(2π

k

Nn)

)

︸ ︷︷ ︸=0

=1√N

N−1

∑k=0

xkej2π k

N n.

→ On the interval 0, . . . ,N−1, the transmit signal multiplex s(n) isgenerated by the scaled N-point Inverse Discrete Fourier Transform(IDFT) of the data symbols xk .

DMT and OFDM

DMT: real-valued baseband multiplex s(n), OFDM: complex-valuedbaseband multiplex s (n)

DMT: wireline communications (lowpass channel), OFDM: wirelesscommunications (passband channel)

DMT: channel known at the transmitter, OFDM: channel unknownat transmitter

Examples of OFDM/DMT systems and parameters

HiperLAN/2 DVB-T 8k ADSL DS VDSL DS DAB

multiplex2: complex complex real real complexN: 64 8192 512 8192 2048L: 16 256 32 or3 40 640 504

bandwidth: 20 MHz 7.61 MHz 966 kHz ≈ 8 MHz 1.536 MHzband: 5.15-5.35 GHz UHF 0-1.1 MHz 0-12 MHz III

FsN : 0.3125 MHz 1.116 kHz 4.3125 kHz 4.3125 kHz 1 kHz

datarate: 6-54 Mbps 5-30 Mbps 0.5-8 Mbps ≤ 54 Mbps ≤ 348 kbpsmodulation: B/Q/8PSK 4/16/64QAM BPSK BPSK diff. QPSK

16/64QAM 4-215QAM 4-215QAMmobility: pedestrian vehicular none none vehicularchannel: wireless wireless wire wire wireless

reach: ≤ 150 m ≈ km ≤ 5 km ≤ 1.5 km ≈ km

III band: 174MHz - 240MHz; UHF band: 790MHz - 806MHz

Note that many standards include various modes with different parameter sets.2complex =̂ OFDM. real =̂ DMT.3L = 32 in case a synch-symbol is used, otherwise L = 40

Matrix representation of DMT and OFDM

Comes in handy since OFDM/DMT is a block transmission scheme

More compact and less painful than scalar description

Yields another interpretation of cyclic prefixing, ISI-free and ICI-freetransmission

Notation:

lowercase boldface: (usually a column4) vector (e.g. a ∈ RN)uppercase boldface: matrix (e.g. A ∈ CN×(N+L))AT: transpose. AH: Hermitian transpose (transposed conjugate).

4In channel coding, they like to use a as row vector.

Channel: linear convolution matrix

We describe the channel dispersion using a convolution matrix H, wherewe assume L = M:

r(0)r(1)

...r(N−1)

=

h(L) · · · h(1) h(0) 0 · · · · · · · · · · · · 0

0. . . h(1) h(0)

. . ....

... h(L)... h(1)

. . ....

... h(L)...

. . .. . .

. . ....

... 0 h(L). . .

. . .. . .

......

.... . .

. . .. . .

. . . 00 · · · · · · 0 · · · 0 h(L) · · · h(1) h(0)

s(−L)...

s(−1)s(0)s(1)

...s(N−1)

(4)

Both DMT (H real-valued) or OFDM (H complex-valued) can berepresented

The first L received samples r(n), n = −L, . . . ,−1 are implicitly ignored

Channel + CP: circulant convolution matrix

Exploiting the fact that s(n) = s(N − n), n = −L, . . . ,−1, we canrewrite (4) as

r(0)r(1)

...r(N−1)

︸ ︷︷ ︸r

=

h(0) 0 · · · 0 h(L) · · · h(1)

h(1) h(0). . . 0

. . ....

... h(1). . .

. . .. . . h(L)

h(L)...

. . .. . . 0

0 h(L). . .

. . ....

.... . .

. . .. . .

. . ....

0 · · · 0 h(L) h(1) h(0)

︸ ︷︷ ︸H̃

s(0)s(1)

...s(N−1)

︸ ︷︷ ︸s

.

Only possible for L ≥ M

Addition of the CP and removal of the CP are implicitly included

Receive signal processing

y0

y1...

yN−1

︸ ︷︷ ︸y

=1√N

1 1 1 . . . 1

1 w w2 . . . wN−1

1 w2 w4 . . . w2(N−1)

......

...

1 wN−1 w2(N−1) . . . w (N−1)2

︸ ︷︷ ︸R

r(0)r(1)

...r(N − 1)

︸ ︷︷ ︸r

where w = e−j2π 1N .

R is the normalised DFT matrix

Rows of R contain the receive signal components

Multiplication of R with r corresponds to computing N correlations(one per subcarrier/subchannel)

Can be implemented in O(N logN) using the FFT

Transmit signal processing

s(0)s(1)

...s(N − 1)

︸ ︷︷ ︸s

=1√N

1 1 1 . . . 1

1 w−1 w−2 . . . w−(N−1)

1 w−2 w−4 . . . w−2(N−1)

......

...

1 w−(N−1) w−2(N−1) . . . w−(N−1)2

︸ ︷︷ ︸RH

x0x1...

xN−1

︸ ︷︷ ︸x

RH is the normalised IDFT matrix

The columns of RH are the transmit signal components

kth transmit symbol (xk−1) scales the kth column of RH

Transmit multiplex is the sum of N scaled complex exponentials(scaled by the data)

Can be implemented in O(N logN) using the IFFT

“Diagonalisation” of the circulant channel matrix

Note:

R−1 = RH (R is unitary matrix) −→ RRH = RHR = IAny circulant matrix H̃ can be written as H̃ = RHΛR, where Λ is adiagonal matrix

Consequently, we can write

y , Rr = RH̃s = RH̃RHx = R H̃︸︷︷︸RHΛR

RHx = Λx.

The DFT/IDFT matrix-pair diagonalises any circulant channelmatrix

As long as L ≥ M (proper cyclic prefix), we obtain parallel,independent subchannels (no ISI, no ICI)

Summary

1 Transmit signal processing: linear combination of transmit signalcomponents (complex exponentials) −→ IFFT

2 Receive signal processing: correlation with complex exponentials −→ FFT

3 OFDM vs DMT

4 Three interpretations of cyclic prefixing

5 Matrix notation