EE6604

Personal & Mobile Communications

Week 16

Multi-carrier Techniques

1

OFDM IMPAIRMENTS

• AWGN

• Timing offset *

• Doppler

• Carrier frequency offset *

• Intersymbol interference (ISI) *

• Large Peak-to-Average Power Ratio (PAPR)

2

XI n{ }

XQ n{ }

~sI ( )t

X{ }n

D/A

D/A~s ( )t

Q

X{ }ng

g

g

N- 1N- 2x x

0 1 1N- 2 N-

IDFT

x

0 1 x x

X X X X

X

guardinsert

x k{ }

FFT-based OFDM Transmitter

3

A/D

A/D

~r ( )tQ

~r ( )tI

z 0

{ }

{ }R

RQ,n

I,n

removeguard N- 2 1N-

FFT

µ ( )vm2

R0 R1 R R

z 1 N-z z N- 1

R

z

{ }z

computermetricserialk

FFT-based OFDM Receiver

4

Performance in AWGN

• Suppose that the discrete-time OFDM time-domain sequence with a cyclic

suffix, Xgn = {Xg

n,m}N+G−1m=0 , is passed through a balanced pair of digital-to-

analog converters (DACs), and the resulting complex envelope is transmit-

ted over a quasi-static flat fading channel with complex gain g.

• The receiver uses a quadrature demodulator to extract the received complex

envelope r(t) = rI(t) + jrQ(t).

• Suppose that the quadrature components rI(t) and rQ(t) are each passed

through an ideal anti-aliasing filter (ideal low-pass filter) having a cutoff

frequency 1/(2T gs ) followed by an analog-to-digital converter (ADC)

• This produces the received complex-valued sample sequence Rgn = {Rg

n,m}N+G−1m=0 ,

where

Rgn,m = gXg

n,m + nn,m ,

g = αejφ is the complex channel gain, and the nn,m are the complex-valued

Gaussian noise samples.

• For an ideal anti-aliasing filter having a cutoff frequency 1/(2T gs ), the nn,m are

independent zero-mean complex Gaussian random variables with variance

σ2 = 12E[|nn,m|2] = No/T

gs , where T g

s = NTs/(N +G).

5

Performance in AWGN

• To be consistent with our earlier results for PSK and QAM signals, we can

multiply the zn,k for convenience by the scalar√NT g

s . Such scaling gives

zn,k = g√

2EhN/(N +G)xn,k + νn,k ,

where the νn,k are i.i.d. zero-mean Gaussian random variables with variance

No.

• Notice that√

2EhN/(N +G)xn,k = sn,k is equal to the complex signal vector

that is transmitted on the ith sub-carrier, where the term N/(N +G) repre-

sents the loss in effective symbol energy due to the insertion of the cyclic

guard interval.

• For each of the zn,k, the receiver decides in favor of the signal vector sn,k that

minimizes the squared Euclidean distance

µ(sn,k) = ‖zn,k − gsn,k‖2 , k = 0, . . . , N − 1 .

• It is apparent that the probability of symbol error is identical to that

achieved with independent modulation on each of the sub-carriers. This

is expected, because the sub-carriers are mutually orthogonal in time.

6

Timing Errors

• Timing offset smaller than the guard interval results in a phase shift

• Consider and OFDM waveform with cyclic prefix on an ideal channel such

that Rn = Xn.

• Suppose that the receiver has a timing offset, ℓ, that lies in the guard

interval. Then instead of taking an FFT on the block {Rn}N−1n=0 , the receiver

takes an FFT on the block {Rn}N−1n=0 instead, where Rn = R(n−ℓ)N , where (n)N

is the residue of n modulo N .

NG

N

` Timing offset

7

Timing Errors

• The FFT output becomes

Zk =1

N

N−1∑

n=0Rne

−j 2πknN

=1

N

N−1∑

n=0R(n−ℓ)Ne

−j 2πknN

=1

N

N−ℓ−1∑

m=−ℓRme

−j2πk(m+ℓ)

N

=

1

N

N−1∑

m=0Rme

−j 2πkmN

e−j 2πkℓN

= Zke−j 2πkℓN , k = 0, . . . , N − 1 .

• Timing offset smaller than the guard interval results in a phase shift on

each sub-carrier.

– The effects of timing offset are easy to remove with adaptive channel

estimation.

• If the timing offset is greater than the guard interval, additional interference

is generated.

– Best solution is to choose sufficient guard interval and make sure the

timing offset lies in the guard interval.

8

Carrier Frequency Offset

• Carrier frequency offset also causes interchannel interference (ICI).

• The transmitted complex envelope with carrier frequency offset ∆f is

s(t) = AN−1∑

n=0xn exp

j2π

n

NTs+∆f

t

uT (t)

• The corresponding IFFT coefficients are

Xk = AN−1∑

n=0xn exp

j2π

nk

N+ k∆fTs

, k = 0, 1, . . . , N +G− 1

• The demodulated sequence (in absence of noise) can be written as

Zl = FFT{Xn} = ηlxl + cl

where

ηl = A

sin (πNTs∆f)

πNTs∆f

ejπNTs∆f

and

cl = A∑

n6=lanH(n, l)

is the random ICI term, where

H(n, l) =

sin [π (n− l −NTs∆f)]

π [n− l −NTs∆f ]

ejπ(n−l−NTs∆f)

9

Effect of Carrier Frequency Offset

• Assume that the receiver can determine πNTs∆f = arg(ηl), the demodulated

sequence in the absence of noise can be written as

Zl = Zle−jπNTs∆f = ηlxl + cl

where

ηl = A

sin (πNTs∆f)

πNTs∆f

and

cl = A∑

n6=lanH(n, l)

is the random ICI term, where

H(n, l) =

sin [π (n− l − sinNTs∆f)]

π (n− l −NTs∆f)

ejπ(n−l−1−NTs∆f)

• Observe that carrier frequency offset has two effects

1. It reduces the useful signal energy by the factor

10log10

sin (πNTs∆f)

πNTs∆f

2

2. Introduces an additional additive noise term cl

10

Combating ISI with OFDM

• Suppose that the IFFT output vector Xn = {Xn,m}N−1m=0 is appended with a

cyclic suffix to yield the vector Xgn = {Xg

n,m}N+G−1m=0 , where

Xgn,m = Xn,(m)N

= AN−1∑

k=0xn,ke

j2πkmN , m = 0, 1, . . . , N +G− 1 ,

G is the length of the guard interval in samples, and (m)N is the residue of m

modulo N . To maintain the data rate Rs = 1/Ts, the DAC in the transmitter

is clocked with rate Rgs = N+G

N Rs, due to the insertion of the cyclic guard

interval.

• Consider a time-invariant ISI channel with impulse response g(t). The

combination of the DAC, waveform channel g(t), anti aliasing filter, and

DAC yields an overall discrete-time channel with sampled impulse response

g = {gm}Lm=0, where L is the length of the discrete-time channel impulse

response.

• The discrete-time linear convolution of the transmitted sequence {Xgn} with

the discrete-time channel produces the discrete-time received sequence {Rgn,m},

where

Rgn,m =

∑mi=0 giX

gn,m−i +

∑Li=m+1 giX

gn−1,N+G+m−i + nn,m , 0 ≤ m < L

∑Li=0 giX

gn,m−i + nn,m , L ≤ m ≤ N +G− 1

.

11

Removal of Guard Interval

• To remove the ISI introduced by the channel, the first G received sam-

ples {Rgn,m}G−1

m=0 are discarded and replaced with the last G received samples

{Rgn,m}N+G−1

m=N .

• If the length of the guard interval satisfies G ≥ L, then we obtain the received

sequence

Rn,m = Rgn,G+(m−G)N

=L∑

i=0giXn,(m−i)N + nn,(m−i)N , 0 ≤ m ≤ N − 1 .

• Note that the first term represents a circular convolution of the transmitted

sequence Xn = {Xn,m} with the discrete-time channel g = {gm}Lm=0.

block blockblock-1 +1n

G ISI G ISI ISIG

G G G

n n

Removal of ISI using a cyclic suffix

12

• The OFDM baseband demodulator computes the DFT of the vector Rn.

This yields the output vector

zn,i =1

N

N−1∑

m=0Rn,me

−j 2πmiN

= TiAxn,i + νn,i , 0 ≤ i ≤ N − 1 ,

where

Ti =L∑

m=0gme

−j 2πmiN

and the noise samples {νn,i} are i.i.d with zero-mean and variance No/(NT gs ).

• Note that T = {Ti}N−1i=0 is the DFT of the zero padded sequence g = {gm}N−1

m=0

and is equal to the sampled frequency response of the channel.

• To be consistent with our earlier results, we can multiply the zn,i for conve-

nience by the scalar√NT g

s , giving

zn,i = TiAxn,i + νn,i i = 0, . . . , N − 1 ,

where A =√

2EhN/(N +G) and the νn,i are i.i.d. zero-mean Gaussian random

variables with variance No.

• Observe that each zn,i depends only on the corresponding data symbol xn,iand, therefore, the ISI has been completely removed.

• Once again, for each of the zn,i, the receiver decides in favor of the signal

vector sm that minimizes the squared Euclidean distance

µ(sm) = ‖zn,i − TiAxn,i‖2 .

13

OFDMA

• OFDMA achieves multiple access by assigning different users disjoint sets

of sub-carriers.

• Assume that there are a total of M sub-carriers that are evenly distributed

among Q users, such that each user is allocated N = M/Q sub-carriers. The

overall sub-carriers are labeled with indices from 0 to M − 1, while the N

sub-carriers allocated to the jth MS have indices that belong to the set Tj.Clearly, the sets Tj must be disjoint such that each sub-carrier is assigned

to at most one MS.

• The sub-carrier allocation can be performed by extending the nth data

vector for the jth MS, denoted, by aj,n with the insertion of M −N zeros in

the sub-carriers belonging the set Tj which is the complement of Tj, i.e.,

xj,n,i =

aj,n,i , if i ∈ Tj0 , otherwise

,

where aj,n,i is the data symbol transmitted to the jth MS in block n on the

ith sub-carrier.

14

OFDMA - Forward Link Transmitter

• On the forward link, the vectors xj,n = {xj,n,i}M−1i=0 are summed up to produce

the nth data block

xn =Q∑

j=1xj,n

that is subsequently applied to an M-point IDFT to produce the length-M

time-domain sequence Xn.

• After the IDFT, a length-G guard interval is appended to each block in the

form of a cyclic prefix or cyclic suffix, to yield the transmitted time-domain

sequence Xgn.

• In the case of a cyclic prefix, the last G symbols of the sequence Xn =

{Xn,m, m = 0, . . . ,M − 1} are copied and appended to the beginning of

Xn. The transmitted time-domain sequence for the nth block with guard

interval, denoted as Xgn, is

Xgn = {Xn,(m)M , m = −G,−G+ 1, · · · ,−1, 0, 1, · · · ,M − 1} ,

where (m)M is the residue of m modulo-M .

15

Subcarriera1,n x1,nSubcarrier

Mapping

IDFT Cyclic

1,n

xn Xn Xgn

1,n

IDFT

(M point)

Cyclic

guard

n n

SubcarrieraQ,n

+

Subcarrier

Mapping

Q,

xQ,n

Baseband OFDMA forward link BS transmitter.

16

OFDMA - Sub-carrier Allocation

• Clustered Carrier (CC-OFDMA): With CC-OFDMA, the M sub-carriers

are divided into Q groups where each group consists of N contiguous sub-

carriers called clusters. The set of sub-carrier indices allocated to the kth

user is {kN, kN + 1, . . . , kN + N − 1}, where 0 ≤ k < Q. CC-OFDMA is

sensitive to frequency-selective fading, because all sub-carriers assigned to

a particular user may fade simultaneously.

• Spaced Carrier (SC-OFDMA): With SC-OFDMA, the M sub-carriers are

partitioned into N groups, where each group has Q contiguous sub-carriers.

Then the kth sub-carrier of each group is assigned to the kth user. That is,

the kth user is assigned the set of sub-carrier indices {k,Q+k, . . . , (N−1)Q+k},where 0 ≤ k < Q. SC-OFDMA is less sensitive to frequency-selective fading,

since the sub-carriers assigned to each user span the entire bandwidth.

• Random Interleaving (RI-OFDMA): RI-OFDMA is used in IEEE802.16a.

While the sub-carriers are partitioned into N groups as in SC-OFDMA, the

sub-carrier index in each of the N groups that is assigned to a particular

user is a random variable. The sub-carrier indices allocated to the kth user

are {ǫk,1, Q + ǫk,2, . . . , (M − 1)Q + ǫk,M−1}, where the ǫk,i are independent and

identically distributed uniform random variables on the set {0, 1, . . . , Q− 1}.

17

OFDMA - Forward Link Receiver

• To remove the ISI introduced by the channel, the guard interval is removed.

If the length of the cyclic prefix is at least as long as the discrete-time

channel length, i.e., G ≥ L, then we obtain the received sequence

Rn,m = Rgn,m

=L∑

i=0giXn,(m−i)M + nn,m , 0 ≤ m ≤ M − 1 ,

• Afterwards, an M-point IDFT is taken to transform to the frequency do-

main. This yields the output vector

zn,i =1

M

M−1∑

m=0Rn,me

−j 2πmiM

= TiAxn,i + νn,i , 0 ≤ i ≤ M − 1 ,

where

Ti =L∑

m=0gme

−j 2πmiM

and the noise samples {νn,i} are i.i.d with zero-mean and variance No/(MT gs ).

• On the forward link each MS will only be interested in the N data symbols

that are transmitted by the BS on its allocated sub-carriers. Hence, only

the DFT outputs with indices in the set Tj are used by the jth MS for data

detection.

18

SubcarrierIDFTznznRemoveR

gn Rn Subcarrier

Demapping

IDFT

(M point)Cyclic

guard

n n

n Tj

Baseband OFDMA forward link receiver.

19

OFDMA - Reverse Link

• On the OFDMA reverse link, Q users transmit their signals to a central BS.

• Each MS transmitter only transmits its own data stream.

• Similar to the OFDMA forward link, the jth MS performs sub-carrier al-

location, and the resulting vector xj,n is applied to an M-point IDFT, and

appended with a length-G cyclic guard interval.

• One of the biggest drawbacks of OFDMA is its high PAPR. A high PAPR

may be acceptable on the forward link; however, a high PAPR is undesirable

on the reverse link since the MS is often is often battery powered and

amplifier back-off is required.

Subcarrier IDFTaj n Xj n Cyclic Xg

j nxj nSubcarrier

Mapping

IDFT

(M point)

j,n j,n Cyclic

guard

j,nxj,n

Baseband OFDMA reverse link MS transmitter.

20

SC-FDMA

• The SC-FDMA transmitter groups the modulation symbols into blocks of

N symbols. Let

xn = (xn,1, xn,2, . . . , xn,N)

denote the nth block of modulation symbols. An N-point DFT (N-DFT) is

taken on each block xn, to yield length-N vectors

Xn = (Xn,1, Xn,2, . . . , Xn,N)

that are the frequency domain representation of the blocks of input symbols.

• The sub-carrier mapper then maps the N components of the vector Xn onto

a larger set of M sub-carriers such that M = NQ, where Q is an integer.

There are several different types of sub-carrier mappings, including the

interleaved (I-FDMA) and localized (L-FDMA) mappings. The sub-carrier

mapping generates the sequence Xn.

• An M-point IDFT is then taken of the sequence Xn to produce the output

sequence xn. The time domain input symbols xn,k have duration Ts seconds.

However, after going through the SC-FDMA modulator the time-domain

output symbols xn,k are compressed and have duration Ts = (N/M)Ts seconds.

21

SC-FDMA Transmitter

DFT Subcarrier IDFTxn Xn Xn xn Cyclic

xgnDFT

(N point)

Subcarrier

Mapping

IDFT

(M point)

Cyclic

guard

n

Baseband SC-FDMA transmitter. There are M sub-carriers of which N are occupied by

the input data.

22

SC-FDMA Receiver

• The SC-FDMA receiver first the cyclic guard interval.

• Afterwards, an M-point DFT is taken to transform to the frequency domain.

• Sub-carrier demapping and equalization is then performed in the frequency

domain.

• Finally, an N-point IDFT is used to convert the samples back to the time-

domain for detection and further processing.

IDFTSubcarrier

DFTznRn xnRemover

gn rn IDFT

(N point)Demapping/

Equalization

DFT

(M point)Cyclic

guard

n

Baseband SC-FDMA receiver with SC-FDE.

23

SC-FDMA - Subcarrier Mapping

input

Xn,0 Xn,1 Xn,2 Xn,N 1

Q 1!

zeros

Q 1!

zeros

Q 1!

zeros

Q 1!

zeros

input

X 0 X Q X 2Q X ( )Q X M

zeros zeroszeros zeros

Xn,0 Xn,Q Xn,2Q Xn,(N 1)Q Xn,M

output

Interleaved FDMA (I-FDMA) subcarrier mapping.

Xn,1 Xn,2 Xn,N 1Xn,0M N!

zeros

X 0 X 1 X 2 X ( ) X MXn,0 Xn,1 Xn,2 Xn,(N 1) Xn,M

Localized FDMA (L-FDMA) subcarrier mapping.

24