Faculty of Engineering
Introduction to
Orthogonal Frequency Division Multiplexing
(OFDM)
EE256
Digital Communications (II)
Lecturer
Assoc. Prof. Mohamed Abdelkarim
Introduction
• Orthogonal Frequency Division Multiplexing is a
multiplexing/modulation/multiple access technique used in
most modern communications systems
• Its basic idea depends mainly on using a large number of
parallel narrow-band subcarriers instead of a single wideband carrier to transport information
• High transmission rates over wireline and wireless channels
with protection from multipath fading
• Used technique in fourth Generation (4G) mobile phones,
(5G) and hogging applications like Video Conferencing, DAB,
DVB,..etc
Channel problems
• Signal in communication channel is affected by three models
• Path loss model
• Shadowing
• Mutlipath effect
Channel problems
Frequency Selective Fading
Fading
2.0 secs
Time 3.0
2.5 secs
secs
Frequency Selective Fading Channels can provide
-- time diversity (can be exploited in DS-CDMA)
-- frequency diversity (can be exploited in OFDM)
Frequency selective fading
• Only a part of OFDM signal is affected in frequency selective channel
unlike single carrier signal
Multiple access techniques
• General wireless cellular systems are multi-users
systems
• Radio resource are limited
• Limited Bandwidth
• Limited number of channels
• The radio resource must be shared among multiple
users
TDMA, CDMA, and OFDMA
Wireless Systems
• Time Division Multiple Access (TDMA) is the most prevalent wireless
access system to date
• GSM, ANSI-136, EDGE, DECT, PHS, Tetra
• Direct Sequence Code Division Multiple Access (DS-CDMA) became
commercial only in the mid 90’s
• IS-95 (A,B, HDR,1x,3x,...), cdma-2000 (3GPP2), W-CDMA (3GPP)
• Orthogonal Frequency Division Multiplexing (OFDM) is perhaps the
least well known
• can be viewed as a spectrally efficient FDMA technique
• IEEE 802.11A, .11G, HiperLAN, IEEE 802.16 OFDM/OFDMA options
TDMA (with FDMA) Principle
Carriers
Power
Freq.
Time-slots
Time
Direct Sequence CDMA Principle
(with FDMA)
User Code
Waveforms
Power
Freq.
Time
OFDM (with TDMA & FDMA) Principle
Tones
Carriers
Power
Freq.
Time-slots
Time
Other Multiple Access Techniques
• Multi-Carrier TDMA
• DECT, PACS
• Frequency Hopped Spread Spectrum
• Bluetooth
• CSMA/CA
• IEEE 802.11 (1 or 2 Mbps standard)
• DS-CDMA with Time Slotting
• 3GPP W-CDMA TDD (Time Division Duplex)
Packet Switched Air Interface is vital for high bit-rates
and high capacity (for data users) -- GPRS, DPRS, etc.
Comparing Performance of TDMA, DS-CDMA,
& OFDM Transceivers
TDMA
Fade Margin
(for mobile apps.)
Range
Re-use & Capacity
FEC Requirements
Variable Bit-rate
Support
Spectral Efficiency
CDMA
OFDM
Modest requirement
(RAKE gain vs powercontrol problems)
Required for mobile
applications
Range increase by reducing
allowed noise rise (capacity)
Difficult to support large
cells (PA , AGC limitations)
Modest (in TDMA) and
High in MC-TDMA
Modest
Re-use planning is
crucial here
FEC optional for voice
FEC is usually inherent (to
increase code decorrelation)
FEC is vital even for
fixed wireless access
Required for mobile
applications
Very easy to increase
cell sizes
Low to modest support
Very elegant methods
to support VBR & VAD
Modest
Poor to Low
Powerful methods
to support VBR
(for fixed access)
Very High
(& Higher Peak Bit-rates)
Generic OFDM Receiver
Slot &
Timing
Sync.
AGC
Sampler
FFT
fc
VCO
fine offset
P/S and
Detection
Error
Recovery
gross offset
Freq. Offset
Estimation
(of all tones sent in one OFDM symbol)
Advantages of OFDM
• Allows carriers to overlap (no guard band), resulting in lesser
wasted bandwidth without any Inter Carrier Interference
(ICI)
• High data rate distributed over multiple carriers resulting in
lower symbol rate (more immune to ISI)
• Permits higher data rate as compared to FDM
• Increased security and bandwidth efficiency possible using
CDMA – OFDM (MC-CDMA)
• Simple guard intervals make the system more robust to
multipath effects.
Orthogonality Principle
• OFDM depends on the
orthogonality between
subcarriers
• Orthgonality principle in
Vector space
B
• A, B and C vectors in space are
orthogonal to each other
A
C
16
Orthogonality Principle
• Real Function space
f1 (t ) = A sin( wt )
f 2 (t ) = B cos(wt )
 +T

f1 (t ) f 2 (t )dt = 0
f m (t ) = M sin( mwt )
f n (t ) = N cos(nwt )
 +T

f m (t ) f n (t ) dt = 0
Orthogonality Principle cont..
f (t ) = sin( wt ) sin(2wt )
m.n Ν
T
 sin(mwt)sin(nwt)dt = 0 where m  n
0
T
 sin(mwt)cos(nwt)dt = 0
0
Here mw and nw are called
m-th and n-th harmonics of w
respectively
What is OFDM ??
Orthogonality in time domain…
What is OFDM ??
Orthogonality in frequency domain…
• Each carrier is modulated using BPSK / QPSK / M-ary QAM
• Frequency response for each carrier is a Sinc(X)
• Overlap of frequency response is possible as against FDM where inter-carrier
spacing is a must
• Frequency responses of the carriers overlap at zero crossings avoiding
Carrier Interference
Inter
Difference between FDM an OFDM
1
OFDM achieves
better bandwidth
efficiency over
conventional FDM
which makes it
suitable for higher
data rates
communication
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-6
-4
-2
0
2
4
6
-4
-2
0
2
4
6
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-6
OFDM general block diagram
Modulation
1-1-11…
i/p
Bitstream
S/P
Modulation
symbol
Add
cyclic
prefix
IFFT
Xn(k)
P/S
xn(t)
Transmitter
x(t)
noise
o/p
Bitstream
De-
Modulation
1-1-11…
P/S
FFT
Remove
cyclic
prefix
S/P
Receiver
Channel
Basic OFDM
• The OFDM symbol can be represented as
x(t ) =
N / 2 −1

X ( k )e
j 2
k
t
T
-T  t  T
k =− N / 2
x(t) is obtained using IFFT operation in the transmitter
At the receiver
1
X (k ) =
N
N / 2 −1

x(t )e
j 2
k
t
T
-T  t  T
t =− N / 2
• T: total symbol duration
N.B: x(t) and X(k) are time/frequency domain samples after
sampling process
IFFT and DFT concept
• To implement OFDM, we need N oscillators
• As N increases, the hardware implementation process becomes
inefficient
• Thus modeling OFDM process as a Discrete Fourier Transform (DFT)
can help for more efficient implementation
• Fast Fourier Transform (FFT) is the practical and efficient method for
implementation of DFT
• Thus, FFT and IFFT is used in all ODFM systems
Cyclic prefix
• Effect of multipath effect on OFDM symbols, delayed version of
symbol 1 due multipath effect will interfere with symbol 2
Cyclic prefix
• To solve this problem we need to use guard band
• Guard band is not efficient due the blanking period and also will
affect IFFT operation
• Therefore instead of blanking the guard period we repeat a part of
the symbol
Cyclic prefix
• the signal is periodic frequency domain, the FFT representation of
delayed signal in time domain is corresponding to phase shift (
rotation ) in frequency domain
FT {x (t − t0 )} = e jk 2t0 X (k )
• Since X(k) is periodic, we can obtain the correct in the frequency
domain
Cyclic prefix
• Another benefit of CP is that the signal can be decoded if the packet
is detected after some delay
OFDM disadvantages
• Carrier frequency offset
• High Peak –to-average power ratio
• Time synchronization
Frequency Offset
• Carrier recovery and tracking critical for OFDM
• Offsets can be comparable to sub-carrier spacing in OFDM
• DAC (at Tx) and ADC (at Rx) never have exactly the sampling period
• Can lead to loss on orthogonality
• OFDM can easily compensate for gross freq. offsets (offsets which are an
integral multiple of sub-carrier width)
Timing Synchronisation
• Timing recovery (at symbol level) is easily achieved in OFDM systems
• Can easily overcome distortions from delay spread
• Can employ non-coherent timing recovery techniques by introducing selfsimilarity
• => very robust to uncompensated frequency offsets
• If cyclic prefix is larger than the rms delay spread, range of (equally good) timing
phases become available
• => robust to estimation errors
PAPR problem
• Peak to average power ratio (PAPR)
| x(t ) |2
PAPR =
Pavg
• The large amplitude variation increases in-band noise and
increases the BER when the signal has to go through
amplifier nonlinearities.
OFDM -- PHY layer tasks
• Signals sent thro’ wireless channels encounter one or more of the
following distortions:
•
•
•
•
•
•
•
additive white noise
frequency and phase offset
timing offset, slip
delay spread
fading (with or without LoS component)
co-channel interference
non-linear distortion, impulse noise, etc
• OFDM is well suited for high-bit rate applications
IEEE Symp./ IISc -2001
IIT Madras
33
Proprietary OFDM Flavours
Wireless Access (Macro-cellular)
Wideband-OFDM
(W-OFDM) of Wi-LAN
www.wi-lan.com
-- 2.4 GHz band
-- 30-45Mbps in 40MHz
-- large tone-width
(for mobility, overlay)
Flash OFDM
from Flarion
www.flarion.com
-- Freq. Hopping for
CCI reduction, reuse
-- 1.25 to 5.0MHz BW
-- mobility support
Vector OFDM
(V-OFDM) of Cisco, Iospan,etc.
www.iospan.com
-- MIMO Technology
-- non-LoS coverage,
mainly for fixed access
-- upto 20 Mbps in MMDS
Wi-LAN leads the OFDM Forum -- many proposals submitted to
IEEE 802.16 Wireless MAN
Cisco leads the Broadand Wireless Internet Forum (BWIF)
Differences from other multiplexing
techniques
• OFDM versus TDM
-number of carriers
-synchronization
-sensitivity
-capacity/efficiency advantages
-complexity and cost issues
Applications
• Digital Audio and Video Broadcasting
• Asymmetric Digital Subscriber Line (ADSL)
• Wireless Networking
• Power-Line Communication Technology
• LTE mobile systems and expected to be used in future 5G mobile
systems
DFT and IDFT in OFDM
• Discrete Fourier Transform (DFT) and Inverse Discrete
Fourier Transform (IDFT) are the main operations of
OFDM system
• The continuous time is sampled at
• The DFT of finite signal is defined as
N −1
X [k ] =  x[n]e
n =0
− j 2
k
n
N
f =
k
N
DFT and IDFT in OFDM
• Let
WN = e
− j 2
1
N
• Then DFT signal can be represented as
N −1
X [k ] =  x[n]WNkn
n =0
• We can represent IDFT signal as
1
x[n] =
N
N −1
− kn
X
[
k
]
W

N
k =0
Matrix representation
• Therefore we can represent DFT relation as follows
X=Wx
1
 X [0]  1
 X [1]  1 W 1
N

 
 X [2]  = 1 WN2

 



 
 X [ N − 1] 1 WNN −1
1
WN2
WN4

( N −1) 2
N

1
  x[0] 

WNN −1   x[1] 
 WN2( N −1)   x[2] 







 WN( N −1)( N −1)   x[ N − 1]
Fast Fourier Transform
• DFT requires N2
complex multiplications
• FFT Complexity:
• Nlog2N complex multiplications and additions
Band limited and Time limited signals
• A time limited signal is one that is nonzero only for a finite length
time interval.
• A band limited is nonzero only for a finite bandwidth
•.
• if a signal is time limited, it cannot be band limited and vice versa.
• Signal can be simultaneously non time limited and non band limited.
Band limited and Time limited signals
• Time Limited signal in time
domain
• In frequency domain
Zero-to zero Bandwidth equals to
2/Ts
Raised cosine pulse shaping filters
• Raised cosine pulses are an example for bandwidth limited pulse
shapes
• Raised cosine pulse shaping filters are used to obtain more practical
and efficient shape for digital pulses
• It can minimize ISI
Raised cosine pulse shaping filters
• Raised cosine filter time response

1

sinc(
)

4T
2


t
h(t ) = 
cos(
)
1
t
T
 sinc( )
T 1 - ( 2t )
T

T

: is called rollt=
T
2
otherwise
off factor and it
affects the
pulse shape
0 1
Raised cosine pulse shaping filters
• Raised cosine filter frequency response
1− 

1
|
f
|


2T

T
1− 
1− 
1+ 
1
H ( f ) =  [1 + cos(
[| f | −
])]
| f |

2T
2T
2T
2
0
otherwise



Choice of ODFM parameters
• Main parameters to be considered in OFDM system
•
•
•
•
•
•
•
•
•
•
Bit rate
Tolerable delay Spread
Total Bandwidth
Subcarrier spacing
Number of subcarriers
Number of bits per symbol
Ratio between OFDM symbol and guard time
Relation between guard time and delay spread
Type of coding and modulation
Existence of null subcarriers ( Subcarriers carrying no data)
Example 1
• Find the transmitted ODFM baseband expression of the following bit
stream 11010111 if N=4 and QPSK is used
Example 2
• We want to design an OFDM system with the following parameters
• Bit rate = 20 Mbps
• Tolerable delay spread = 200 ns
• Bandwidth <15 MHz
Example 2 : solution
If delay spread is 200 ns then we take 800 ns as a safe value for guard
time
- We take OFDM symbol 6 times greater that guard time = 4.8 s
- Subcarrier spacing equals to 1/Ts = 1/(4.8 s -0.8 s)= 250 KHz
- To achieve 20 Mbps, no. of bits per symbol = bit rate x symbol time =
20x106 x 4.8x10-6 = 96 bits per symbol
- We have two options, using 16 QAM , coding rate ½ which gives 2
bits per symbol per subcarrier with total numbers of subcarrier
equals 48 subcarrier
Or QPSK with rate ¾ which gives 1.5 bits per symbol per subcarrier
with total number of subcarriers equals to 64
Example 2 : solution
• Total bandwidth in case 2 equals to 250 x 103 x 64 = 16 MHz which
cannot be used
• Therefore we can use case 1 with 48 subcarriers to achieve BW < 15
MHz which gives 60 subcarriers
• Therefore we can use case 1 with 16 zero subcarriers