Multipath can be described in two domains: time and frequency
Time domain: Impulse response
time
time
time
Impulse response
Frequency domain: Frequency response
time
time
time
Sinusoidal signal as input
f
Frequency response
time
Sinusoidal signal as output
Introduction to OFDM modulation
N carriers
Symbol: 2 periods of f0
Transmit
+
f
f
Symbol: 4 periods of f0
B
Symbol: 8 periods of f0
Data coded in frequency domain
Channel frequency
response
Transformation to time domain:
each frequency is a sine wave
in time, all added up.
N carriers
Receive
time
f
B
Time-domain signal
Frequency-domain signal
Decode each frequency
bin separately
Introduction to OFDM (Orthogonal frequency division multiplex)
Time-frequency grid
Data
Frequency
N carriers
B
Carrier
f0
B
One OFDM symbol
Features
T=1/f0
Time
– No intercarrier guard bands
– Controlled overlapping of bands
– Maximum spectral efficiency (Nyquist rate)
Intercarrier Separation =
Any integer Multiple of 1/(symbol duration)
– Easy implementation using IFFTs
– Very sensitive to time-freq. synchronization
Modulation technique
A user utilizes all carriers simultaneously to transmit its data as coded quantity at each
frequency carrier, which can be quadrature-amplitude modulated (QAM).
OFDM Modulation and Demodulation using FFTs
f
b0
b1
b2
.
.
.
.
bN-1
IFFT
(Inverse fast
Fourier transform)
time
Data coded in
frequency domain:
one symbol at a time
d0
d1
d2
d3
.
.
.
.
dN-1
P/S
d0, d1, d2, …., dN-1
(Parallel to
serial converter)
Transmit time-domain
samples of one symbol
Data in time domain:
one symbol at a time
S/P
d0’, d1’, d2’, …., dN-1’
(Serial to
parallel converter)
Receive time-domain
samples of one symbol
time
d0’
d1’
d2’
d3’
.
.
.
.
dN-1’
FFT
(Fast Fourier
transform)
f
b0’ Decode each
b1’ frequency bin
b2’ independently
.
.
.
.
bN-1’
Loss of orthogonality (by frequency offset)
ψ k (t) = exp( jk2π t / T ) y ψ k +m ( t) = exp ( j2 π (k + m)t / T )
Transmission pulses
ψ δk +m (t) = exp (j2 π ( k + m + δ ) / T ) con δ ≤ 1 / 2
Reception pulse with offset δ
I m (δ ) =
Interference between
channels k and k+m
Im (δ ) =
T sin πδ
π m+δ
∫
T
0
Summing up
∀m
exp( jk 2π t / T ) exp(− j(k + m + δ )2πt / T )dt =
N −1
1
23
T 2
2 ≈( δ)
14
m =1 m
∑ I 2m (δ ) ≈ (Tδ )2 ∑
m
T (1 − exp(− j2πδ ))
j2π (m + δ )
N >> 1 (N > 5 Is enough )
for
Total ICI due to loss of orthogonality
Interference in dB
0
-10
m=1
-20
m=3
m=5
m=7
-30
ICI in dB
Loss for 8 carriers
-40
-50
-60
-70
-0.4
-0.3
-0.2
-0.1
0
0.1
Frequency offset
0.2
0.3
0.4
-10
-15
-20
-25
-30
-35
-40
-45
-50
-55
-60
δ =0.05
δ =0.02
δ =0.01
δ =0.005
Practical limit
δ =0.002
δ =0.001
2
4
6
8
10
12
14
Carrier position within the band (N=16)
δ assumed r.v.
Gaussian σ=δ
16
Loss of orthogonality (time)
Let us assume
a misadjustment τ
Then
if m=k-l
− T / 2+ τ
Xi = c0 ∫
−T
/2
τ

senmπ
2T
T , c ≠c
0
1
Xi = 
mπ

c0 = c1
0,

In average, the interfering
power in any carrier is
 Xi 2
E 2
T
independent
on m
τ

ICI ≈ 20 log  2  , τ << T
T
Per carrier
2
2

1
τ 1
 τ
=
+
=
 4  T  2 0 2 2 T 

ICI due to loss of orthogonaliy
0
-45
-5
-40
m=1
-10
-35
-15
-20
ICI in dB
Interference in dB
τ
Xi 2mπ T
τ
≈
=2
T
mπ
T
Or approximately,
when τ<<T
Loss for 16 carriers
m=5
-25
m=10
-30
-35
assumed a uniform r.v.
-30
-25
N=8
Max. practical limit
-20
-40
-15
-45
-50
2 consecutive
symbols
T/ 2
ψ k (t)ψ *l (t − τ )dt + c 1 ∫−T / 2 +τ ψ k (t)ψ l* (t − τ )dt
0
0.1
Zone of interest
0.2
0.3
0.4
0.5
0.6
0.7
Relative misadjustment τ
0.8
0.9
1
-10
N=64
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Standard deviation of the relative misadjustment
0.08
Including a cyclic prefix to each OFDM symbol
To combat the multipath: including time guards between the symbols
copy
CP
τ
T
Tc
Including the Cyclic Prefix
Without the Cyclic Prefix
Symbol: 8 periods of fi
Initial transient
Ψi(t)
Channel: h(n)=(1 ) –n / n
n=0 ,…,2 3
≠Ψi(t)
Loss of orthogonality
Decaying transient
Symbol: 8 periods of fi
Passing the channel h(n)
Passing the channel h(n)
CP
Ψi(t)
Initial transient
remains within
the CP
The inclusion of a CP
maintains the orthogonality
Ψj(t)
Ψj(t)
Symbol: 4 periods of fi
Final transient
remains within
the CP
Symbol: 4 periods of fi
CP functions:
– It acomodates the decaying transient of the previous symbol
– It avoids the initial transient reachs the current symbol
..
Symplified scheme of an OFDM transceiver
Transmitter
.
S
P
S
IFFT
BITS
CODER
Cyclic
prefix (CP)
DAC
RF
P
f0
RF
ADC
Filter
S
f0
PLL,
symbol
timing
DECOD.
P
FFT
Receiver
S
P
Channel estimation
frequency offset
BITS
Windowing of the OFDM symbol
Total band used by OFDM: it depends on the number or carriers
ACI
ACI
Wide separation
Maintainig a fix bandwidth, if N increases
Narrow
separation
Adjacent channel interference decreases
BUT
It is interesting to have few carriers as well:
To introduce short delay in data gathering and signal processing (FFTs)
•
• To have a bigger intercarrier separation --> It reduces the relative frequency offset
Compromise
Need to shape the OFDM symbols
OFDM modulators with symbol shaping
p(t)
ejωnt
Equivalent
architectures
an
………
ejωnt
p(t)
an
Σan p(t) ejωnt
p(t) Σanejωnt
………
ak
ak
p(t)
ejωkt
ejωkt
Implemented with FFT
After the synchronous reception
p(t) even
I=∫
T/2
−T / 2
p(t)e
j2 π (k− n)t / T
T/2
0 , k ≠ n
dt =2 ∫ p(t)cos[2π (n − k)t / T]dt = 
0
1 , k = n
p(t)
T/2
0
cos
The simplest way to maintain symmetry within -T/2<t<T/2 is p(t)=k
❏ Symbol shaping has to be carried out as part
❏
of the symbol duration + CP
The total ACI can be condiderably reduced
even
odd
PC+T
Robustness against the channel and ACI improvement
N
……
Virtual OFDM symbols within the slot
• With guards (Cyclic prefixes), the channel’s time dispersion is avoided
L=N+CP
PC
PC
……
PC
PC
PC
OFDM symbols with time guards (CPs)
• With smooth transitions between symbols, the adjacent channel interference is minimized
PC
PC
PC
……
OFDM symbols with time guards and symbol shaping
PC
802.11a Physical Layer Data Symbol Format
t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 GI2
Short training sequence:
AGC and frequency offset
T1
T2
GI
OFDM Symbol
GI
OFDM Symbol
Long training sequence:
Channel estimation
Training symbols: 4 us each
t: 0.8 us, 16 samples
GI2: 1.6 us, 32 samples
T: 3.2 us, 64 samples
Data Symbols: 4 us each
GI: 0.8 us, 16 samples
OFDM Symbol: 3.2 us, 64 samples
* Only 52 of the 64 carriers are used.
* 4 of the 52 carriers are used for
pilot carriers (no data).
Data rate for each 20 Mhz channel:
20 Msamples per second.
250 Ksymbols per second.
48 data carriers per symbol.
1/2 or 3/4 convolutional code.
1 bit/carrier (BPSK) to 6 bits/carrier (64 QAM).
Overall:
Lowest: 48 * 1 * 1/2 * 250K = 6 Mbps.
Highest: 48 * 6 * 3/4 * 250K = 54 Mbps.
Turbo mode supports 108 Mbps using 40 Mhz channel.
Robustness against errors: random noise and channel-selected errors
Random noise: primarily introduced by thermal and circuit noise.
Channel-selected errors: introduced by magnitude distortion in
channel frequency response.
Frequency
Time-frequency grid
B
Bad carriers
f0
f
Frequency response
Data bits
T=1/f0
Time
Errors are no longer random. Interleaving is often used to scramble
the data bits so that standard error correcting codes can be applied.
Spectrum Mask
Power Spectral Density
-20 dB
-28 dB
-40 dB
-30
-20
-11 -9
f carrier
9
11
20
30
Frequency (MHz)
• Requires extremely linear power amplifier design.
Adjacent Channel and Alternate Adjacent Channel Rejection
D a te
ra te
6 M bps
M in im u m
S e n s ib ility
-8 2 d B m
A d ja c e n t C h a n n e l
R e je c tio n
16 dB
A lte r n a te
C h a n n e l re je c tio n
32 dB
12M bps
-7 9 d B m
13 dB
29 dB
24M bps
-7 4 d B m
8 dB
24 dB
36M bps
-7 0 d B m
4 dB
20 dB
54M bps
-6 5 d B m
0 dB
15 dB
32 dB
16 dB
Signal
Frequency
• Requires joint design of the anti-aliasing filter and ADC.