OFDM (Orthogonal Frequency Division Multiplexing) is a digital multi-carrier modulation technique that divides the available spectrum into multiple orthogonal subcarriers. It provides advantages like efficient handling of multi-path fading and channel delay spread. Key aspects of OFDM include the insertion of a guard interval between symbols to suppress adjacent symbol interference and the use of cyclic prefixing to suppress inter-carrier interference. The FFT algorithm is used for modulation and demodulation of OFDM signals.

1. Orthogonal Frequency
Division Multiplexing
OFDM
fred harris
Cubic Signal Processing Chair
San Diego State University
fred.harris@sdsu.edu
Vehicular Technology Conference - 2004

2. Textbooks and References
 “Wireless OFDM Systems: How to Make Them Work”
Marc Engels, Editor
 “OFDM Wireless LANs: A Theoretical and Practical Guide”
Juha Heiskala and John Terry
 “OFDM for Wireless Multimedia Communications”
Richard Van Nee and Ramjee Prasad
 “Single and Multi-Carrier Quadrature Amplitude Modulation”
Lajos Hanzo, William Webb, and Thomas Keller
 “ADSL, VDSL, and Multicarrier Modulation”
John Bingham
 “Implementing ADSL”
David Ginsburg
 “DSL Advances
Massimo Sorbara, John Cioffi, and Peter Silverman

3. OFDM
 OFDM also known as
Multi-Carrier or Multi-Tone Modulation
 DAB-OFDM
Digital Audio Broadcasting
 DVD-OFDM
Digital Video Broadcasting
 ADSL-OFDM
Asynchronous Digital Subscriber Line
 Wireless Local Area Network
IEEE-802.11a, IEEE-802.11g
ETSI BRAN (Hyperlan/2)

4. OFDM Systems
System Transform
Size
Number
Carriers
Channel
Spacing
kHz
Bandwidth
MHz
Sample
Rate
MHz
Symbol
Duration
sec
Data
Rate
Mbits/s
HyperLAN/2 64 52
4
312.5 16.25 20 3.2
0.8
6-54
802.11a 64 52
4
312.5 16.56 20 3.2
0.8
6-54
DVB-T 2048
1024
1712
842
4.464 7.643 9.174 224 0.68-14.92
DAB 2048
8192
1536 1.00 1.536 2.048 24/48/96
msec
3.072
ADSL 256 (down)
64 (up)
36-127
7-28
4.3125 1.104 1.104 231.9 0.64-8.192

5. OFDM Advantages
 Efficiently Deals With Multi-path Fading
 Efficiently Deals With Channel Delay Spread
 Enhanced Channel Capacity
 Adaptively Modifies Modulation Density
 Robustness to Narrowband Interference

6. OFDM Disadvantages
 OFDM Sensitive to
Small Carrier Frequency Offsets
 OFDM Exhibits
High Peak to Average Power Ratio
 OFDM Sensitive to
High Frequency Phase Noise
 OFDM Sensitive to
Sampling Clock Offsets

7. Single Carrier System
Sequential Transmission
of Waveforms
Waveforms are
Short Duration T
Waveforms Occupy
Full System Bandwidth 1/T

8. Multi-Carrier System
Parallel Transmission
of Waveforms
Waveforms are
Long Duration MT
Waveforms Occupy 1/M th
Of System Bandwidth 1/T

9. OFDM: Dense Multichannel System
Conventional Multichannel System
Non Overlapping Adjacent Channels.
Channels separated by More
Than Their Two Sided bandwidth
OFDM Multichannel System
50% Overlap of Adjacent Channels
Available bandwidth is Used Twice
Channels separated by Half
Their Two Sided bandwidth

10. Standard Digital
Communication System

11. Bandlimited Channel
Nyquist Spectrum Nyquist Spectrum
With Cosine Taper
Infinite Duration Nyquist Pulse Finite Duration Nyquist Pulse

12. Translation in Time Domain
Phase Slope in Frequency Domain
Translated Signals Are Orthogonal When Peak is Translated to Zero Crossings of Original

13. Channel Distortion
Modifies Received Wave Shape
-6 -4 -2 0 2 4 6
-0.2
0
0.2
0.4
0.6
0.8
1
Matched Filter Output For SQRT Nyquist Pulse
-6 -4 -2 0 2 4 6
-0.2
0
0.2
0.4
0.6
0.8
1
Matched Filter Output For Channel Distorted SQRT Nyquist Pulse

14. Inter Symbol Interference (ISI)
Due to Channel Distorted Signal
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-2
-1
0
1
2
Eye Diagram No Channel Distortion
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-2
-1
0
1
2
Eye Diagram With Channel Distortion

15. Steady State Response of a Filter
to a Sine Wave is a Sine Wave

16. Rectangle Pulse: DC Centered Spectrum
with Equally Spaced Zeros
0
sin(2 )
2
( )
(2 )
2
P
P
P
T
f
H f AT
T
f




17. Shift Spectrum with Linear Phase on DC
Pulse: Move Spectrum to First Spectral Zero
)
2
)
1
(
2
(
)
2
)
1
(
2
sin(
)
(
P
p
P
p
P
k
T
T
k
f
T
T
k
f
AT
f
H






18. Real Part of Complex Exponential Time Series:
Integer Number of Cycles per Interval
0 0.5 1
-1
0
1
0 0.5 1
-1
0
1
0 0.5 1
-1
0
1
0 0.5 1
-1
0
1
0 0.5 1
-1
0
1
0 0.5 1
-1
0
1
0 0.5 1
-1
0
1
0 0.5 1
-1
0
1
0 0.5 1
-1
0
1
0 0.5 1
-1
0
1
0 0.5 1
-1
0
1
0 0.5 1
-1
0
1
0 0.5 1
-1
0
1
0 0.5 1
-1
0
1
0 0.5 1
-1
0
1
0 0.5 1
-1
0
1

19. Imaginary Part of Complex Exponential Time Series:
Integer Number of Cycles per Interval
0 0.5 1
-1
0
1
0 0.5 1
-1
0
1
0 0.5 1
-1
0
1
0 0.5 1
-1
0
1
0 0.5 1
-1
0
1
0 0.5 1
-1
0
1
0 0.5 1
-1
0
1
0 0.5 1
-1
0
1
0 0.5 1
-1
0
1
0 0.5 1
-1
0
1
0 0.5 1
-1
0
1
0 0.5 1
-1
0
1
0 0.5 1
-1
0
1
0 0.5 1
-1
0
1
0 0.5 1
-1
0
1
0 0.5 1
-1
0
1

20. Spectra Of Complex Exponential Time Series:
Integer Number of Cycles per Interval
0 0.5
0
0.5
1
0 0.5
0
0.5
1
0 0.5
0
0.5
1
0 0.5
0
0.5
1
0 0.5
0
0.5
1
0 0.5
0
0.5
1
0 0.5
0
0.5
1
0 0.5
0
0.5
1
0 0.5
0
0.5
1
0 0.5
0
0.5
1
0 0.5
0
0.5
1
0 0.5
0
0.5
1
0 0.5
0
0.5
1
0 0.5
0
0.5
1
0 0.5
0
0.5
1
0 0.5
0
0.5
1

21. Continuous Time: Orthogonal Time Signal Set
 
















T
m
n
k
m
n
if
T
m
n
if
dt
t
t
T
t
k
t
k
T
j
t
T
t
k
t
0
k
0
)
(
)
(
0
,
,
2
,
1
,
0
,
1
,
2
,
:
)
2
exp(
)
(
0
,
,
2
,
1
,
0
,
1
,
2
,
:
)
(










22. Discrete Time: Orthogonal Time Signal Set
k
1
0
..., 2, 1, 0 ,1, 2,....,
( ) :
0 1
0 ,1, 2,...., 1
2
( ) exp( ) :
0
0 ,1, 2,...., 1
2
exp( ) :
0
0
( ) ( )
: ( ) ( ) ( ) ( )
k
N
n m
n
k N k k N k
k
n
n N
k N
n j k nT
NT nT NT
k N
j k n
N n N
if n m
n n
N if n m
NOTE n n n n




 
   



  
  
  
 
  
 
 
  
 



 


23. OFDM Modulator

24. OFDM Demodulator

25. OFDM is a Block Process

26. Adjacent Symbol Interference (ASI)
Symbol Smearing Due to Channel

27. Guard Interval Inserted Between Adjacent
Symbols to Suppress ASI

28. Cyclic Prefix Inserted in Guard Interval to
Suppress Adjacent Channel Interference (ACI)

29. Data Length Defines Sinc Width:
Spectral Spacing Matches Width

30. Extended Data Length Reduces Sinc
Width: Spectral Spacing Preserved

31. OFDM Symbol: Time and Spectra
Channel Input and Output
20 40 60 80 100 120 140 160 180
-0.4
-0.2
0
0.2
0.4
0.6
Real Part of Time Series, Input to Channel
20 40 60 80 100 120 140 160 180
-0.4
-0.2
0
0.2
0.4
0.6
Real Part of Time Series, Output of Channel
-0.5 0 0.5
-30
-25
-20
-15
-10
-5
0
5
10
Spectrum
-0.5 0 0.5
-30
-25
-20
-15
-10
-5
0
5
10
Spectrum

32. OFDM Spectra
Without and with Cyclic Prefix
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-1.5
-1
-0.5
0
0.5
1
1.5
OFDM spectral lines With Channel Without Cyclic Prefix
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-1.5
-1
-0.5
0
0.5
1
1.5
OFDM spectral lines With Channel With Cyclic Prefix

33. Overlaid Constellations , All Frequencies,
Without and With Cyclic Prefix
-1.5 -1 -0.5 0 0.5 1 1.5
-1.5
-1
-0.5
0
0.5
1
1.5
OFDM Constellations With Channel Without Cyclic Prefix
-1.5 -1 -0.5 0 0.5 1 1.5
-1.5
-1
-0.5
0
0.5
1
1.5
OFDM Constellations With Channel With Cyclic Prefix

34. Constellations: Different OFDM Bins
Without Cyclic Prefix
-1 0 1
-1
0
1
-1 0 1
-1
0
1
-1 0 1
-1
0
1
-1 0 1
-1
0
1
-1 0 1
-1
0
1
-1 0 1
-1
0
1

35. Constellations: Different OFDM Bins
With Cyclic Prefix
-1 0 1
-1
0
1
-1 0 1
-1
0
1
-1 0 1
-1
0
1
-1 0 1
-1
0
1
-1 0 1
-1
0
1
-1 0 1
-1
0
1

36. Channel Estimate with Pilots
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
0
0.5
1
Channel, Bandwidth, and Samples
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
0
0.5
1
Zero Packed
Spectral Samples
and
Extended Reflection
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
0
0.5
1
Interpolated
Spectral Points
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
0
0.005
0.01
Magnitude of Interpolation Error For In-Band Frequencies
Normalized Frequency

37. DFT (FFT) as Signal Generator
for Complex Sinusoids

38. DFT (FFT) As Signal Analyzer
for Complex Sinusoids
1
,
...
,
2
,
1
,
0
:
)
(
)
(
1
0
2


 



N
k
e
n
h
k
H
N
n
nk
N
j


39. Radix-2 FFT Flow Diagrams

40. Input Vector FFT Mapped to Output Time Series,
Up-Sampled, Converted Via DAC to Waveform,
and I-Q Up-Converted

41. The FFT as Signal Generator
and Interpolator

42. OFDM Modulation With IFFT
and Interpolator

43. OFDM Demodulation With FFT

44. OFDM Transceiver

45. Time and Spectra of Sparse OFDM Symbol
0 10 20 30 40 50 60 70 80 90 100
-1
-0.5
0
0.5
1
Real Part OFDM Time Series
Normalized Time
Amplitude
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-60
-50
-40
-30
-20
-10
0
10
Spectrum
Normalized Frequency
Log
Magnitude
(dB)

46. Time and Spectra With Frequency Offset = 0.1 Bin
0 10 20 30 40 50 60 70 80 90 100
-1
-0.5
0
0.5
1
Real Part OFDM Time Series with Offset Frequency = 0.1 Bin Width
Normalized Time
Amplitude
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-60
-50
-40
-30
-20
-10
0
10
Spectrum With Frequency Offset = 0.1 Bin Width
Normalized Frequency
Log
Magnitude
(dB)

47. Time and Spectra With Sample Clock Offset = 1.02 fs
0 10 20 30 40 50 60 70 80 90 100
-1
-0.5
0
0.5
1
Real Part OFDM Time Series with Sampling Clock = 1.02 fs
Normalized Time
Amplitude
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-60
-50
-40
-30
-20
-10
0
10
Spectrum With Sampling Clock = 1.02 fs
Normalized Frequency
Log
Magnitude
(dB)

48. Time and Spectra With Sample Clock Offset = 0.98 fs
0 10 20 30 40 50 60 70 80 90 100
-1
-0.5
0
0.5
1
Real Part OFDM Time Series with Sampling Clock = 0.98 fs
Normalized Time
Amplitude
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-60
-50
-40
-30
-20
-10
0
10
Spectrum With Sampling Clock = 0.98 fs
Normalized Frequency
Log
Magnitude
(dB)

49. Ideal I-Q Up and Down Conversion
S
hape
S
hape
Match
Match
CHANNEL
cos( t)

cos( t)

-sin( t)

-sin( t)

n(t)
I(t) I(t)
Q(t)
Q(t)
^
^

50. Spectral and Time Description of
Real Sinusoids

51. Complex Sinusoids-I

52. Complex Sinusoids-II

53. Complex Baseband and Complex
Band-Centered Spectra

54. Complex Baseband and Real
Band-Centered Spectra

55. Complex Down Conversion

56. Gain and Phase Imbalance in
I-Q Mixers

57. Spectral Image Due to Gain Imbalance

58. Spectral Image Due to Phase Imbalance

59. Line Spectral Images Due to I-Q Mismatch

60. Coupling Between Positive and Negative FFT Indices Due to
I-Q Imbalance and First Order Correction Mechanism
(1 ) ( )
( ) ( )
2 2 2
( ) ( )
( ) (1 )
2 2 2
(1 ) ( )
( ) ( )
2 2 2
( ) ( )
( ) (1 )
2 2 2
j j
G k H k
G k H k
j j
j j
H k G k
H k G k
j j
  
  
  
  
 
  
 
   
  
   
 
   
 
  
 
 
 
  
 
   
  
   
 
   
 
  
 
 

61. Test Bench: Demonstration of Receiver I-Q
Imbalances, Carrier Offset, and Timing Offset

62. Carrier Offset: 4% of FFT Bin Width

63. Timing Offset: 10% of Sampling Time Period

64. Timing Clock Offset: 5% of Sampling Time Period per Frame

65. Gain Imbalance: 10% Error

66. Phase Imbalance: 0.1 Radian Error

67. I-Q Mixer Imbalance; 20% Gain, 0.2 Radians

68. Differential Delay to I/Q Mixers,
10% of Sample Interval

69. Periodic Time Segments in OFDM Frame
Obtained by Zero Packing Spectrum

70. Probe Mismatch During Short Repeated Preamble

71. Power Amplifier Non-Linearity
0 1 2 3 4
0
0.5
1
1.5
2
2.5
3
3.5
4
Nonlinear Transfer Function of Amplifier
1-dB Compression Point
0 2 4 6 8 10
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Input and Output of Non-Linear Amplifier
-0.5 0 0.5
-60
-50
-40
-30
-20
-10
0
10
Spectrum of Two Input Sinusoids
Normalized Frequency
-0.5 0 0.5
-60
-50
-40
-30
-20
-10
0
10
Spectrum of Two Output Sinusoids
Normalized Frequency

72. 16-QAM Input and Output Envelopes.
Saturation and 1-dB Compression Circles
-2 -1 0 1 2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Envelope at Output of Amplifier
-2 -1 0 1 2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Envelope at Input to Amplifier
Saturation at 2-Times RMS Signal Level
Saturation Saturation
1-dB
Compression
1-dB
Compression

73. Limiting Amplifier Effect on Received QAM Constellation
-1 -0.5 0 0.5 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Matched Filter Applied to Input of Amplifier
-1 -0.5 0 0.5 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Matched Filter Applied to Output of Amplifier

74. Limiting Amplifier Effect on Signal Spectra
-4 -3 -2 -1 0 1 2 3 4
-60
-50
-40
-30
-20
-10
0
10
Spectrum at Input to Amplifier
Normalized Frequency (f/fsym
)
Log
Magnitude
(dB)
-4 -3 -2 -1 0 1 2 3 4
-60
-50
-40
-30
-20
-10
0
10
Spectrum at Output of Amplifier
Normalized Frequency (f/fsym
)
Log
Magnitude
(dB)
SPECTRAL REGROWTH

75. 16-QAM (=0.2) Envelope Statistics
0 0.5 1 1.5 2 2.5
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
16-QAM Histogram at Amplifier Input
Normalized Amplitude (x/x
)
0 0.5 1 1.5 2 2.5
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016 std dev =1.03 clip level
16-QAM Histogram at Amplifier Output
Normalized Amplitude (x/x
)
0 0.5 1 1.5 2 2.5 3
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
Probabilty of Level Crossing
Normalized Amplitude (x/x
)

76. OFDM Input and Output Envelopes:
Saturation and 1-dB Compression Circles
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
Envelope at Input to Amplifier
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
Envelope at Output of Amplifier
Saturation at 2-Times RMS Signal Level
Saturation
Saturation
1-dB
Compression
1-dB
Compression

77. Limiting Amplifier Effect on OFDM Constellation
-1 -0.5 0 0.5 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
OFDM Constellation at Input to Amplifier
-1 -0.5 0 0.5 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
OFDM Constellation at Output of Amplifier

78. OFDM Envelope Statistics
0 1 2 3 4
0
0.005
0.01
0.015
OFDM Histogram at Amplifier Input
Normalized Amplitude (x/x
)
0 1 2 3 4
0
0.005
0.01
0.015
std dev =1.06 clip level
OFDM Histogram at Amplifier Output
Normalized Amplitude (x/x
)
0 1 2 3 4
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
Probabilty of Level Crossing
Normalized Amplitude (x/x
)

79. OFDM Envelope Statistics with
Selected Alternate Mapping
0 1 2 3 4
0
0.005
0.01
0.015
0.02
OFDM Histogram: One FFT
0 1 2 3 4
0
0.005
0.01
0.015
0.02
OFDM Histogram: Two FFTs
0 1 2 3 4
0
0.005
0.01
0.015
0.02
OFDM Histogram: Four FFTs
Normalized Amplitude (x/x
)
0 1 2 3 4
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
Probabilty of Level Crossing
Normalized Amplitude (x/x
)
One FFT
Two FFTs
Four FFTs

80. Clipping

81. Smart Clipping

82. Reserve Frequency Bins Form Clipping
Pulses

83. Selecting Reserve Frequency Bins
-60 -40 -20 0 20 40 60
0
0.2
0.4
0.6
0.8
1
Spectrum 11-Adjacent Frequencies
-0.5 0 0.5
0
0.2
0.4
0.6
0.8
1
Time Series for 11-Adjacent Frequencies
-60 -40 -20 0 20 40 60
0
0.2
0.4
0.6
0.8
1
Spectrum 11-Equally Spaced Frequencies
-0.5 0 0.5
0
0.2
0.4
0.6
0.8
1
Time Series for 11-Equally Spaced Frequencies
-60 -40 -20 0 20 40 60
0
0.2
0.4
0.6
0.8
1
Spectrum 11-Randomly Spaced Frequencies
-0.5 0 0.5
0
0.2
0.4
0.6
0.8
1
Time Series for 11-Randomly Spaced Frequencies

84. Reserve Bin Canceller Clipping at 2.5  (8 dB)
0 50 100 150 200 250
0
1
2
3
4
5
Peak envelope, input to PAR control
0 50 100 150 200 250
0
1
2
3
4
5
Peak envelope, output of first pass PAR control
data
clip level
data std dev
average peak
0 50 100 150 200 250
0
1
2
3
4
5
Peak envelope, output of second pass PAR control
data
clip level
data std dev
average peak
0 50 100 150 200 250
0
1
2
3
4
5
Peak envelope, output of third pass PAR control
data
clip level
data std dev
average peak

85. Statistics for Clip at 2.5 (8 dB)
0 1 2 3 4
0
0.005
0.01
0.015
0.02
0.025
input histogram
0 1 2 3 4
0
0.005
0.01
0.015
0.02
0.025
std dev =0.928
clip level
output histogram
-5 0 5 10
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
average =-0.648 dB
prob of level crossing
PAR (dB)
input
pass-1
pass-2
pass-3

86. Reserve Bin Canceller Clipping at 2.2  (6.9 dB)
0 50 100 150 200 250
0
1
2
3
4
5
Peak envelope, input to PAR control
0 50 100 150 200 250
0
1
2
3
4
5
Peak envelope, output of first pass PAR control
data
clip level
data std dev
average peak
0 50 100 150 200 250
0
1
2
3
4
5
Peak envelope, output of second pass PAR control
data
clip level
data std dev
average peak
0 50 100 150 200 250
0
1
2
3
4
5
Peak envelope, output of third pass PAR control
data
clip level
data std dev
average peak

87. Statistics for Clip at 2.2 (6.9 dB)
0 1 2 3 4
0
0.005
0.01
0.015
0.02
0.025
input histogram
0 1 2 3 4
0
0.005
0.01
0.015
0.02
0.025
std dev =0.928
clip level
output histogram
-5 0 5 10
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
average =-0.653 dB
prob of level crossing
PAR (dB)
input
pass-1
pass-2
pass-3

88. Reserve Bin Canceller Clipping at 2.0  (6 dB)
0 50 100 150 200 250
0
1
2
3
4
5
Peak envelope, input to PAR control
0 50 100 150 200 250
0
1
2
3
4
5
Peak envelope, output of first pass PAR control
data
clip level
data std dev
average peak
0 50 100 150 200 250
0
1
2
3
4
5
Peak envelope, output of second pass PAR control
data
clip level
data std dev
average peak
0 50 100 150 200 250
0
1
2
3
4
5
Peak envelope, output of third pass PAR control
data
clip level
data std dev
average peak

89. Statistics for Clip at 2.0 (6 dB)
0 1 2 3 4
0
0.005
0.01
0.015
0.02
0.025
input histogram
0 1 2 3 4
0
0.005
0.01
0.015
0.02
0.025
std dev =0.927
clip level
output histogram
-5 0 5 10
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
average =-0.659 dB
prob of level crossing
PAR (dB)
input
pass-1
pass-2
pass-3

90. OFDM 802.11a

91. Time-Frequency Profile of 802.11a Tones
Pilot Tones Shown in Yellow

92. Preamble and Pilot Structure
 Short Symbols
 Start of Frame Detection
 Signal Strength Indication
 Frequency Offset Resolution
 Long Symbols
 Channel Estimate
 Fine Time Resolution
 Distributed Pilots
 Carrier Tracking
 Sample Clock Tracking

93. Preamble Time Structure
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
0.2
0.4
0.6
0.8
Magnitude IEEE 802.11a Preamble
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-1
-0.5
0
0.5
1
Real Part
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-0.5
0
0.5
1
Imaginary Part

94. Detecting Frame Start with
Repeated Short Symbols

95. Signals in Preamble Detector
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.2
0.4
0.6
0.8
Envelope of Input Signal
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.2
0.4
0.6
0.8
Delayed Envelope of Input Signal
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.2
0.4
0.6
0.8
cross and auto correlations of Input Signal
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.5
1
Ratio of Cross to Auto Correlation
Detection
Threshold
Cross Correlation

96. Detail of Signal in Preamble Detector
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6
0
0.2
0.4
0.6
0.8
Envelope of Input Signal
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6
0
0.2
0.4
0.6
0.8
Delayed Envelope of Input Signal
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6
0
0.2
0.4
0.6
0.8
cross and auto correlations of Input Signal
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6
0
0.5
1
Ratio of Cross to Auto Correlation
Detection
Threshold
Cross Correlation
Auto Correlation

97. Maximum Likelihood Estimator for
Frequency Offset

98. Frequency and Signal Strength Estimates
0 0.5 1 1.5 2 2.5 3 3.5 4
-10
-5
0
5
10
Estimate of Frequency Offset
Sample Time
Frequency
Offset
in
FFT
Bins
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.02
0.04
0.06
0.08
0.1
Estimate of Signal Strength
Sample Time
Mag
Square
Known Offset
3.3 FFT Bins

99. Cross Correlation of Long Preamble
0 1 2 3 4 5 6 7
0
2
4
6
8
10
12
14
Cross Correlation of Input Signal With Long Preamble Section
Time Samples
Amplitude
4.3 4.4 4.5 4.6 4.7 4.8
0
5
10
15
Zoom to First Correlation Peak
Time Samples
Amplitude
5.3 5.4 5.5 5.6 5.7 5.8
0
5
10
15
Zoom to Second Correlation Peak
Time Samples
Amplitude
Expected Peak Position Expected Peak Position

100. Clipped Cross Correlation of Long Preamble
0 100 200 300 400 500 600 700 800 900
0
10
20
30
40
50
Clipped Cross Correlation of Input Signal With Long Preamble Section
Time Samples
Amplitude
4.3 4.4 4.5 4.6 4.7 4.8
0
10
20
30
40
50
Zoom to First Correlation Peak
Time Samples
Amplitude
5.3 5.4 5.5 5.6 5.7 5.8
0
10
20
30
40
50
Zoom to Second Correlation Peak
Time Samples
Amplitude
Clipped Correlator
Replica Signal Clipped Version of Template Signal
Sign[Real(Template)]+j*sign[Imag(Template)]

101. Channel Probe With Long Preamble
-0.5 0 0.5
0
1
2
3
4
5
6
7
Channel Probe, Long Segment of Preamble
Normalized Frequency
Amplitude
-0.5 0 0.5
0
1
2
3
4
5
6
7
Response of Channel Probe, One Look
Normalized Frequency
Amplitude
-0.5 0 0.5
0
1
2
3
4
5
6
7
Response of Channel Probe, No Noise
Normalized Frequency
Amplitude
-0.5 0 0.5
0
1
2
3
4
5
6
7
Response of Channel Probe, Average of Two Looks
Normalized Frequency
Amplitude

102. Constellation with Residual Carrier Offset
-1.5 -1 -0.5 0 0.5 1 1.5
-1.5
-1
-0.5
0
0.5
1
1.5
5000 Constellations, Zero Carrier Frequency Offset
-1.5 -1 -0.5 0 0.5 1 1.5
-1.5
-1
-0.5
0
0.5
1
1.5
400 pilot, Zero Carrier Frequency Offset
-1.5 -1 -0.5 0 0.5 1 1.5
-1.5
-1
-0.5
0
0.5
1
1.5
5000 Constellations, 5 ppm Carrier Frequency Offset
-1.5 -1 -0.5 0 0.5 1 1.5
-1.5
-1
-0.5
0
0.5
1
1.5
400 pilot, 5 ppm Carrier Frequency Offset

103. Frequency Domain Residual Carrier Offset
-30 -20 -10 0 10 20 30
-1.5
-1
-0.5
0
0.5
1
1.5
60 Frames Real Part FFT: Zero Carrier Frequency Offset
Frequency Index
Amplitude
-30 -20 -10 0 10 20 30
-1.5
-1
-0.5
0
0.5
1
1.5
60 Frames Real Part FFT: 5 ppm Carrier Frequency Offset
Frequency Index
Amplitude

104. Constellations with Sample Clock Offset
-1.5 -1 -0.5 0 0.5 1 1.5
-1.5
-1
-0.5
0
0.5
1
1.5
5000 Constellations, Zero Clock Frequency Offset
-1.5 -1 -0.5 0 0.5 1 1.5
-1.5
-1
-0.5
0
0.5
1
1.5
400 pilot, Zero Clock Frequency Offset
-1.5 -1 -0.5 0 0.5 1 1.5
-1.5
-1
-0.5
0
0.5
1
1.5
5000 Constellations, 300 ppm Clock Frequency Offset
-1.5 -1 -0.5 0 0.5 1 1.5
-1.5
-1
-0.5
0
0.5
1
1.5
400 pilot, 300 ppm Clock Frequency Offset

105. Frequency Domain With Sample Clock Offset
-30 -20 -10 0 10 20 30
-1.5
-1
-0.5
0
0.5
1
1.5
80 Frames Real Part FFT: Zero Clock Frequency Offset
Frequency Index
Amplitude
-30 -20 -10 0 10 20 30
-1.5
-1
-0.5
0
0.5
1
1.5
80 Frames Real Part FFT: 300 ppm Clock Frequency Offset
Frequency Index
Amplitude

106. Other Variants of OFDM
Amplitude and Phase Overlays
 Shaped OFDM
 OQAM OFDM
 Coded OFDM
 CI OFDM

107. Shape to Control Spectral Side Lobes

108. Overlapped OFDM Frames

109. Polyphase Filter For Shaped OFDM
Shaping and Matched Filter

110. Impulse Response of Shaped OFDM
Modulator and Demodulator
0 1 2 3 4 5 6 7 8
0
0.5
1
Impulse at Input to IFFT (DC-bin)
0 1 2 3 4 5 6 7 8
0
0.5
1
Impulse Response at Output of IFFT
0 1 2 3 4 5 6 7 8
0
0.5
1
Impulse Response at Output of Polyphase Shaping Filter
0 1 2 3 4 5 6 7 8
0
0.5
1
1.5
Impulse Response at Output of
Polyphase Matched Filter
0 1 2 3 4 5 6 7 8
0
0.5
1
Impulse Response at Output of FFT

111. Orthogonal: Adjacent Time Slots Non
Adjacent Frequency Bins
0 1 2 3 4 5
-0.2
0
0.2
0.4
0.6
0.8
1
Impulse Response Shaping Filter
0 2 4 6 8 10
-0.2
0
0.2
0.4
0.6
0.8
1
Auto Correlation Response
-4 -3 -2 -1 0 1 2 3 4
-60
-40
-20
0
Adjacent Spectral Bins Correlated
Alternate Spectral Bins Not Correlated
Spectrum: Shaping Filter Centered on IFFT Spectral Bins

112. Impulse Response Time-Frequency Profile
-10
-5
0
5
10
0
2
4
6
8
0
0.2
0.4
0.6
0.8
1
frequency bin
OFDM Frame Number

113. Orthogonality Between Real and Imaginary
Part of Shaped OFDM Frequency Bins

114. Even and Odd Symmetric Wave Shapes from
Adjacent Bins are Orthogonal in Shaped OFDM

115. Symmetry Considerations in Real and Imaginary
Components of Offset Shaped OFDM Frames

116. Offset OFDM

117. Compare Spectra
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-60
-40
-20
0
Spectrum of Standard OFDM With Cyclic Prefix
Normalized Frequency
Log
Magnitude
(dB)
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-60
-40
-20
0
Spectrum of OFDM/OQAM Without Cyclic Prefix
Normalized Frequency
Log
Magnitude
(dB)

118. OFDM and Shaped OFDM PAR
0 1 2 3 4
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
Histogram: Standard OFDM
Normalized Amplitude (x/x
)
0 1 2 3 4
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
Histogram: Shaped OFDM/OQAM
Normalized Amplitude (x/x
)
0 1 2 3 4
10
-4
10
-3
10
-2
10
-1
10
0
Probabilty of Level Crossing
Normalized Amplitude (x/x
)
Standard OFDM
Shaped OFDM/OQAM

119. Complementary Codes

120. Canceling Correlation Side Lobes

121. Inserting CC in OFDM
2 2
2
( ) ( ) ( ) ( ) 2 ( )
( ) ( ) 2 (A Constant Power Level)
Since Sample Values ( ) are equal to 1
Average Power in ( ) = N
2N
Thus Peak to Average Power Ratio 2
N
N N N N
N N
N
N
A n A n A n A n N n
A B N
A n
A

 

   
 
 
Now Reverse Domains
Use Complementary Code Sequence
as amplitude of Carriers in Frequency Domain
Then time series has
Peak Squared Magnitude = 2N
Average Magnitude = N
for Peak to Average Power Ratio = 2

122. CC and Digital Filters

123. Equivalent Phase Coding

124. PAR in CCK OFDM and Standard OFDM
0 500 1000 1500 2000 2500 3000 3500
0
0.5
1
1.5
2
2.5
Mag Square of CC Rate 1/2 OFDM
0 500 1000 1500 2000 2500 3000 3500
0
2
4
6
8
Mag Square of Coded Rate 1/2 OFDM
Mean = 1
Mean = 1
Peak = 2
Peak = 6.26

125. OFDM and CC-OFDM PAR

126. C-I OFDM
Carrier Interferometry
 OFDM with Phase Overlay
 In Conventional OFDM
Rectangle Envelope in Time
Dirichlet Kernel in Frequency
 In CI-OFDM
Rectangle Envelope in Frequency
Dirichlet Kernel in Time
 Sin(x)/x in Time Domain Without Excess Bandwidth,
No Square-Root Nyquist Shaping Filter

127. Frequency Domain Phase Slope in
Continuous and in Sampled Data Domains

128. Circularly Shifted Time Domain Dirichlet Kernels
-10 0 10
-0.5
0
0.5
1
-10 0 10
-0.5
0
0.5
1
-10 0 10
-0.5
0
0.5
1
-10 0 10
-0.5
0
0.5
1
-10 0 10
-0.5
0
0.5
1
-10 0 10
-0.5
0
0.5
1
-10 0 10
-0.5
0
0.5
1
-10 0 10
-0.5
0
0.5
1
-10 0 10
-0.5
0
0.5
1
-10 0 10
-0.5
0
0.5
1
-10 0 10
-0.5
0
0.5
1
-10 0 10
-0.5
0
0.5
1
-10 0 10
-0.5
0
0.5
1
-10 0 10
-0.5
0
0.5
1
-10 0 10
-0.5
0
0.5
1
exp(-j 2 k 0/N) exp(-j 2 k 1/N) exp(-j 2 k 2/N) exp(-j 2 k 3/N) exp(-j 2 k 4/N)
exp(-j 2 k 9/N)
exp(-j 2 k 8/N)
exp(-j 2 k 7/N)
exp(-j 2 k 6/N)
exp(-j 2 k 5/N)
exp(-j 2 k 10/N) exp(-j 2 k 11/N) exp(-j 2 k 12/N) exp(-j 2 k 13/N) exp(-j 2 k 14/N)

129. Linear Versus Circular Convolution

130. Fast Circular Convolution with the FFT

131. Single Symbol in CI-OFDM

132. M-Symbols in CI-OFDM

133. 1-to-2 Interpolated Time Domain Data Points
-20 -15 -10 -5 0 5 10 15 20
-0.2
0
0.2
0.4
0.6
0.8
1
CI-OFDM Real Time Series and 1-Modulation Sample
Time
Amplitude
-20 -15 -10 -5 0 5 10 15 20
-0.2
0
0.2
0.4
0.6
0.8
1
CI-OFDM Real Time Series and 4-Modulation Samples
Time
Amplitude

134. CI-OFDM Data Frame
-25 -20 -15 -10 -5 0 5 10 15 20
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
CI-OFDM Real Time Series and 32-Modulation Samples
Time
Amplitude
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-60
-50
-40
-30
-20
-10
0
10
Spectrum
Normalized Frequency
Log
Magnitude
(dB)
Cyclic Prefix

135. CI-OFDM Statistics
0 1 2 3 4
0
0.005
0.01
0.015
Histogram: Standard OFDM (QPSK)
Normalized Amplitude (x/x
)
0 1 2 3 4
0
0.05
0.1
Histogram: CI-OFDM (QPSK)
Normalized Amplitude (x/x
)
0 1 2 3 4
10
-4
10
-3
10
-2
10
-1
10
0
Probabilty of Level Crossing
Normalized Amplitude (x/x
)

136. CI-OFDM Statistics
0 1 2 3 4
0
0.005
0.01
0.015
Histogram: Standard OFDM
Normalized Amplitude (x/x
)
0 1 2 3 4
0
0.02
0.04
0.06
0.08
Histogram: CI-OFDM (16-QAM)
Normalized Amplitude (x/x
)
0 1 2 3 4
10
-4
10
-3
10
-2
10
-1
10
0
Probabilty of Level Crossing
Normalized Amplitude (x/x
)

137. That’s all Folks
Professor harris, may I be excused?
My brain is full.