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OFDM Matlab 实现.doc
1INTRODUCTION
1.1Purpose
1.2OFDM Overview
2OFDM OPERATION
2.1Preliminary Concepts
2.2Definition of Carriers
2.3Modulation
2.4Transmission
2.5Reception and Demodulation
3ANALYSIS
3.1Guard Period
3.2Windowing
3.3Multipath Characteristics
3.4Bandwidth
3.5Physical Implementation
3.6Applications
4REFERENCES
5MATLAB
ECE 5664 Project:
Orthogonal Frequency Division Multiplexing (OFDM):
Tutorial and Analysis
December 11, 2001
Erich Cosby
ecosby@vt.edu
Virginia Tech.
Northern Virginia Center
1
1
INTRODUCTION......................................................................................................................................... 3
1.1
Purpose ............................................................................................................................................... 3
1.2
OFDM Overview..................................................................................................................................3
2 OFDM OPERATION ....................................................................................................................................4
Preliminary Concepts ........................................................................................................................ 4
2.1
2.2
Definition of Carriers ..........................................................................................................................5
Modulation.......................................................................................................................................... 5
2.3
2.4
Transmission...................................................................................................................................... 9
2.5
Reception and Demodulation ........................................................................................................... 9
ANALYSIS.................................................................................................................................................12
3
3.1
Guard Period.....................................................................................................................................12
Windowing........................................................................................................................................ 13
3.2
Multipath Characteristics ................................................................................................................ 13
3.3
3.4
Bandwidth......................................................................................................................................... 13
Physical Implementation .................................................................................................................14
3.5
3.6
Applications ......................................................................................................................................14
REFERENCES.......................................................................................................................................... 15
4
5 MATLAB ....................................................................................................................................................16
2
Purpose
INTRODUCTION
1
1.1
Efficient use of radio spectrum includes placing modulated carriers as close as possible without causing
Inter-Carrier Interference (ICI). Optimally, the bandwidth of each carrier would be adjacent to its neighbors,
so there would be no wasted spectrum.
In practice, a guard band must be placed between each carrier
bandwidth to provide a space where a filter can attenuate an adjacent carrier’s signal. These guard bands
are wasted bandwidth.
In order to transmit high data rates, short symbol periods must be used. The symbol period is the inverse of
the baseband data rate (T = 1/R), so as R increases, T must decrease.
In a multi-path environment, a
shorter symbol period leads to a greater chance for Inter-Symbol Interference (ISI). This occurs when a
delayed version of symbol ‘n’ arrives during the processing period of symbol ‘n+1’.
Orthogonal Frequency Division Multiplexing (OFDM) addresses both of these problems. OFDM provides a
technique allowing the bandwidths of modulated carriers to overlap without interference (no ICI).
It also
provides a high date rate with a long symbol duration, thus helping to eliminate ISI. OFDM may therefore be
considered as a candidate modulation technique in a broadband, multi-path environment.
The purpose of this report is to provide the following information concerning OFDM:
theory of operation
analysis of important characteristics
implementation example (matlab)
1.2 OFDM Overview
OFDM is a modulation technique where multiple low data rate carriers are combined by a transmitter to form
a composite high data rate transmission. Digital signal processing makes OFDM possible. To implement the
multiple carrier scheme using a bank of parallel modulators would not be very efficient in analog hardware.
However, in the digital domain, multi-carrier modulation can be done efficiently with currently available DSP
hardware and software. Not only can it be done, but it can also be made very flexible and programmable.
This allows OFDM to make maximum use of available bandwidth and to be able to adapt to changing system
requirements.
Each carrier in an OFDM system is a sinusoid with a frequency that is an integer multiple of a base or
fundamental sinusoid frequency. Therefore, each carrier is like a Fourier series component of the composite
signal.
In fact, it will be shown later that an OFDM signal is created in the frequency domain, and then
transformed into the time domain via the Discrete Fourier Transform (DFT).
Two periodic signals are orthogonal when the integral of their product, over one period, is equal to zero.
This is true of certain sinusoids as illustrated in Equation 1.
Equation 1 : Definition of Orthogonal
Time
:
Continuous
T
cos(
0
2
tnf
0
)
cos(
2
tmf
0
)
dt
0
(n
m)
Discrete
Time
1-N
n2
k
N
0k
cos
:
cos
m2
k
N
0
(n
m)
The carriers of an OFDM system are sinusoids that meet this requirement because each one is a multiple of
a fundamental frequency. Each one has an integer number of cycles in the fundamental period.
3
OFDM OPERATION
the DFT points. All
Preliminary Concepts
2
2.1
When the DFT (Discrete Fourier Transform) of a time signal is taken, the frequency domain results are a
function of the time sampling period and the number of samples as shown in Figure 1. The fundamental
frequency of the DFT is equal to 1/NT (1/total sample time). Each frequency represented in the DFT is an
integer multiple of the fundamental frequency. The maximum frequency that can be represented by a time
signal sampled at rate 1/T is fmax = 1/2T as given by the Nyquist sampling theorem. This frequency is located
in the center of
the representative
frequencies. The maximum frequency bin of the DFT is equal to the sampling frequency (1/T) minus one
fundamental (1/NT).
The IDFT (Inverse Discrete Fourier Transform) performs the opposite operation to the DFT.
It takes a signal
defined by frequency components and converts them to a time signal. The parameter mapping is the same
as for the DFT. The time duration of the IDFT time signal is equal to the number of DFT bins (N) times the
sampling period (T).
It is perfectly valid to generate a signal in the frequency domain, and convert it to a time domain equivalent
for practical use*. This is how modulation is applied in OFDM.
frequencies beyond that point are images of
* The frequency domain is a mathematical tool used for analysis. Anything usable by the real world
must be converted into a real, time domain signal.
1
2
3
. . . . . . . .
T (sample period)
(total time used for the DFT is the product
of the sample period times the number of samples)
NT
DFT
IDFT
s(t)
| S(f) |
. . . . . . . .
. . . . . . . .
0 1/NT 2/NT 3/NT …………
1/2T
(Nyquist bin)
DFT bins representing discrete
frequency components of f(t).
……….. (N-1)/NT
(N/NT = 1/T =
sampling frequency)
Figure 1: Parameter Mapping from Time to Frequency for the DFT
4
N
(number of samples)
t
f
Definition of Carriers
In practice the Fast Fourier Transform (FFT) and IFFT are used in place of the DFT and IDFT, so all further
references will be to FFT and IFFT.
2.2
The maximum number of carriers used by OFDM is limited by the size of the IFFT. This is determined as
follows in Equation 2:
Equation 2 : OFDM Carrier Count
N
N
carriers
carriers
IFFTsize
2
IFFTsize
2
(real
-
valued
time
signal)
1
(complex
-
signal)
time
valued
In order to generate a real-valued time signal, OFDM (frequency) carriers must be defined in complex
conjugate pairs, which are symmetric about the Nyquist frequency (fmax). This puts the number of potential
carriers equal to the IFFT size/2. The Nyquist frequency is the symmetry point, so it cannot be part of a
complex conjugate pair. The DC component also has no complex conjugate. These two points cannot be
used as carriers so they are subtracted from the total available.
If the carriers are not defined in conjugate pairs, then the IFFT will result in a time domain signal that has
imaginary components. This must be a viable option as there are OFDM systems defined with carrier counts
that exceed the limit for real-valued time signals given in Equation 2. Reference [1] describes a system with
IFFT size 256 and carrier count 216. This design must result in a complex time waveform. Further
processing would require some sort of quadrature technique (use of parallel sine and cosine processing
paths).
In this report, only real-value time signals will be treated, but in order to obtain maximum bandwidth
efficiency from OFDM, the complex time signal may be preferred (possibly an analagous situation to QPSK
vs. BPSK). Equation 2, for the complex time waveform, has all IFFT bins available as carriers except the DC
bin.
Both IFFT size and assignment (selection) of carriers can be dynamic. The transmitter and receiver just
have to use the same parameters. This is one of the advantages of OFDM.
Its bandwidth usage (and bit
rate) can be varied according to varying user requirements. A simple control message from a base station
can change a mobile unit’s IFFT size and carrier selection.
2.3 Modulation
Binary data from a memory device or from a digital processing stream is used as the modulating (baseband)
signal. The following steps may be carried out in order to apply modulation to the carriers in OFDM:
combine the binary data into symbols according to the number of bits/symbol selected
convert the serial symbol stream into parallel segments according to the number of carriers, and
form carrier symbol sequences
apply differential coding to each carrier symbol sequence
convert each symbol into a complex phase representation
assign each carrier sequence to the appropriate IFFT bin, including the complex conjugates
take the IFFT of the result
This is the same modulation technique described in Reference [3]. The Reference [2] matlab program
carries out these steps and provides detailed commentary and examples for each one.
OFDM modulation is applied in the frequency domain. Figure 2 and Figure 3 give an example of modulated
OFDM carriers for one symbol period, prior to IFFT. For this example, there are 4 carriers, the IFFT bin size
is 64, and there is only 1 bit per symbol. The magnitude of each carrier is 1, but it could be scaled to any
value. The phase for each carrier is either 0 or 180 degrees, according to the symbol being sent. The phase
determines the value of the symbol (binary in this case, either a 1 or a 0).
In the example, the first 3 bits (the
first 3 carriers) are 0, and the 4th bit (4th carrier) is a 1.
5
1.5
1
e
d
u
t
i
n
g
a
M
0.5
0
-0.5
0
OFDM Carrier Frequency Magnitude
10
20
30
IFFT Bin
40
50
60
Figure 2: OFDM Carrier Magnitude prior to IFFT
)
s
e
e
r
g
e
d
(
e
s
a
h
P
200
150
100
50
0
-50
-100
-150
-200
0
OFDM Carrier Phase
10
20
30
IFFT Bin
40
50
60
Figure 3: OFDM Carrier Phase prior to IFFT
6
Note that the modulated OFDM signal is nothing more than a group of delta (impulse) functions, each with a
phase determined by the modulating symbol.
In addition, note that the frequency separation between each
delta is proportional to 1/N where N is the number of IFFT bins. The frequency domain representation of the
OFDM is described in Equation 3.
Equation 3 : OFDM Frequency Domain Representation (one symbol period)
)(
kS
e
j
m
Nmk
2
j
m
e
Nmk
2
single
(real)
OFDM
modulated
carrier
-N to (0
1)
frequency
k
m
OFDM
carrier
N
IFFT
bin
size
frequency
)(
kS
ofdm
last
c
cm
first
e
j
m
Nmk
2
j
m
e
Nmk
2
composite
(real)
OFDM
modulated
carriers
c
OFDM
carrier
(first
through
last)
It is clear that the OFDM signal has a varying amplitude.
After the modulation is applied, an IFFT is performed to generate one symbol period in the time domain. The
It is very
IFFT result is shown in Figure 4.
important that the amplitude variations be kept intact as they define the content of the signal.
If the
amplitude is clipped or modified, then an FFT of the signal would no longer result in the original frequency
characteristics, and the modulation may be lost.
This is one of the drawbacks of OFDM, the fact that it requires linear amplification.
In addition, very large
amplitude peaks may occur depending on how the sinusoids line up, so the peak-to-average power ratio is
high. This means that the linear amplifier has to have a large dynamic range to avoid distorting the peaks.
The result is a linear amplifier with a constant, high bias current resulting in very poor power efficiency. For a
detailed treatment of the peak-to-average power ratio problem in OFDM, see Reference [4].
Figure 5 is provided to illustrate the time components of the OFDM signal. The IFFT transforms each
complex conjugate pair of delta functions (each carrier) into a real-valued, pure sinusoid. Figure 5 shows the
separate sinusoids that make up the composite OFDM waveform given in Figure 4. The one sinusoid with
180 phase shift is clearly visible as is the frequency difference between each of the 4 sinusoids. Note that
this figure is ‘zoomed’ i.e. all 64 point of the IFFT are not shown.
In addition, note that the waveform plots
are not very smooth. This is because there are not many samples per cycle for any of the sinusoids.
The time domain representation of the OFDM signal is given in Equation 4.
Equation 4: OFDM Time Domain Representation (one symbol period)
)(
ns
c
last
1
N
cos
0
n
cm
first
2
mn
N
m
sample
time
n
m
OFDM
carrier
N
IFFT
bin
size
modulation
for
OFDM
(m)
carrier
phase
m
c
c,
OFDM
carriers
and
last)
(first
first
last
7
OFDM Time Signal, One Symbol Period
0.08
0.06
0.04
0.02
0
-0.02
-0.04
-0.06
-0.08
e
d
u
t
i
l
p
m
A
-0.1
0
10
20
30
40
50
60
70
Time
Figure 4: OFDM Signal, 1 Symbol Period
Separated Time Waveforms Carriers
0.03
0.02
0.01
0
-0.01
-0.02
-0.03
e
d
u
t
i
l
p
m
A
0
2
4
6
8
Time
10
12
14
16
18
Figure 5: Separated Components of the OFDM Time Waveform
8
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