Notes OFDM – HTE March 2013 Sayfa 1 Notes on Orthogonal Frequency Division Multiplexing (OFDM) 1. Discrete Fourier Transform As a reminder, the analytic forms of Fourier and inverse Fourier transforms are , exp 2 (1.1) X f xt tf dt xt j ft dt where we have assumed that Fourier transform direction is from time axis (domain) to frequency axis (domain). In this sense xt is our time signal, while X f is the frequency signal or rather the frequency spectrum of xt . Finally , exp 2 tf j ft is the operator that enables the act of transformation from time axis to frequency axis. The inverse Fourier transform will be given by , exp 2 (1.2) xt X f ft df X f j ft df (1.2) allows the return to the time axis from the frequency axis, where the operator is now * , , exp 2 ft tf j ft , i.e. the conjugate of the operator in (1.1). Symbolically (1.1) and (1.2) can be represented as below 1 , (1.3) X f F xt xt F X f Some comments are in order regarding Fourier transform and its inverse In general Fourier transform is applicable to transformations between any axes and domains, for instance to go from spatial axis to spatial frequency axis, again Fourier transform and its inverse can be used. In these lecture notes, our axes will continue to be time and frequency as given in (1.1) and (1.2). Fourier transform can be multidimensional. It means that the single integrals in (1.1) and (1.2) can be extended to double or triple or N number of integrals. As shown by (1.1) and (1.2), to include all behaviours along one axis, the integral limits range from to . In general these limits will be finite. The transformation operators have the property that , , 1 ft tf . Thus no scaling is applied during transformations. This way, we can expect that the energies or the powers (of the same signal) in the two domains to be exactly equivalent. We now introduce the discrete Fourier transform by associating the time and frequency functions, variables and the operators as follows , , , , , , , , (1.4) t n f k xt xn X f X k tf nk ft kn
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Notes on Orthogonal Frequency Division Multiplexing (OFDM) · Notes on Orthogonal Frequency Division Multiplexing (OFDM) 1. Discrete Fourier Transform As a reminder, the analytic
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Notes OFDM – HTE March 2013 Sayfa 1
Notes on Orthogonal Frequency Division Multiplexing (OFDM)
1. Discrete Fourier Transform
As a reminder, the analytic forms of Fourier and inverse Fourier transforms are
, exp 2 (1.1)X f x t t f dt x t j ft dt
where we have assumed that Fourier transform direction is from time axis (domain) to frequency
axis (domain). In this sense x t is our time signal, while X f is the frequency signal or rather
the frequency spectrum of x t . Finally , exp 2t f j ft is the operator that enables the
act of transformation from time axis to frequency axis. The inverse Fourier transform will be given
by
, exp 2 (1.2)x t X f f t df X f j ft df
(1.2) allows the return to the time axis from the frequency axis, where the operator is now
*, , exp 2f t t f j ft , i.e. the conjugate of the operator in (1.1). Symbolically (1.1)
and (1.2) can be represented as below
1 , (1.3)X f F x t x t F X f
Some comments are in order regarding Fourier transform and its inverse
In general Fourier transform is applicable to transformations between any axes and domains,
for instance to go from spatial axis to spatial frequency axis, again Fourier transform and its
inverse can be used. In these lecture notes, our axes will continue to be time and frequency as
given in (1.1) and (1.2).
Fourier transform can be multidimensional. It means that the single integrals in (1.1) and (1.2)
can be extended to double or triple or N number of integrals.
As shown by (1.1) and (1.2), to include all behaviours along one axis, the integral limits range
from to . In general these limits will be finite.
The transformation operators have the property that , , 1f t t f . Thus no scaling is
applied during transformations. This way, we can expect that the energies or the powers (of the
same signal) in the two domains to be exactly equivalent.
We now introduce the discrete Fourier transform by associating the time and frequency functions,
variables and the operators as follows
, , ,
, , , , , (1.4)
t n f k x t x n X f X k
t f n k f t k n
Notes OFDM – HTE March 2013 Sayfa 2
n and k will be integer variables, limited to the range
0 1 , 0 1 (1.5)n N k N
Therefore N may be regarded to represent the infinity of the discrete world. Bearing in mind that
integration of the analytic world will correspond to summation in discrete domain, the discrete
Fourier transform (DFT) of x n into X k will be
1
0
exp 2 / (1.6)N
n
X k x n j kn N
1/ N in the argument of the exponential operator in (1.6) acts as normalization factor. The inverse
discrete Fourier transform (IDFT) of X k into x n will be
1
0
1exp 2 / (1.7)
N
n
x n X k j kn NN
Then symbolically, (1.6) and (1.7) can be combined as
(1.8)DFT
IDFTx n X k
In analytic world, (1.1) and (1.2) work without any problems. When switching to discrete case,
care must be taken, because there are three main points to observe there
a) Infinity of analytic world must suitably be converted to the appropriate lengths in discrete
world.
b) Grid spacing of the two domains of the discrete world must be taken into account. In analytic
case this is properly covered since dt and df correspond to the same infinitesimally small
intervals, thus creating no scaling problems.
c) The rules of sampling theorem should be properly adhered to. This is because n and k are
effectively the sampling rates in time and frequency domains. Hence the smallest time
resolution that can be achieved is 2n and correspondingly the smallest frequency resolution that
we can attain is 2k .
Example 1.1 : Let x n be a delta (impulse) function such that
1 0
(1.9)0 1 1
nx n
n N
Find X k for this discrete time delta function.
Solution : Graphically, (1.9) will be as shown in Fig. 1.1. Using (1.6), we get
Notes OFDM – HTE March 2013 Sayfa 3
1
0
0 01
0
exp 2 /
0 exp 0 1 exp 2 / 2 exp 4 /
1 exp 1 (
N
n
X k x n j kn N
x n x n j k N x n j k N
x n N j k
1.10)
So the result is 1X k which agrees with the analytic result of
1 (1.11)X f F x t t
Fig. 1.1 The graph of discrete (time) delta function given by (1.9).
Fig. 1.2 displays the graph of 1X k .
Fig 1.2 Discrete Fourier transform of discrete time delta function of (1.9).
Example 1.2 : Let x n be time shifted a delta function such that
0
x ( n )
1
0 0
n
n = N - 1
X ( k )
k
1
k = N - 1
0
1 1
Notes OFDM – HTE March 2013 Sayfa 4
1
(1.12)0 elsewhere
n mx n
Find X k for this discrete time delta function.
Solution 1.2 : Graphically (1.12) will be as shown in Fig. 1.3. Again using the general formulation
given in (1.6), we get
1
0
0 0 0
0 01
0
exp 2 /
0 exp 0 1 exp 2 / 2 exp 4 /
1 exp 2 1 / exp 2 / 1 exp 2 1 /
1 exp exp 2 /
N
n
X k x n j kn N
x x j k N x j k N
x m j k m N x m j km N x m j k m N
x N j k j km N
(1.13)
Fig. 1.3 The graph of discrete time shifted delta function given by (1.12).
Again the finding of (1.13) agrees with the analytic result of
1 1exp 2 (1.14)X f F x t t t j ft
Since (1.13) is complex, it can be plotted in two parts as split below
1exp 2 / (1.15)
2 /
X kX k j km N
k km N
The plot associated with (1.15) is given in Fig. 1.4.
x ( n )
1
0 0 0 00
n = m
n
n = N - 1
Notes OFDM – HTE March 2013 Sayfa 5
Fig. 1.4 Discrete Fourier transform of time shifted delta function (from (1.15)).
Example 1.3 : Now take a single exponential at a specific frequency k m , thus we may write
So we try to find the discrete Fourier transform of x n as given in (1.16).
Solution 1.3 : Again by using (1.6), we get
1 1
0 0
1
0
exp 2 / exp 2 / exp 2 /
exp 2 / (1.17)
N N
n n
N
n
X k x n j kn N j mn N j kn N
j n m k N
By using the identity
1
0
if , integerexp 2 / (1.18)
0 if , integer , integer
N
n
N m k pN pj n m k N
m k m k
X k given by (1.17) becomes
if
(1.19)0 otherwise
N k mX k
(1.19) means a delta function at k m . The corresponding relation of the analytic world is
1 1exp 2 (1.20)X f F x t j f t f f
Fig. 1.5 displays the graph of (1.19).
( k )
k = N - 1
0
| X ( k ) |
1
k
k = N - 1k
- 2 m ( N - 1 ) / N
exp 2 / (1.16)x n j mn N
Notes OFDM – HTE March 2013 Sayfa 6
Fig. 1.5 Discrete Fourier transform of a single exponential as given by (1.16).
Exercise 1.1 : Prove the identity in (1.18). Note that if and m k are limited to a single N range,
so that 0 1m N and 0 1k N , then (1.18) can be converted into
1
0
if , 0 1exp 2 / (1.21)
0 if , 0 1
N
n
N m k m Nj n m k N
m k k N
Exercise 1.2 : Find the discrete Fourier Transform of the sum of two exponentials as given below.
1 1 2 2exp 2 / exp 2 / (1.22)x n A j m n N A j m n N
Check that the discrete Fourier Transform that you have found for (1.22) resembles the
corresponding analytic expression.
2. Basis for Orthogonal Frequency Division Multiplexing (OFDM)
Conventionally we place a message signal on a single carrier. For instance, to have double side
band AM type of modulation, if s t is the message signal, we simply multiply by a sinusoidal at
the frequency of c
f to obtain the modulated signal as
exp 2 or cos 2 or sin 2 (2.1)c c c
y t s t j f t y t s t f t y t s t f t
Due to the frequency spectrum of s t , i.e., S f occupying a finite bandwidth, we have to take
into account, the frequency response of the channel for the transmitted signal to arrive at the
receiver without distortions.
Suppose that S f has a bandwidth of 2W , then Y f F y t has the two sided spectrum of
2W each centered around c
f f as shown in Fig. 2.1. Assuming that the communication channel
that our modulated signal is going to pass through has the following frequency response
0 0 0 00
X ( k )
k
N
k = N - 1k = m
Notes OFDM – HTE March 2013 Sayfa 7
1 or any constant 1 if
(2.1)arbitrary elsewhere
cf f W
C f
Fig. 2.1 A communication channel that introduces no distortion.
then confined to the frequency range of c
f f W at the output of the channel, we will have
1 1 1 (2.2)r t F C f Y f F Y f y t
The implication of (2.2) is that we have been able to receive the exact copy of the transmitted
signal and no distortions due to channel impairments have been experienced. Note (2.2) is as a
result of the channel response being frequency independent within the band of our transmitted
signal.
Next, let’s assume that our transmitted signal remains the same, but the channel response turns
into the one shown in Fig. 2.2.
Fig. 2.2 A communication channel that will introduce distortion.