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Orthogonal Frequency-Division
Multiplexing
7.1 Introduction
Orthogonal Frequency-Division Multiplexing (OFDM) forms the basis for 4G, i.e., Fourth Generation wireless communication systems. OFDM is used in 4G wireless cellular standards such as Long-Term Evolution (LTE) and WiMAX (Worldwide Interoperability for Microwave Access). OFDM is a key broadband wireless technology which supports data rates in excess of 100 Mbps. Similarly, the wireless local area (LAN) standards such as 802.11 a/g/n are based on OFDM. Next we describe multicarrier transmission, which is the motivation and key idea behind OFDM.
7.2 Motivation and Multicarrier Basics
Consider a bandwidth B = 2W available for communication, where W is the one-sided bandwidth, or, in other words, the maximum frequency. For a single carrier communication system, the symbol time T is given as
T =
B
basically implying that symbols can be transmitted at intervals of (^) B^1 seconds each. Therefore, the symbol rate is given as
Rate =
1 /B =^ B^ (7.1)
Orthogonal Frequency-Division Multiplexing 231
Such a system is termed a single-carrier communication system. In such a system, a single carrier is employed for the entire baseband bandwidth of B. Therefore, roughly speaking, the symbols are transmitted as symbol X(0) from 0 ≤ t < T , symbol X(1) from T ≤ t < 2 T , and so on, i.e., roughly one symbol transmitted every T = (^) B^1 seconds. Consider now dividing the total bandwidth B into N sub-bands of bandwidth B/N each as shown in Figure 7.1. Each subcarrier can now be represented by a subcarrier. Therefore, the subcarriers are placed at... , − B N , 0 , B N ,... , as shown in the figure. For instance, consider the bandwidth B = 256 kHz with N = 64 subcarriers. The bandwidth per sub-band is equal to 256 64 = 4^ kHz, which is also the frequency spacing between the subcarriers. We now implement a multi-carrier transmission system as follows. Consider the i th^ subcarrier at the frequency fi = i (^) NB , with − N 2 − 1 ≤ i ≤ N 2. Let X (^) i denote the data transmitted on the i th^ subcarrier. Then, the signal si (t) corresponding to the i th^ subcarrier is given as
si (t) = X (^) i ej^2 πf^ i^ t^ = X (^) i ej^2 πi^ NB t
where fi is the i th^ subcarrier centre frequency, as described above, and ej^2 πf^ i^ t^ is the i th subcarrier. The above equation shows the data modulation process over the i th^ subcarrier. The N different data symbols X (^) i are modulated over the N different subcarriers with centre frequencies fi. Hence, there are a total of N data streams. Next we illustrate the scheme for multicarrier transmission.
Total bandwidth B
-( N/2-1) B N/ - B N/ B N/ 2 B N/ ( N/2) B N/
Sub-band
Subcarrier spacing (^) N subcarriers
Figure 7.1 Multi-carrier concept
Orthogonal Frequency-Division Multiplexing 233
= B
N
N B
0
X (^) l dt
i=l
+ B
N (^) i=l
N B
0
X (^) i ej^2 π(i−^ l)f^0 t^ dt
= X (^) l + B N i=l
X (^) i
N B
0
ej^2 π(i−^ l)f^0 t^ dt
= X (^) l
where we have used the fact that 0 T 0 ej^2 π(i−^ l)f^0 t^ dt = 0 for i = l , since this is basically integrating a sinusoid of frequency (i − l) f 0 , which is a multiple of the fundamental frequency f 0 over the period T 0. Therefore, since there are an integer number of cycles of the sinusoid of frequency (i − l) f 0 , this integral is 0. In fact, this basically implies that the different sinusoids ej^2 πif^0 t^ and ej^2 πlf^0 t^ are orthogonal. It is this key property of orthogonality which helps extract the different streams X (^) i modulated over the different subcarriers. This property of orthogonality can be summarized as
N/B 0
ej^2 π(i−^ l)^ B^ N^ t^ =
0 i = l N B i^ =^ l
Therefore, all the subcarriers other than the l th^ subcarrier are orthogonal to the l th subcarrier. Further, observe that multiplying with ej^2 πf^ l^ t^ ∗^ and integrating is basically coherent demodulation, i.e., demodulation with the carrier matched to the subcarrier frequency fl = l B N. Thus, X (^) l , the data modulated on the different subcarriers, can be conveniently recovered by coherently demoudulating with each of the subcarriers corresponding to l = − N 2 − 1 ,... , N 2. The above scheme of transmission on multiple orthogonal subcarriers and the associated data recovery at the receiver is termed MultiCarrier Modulation (MCM). Also observe that the window of time associated with detection of this multicarrier signal is N B = (^) f^1 0 = T 0 , which is basically the time period of integration. Hence, MCM basically transmits N symbols using N subcarriers in a time period of N B. The symbol rate is, therefore, N N/B =^ B^. Thus, the overall symbol rate in single carrier vs multicarrier systems is unchanged. The transmitter and receiver block schematics for this MCM system are shown in figures 7. and 7.3 respectively.
234 Principles of Modern Wireless Communication Systems
Serial to parallel conversion because we are transmitting symbols in parallel
N
i th^ data stream is modulated onto the i thsubcarrier
Sum all the subcarriers
S/P Symbols Demux
Bank of modulators
Summer
Composite signal
To channel
Figure 7.2 Multicarrier modulation transmitter
Coherent demodulation with exp ( - j 2 pf t (^) l)
Parallel to serial multiplexing
Bank of correlators or demodulators
y t( ) Repeater from channel
P/S Mux (^) Serail symbol stream
N Information symbols
Figure 7.3 Multi-carrier modulation receiver
It is very important now to note the following fact. Observe from Eq. (7.1) and the above rate for an MCM system. It is clear that the symbol rate in both these systems is exactly identical, i.e., B. The single-carrier system transmits each symbol in time (^) B^1 , while the MCM system transmits N symbols in parallel in time N B. What then is the advantage of an MCM system over the single-carrier system? To understand this, consider an example with a transmission bandwidth of B = 1. 024 MHz, i.e., 1024 kHz. As seen in an earlier chapter, notice that this bandwidth B is much greater than the coherence bandwidth Bc which is typically around 250 kHz, i.e., Bc ≈ 250 kHz. Therefore, since the transmission bandwidth B >> Bc , the single-carrier system experiences frequency-selective fading and inter-symbol interference. However, consider an OFDM system with employs N = 256 subcarriers in the same bandwidth. The bandwidth per subcarrier is Bs = 1024256 = 4 kHz. It can be readily seen that the subcarrier bandwidth of 4 kHz is significantly lower than the coherence
236 Principles of Modern Wireless Communication Systems
Figure 7.4 OFDM transmitter schematic with IFFT
Figure 7.5 OFDM receiver schematic with FFT
7.2.2 Cyclic Prefix in OFDM
In this section, we explain the concept of cyclic prefix, which is an important component of an OFDM system. Consider a frequency-selective channel modelled with channel taps h (0) , h (1) ,... , h (L − 1). Thus, the received symbol y at a given time instant n can be expressed as
y (n) = h (0) x (n) + h (1) x (n − 1) +... + h (L − 1) x (n − L + 1) ISI component
from which it can be seen that the received symbol y (n) at the time instant n experiences inter symbol interference from the previous L − 1 transmitted symbols. Consider now two OFDM symbols as follows. Let x (0) , x (1) ,... , x (N − 1) denote the IFFT samples of the modu- lated symbols X (0) , X (1) ,... , X (N − 1), while x˜ (0) , ˜x (1) ,... , ˜x (N − 1) denote the
Orthogonal Frequency-Division Multiplexing 237
IFFT samples of the previous modulated symbol block X˜ (0) , X˜ (1) ,... , X˜ (N − 1). Thus, the samples corresponding to these two blocks of OFDM symbols are transmitted sequentially as
x ˜ (0) , x˜ (1) ,... , ˜x (N − 1) , Previous block
x (0) , x (1) ,... , x (N − 1) Current block
Now, consider the received symbol y (0) corresponding to the transmission of x (0). This can be expressed as
y (0) = h (0) x (0) + h (1) ˜x (N − 1) +... + h (L − 1) ˜x (N − L + 1) ISI from previous OFDM symbol
It can be seen from the above equation that the received symbol y (0) experiences inter-symbol interference from ˜x (N − 1) , x˜ (N − 2) ,... , x˜ (N − (L − 1)). Thus, there is inter-OFDM symbol interference in this new OFDM system. The initial samples of the current OFDM symbol block are being subject to interference from the N − 1 samples of the previous OFDM block. This is shown in Figure 7.6. Similarly, the received symbol y (1) is given as
y (1) = h (0) x (1) + h (1) x (0) h (2) ˜x (N − 1) +... + h (L − 1) ˜x (N − L + 2) ISI from previous OFDM symbol
which can again be seen to experience inter-OFDM symbol interference from the previous OFDM block symbols x˜ (N − 1) , ˜x (N − 2) ,... , ˜x (N − L + 2). Let us now consider a modified transmission scheme as follows. To each transmitted OFDM sample stream, we pad the last L (^) c symbols to make the transmitted stream as follows.
˜x | (0) , ˜x (1) ,... ,{z ˜x (N − 1) (^) }, Previous block
x | (N − Lc ) , x (N − L{zc + 1) ,... , x (N − 1)} Cyclic prefix
x | (0) , x (1) ,... , x{z (N − 1)} Current block
.
OFDM symbol size = N B/ samples
Initial samples, of subject to inter OFDM symbol interference
Figure 7.6 Inter-OFDM symbol interference
Orthogonal Frequency-Division Multiplexing 239
where Y (k) , 0 ≤ k ≤ N − 1 , denotes the N -point DFT of y (n). Similarly, X (k) denotes the N -point DFT of x (n). Further, observe that the samples x (n) have been generated as the IDFT of X (n). Therefore, the DFT of the samples x (n) yields back the original transmitted symbols X (n). The coefficients H (k) denotes the DFT of the zero-padded channel filter,
h (0) , h (1) ,... , h (L − 1) , 0 ,... , 0 (N − L)
Thus, observe that Eq. (7.3) represents the flat-fading channel across the k th^ subcarrier in the OFDM system. The quantity Y (k) represents the output symbol, while H (k) denotes the equivalent flat-fading channel coefficient. This holds true for each subcarrier k , i.e., for 0 ≤ k ≤ N − 1. Thus, the frequency-selective fading channel is converted into a group of narrowband flat-fading channels, one channel across each subcarrier. Observe that if a single carrier system was used, and the symbols X (0) , X (1) ,... , X (N − 1) were transmitted directly then the received symbol y (n) would be given as
y (n) = h (0) X (n) + h (1) X (n − 1) +... + h (L − 1) X (n − L + 1)
Each symbol X (n) would experience inter-symbol interference of L − 1 past symbols. Therefore, using this novel scheme of OFDM, we have been able to totally eliminate the inter-symbol interference arising out of the frequency-selective nature of the channel. The set of parallel flat-fading channels can be summarized by the expressions
Y (0) = H (0) X (0)
Y (1) = H (1) X (1) .. . Y (N − 1) = H (n − 1) X (N − 1)
This conversion of the frequency-selective wideband channel into N narrowband flat-fading channels is shown schematically in Figure 7.7. Also, the modified transmitter and receiver schematics with the blocks corresponding to the cyclic prefix are given in Figures 7.8 and 7.9 respectively. Now, considering the noise at the receiver, the received symbol Y (k) can be expressed as
Y (k) = H (k) X (k) + N (k) (7.5)
240 Principles of Modern Wireless Communication Systems
Figure 7.7 OFDM parallel subchannels
where N (k) denotes the noise across the k th^ subcarrier. A simple detection scheme for X (k) is to use the zero-forcing detector for the subcarrier as
X^ ˆ (k) = 1 H (k)Y^ (k) =^ X^ (k) +^
N (k) H (k) N^ ˜ (k)
Further, for a simplistic BPSK or QPSK-modulated transmission, the coherent or matched filter detector can be simply obtained by multiplying with H ∗^ (k) , i.e., the complex conjugate of H (k) as
H ∗^ (k) Y (k) = | H (k) | 2 X (k) + H ∗^ (k) N (k) N (k)
Serial to parallel conversion
N = # of Subcarriers
Block for addition of cyclic prefix (CP)
X(0) x(0)
S/P Demux
IFFT N
P/S Mux
Add Symbols CP
X N( -1) x N( -1)
To channel
Samples at rate B
Figure 7.8 OFDM transmitter schematic with CP
242 Principles of Modern Wireless Communication Systems
the loss in efficiency can be calculated as
Loss in effciency = Cyclic prefix Total OFDM symbol length
= L^ −^1 N + L − 1
= L^ −^1 N + L − 1 However, as the block length N becomes very large, we have
N^ lim →∞
L − 1
N + L − 1 →^0
Thus, the loss in throughput approaches 0 as the number of subcarriers N increases, for a fixed length of the delay spread L. Also observe that as the number of subcarriers N increases, the symbol time N B increases as shown in Figure 7.10. Increasing N results in increasing OFDM symbol time, thus restricting the ISI to a small fraction of the OFDM symbol block, i.e., the fraction (^) NL is progressively smaller. However, as the block length N increases, the decoding delay at the receiver also increases as one has to wait for arrival of the entire block of N samples before it can be demodulated. Hence, there is a trade-off for increasing N vs decoding delay. Now, we present another intuitive framework to understand the effect of various parameters. As we have said previously, the duration of the cyclic prefix has to be greater than the delay spread.
Figure 7.10 Inter OFDM symbol interference with increasing OFDM symbol time
Orthogonal Frequency-Division Multiplexing 243
Therefore, we need
L (^) c × T (^) s ≥ T (^) d
where T (^) s denotes the sample time and T (^) d denotes the delay spread. Also, the sample time T (^) s = (^) B^1 , where B is the total bandwidth of the system and T (^) d = (^) B^1 c , where Bc is the coherence bandwidth of the system. The above condition implies
L (^) c ≥ T^ d T (^) s
= B
Bs Combining this with the earlier condition that N >> L (^) c for efficiency in terms of the effective data rate, we have
N >> L (^) c ≥ B Bc
This can also be recast as Bc >> B N. Interestingly, this is the same condition for frequency flat fading across each subcarrier since this implies that the subcarrier bandwidth (^) NB is required to be much less than the coherence bandwidth Bc. Thus, an appropriately designed OFDM system converts a frequency-selective fading channel into a set of parallel narrowband flat-fading channels across the subcarriers. The example next illustrates an OFDM system design example.
7.3 OFDM Example
In this section, we consider a practical WiMAX example to illustrate the impact of the various parameters in the design of a complete OFDM system. As already stated in the beginning, WiMAX, which stands for Worldwide Interoperability for Microwave Access, is a prominent 4G wireless standard. The total number of subcarriers N = 256 , with a bandwidth of 15. 625 kHz per subcarrier. Therefore,
B N = 15. 625 kHz
⇒ B = N × 15 .625 = 256 × 15. 625
= 4 MHz
Orthogonal Frequency-Division Multiplexing 245
Figure 7.11 WiMAX OFDM symbol with cyclic prefix
7.4 Bit-Error Rate (BER) for OFDM
Consider the OFDM subcarrier system model given in Eq. (7.5), i.e.,
Y (k) = H (k) X (k) + N (k) (7.6)
where N (k) is the subcarrier noise obtained from the FFT of the noise samples at the output of the receiver as
N (k) =
N − 1
m=
n (m) e−^ j^2 π^ km N
where N is the number of subcarriers, and n (0) , n (1) ,... , n (N − 1) are additive noise samples for each of the output samples y (0) , y (1) ,... , y (N − 1). We now deduce the statistical properties of these noise samples N (k) , which are required to characterize the BER performance of the OFDM system. Firstly, observe that the noise N (k) is the linear combination of Gaussian noise samples n (0) , n (1) ,... , n (N − 1). Hence, it is Gaussian in nature. Further, the mean or expected value of N (k) is given as
E { N (k)} = E
N − 1
m=
n (m) e−^ j^2 π^ km^ N
N − 1
m=
E { n (m)} e−^ j^2 π^ km N
246 Principles of Modern Wireless Communication Systems
Further, the variance σ (^) N^2 of the noise sample N (k) is given as σ (^) N^2 = E | N (k)| 2
= E
N − 1
m=
n (m) e−^ j^2 π^ km N^ N^ −^1 l=
n (l) e−^ j^2 π^ kl N
∗
= E
N − 1
m=
N − 1
l=
n (m) n∗^ (l) e−^ j^2 π(m−^ l)^ Nk^ (7.7)
Observe that since the noise samples n (m) are independent identically distributed Gaussian of variance σ (^) n^2 , it follows that E { n (m) n∗^ (l)} = 0 if m = l and σ (^) n^2 if m = l. Therefore, the above expression for the noise variance can be simplified as
σ (^) N^2 =
N − 1
m=
N − 1
l=
E { n (m) n∗^ (l)} e−^ j^2 π(m−^ l)^ Nk
N − 1
m=
σ (^) n^2
= N σ (^) n^2 Further, let us assume that each of the channel taps h (0) , h (1) ,... , h (L − 1) is Rayleigh fading in nature, i.e., has a complex symmetric Gaussian distribution of mean 0 and variance
- Therefore, the channel coefficient across the k th^ subcarrier is given as
H (k) =
N − 1
m=
h (m) e−^ j^2 π^ km^ N
As can be seen, H (k) is a linear combination of Gaussian random variables h (k) , 0 ≤ k ≤ L − 1. Therefore, H (k) is indeed complex Gaussian i.e. has a Rayleigh fading envelope. Further, since each h (k) is zero mean, H (k) also has mean zero. Further, identical to the development of the noise variance above, it follows on similar lines that the channel power
248 Principles of Modern Wireless Communication Systems
Figure 7.12 MIMO OFDM transmitter schematic
where w (n) denotes the noise. Hence, a MIMO frequency-selective channel can be modelled as a MIMO FIR filter, which can be described as
y (n) =
L− 1
l=
H (l) x (n − l) + w (n)
= H (0) x (n) + H (1) x (n − 1) +... + H (L − 1) x (n − L + 1) ISI from previous symbol vectors
+w (n)
Therefore, the symbol vector y (n) at the time instant n is affected by inter-symbol vector inter- ference from x (n − 1) , x (n − 2) ,... , x (n − L + 1). This is an L -tap frequency-selective MIMO channel. As can be seen, in a MIMO frequency-selective channel, the interference occurs between current and previous transmit symbol vectors. In a MIMO-OFDM system, one needs to perform the IFFT operation at each transmit antenna. The schematic figures showing clearly the processing at the transmitter and receiver of the MIMO-OFDM system are shown in figures 7.12 and 7.13 respectively. Hence, employing MIMO-OFDM, the MIMO frequency-selective channel can be converted into a set of parallel flat-fading MIMO channels.
Orthogonal Frequency-Division Multiplexing 249
These can be described as,
y ˜ (0) = H˜ (0) ˜x (0)
y˜ (1) = H˜ (1) ˜x (1) .. . y ˜ (N − 1) = ˜H (N − 1) ˜x (N − 1)
The model across the k th^ subcarrier is y˜ (k) = H˜ (k) ˜x (k) , where ˜y (k) and x˜ (k) are the received and transmitted symbol vectors corresponding to the k th^ subcarrier, and H˜ (k) is the flat-fading channel matrix corresponding to the subcarrier k. Each of the received vectors ˜y (0) , ˜y (1) ,... , y˜ (N − 1) can be processed by a simple MIMO zero-forcing receiver or a MIMO-MMSE receiver for detection of the vectors ˜x (0) , ˜x (1) ,... , x˜ (N − 1). The zero- forcing MIMO receiver is given as
x^ ˆ˜ (^) ZF (k) = H˜ (k) †y^ ˜ (k)
= H˜ H^ (k) H˜ (k)
H H^ (k) ˜y (k)
Also, the MMSE receiver for the subcarrier k of the MIMO-OFDM system is given as
x^ ˆ˜ (^) MMSE (k) = H˜ (k) † ˜y (k)
= Pd Pd H˜ H^ (k) ˜H (k) + σ (^2) w I
H H^ (k) ˜y (k)
where Pd denotes the data power. The channel matrices H˜ (0) , H˜ (1) ,... , H˜ (N − 1) corresponding to the OFDM subcarriers are given as follows. Let hu,v (k) , ˜hu,v (k) denote the (u, v)th^ entries of the matrices H (k) , H˜ (k) respectively. Then, ˜hu,v (k) is given as the N -point DFT of the zero-padded coefficients
hu,v (0) , hu,v (1) ,... , hu,v (L − 1) , 0 ,... 0 (N − L)
In effect, the channel matrix H˜ (k) is the k th^ frequency point corresponding to the FFT of the zero padded channel matrices [H (0) , H (1)... , H (L − 1) , (^0) r × t ,... , (^0) r × t ].
