Orthogonal Frequency Division Multiplexing

Orthogonal Frequency Division Multiplexing John MacLaren Walsh ECES 421. Spring, 2013 May 7 & 9, 2013 1 Introduction OFDM has several significant benefits over single carrier QAM modulation which has lead to its adoption in many of the modern wireless standards, including ADSL, European Digital Video Broadcast, IEEE 802.11a/g/n (WiFi), WiMax, and 3GPP Long Term Evolution (LTE). First of all, OFDM enables easy equalization, solving many of the equalization complexity issues we encountered for single carrier QAM by working instead in the frequency domain. Second of all, the ability to control the size of the constellation on each subcarrier in OFDM also allows it to mimic the \waterfilling” construction which maximizes the amount of information which can be reliably transmitted over the channel: subcarriers (frequencies) with better channel gains can utilize higher order modulations and coding rates. A final significant benefit that we will encounter later in the course is that OFDM allows for an especially easy to implement and understand multiple access strategy OFDMA, in which different subcarriers (frequencies) are allocated to different users at different times. To understand how OFDM modems work, we must first revisit several properties of the discrete Fourier Transform, and its efficient implementation the Fast Fourier Transform. 2 Review: Circular Convolution and the FFT The circular convolution between two length N signals x1[n] and x2[n], n 0; 1;:::;N 1 , is defined as 2 f − g N−1 X x1[n] x2[n] = x1[`]x2[((n `)) ] (1) ◦ − N `=0 where N n n < 0 ((n)) = (2) N n− n 0 ≥ Circularly convolving two length N signals x1[n] and x2[n] is the same thing as multiplying their discrete Fourier transforms DF T x1[n] x2[n] X1[k]X2[k] (3) ◦ ! The underlying principle of OFDM is based on this property of the DFT. 3 Architecture of an OFDM Transmitter and Receiver The architecture of an OFDM transmitter and receiver pair is depicted in Fig. 1. First, during the `th block, N QAM symbols c`[k]; k 0;:::;N 1 are mapped to \sub-carriers", one per subcarrier, which are the inputs to a size N IFFT, which2 f computes− g the DFT shown below efficiently N−1 X j 2π kn q [n] = c [k]e N ; n 0;:::;N (4) ` ` 2 f g k=0 1 ... cyclic ... cyclic prefix is prefix removed at receiver FEQ w[n] FEQ ... ... ... ... ... P/S h[n] + S/P IFFT FFT ... Channel Model FEQ FEQ last P samples b[n] cˆ[k] c[k] a[n] QAM symbols z[k]=H[k]c[k]+W[k] (if Lh P + 1) Transmitter ≤ Receiver Figure 1: An OFDM transmitter, channel model, and receiver. Next, the last P samples of the result of this IFFT are prepended to the beginning of the block, forming the so-called cyclic prefix, and collectively forming a block of size N + P samples which is called one OFDM symbol. q`[N P + n] n < P a`[n] = − ; n 0;:::;N + P 1 (5) q`[n P ] n P 2 f − g − ≥ These OFDM blocks are then serialized to form b n c xc[n] = a N+P [n mod (N + P )] (6) which is then put through an digital to analog convertor to form the complex baseband signal to be modulated by an ordinary quadrature modulator. As discussed in the previous notes on equalization, on its way from the transmitter to the receiver, the signal is subject to a multipath channel, as well as potentially some additive interference, and at the receiver there is additive white Gaussian noise in the electronics. As discussed in the lecture on quadrature modulation, we can model the effects of this multipath channel by modeling the signal at the receiver, after demodulation and sampling as L −1 Xh y [n] = h[`]x[n ∆ `] + w[n] (7) c − − `=0 where ∆ indicates the delay before the channel is perceived as having any significant energy, and h[n] is the sampled impulse response of the channel, whose delay spread is Lh samples long, and w[n] is the additive white Gaussian noise. The receiver then proceeds to process this signal yc[n] in the exact opposite manner as at the transmitter. First the blocks are collected b [n] = y [n + ∆ + `(N + P )]; n 0; 1 :::;N + P 1 (8) ` c 2 f − g then the cyclic prefix is removed r [n] = b [n + P ]; n 0; 1;:::;N 1 (9) ` ` 2 f − g The Fast Fourier transform is then taken, which calculates the DFT shown below efficiently N X −j 2π kn z`[k] = r`[n]e N (10) n=0 2 { { { { sent CP DATA CP DATA CP DATA CP DATA convolutional channel received CP DATA CP DATA CP DATA CP DATA Figure 2: If the channel length is less than one plus the length of the cyclic prefix, then dropping the cyclic prefix at the receiver will remove all of samples containing inter-block interference, allowing the blocks to be processed separately at the receiver in the same manner that they were at the transmitter. Additionally, the blocks of remaining samples (labeled DATA in the received signal above) will each be the circular convolution of the channel with the transmitted blocks. 3.1 The Purpose of the Cyclic Prefix: Removing Inter-Block & Inter-Carrier Interference The cyclic prefix allows a OFDM transmitter receiver pair to accomplish to important objectives, provided that the length of the multipath channel is less than or equal to the length of the cyclic prefix plus one: 1) it eliminates interblock interference, and 2) it enables easy equalization, as for the remainder of the block it makes the linear convolution with the ISI channel into a circular convolution. Indeed, as is indicated in Fig. 2, if we consider the output of the channel at times n ( 0;:::;P 1 + k(N + P ), this will contain a contribution from both the previous OFDM symbol and2 thef next one.− g However, if the channel length Lh P + 1 then these are the only samples containing the inter-block interference, and hence, once we drop≤ these samples at the receiver (when we remove the cyclic prefix), we will have no interblock interference. Additionally, under the same criterion Lh P + 1 the remaining samples in each block after dropping the cyclic prefix will be the circular convolution≤ of the channel with the associated transmitted block. Hence, by the circular convolution property of the DFT, when the FFT is taken of these received blocks, they will be the product of the FFT of the channel H[k] and the original QAM symbols, plus the FFT of the additive noise, as depicted in Fig 1. z`[k] = H[k]c`[k] + w`[k] (11) The estimated QAM symbols can then be reconstructed by processing the subcarriers k individually by multiplying them with a frequency domain equalizer F [k] c^`[k] = F [k]z`[k] (12) H[k]∗ where the F [k] can selected as jH[k]j2+σ2 , which minimizes the average magnitude squared error between 2 c^`[k] and c`[k], where σ is the variance w`[k]. These FEQs F [k] can be estimated by placing training information, also called pilot signals or reference signals, agreed upon in advance between the transmitter and receiver, on various subcarriers k which are spread out in frequency. Because the channel is shorter than the cyclic prefix, and the cyclic prefix is typically far shorter than the FFT size, the channel coeffi- cients H[k] vary slowly with frequency k, and hence the channel estimates from the pilot signals can be interpolated to yield improved channel estimates for other carriers. 3.2 Considerations when Selecting a Cyclic Prefix Length and FFT Size There are two important considerations when selecting a cyclic prefix length and FFT size for an OFDM system in practice. We already noted that the cyclic prefix needs to be longer than one less than the length of the channel (its delay spread). As the cyclic prefix does not bear any extra information itself, it is desirable to select a very large FFT size so that the redundancy is introduces is low. However, the multipath channel must remain fixed over the entire duration of the OFDM symbol (i.e. the entire cyclic prefix and received data block), and this places an upper limit on the length of the FFT based on how quickly the channel is varying. 3 3.3 Bit-Loading & OFDMA Because the OFDM construction turns the wideband multipath channel into multiple parallel lower rate channels associated with evenly spaced frequencies within the band, it enables several intuitive other jH[k]j2 technologies. As mentioned in the introduction, those carriers k with higher signal to noise ratios σ2 can be given higher order QAM symbols and higher code rates. This process is called bit-loading and adaptive modulation and coding in different contexts. Additionally, using OFDM modems also enable an especially intuitive form of multiple access. If the transmitter wishes to send individual messages to multiple receivers (as in a cellular system) listening to the same wideband channel, one can place information to be transmitted to different users on different subcarriers. Furthermore, since the multipath profiles H[k] of different receivers listening to the same transmitter will be different, if these multipath profiles are reliably fed back to the transmitter, it can also select those subcarriers for different users which they are receiving with high signal to noise ratios. On the other hand, if the multipath profiles are not fed back, or are varying too quickly, the transmitter can diversify which frequencies are used for different users by spreading the carriers/blocks of carriers given to each particular user out in frequency.

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