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lecture16

# lecture16 - ECE 5670 Digital Communications Lecture 16...

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ECE 5670 : Digital Communications Lecture 16: Orthogonal Frequency Division Modulation (OFDM) and Capacity of the Wireline Channel 1 3/10/2011 Instructor: Salman Avestimehr Introduction In this lecture we will see in detail the OFDM method to convert the wireline channel into a parallel AWGN channel. We also see that this achieves the capacity of the wireline channel – in other words, the largest possible data rate of communication is achieved by the OFDM method. OFDM Consider the frequency selective model that we have been working with as a good approxi- mation of the wireline channel: y [ m ] = L - 1 summationdisplay =0 h x [ m ] + w [ m ] , m 1 . (1) We will convert the ISI channel in Equation (1) into a collection of AWGN channels, each of different noise energy level: ˆ y [ N c k + n ] = ˆ h n ˆ x [ N c k + n ] + ˆ w [ N c k + n ] , k 0 , n = 0 . . . N c 1 . (2) We will be able to make this transition by some very simple signal processing techniques. Interestingly, these signal processing techniques are universally applicable to every wireline channel, i.e., they do not depend on the exact values of channel coefficients h 0 , . . . , h L - 1 . This makes OFDM a very robust communication scheme over the frequency-selective channel. Cyclic Prefix Suppose we have mapped our information bits into N c voltages. We will revisit the issue of how these voltages were created from the information bits at a slightly later point in this lecture. For now, we write them as a vector: d = [ d [0] , d [1] , . . . , d [ N c 1]] t . 1 Based on lecture notes of Professor Pramod Viswanath at UIUC. 1

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We use these N c voltages to create an N c + L 1 block of transmit voltages as: x = [ d [ N c L + 1] , d [ N c L + 2] , . . . , d [ N c 1] , d [0] , d [1] , . . ., d [ N c 1]] t , (3) i.e., we add a prefix of length L 1 consisting of data symbols rotated cyclically (Figure 1). The first L 1 transmitted symbols contain the “data” symbols d [ N c ( L 1)] , . . . , d [ N c 1]. The next N c transmitted voltages or symbols contain the “data” symbols d [0] , d [1] , . . ., d [ N c 1]. In particular, for a 2-tap frequency-selective channel we have the following result of cyclic precoding: x [1] = d [ N c 1] x [2] = d [0] x [3] = d [1] . . . x [ N c + 1] = d [ N c 1] With this input to the channel (1), consider the output y [ m ] = L - 1 summationdisplay =0 h x [ m ] + w [ m ] , m = 1 , . . . , N c + 2( L 1) . The first L 1 elements of the transmitted vector x were constructed from circularly wrapped elements of the vector d , which are included in the last N c 1 elements of x . The receiver hence ignores the first L 1 received symbols y [1] , . . . , y [ L 1]. The ISI extends over the first L 1 symbols and the receiver ignores it by considering only the output over the time interval m [ L, N c + L 1]. Let us take a careful look at how the N receive voltages (received at times L through N c + L 1) depend on the transmit voltages d [0] , . . . , d [ N c 1]: y [ m ] = L - 1 summationdisplay =0 h d [( m L ) modulo N c ] + w [ m ] . (4) See Figure (1).
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lecture16 - ECE 5670 Digital Communications Lecture 16...

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