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Unformatted text preview: ECE 5670 : Digital Communications Lecture 22: Time, Frequency and Antenna Diversity 1 4/7/2011 and 4/12/2011 Instructor: Salman Avestimehr Introduction We have seen that the communication (even with coding) over a slow fading flat wireless channel has very poor reliability. This is because there is a significant probability that the channel is in outage and this event dominates the total error event (which also include the effect of additive noise). The key to improving the reliability of reception (and this is really required for wireless systems to work as desired in practice) is to reduce the chance that the channel is in outage. This is done by communicating over different channels and the name of the game is to harness the diversity available in the wireless channel. There are three main types: temporal, frequency and antennas. We will see each one in this lecture, starting time diversity. Time Diversity Channel The temporal diversity occurs in a wireless channel due to mobility. The idea is to code and communicate over the order of time over which the channel changes (called the coherence time). This means that we need enough delay tolerance by the data being communicated. A simple single-tap model that captures time diversity is the following: y = h x + w , = 1 ,...,L. (1) Here the index represents a single time sample over different coherence time intervals. By interleaving across different coherence time intervals, we can get to the time diversity channel in Equation (1). A good statistical model for the channel coefficients h 1 ,...,h L is that they are all independent and identically distributed. Further more, in a environment with lots of multipath we can suppose that they are complex Gaussian random variables (the so-called Rayleigh fading). 1 Based on lecture notes of Professor Pramod Viswanath at UIUC. 1 A Single Bit Over a Time Diversity Channel For ease of notation, we start out with L = 2 and just a single bit to transmit. The simplest way to do this is to repeat the same symbol: i.e., we set x 1 = x 2 = x E. (2) At the receiver, we have four (real) voltages: [ y 1 ] = [ h 1 ] x + [ w 1 ] , (3) [ y 1 ] = [ h 1 ] x + [ w 1 ] , (4) [ y 2 ] = [ h 2 ] x + [ w 2 ] , (5) [ y 2 ] = [ h 2 ] x + [ w 2 ] . (6) As usual, we suppose coherent reception, i.e., the receiver has full knowledge of the exact channel coefficients h 1 ,h 2 . By now, it is quite clear that the receiver might as well take the appropriate weighted linear combination to generate a single real voltage and then make the decision with respect to the single transmit voltage x . This is the matched filter operation: y MF = [ y 1 ] [ h 1 ] + [ y 1 ] [ h 1 ] + [ y 2 ] [ h 2 ] + [ y 2 ] [ h 2 ] . (7) Using the complex number notation, y MF = [ h * 1 y 1 ] + [ h * 2 y 2 ] , (8) = ( | h 1 | 2 + | h 2 | 2 ) x + w. (9) Here w is a real Gaussian random variable because it is the sum of four independent and...
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This note was uploaded on 10/02/2011 for the course ECE 5670 taught by Professor Scaglione during the Spring '11 term at Cornell University (Engineering School).
- Spring '11