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idft - ECE 421 Sum 2010 Notes Set 12 Inverse Discrete...

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ECE 421 - Sum 2010 Notes Set 12 : Inverse Discrete Fourier Transform 1 INTRODUCTION Previously we learned to compute the spectrum of a set of signal samples using the Discrete Fourier Transform (DFT). Recently the Inverse Discrete Fourier Transform (IDFT) has also become important. The IDFT computes a set of time-domain samples which has a specific frequency domain spectrum. This is a valuable technique used in many high-speed access links which are frequency-dependent. Two of these access links are: Digital Subscriber Line (DSL): DSL access allows the residential subscriber to use the same twisted-pair copper line for simultaneous telephone conversations and high-speed internet access. For economic viability, DSL was engineered to work in the existing physical plant. This means it must work in an environment which has reflections and interference due to the aggregation of individual lines into bundles. Wireless Access: There are now a large number of wireless access platforms which must all share a common environment: laptop computers, tablet computers, telephones supporting voice, audio, and video, etc. The energy from each of these applications must ad- here to stringent frequency domain constraints for compatibility. Additionally, these devices must perform in an environment which has substantial reflections. The end effect on these applications is that the access channels may have some frequencies which will propagate energy well, but other frequencies which will not propagate very well. Thus, an efficient transmitter must send energy only at the appropriate “good” frequencies. But since a transmittter must work in the “time” domain, how does a transmitter we create a time domain signal which occupies only the “good” frequencies? One approach which has been quite useful for this application uses the IDFT to create the transmitter signal. This will be the motivating application for studying the IDFT in this set of Notes. In actual communication systems, this approach goes by the names Discrete Multi-Tone (DMT), Multi-Carrier Modulation (MCM), and Orthogonal Frequency Divi- sion Multiplexing (OFDM).
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ECE 421 - Sum 2010 Notes Set 12 : Inverse Discrete Fourier Transform 2 DFT AS A MATRIX OPERATION The derivation of the IDFT is facilitated by considering the matrix formulation for the DFT. Since we have previously studied matrix operations, let’s consider how “taking the DFT” of a time signal can be interpreted as a matrix-vector operation. Assume an M -length block of digital samples has been acquired and is given by x m 0 m M 1. Then the DFT coefficients X k are given by the summation X k M 1 m 0 x m e j 2 π km M 0 k M 1 1 Note that there are M time-domain samples in x m and there are M frequency domain
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