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DFT - ECE 3250 THE DFT Fall 2008 Real-world discrete-time...

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ECE 3250 THE DFT Fall 2008 Real-world discrete-time signal processing deals exclusively with finite-duration signals. Infinite-duration signals are useful mathematically, but nobody has ever watched such a signal play out in its entirety. When addressing signals and systems problems involving infinite-duration signals, one’s goal is often to generate useful approximate results by manipulating finite-duration signals in a computationally efficient way. The DFT is a tool that facilitates such manipulations. In essence, the DFT reduces a variety of signal- processing calculations to finite-dimensional linear algebra. For ease of exposition, I’ll assume that all the signals we’re dealing with here are complex-valued. This doesn’t cost us any generality, since a real-valued signal is just a special kind of complex-valued signal. Definition 1: Given a positive natural number N , an N -point signal is a discrete-time signal x C Z that satisfies x ( n ) = 0 for n < 0 and x ( n ) = 0 for n N . An N -point signal is simply a finite-duration signal x whose “duration interval” is contained in the range 0 n < N . Observe that x ( n ) could still be zero for some n - values in that range. In particular, if M > N , we can regard any N -point signal x as an M -point signal that just happens to satisfy x ( n ) = 0 for N n < M . The important thing to recognize is that an N -point signal x is specified completely by N numbers x (0), x (1), . . . , x ( N - 1). In this way, the set of all C -valued N -point signals is in one-to-one correspondence with C N , the set of all N -vectors with entries in C . If x is an N -point signal, I’ll denote by x the column N -vector ˆ x (0) x (1) x (2) . . . x ( N - 1) ˜ T . The set of all N -point signals is closed under the taking of linear combinations in C Z , so it forms a vector space under the usual vector operations on C Z . It is easy to check that those vector operations map nicely to the usual vector operations on C N under the correspondence I alluded to in the preceding paragraph. In other words, if x 1 and x 2 are the vectors corresponding to N -point signals x 1 and x 2 , then for any c 1 and c 2 in C the vector c 1 x 1 + c 2 x 2 corresponds to the N -point signal c 1 x 1 + c 2 x 2 . I’ll be making considerable use of the notation l N for l Z and nonzero N N . Read that notation as “ l mod N .” It denotes the unique natural number m such that 0 m < N and m = pN + l for some p Z . Technically, l N is the remainder you obtain after dividing l by N . For example, 7 5 = 2 whereas 15 5 = 0. Note that l N = l if and only if 0 l < N . Definition 2: Given a positive N Z and an N -point signal x and any n o with 0 n o < N , the cyclic shift of x by n o is the N -point signal CShift n o ( x ) with specification CShift n o ( x )( n ) = x ( n - n o N ) = x ( n - n o ) if n o n < N x ( N + n - n o ) if 0 n < n o .
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