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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 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, ones 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, Ill assume that all the signals were dealing with here are complex-valued. This doesnt 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, Ill denote by x the column N-vector 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 . Ill be making considerable use of the notation hh l ii N for l Z and nonzero N N . Read that notation as l mod N . It denotes the unique natural number m such that m < N and m = pN + l for some p Z . Technically, hh l ii N is the remainder you obtain after dividing l by N . For example, hh 7 ii 5 = 2 whereas hh 15 ii 5 = 0. Note that hh l ii 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 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 ( hh n- n o ii N ) = x ( n- n o ) if n o n < N x ( N + n- n o ) if 0 n < n o ....
View Full Document

This note was uploaded on 08/10/2010 for the course ECE 4370 at Cornell University (Engineering School).

Page1 / 10

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online