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Normalized dtft as an operator

Note that by taking the DTFT of a sequence we get a function defined on [ - π , π ] . In vector space notation we can view the DTFT as an operator (transformation). In this context it is useful to consider the normalizedDTFT

F ( x ) : = X ( e j ω ) = 1 2 π n = - x [ n ] e - j ω n .

One can show that the summation converges for any x 2 ( π ) , and yields a function X ( e j ω ) L 2 [ - π , π ] . Thus,

F : 2 ( Z ) L 2 [ - π , π ]

can be viewed as a linear operator!

Note: It is not at all obvious that F can be defined for all x 2 ( Z ) . To show this, one can first argue that if x 1 ( Z ) , then

X ( e j ω ) 1 2 π n = - x [ n ] e - j ω n 1 2 π n = - x [ n ] e - j w n = 1 2 π n = - x [ n ] <

For an x 2 ( Z ) 1 ( Z ) , one must show that it is always possible to construct a sequence x k 2 ( Z ) 1 ( Z ) such that

lim k x k - x 2 = 0 .

This means { x k } is a Cauchy sequence, so that since 2 ( Z ) is a Hilbert space, the limit exists (and is x ). In this case

X ( e j ω ) = lim k X k ( e j ω ) .

So for any x 2 ( Z ) , we can define F ( x ) = X ( e j ω ) , where X ( e j ω ) L 2 [ - π , π ] .

Can we always get the original x back? Yes, the DTFT is invertible

F - 1 ( X ) = 1 2 π - π π X ( e j ω ) · e j ω n d ω

To verify that F - 1 ( F ( x ) ) = x , observe that

1 2 π - π π 1 2 π k = - x [ k ] e - j ω k e j ω n d ω = 1 2 π k = - x [ k ] - π π e - j ω ( k - n ) d ω = 1 2 π k = - x [ k ] · 2 π δ [ n - k ] = x [ n ]

One can also show that for any X L 2 [ - π , π ] , F ( F - 1 ( X ) ) = X .

Operators that satisfy this property are called unitary operators or unitary transformations . Unitary operators are nice! In fact, if A = X Y is a unitary operator between two Hilbert spaces, then one can show that

x 1 , x 2 = A x 1 , A x 2 x 1 , x 2 X ,

i.e., unitary operators obey Plancherel's and Parseval's theorems!

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Source:  OpenStax, Digital signal processing. OpenStax CNX. Dec 16, 2011 Download for free at http://cnx.org/content/col11172/1.4
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