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Normalized DTFT as an Operator

Module by: Mark A. Davenport. E-mail the author

Normalized DTFT as an operator

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

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

One can show that the summation converges for any x2(π)x2(π), and yields a function X(ejω)L2[-π,π]X(ejω)L2[-π,π]. Thus,

F : 2 ( Z ) L 2 [ - π , π ] F : 2 ( Z ) L 2 [ - π , π ]
(2)

can be viewed as a linear operator!

Note: It is not at all obvious that FF can be defined for all x2(Z)x2(Z). To show this, one can first argue that if x1(Z)x1(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 ] < 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 ] <
(3)

For an x2(Z)1(Z)x2(Z)1(Z), one must show that it is always possible to construct a sequence xk2(Z)1(Z)xk2(Z)1(Z) such that

lim k x k - x 2 = 0 . lim k x k - x 2 = 0 .
(4)

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

X ( e j ω ) = lim k X k ( e j ω ) . X ( e j ω ) = lim k X k ( e j ω ) .
(5)

So for any x2(Z)x2(Z), we can define F(x)=X(ejω)F(x)=X(ejω), where X(ejω)L2[-π,π]X(ejω)L2[-π,π].

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

F - 1 ( X ) = 1 2 π - π π X ( e j ω ) · e j ω n d ω F - 1 ( X ) = 1 2 π - π π X ( e j ω ) · e j ω n d ω
(6)

To verify that F-1(F(x))=xF-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 ] 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 ]
(7)

One can also show that for any XL2[-π,π]XL2[-π,π], F(F-1(X))=XF(F-1(X))=X.

Operators that satisfy this property are called unitary operators or unitary transformations. Unitary operators are nice! In fact, if A=XYA=XY 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 , x 1 , x 2 = A x 1 , A x 2 x 1 , x 2 X ,
(8)

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

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