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Our next job should be to come up with an inverse transformation. If inverse transformation exists, then there is no loss of information from discrete (time) domain to frequency domain and vice-versa. Existence of inverse will establish, transform nature of (4). If (2) defines IFT in continuous domain, in the discrete domain, by analogy of (1) and (4) we can hypothesize following inverse transform. |
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Where K is a suitable scaling factor.
Our next job is to verify that indeed (4) and (5) define a transformation pair. Substituting (4) in (5), we get following expression for right hand side of (5). |
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[Note the use of dummy subscript  ] |
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Let us work this expression out in a long hand fashion; for compactness we use notation  |
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In the above expression, for the first row  is set to zero, for the second row it is set to one and for the last row  . |
