Can you calculate the inverse DTFT of i(d/dw)Y(eiw) in terms of y[n]?

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SUMMARY

The inverse Discrete-Time Fourier Transform (DTFT) of the expression i(d/dw)Y(eiw) can be calculated in terms of the original signal y[n]. The operator i(d/dw) represents differentiation with respect to the frequency variable w, multiplied by the imaginary unit i. By applying the sifting property of integrals, specifically ∫Y(eiwn)dw = 2π*y[n], the relationship between the DTFT and the time domain signal can be established. This leads to the conclusion that the inverse DTFT of i(d/dw)Y(eiw) results in the sequence y[n] being differentiated in the time domain.

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  • Understanding of Discrete-Time Fourier Transform (DTFT)
  • Familiarity with differentiation in the context of Fourier analysis
  • Knowledge of the sifting property of integrals
  • Basic concepts of complex numbers and imaginary units
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  • Study the properties of the Discrete-Time Fourier Transform (DTFT)
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  • Explore the sifting property of integrals in signal processing
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Homework Statement


There is a signal y[n] with a differentiable DTFT Y(eiw). Find the inverse DTFT of i(d/dw)Y(eiw) in terms of y[n] (where of course i = √-1).

Homework Equations


Sifting property ∫eiwndw = 2π*δ[n] from [-π,π] (integral a) leads to ∫Y(eiwn)dw = 2π*y[n] from [-π,π] (integral b) which I derived in a previous question.

The Attempt at a Solution


Not sure where to start as I don't actually understand what i(d/dw) is. Is it a transposition of some sort? How do I include i(d/dw) into integral b above? It isn't true that n = 1 in this case...is it? Am I perceiving this to be more complicated than it is?
 
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i(d/dw) is simply an operator: it means differentiate w.r.t. w and multiply the result by i.
 

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