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

You can use this operator to find the inverse DTFT by applying it to Y(eiw) and then using the sifting property to isolate y[n]. In summary, the problem is to find the inverse DTFT of i(d/dw)Y(eiw) in terms of y[n] using the sifting property and integral b.
  • #1
nacreous
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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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  • #2
i(d/dw) is simply an operator: it means differentiate w.r.t. w and multiply the result by i.
 

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

What is the purpose of calculating inverse DTFT?

The purpose of calculating inverse DTFT (Discrete-Time Fourier Transform) is to convert a discrete-time signal from the frequency domain back to the time domain. This allows us to analyze and manipulate the signal in the time domain, which is often more intuitive and easier to understand.

How is the inverse DTFT calculated?

The inverse DTFT is calculated using the inverse Fourier transform formula, which involves integrating the signal over all frequencies and multiplying it by a complex exponential function. This process essentially reverses the steps of the DTFT, converting the signal back to the time domain.

What is the relationship between DTFT and inverse DTFT?

The DTFT and inverse DTFT are inverse operations of each other. The DTFT converts a signal from the time domain to the frequency domain, while the inverse DTFT converts it back to the time domain. This means that taking the inverse DTFT of a signal that has been transformed by the DTFT will result in the original signal.

What types of signals can be analyzed using inverse DTFT?

The inverse DTFT can be used to analyze any discrete-time signal that has been transformed into the frequency domain using the DTFT. This includes signals that are periodic, aperiodic, finite, or infinite in length.

What are some common applications of inverse DTFT?

The inverse DTFT has many practical applications in fields such as signal processing, telecommunications, and audio engineering. It is commonly used to analyze and manipulate signals in the time domain, as well as in the design of filters and other digital signal processing techniques.

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