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I am working through Signals and Systems Demystified, but I am at an impasse.

I would like to take the inverse Fourier transform of

[tex]

H(f)=\begin{cases}

-j&\text{if } f > 0\\

j&\text{if } f<0\end{cases}

[/tex]

So

[tex]

h(t) = \int_{-\infty}^{0} je^{j2\pi f t}df + \int_{0}^{\infty} -je^{j2\pi ft}df

= \int_{-\infty}^{0} je^{j2\pi f t}df + \int_{-\infty}^{0} je^{j2\pi f t}df

= 2\int_{-\infty}^{0} je^{j2\pi f t}df

= \frac{j2e^{j2\pi ft}}{j2\pi t}|_{-\infty}^{0}

[/tex]

So

[tex]

h(t) = \frac{1}{\pi t} - \lim_{f \rightarrow -\infty} \frac{e^{j2\pi ft}}{\pi t}

[/tex]

The book says that [tex]h(t) = \frac{1}{\pi t}[/tex] is the answer, but I don't understand how. Isn't [tex]\lim_{f \rightarrow -\infty} \frac{e^{j2\pi ft}}{\pi t}[/tex] periodic and nonconverging?

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# Homework Help: Inverse fourier troubles: e^(-j*infty)

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