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Why do the following two methods give different answers (or are they the same) for the Fourier transform of $Y = \cos(\omega_0 t + \phi)$, with respect to $t \to \omega$ ?

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They are the same.

Using the Dirac impulse sifting property $f(x) \delta(x-a) = f(a)\delta(x-a)$ you can also verify that the second method produces the same output as the first method.

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  • $\begingroup$ Thanks Fat32. I used the sifting property to convert the answer of Method 1 to Method 2 and it is correct. But how I can convert the answer of Method 2 to Method 1, since both exp(j phi) and exp(-j \phi) in Method 2 are constants (i.e., no time or omega dependent terms). $\endgroup$ – Ganth Apr 10 at 2:54
  • $\begingroup$ you can't go from method-2 into method-1. Take for example $$f(x) = 5 \delta(x-1) + 2 \delta(x-2)$$ this has infinitely many possible equivalents such as $$f_1(x) = (x+4)\delta(x-1) + x \delta(x-2)$$ or just another $$f_2(x) = (2x+3)\delta(x-1) + (3x-4) \delta(x-2)$$; after a generalization, as you can see there are infinite number of possible eqivalents... $\endgroup$ – Fat32 Apr 11 at 10:08
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    $\begingroup$ Thank you very much, Fat32 $\endgroup$ – Ganth Apr 11 at 10:17

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