Fourier transform and energy of a convolution

Hi guys i have to find the fourier transform of the convolution: $$sinc(t/2T)*\sum\limits_{n-\infty}^{+\infty} (-1)^{n}\delta(t - nT)$$

i was thinking of express the summatory as : $$\sum\limits_{n-\infty}^{+\infty} (-1)^{n}\delta(t - nT) = \mathrm{III}_{2T}(t) -\mathrm{III}_{T}(t-1)$$ so we have:

$$sinc(t/2T)*[\mathrm{III}_{2T}(t) -\mathrm{III}_{T}(t-1)]$$

So the convolution in the fourier domain became a product: $$rect(f2T)\mathrm{III}_{1/2T}(t)-2rect(f2T)\mathrm{III}_{1/T}(t)e^{ipif}$$

Well is this a good result and how i can find the energy from this formula ?

• Please define your symbols, What is $III$ ? Also in your last formula you seem to mix frequency and time domain functions. – Hilmar Dec 3 '20 at 20:51
• Hi , III is the delta comb ,where i mix them ? – Giovanni Cerciello Dec 3 '20 at 21:15
• You appear to do a Fourrier transform in going from equation 3 to equation 4. However the variable of the $III$ function is still $t$ – Hilmar Dec 4 '20 at 17:46