i'm trying to calculate the frequency spectrum of a signal that has been sampled using natural sampling PAM (pulse amplitude modulation). The message signal is given as $s(t)$, and the sampling signal is a rectangular pulse train with period $T_s$, height 1, and width $\tau$.
I know that the spectrum of the sampled signal will be the convolution between the transform of the pulse train and the message spectrum. However, I am having trouble deriving the correct expression for the series representation of the pulse train. Here is what I have so far:
$$h(t) = \sum_{-\infty}^{\infty} D_n e^{jn2\pi ft}$$
Where $D_n$ is given by
$$D_n = \frac{1}{T_s} \int_{-\tau/2}^{\tau/2} e^{-jn2\pi t/T_s} \ dt$$
Integrating gives:
$$D_n = \frac{1}{T_s} \frac{-T_s}{jn2\pi} e^{-jn2\pi t/T_s} |^{\tau/2}_{-\tau/2}$$
Plugging in bounds gives:
$$D_n = \frac{1}{n\pi} (\sin(\frac{n\pi \tau}{T_s})) = \frac{\tau}{T_s}sinc(\frac{n \pi \tau}{T_s})$$
However every other solution I see online is not in this form (There is no pi term in the numerator of the sinc). Any suggestions on how to proceed would be greatly appreciated.
Thanks,
Matt