Why does $ h_1(t) $ work when sampling with Zero-Order Hold

$ x_0(t) = h_1(t) * (x(t)p(t)) $

$ x_0(t) = h_1(t) * \sum_{n = -\infty}^{\infty}x(t) \delta(t-nT) $

$ x_0(t) = h_1(t) * \sum_{n = -\infty}^{\infty}x(nt) \delta(t-nT) $

$ x_0(t) = \sum_{n = -\infty}^{\infty}x(nt) h_1(t-nT) $

$ h_1(t) $ works because $ p(t) $ is a shifted delta function, and when you convolve a specific function with a shifted delta function, you get back the specific function but shifted in time.

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