# Sampling: Multiplication by Square Wave

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Science Advisor
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We are considering a system in which the input signal x(t) is multiplied by a periodic square wave s(t) in order to produce an output w(t). The input signal is band limited with $|X(j\omega)| = 0 \ \ \textrm{for} \ \ |\omega| \geq \omega_M$, where $\omega_M$ is the bandwidth. We are supposed to find (given a certain width of the periodic square wave, e.g. T/3), the maximum value of T (in terms of $\omega_M$) for which there is no aliasing among the replicas of $X(j\omega)$ in $W(j\omega)$.

I do not know how to approach this problem. This is not simple impulse train sampling. It is not zero order hold sampling. In fact...what the hell is this? Multiplication by a square wave?!?!? Sorry, I don't know where to start.

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## Answers and Replies

berkeman
Mentor
Multiplication by a square wave is a form of modulation. Start off multiplying by a sine wave instead, and work out the images with sum and difference math. Then consider what a square wave's spectra looks like....