Goal is to build a lab exercise for students to illustrate digital IQ-modulation (QPSK, M-QAM). The bit stream for I and Q are generated by e.g. an Arduino. Is it possible to use a square wave as local oscillator (original and 90-phase shifted) or must it be a sine wave? Most books and datasheets assume a sine-shaped local oscillator. How would you implement it (square wave, square wave filtered, sine wave?) for a lab exercise (I'm not asking for a electric schema, although welcome, but more in principle how would you do it)? Goal of the exercise is not to show the most efficient way to transmit data but to make the students see and feel the most important principles of digital IQ modulation.
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From https://en.wikipedia.org/wiki/Frequency_mixer : Switching is the normal operating mode of a packaged double balanced mixer [e.g., in DB Gilbert cell topology], with the local oscillator drive considerably higher than the signal amplitude. Instead of the LO sine wave term, the signum function [square wave] is used. Switching mixer enables to achieve a lower noise figure and larger conversion gain. – V.V.T Jul 16 '20 at 13:36
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Yes, you can use a square wave LO. It makes the multiplication by the data trivial. With digital data, it can all be done with logic, and with analogue IQ data you can do it with 4053-type analogue muxes.
It sounds like you're not expecting the Arduino to produce the LO. Use a 4x LO digital clock, clocking a /4 Johnson counter. That's most easily made with a single dual flip flop HC74 package, Q to D for one, Qbar to D for the other. This will produce all four phases, 0, 90, 180, 270 of your quadrature LO.

Neil_UK
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