Timing is derived from the spec an's own clock system. It does not "cooperate" with the TX, unless you obtain one designed for radio servicing that can demodulate a given application. Odds are for a educational project at the undergraduate level you would not need a demodulating capable unit. If you get access to a modern lab grade one, will not drift any where near enough to cause you any issues. In fact, for the most part, it will be more stable the then spread spectrum transmitter is. You need to give us more data as to the frequency band and radio service you wish to look at, as with what you have said we could specify anything from a modified TV tuner hooked to a ramp generator and oscilloscope to a 40,000$ instrument.. You might wish to get on interlibrary loan a copy of the ARRL Spread Spectrum Handbook, which is outdated but very useful from a beginning point of view.
I'm assuming you mean at 88-108 Mhz FM, or are your looking at Digital Radio Mondial ? Or the UK/EU service at 200-300 Mhz?
Without knowing the target application, It is impossible to suggest a unit without knowing frequency band, desired bandwidth, and noise floor/sensitivity issues for the instrument.
Start with this:
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And This:
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And look at home made systems here:
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and here:
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and here:
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I'm not sending you to the home-made systems to suggest you build one, I'm sending you there to learn how a simple single stage superheterodyne SA works...
Steve