Modeling a signal through the environment I have a signal that is composed of a sinusoid.  I will send it over the air through radios.  Part of the signal will go through processing on a computer, which means that it will be converted from analog to digital, and back again.

I have two questions.  How can I model any frequency shift that will occur in this system?  And, how can I model how the time of the signal may change?  That is, if the signal occurs in bursts (e.g. 200 ms of tone followed by 100 ms of absence of tone), how can I determine how the time may decrease?  Is the time changed only by the digital part of the system (due to sampling), or might the radio path also affect the duration?

Thank you
You can model frequency shift this way:


$s = s \circ exp(j \cdot 2\pi \cdot (f-\hat{f}) \cdot t)$,


where $\circ$ means Hadamard product and $s$ is input signal vector (e.g. frame or part of data stream of interest). You can implement this operation in the baseband or in carrier frequency. The effect will be generally the same. 

As you've told your signal time can increas if multipath exists. You can model it as FIR filter of the order of your impulse response length. Refer to the channel modeling techniques for details.

Also your signal time can vary (increase and decrease) if relative motion of receiver and transmitter exists. This effect appears in the case of so called wideband model, e.g. sonic channel in the underwater communications. For such a case signal bandwidth is comparable to carrier frequency so Doppler shift is very different for low frequencies and high ones in the band of interest. It causes varying of signal duration in the receiver end. If you deal with radio signals your signal model is probably narrowband so this effect is neglectable.