September 2004, Issue 170 Test Your EQ

Answer 2—The phase shift measurement problem can be made easier by shifting the two signals down to a lower frequency by mixing them with a second signal whose frequency is extremely close to the original signal frequency. For example, if the signal frequency w₁ is 83 MHz and the mixing frequency w₂ is 82.9 MHz, the difference frequency is just 100 kHz.

How does this help, exactly? The transmitted signal is

The received signal, delayed by time Dt, is

 

The delay can also be expressed as a phase angle f, where f = w₁Dt.

The transmit mixer output, after filtering, is 

 

The receive mixer output, after filtering, is

Rearranging terms, you get the following:

In other words, the phase angle introduced by the time delay is preserved through the mixing process even though the output frequency of the mixers is several orders of magnitudes lower. This makes sense intuitively if you consider that if the two high-frequency signals slip past each other by one full cycle, the beat frequency outputs of the mixers must do the same thing.

The time delay Dt¢ represented by this phase angle has been multiplied by the ratio of the original signal frequency to the mixer output frequency, making it much easier to measure.

Additional information on this topic can be found at www.uow.edu.au/eng/phys/200labs/phys205/spdoflght.pdf.

Contributor: David Tweed