Frequency or phase shift when using pulsed wave doppler

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mkh004
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Hi,
I'm a medical student who is using ultrasound to measure cardiac strain. I'm trying to read up a bit on the physics behind doppler imaging (I did AP physics in high school), and I have a question for some genius out there. I keep reading that in pulsed wave doppler measurements, one does not measure the change in frequency, but a phase shift of the waves. Why?
 
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To measure a freq. you have to count pulses over specific period of time , "gate" time. This period can't be too long, I'd think at least no more than 100 milliseconds for biological process, to keep update rate 10 Hz or so. This implies error in measurements, +-10 Hz. But if you measure a phase, there is no such limitation, and precision in estimation of deviation a frequency 360 times better for phase error 1 degree