10/4/2002 Phase and Frequency.doc 1/5 Phase and Frequency Consider the trig functions sinx and cosx. Q: What are the units of x ?? A: The units of x must be radians. In other words x is phase φ , i.e., cos φ and sin φ. Phase can of course be a function of time, i.e., cos ( )tφ. For example: 00cos ( + )tωφ In other words, the signal phase ()tφ is 00()ttφωφ=+ ! A: Time for some definitions! Q: What the !?! I alwaysthought “phase” was 0φ,not 00tωφ+ !10/4/2002 Phase and Frequency.doc 2/5 We call 00()ttφωφ=+ the total, or absolute phase of the sinusoidal signal. Note the total phase is a linearly increasing function of time ! The slope of this line is 0ω, while the y-intercept is 0φ. We can define the relative phase ()rtφ as: 0() ()rtttφφω=− Thus, if 00()ttφωφ=+, then 0()rtφφ= . But, the relative phase need not be a constant. In general, we can write: 0cos ()rttωφ+ Therefore, the relative phase is in general some arbitrary function of time. t ()tφ 0φ 00tωφ+10/4/2002 Phase and Frequency.doc 3/5 A: Wrong ! Frequency too is a little more complicated than you might have imagined. Angular frequency is defined as the rate of (total) phase change with respect to time. As a result, it is measured in units of radians/second. How do we determine the rate of phase change with respect to time? We take the derivative of ( )tφ with respect to t ! I.E., ()( ) (radians/sec)dttdtφω= Q: O.K., so you have made phase really complicated, but at least the signal frequency is still 0ω , right ??10/4/2002 Phase and Frequency.doc 4/5 For example, if 00()ttφωφ=+, then: 000()()dttdtωφωω+== A: Not so fast! The frequency (i.e., the rate of phase change) is equal to 0ω only if total phase is 00()ttφωφ=+. In other words, the frequency is equal to 0ω if the relative phase is a constant 0φ. Otherwise: 0000()()()()()()rrrrdt ttdtdtdtdt dtdtdttωφωωφφωωω+==+=+=+ Q: See! I told you! Thefrequency is 0ω after all !10/4/2002 Phase and Frequency.doc 5/5 In other words, the total frequency ()tω is the sum of the carrier frequency 0ω and the relative frequency ()rtω. The signal frequency can change with time ! Remember, we can also express frequency in cycles/second (i.e., Hz) if we divide by 2π. (Hz)2()()tftωπ= Therefore, we can write: 0()
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