of electrical voltage over time, we can represent the voltage U at each time t as v(t). The mathematical concept of a func- tion tells us that there is one f(x) for each x or, in our electrical case, one specific volt- age v(t) for each time t. Fourier analysis tells us that we can think of the same v(t) in another form as V(s), where S is one particular frequency rather than an instant of time. The function V(s) is a little more complicated than v(t) because it contains not only the amplitude of fre- quency S but also its phase. The relationship between time response v(t) and frequency response V(s) is a pair of integrals from college calculus. (1.1) V(s)e-its ds (1.2) Superscript(3)The coauthor who lives in Texas notes that this could mean that 3-in hail interferes in the CDMA band. Frankly, we're a lot more concerned about equipment damage than about radio in- terference when the hail is the size of tennis balls. 4The audible difference between an oboe and a violin playing the same steady note B, is in their response at higher frequencies. Those higher frequencies are called harmonics. The audible dif- ference between an oboe and a piano, on the other hand, is not only their frequency response but the percussive time response of the piano. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. Radio Engineering Concepts Key Radio Concepts While our time and frequency intervals in real life do not go from minus infinity (-00) to plus infinity (+00), an important message from these two integrals is that frequency response V(s) depends on the time response v(t) over an extended period of time and, conversely, that the time response v(t) is determined by knowing enough about the fre- quency response V(s). Equations (1.1) and (1.2) for time and frequency are nearly sym- metric, and they tell us that there is a duality in the time-frequency relationsh