Transcription of Complex Signals - DTU
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Complex Signals - DTU
Chapter 2 Complex SignalsA number of signal processing applications make use of Complex Signals . Someexamples include the characterization of the Fourier transform, blood velocityestimations, and modulation of Signals in telecommunications. Furthermore,a number of signal-processing concepts are easier to derive, explain and un-derstand using Complex notation. It is much easier, for example to add thephases of two Complex exponentials such asx(t) =ej 1e 2, than to manipulatetrigonometric formula, such as cos( 1) cos( 2).We start by introducing Complex Signals in Section , and treating theFourier relations in Sec. Among all Complex Signals , the so-calledanalyticsignals are especially useful, and these will be considered in greater detail inSection Introduction to Complex signalsA Complex analog signalx(t) is formed by the signal pair{xR(t),xI(t)}, wherebothxR(t) andxI(t) are the ordinary real Signals . The relationship betweenthese Signals is given by:x(t) =xR(t) +jxI(t),( )wherej= 1.
2.1.2 Phasors The word phasor is often used bymathematicians to meanany complex number. In engineering, it is frequently used to denote a complex exponential function of constant modulus and linear phase, that is a function of pure harmonic behavior. Here is an example of such a phasor: x(t) = Aej2πf 0t, (2.6)
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