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 .

where x∗(t) is the complex conjugate of x(t), and (↔) denotes a Fourier trans-form pair. Let the complex signal x(t) be expressed in the form: x(t) = x 1(t)+jx 2(t), (2.8) where x 1(t) and x 2(t) are real signals. Let their spectra be X 1(f) and X 2(f), respectively, i.e. x 1(t) ↔ X 1(f) and x 2(t) ↔ X 2(f). The real part of x(t) can be ...

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  Form, Signal, Trans, Fourier, Fourier trans form

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