9Fourier Transform Properties - MIT OpenCourseWare

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9Fourier TransformPropertiesThe fourier Transform is a major cornerstone in the analysis and representa-tion of signals and linear, time-invariant systems, and its elegance and impor-tance cannot be overemphasized. Much of its usefulness stems directly fromthe Properties of the fourier Transform , which we discuss for the continuous-time case in this lecture. Many of the fourier Transform Properties might atfirst appear to be simple (or perhaps not so simple) mathematical manipula-tions of the fourier Transform analysis and synthesis equations. However, inthis and later lectures, as we discuss issues such as filtering, modulation, andsampling, it should become increasingly clear that these Properties all haveimportant interpretations and meaning in the context of signals and first property that we introduce in this lecture is the symmetry prop-erty, specifically the fact that for time functions that are real-valued, the Four-ier Transform is conjugate symmetric, , X( -o) = X*(w).

Fourier Transform Properties The Fourier transform is a major cornerstone in the analysis and representa-tion of signals and linear, time-invariant systems, and its elegance and impor-tance cannot be overemphasized. Much of its usefulness stems directly from the properties of the Fourier transform, which we discuss for the continuous-

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