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ecen314-lec2

ecen314-lec2 - Lecture 2 Linear Transformations on the...

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Lecture 2: Linear Transformations on the Independent Variableof Signals Tie Liu August 30, 2011 1 Transformation of signals in CT The are many ways of transforming a CT signal into another. For instance, we can scale it, shift it in time, differentiate it, or perform a combination of these actions. Later in this course, we will introduce the idea of transforming a signal as a system. To familiarize you with manipulating signals, well examine a particular type of transformation in this subsection: transformation on the independent variable of signals. More formally, let us for now restrict ourselves to transformations of the form: x ( t ) -→ y ( t ) = x ( f ( t )) where x ( t ) is the starting signal given to us, y ( t ) is the signal we end up with after the transformation, and f ( t ) is a function of t . The arrow “ ” denotes the action and direction of transformation. The function f ( t ) can be any well-defined function, of course, but for the study of ECEN 314, we will look at the class of linear functions f ( t ) = at + b where a and b are arbitrary real constants. The resulting transformation of x ( t ) into y ( t ) is hence called “linear transformations on the independent variable.” All such transformations can be decomposed into just three fundamental types of signal transformations on the independent variable: time shift, time scaling, and time reversal. They involve a change of the variable t into something else: Time shift: f ( t ) = t - t 0 for some t 0 R . Time scaling: f ( t ) = at for some a R + . Time reversal (or flip): f ( t ) = - t .

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ecen314-lec2 - Lecture 2 Linear Transformations on the...

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