Lecture 7 Topics More on Linearity Eigenfunctions of Linear Systems Fourier Transforms –As the limit of Fourier Series –Spectra –Convergence of Fourier Transforms –Fourier Transform: Synthesis equation Analysis equation

ax(t) ay(t) a 1 x 1 (t) + a 2 x 2 (t) a 1 y 1 (t) + a 2 y 2 (t) Superposition LINEARITY – another perspective

Consider a circuit with internal source V: R1R1 R2R2 R3R V-+V- + _ For this is not linear, as

+ + If i.e. if there is an Initial charge on C and if x(t) yields output y(t) (with an initial charge on C) Then ax(t) does not yield ay(t). Such systems are said to be linear for “zero initial energy” and are decomposable into the forced response due to the input x(t) and the transient response due to y(0) (or the response due to an internal source). These systems are said to be linear “separably”. Ref: Carlson, Signal & Linear System Analysis

Eigenfunctions: If then where x(t) y(t)

Eigenfunctions: If then where x(t) y(t) SUPERPOSITION If Then the such that are the eigenfunctions of h(t)

For L.T.I. systems where the s k are complex constants. Then: with

For L.T.I. systems where the s k are complex constants. Then: with a complex constant dependent on s k. Ref: Jackson, Signals and Systems, Chapter 4

We defined: Note that: is the Laplace Transform

We defined: Note that: is the Laplace Transform is the Fourier Transform