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2 Ship encountering the superposition of 3 waves.

3 Fourier Series Representation of Periodic Signals Signals can be represented as linear combinations of basic signals with the following 2 properties. The set of basic signals can be used to construct a broad and useful class of signals. The response of an LTI system is a combination of the responses to these basic signals at the input.

4 Fourier Analysis Both properties are provided for by the complex exponential signals in the continuous and discrete time I.e. Signal of the form:-

5 The principle of superposition for linear systems. LTI System x(t) x[n] y(t) y[n]

6 Criteria for choosing a set of basic signals in terms of which to decompose the input

7 Choice for basic signals that led to the convolution integral and convolution sum for LTI Systems

8 Complex exponentials as a set of basic signals

9 Eigenfunction (e st, z n) & Eigenvalues (H(s), H(z)) LTI System x(t) x[n] y(t) y[n]

10 Showing complex exponential as eigenfunction of system

11 Showing complex exponential as eigenfunction of system

12 Example x(t) to be linear combination of 3 complex exponentials

13 Example x(t) is linear combination of 3 complex exponentials

14 Fourier Series Representation of Continuous- time Periodic Signals. A signal is periodic if x(t)=x(t+T), for all t. T is the fundamental period.

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17 Trigonometric forms of Fourier Series.

18 Trigonometric forms of Fourier Series.

19 Solving for Fourier Series Coefficients(Analysis)

20 Solving for Fourier Series Coefficients(Analysis)

21 Summarizing for Fourier Series Pair Representation Fourier Series Synthesis:- eqn 3.38 Fourier Series Analysis or F.S. Coefficients 0r Spectral Coefficents:- eqn 3.39

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23 Magnitude of Fourier Coefficients.

24 Phase of Fourier Coefficients.

25 Example 3.5 T/2 T1T1 -T/2 -T 1 This periodic signal x(t) repeats every T seconds. x(t)=1, for |t|<T 1, and x(t)=0, for T 1 <|t|< T/2 Fundamental period= T, Fundamental frequency    Choosing the period of integration to be between -T/2 and +T/2. Use eqn 3.39 to get at Fourier Series Coefficients.

26 Example 3.5 continued

27 Example 3.5 continued

28 Example 3.5 Continued In this example the coefficients are real values. Generally the coefficients are complex. In this case we can represent a single plot of magnitude of coefficient against k. Generally we will have the magnitude plot and the phase plot of the coefficients.

29 Example 3.5 Continued

30 Plot of the Coefficients with T 1 Fixed and T varied. a) T=4T 1 b) T=8T 1 c) T=16T 1