Fourier series Eigenfunction Approach In the figure below

Fourier series: Eigenfunction Approach In the figure below we have a simple exponential input that yields the following output: • Using this and the fact that H is linear, calculating y(t) for combinations of complex exponentials is also straightforward. Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 1

Fourier series: Eigenfunction Approach • The action of H on an input such as those in the two equations above is easy to explain: H independently scales each exponential component esnt by a different complex number H(sn) C. As such, if we can write a function f(t) as a combination of complex exponentials it allows us to: Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 2

Fourier series : Eigenfunction Approach • easily calculate the output of H given f(t) as an input (provided we know the eigenvalues H(s)) • interpret how H manipulates f(t) Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 3

Fourier series • An arbitrary T-periodic function f(t) can be written as a linear combination of harmonic complex sinusoids where and cn – called the Fourier coefficients. Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 4

Fourier series • Example #1: For the given function below, break it down into its ”simpler” parts and find its Fourier coefficients. The tricky part of the problem is finding a way to represent the above function in terms of its basis, . To do this, we will use our knowledge of Euler’s Relation to represent our cosine function in terms of the exponential. Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 5

Fourier series • Now from this form of our function and from Equation (2), by inspection we can see that our Fourier coefficients will be: Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 6

Fourier series • Example #2: For the given function below, break it down into its ”simpler” parts and find its Fourier coefficients. • As done in the previous example, we will again use Euler’s Relation to represent our sine function in terms of exponential functions. Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 7

Fourier series Now from this form of our function and from Equation (2), by inspection we can see that our Fourier coefficients will be: Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 8

Fourier series • Example #3: For the given function below, break it down into its ”simpler” parts and find its Fourier coefficients. • Once again we will use the same technique as was used in the previous two problems. The break down of our function yields Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 9

Fourier series And from this we can find our Fourier coefficients to be: Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 10

Fourier coefficients • The following general equation is used to compute the Fourier coefficients: • The sequence of complex numbers { n, n Z : cn} is just an alternate representation of the function f(t). Knowing the Fourier coefficients cn is the same as knowing f(t) explicitly and vice versa. • Given a periodic function, we can transform it into its Fourier series representation using Equation (3). Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 11

Fourier coefficients • Likewise, we can inverse transform a given sequence of complex numbers, cn, using Equation (2) to reconstruct the function f(t). • By looking at the Fourier series of a signal f (t), we can infer mathematical properties of f(t) such as smoothness, existence of certain symmetries, as well as the physically meaningful frequency content. Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 12

Trigonometric Fourier series • The trigonometric Fourier series representation of a periodic signal x(t) with fundamental period T 0 is given by where ak, and bk, are the Fourier coefficients given by Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 13

Trigonometric Fourier series • The coefficients ak, and bk, and the complex Fourier coefficients ck, are related by or Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 14

Trigonometric Fourier series • Even and odd signals: If a periodic signal x(t) is even, then bk = 0 and its Fourier series contains only cosine terms: If a periodic signal x(t) is odd, then ak = 0 and its Fourier series contains only sine terms: Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 15

Amplitude and Phase spectra of a periodic signal • Let the complex Fourier coefficients cn in Eq. (2) be expressed as • A plot of lcnl versus the angular frequency w is called the amplitude spectrum of the periodic signal x(t), and a plot of f, versus w is called the phase spectrum of x(t). • Since the index n assumes only integers, the amplitude and phase spectra are not continuous curves but appear only at the discrete frequencies kwo. They are therefore referred to as discrete frequency spectra or line spectra. Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 16

Power content of a periodic signal • The average power of a periodic signal x(t) over any period as • If x(t) is represented by the complex exponential Fourier series in Eq. (2), then it can be shown that • Equation (11) is called Parserval's identity (or Parseval's theorem) for the Fourier series. Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 17