Lecture 11: Introduction to Fourier Series Sections 2.2.3, 2.3

Trigonometric Fourier Series Outline ▫Introduction ▫Visualization ▫Theoretical Concepts ▫Qualitative Analysis ▫Example ▫Class Exercise

Introduction What is Fourier Series? ▫Representation of a periodic function with a weighted, infinite sum of sinusoids. Why Fourier Series? ▫Any arbitrary periodic signal, can be approximated by using some of the computed weights ▫These weights are generally easier to manipulate and analyze than the original signal

Periodic Function What is a periodic Function? ▫A function which remains unchanged when time-shifted by one period  f(t) = f(t + T o ) for all values of t What is T o ToTo ToTo

Properties of a periodic function 1 A periodic function must be everlasting ▫From –∞ to ∞ Why? Periodic or Aperiodic?

Properties of a periodic function You only need one period of the signal to generate the entire signal ▫Why? A periodic signal cam be expressed as a sum of sinusoids of frequency F 0 = 1/T 0 and all its harmonics

Visualization Can you represent this simple function using sinusoids? Single sinusoid representation

Visualization To obtain the exact signal, an infinite number of sinusoids are requiredamplitude Fundamental frequency New amplitude 2 nd Harmonic amplitude 4 th Harmonic

Theoretical Concepts (6) (6) Period Cosine terms Sine terms

Theoretical Concepts (6) (6)

DC Offset What is the difference between these two functions?A A A Average Value = 0 Average Value ?

DC Offset If the function has a DC value:

Qualitative Analysis Is it possible to have an idea of what your solution should be before actually computing it? For Sure

Properties – DC Value If the function has no DC value, then a 0 = ?12 -A A DC?A DC?

Properties – Symmetry A A 0 π/2 π 3π/2 f(-t) = -f(t) Even function Odd function 0 -A A π/2 π 3π/2 f(-t) = f(t)

Properties – Symmetry Note that the integral over a period of an odd function is? If f(t) is even: Even OddX=Odd Even EvenX=Even

Properties – Symmetry Note that the integral over a period of an odd function is zero. If f(t) is odd: Odd EvenX=Odd Odd OddX=Even

Properties – Symmetry If the function has: ▫even symmetry: only the cosine and associated coefficients exist ▫odd symmetry: only the sine and associated coefficients exist ▫even and odd: both terms exist

Properties – Symmetry If the function is half-wave symmetric, then only odd harmonics exist Half wave symmetry: f(t-T 0 /2) = -f(t) 12 -A A

Properties – Discontinuities If the function has ▫Discontinuities: the coefficients will be proportional to 1/n ▫No discontinuities: the coefficients will be proportional to 1/n 2 Rationale:12 -A A A Which is closer to a sinusoid? Which function has discontinuities?

Example Without any calculations, predict the general form of the Fourier series of:12 -A A DC? No, a 0 = 0; Symmetry? Even, b n = 0; Half wave symmetry? Yes, only odd harmonics Discontinuities? No, falls of as 1/n 2 Prediction a n  1/n 2 for n = 1, 3, 5, …;

Example Now perform the calculation zero for n even

Example Now compare your calculated answer with your predicted form DC? No, a 0 = 0; Symmetry? Even, b n = 0; Half wave symmetry? Yes, only odd harmonics Discontinuities? No, falls of as 1/n 2

Class exercise Discuss the general form of the solution of the function below and write it down Compute the Fourier series representation of the function With your partners, compare your calculations with your predictions and comment on your solutionA012 -2

Spectral Lines

Gives the frequency composition of the function ▫ Amplitude, phase of sinusoidal components Could provide information not found in time signal ▫ E.g. Pitch, noise components May help distinguish between signals ▫ E.g speech/speaker recognition

Spectral Lines Example QUESTIONS -- DC Yes____ a o = ? No_____ a o = 0 Symmetry Even____ a n = ? b n = 0 Odd____ a n = 0 b n = ? Nether even nor odd ____ a n = ? b n = ? Halfwave symmetry Yes_____ only odd harmonics No______ all harmonics Discontinuities Yes_____ proportional to1/n No______ proportional to1/n 2 Note ? means find that variable. Comment on the general form of the Fourier Series coefficients [a n and/or b n.] X X X X

Spectral Lines Example