ES 240: Scientific and Engineering Computation. Introduction to Fourier Series Trigonometric Fourier Series Outline Introduction Visualization Theoretical Concepts Qualitative Analysis Example Class Exercise ES 240: Scientific and Engineering Computation. Introduction to Fourier Series 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 ES 240: Scientific and Engineering Computation. Introduction to Fourier Series Periodic Function What is a periodic Function? A function which remains unchanged when time-shifted by one period f(t) = f(t + To) for all values of t To What is To To
ES 240: Scientific and Engineering Computation. Introduction to Fourier Series Properties of a periodic function 1 A periodic function must be everlasting From to Why? Periodic or Aperiodic? ES 240: Scientific and Engineering Computation. Introduction to Fourier Series 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 F0 = 1/T0 and all its harmonics ES 240: Scientific and Engineering Computation. Introduction to Fourier Series Visualization Can you represent this simple function using sinusoids? Single sinusoid representation ES 240: Scientific and Engineering Computation. Introduction to Fourier Series Visualization New amplitude amplitude amplitude 2nd Harmonic Fundamental th 4 Harmonic
frequency cos(5 aaa351cos( 3 cos( t))t ) 00t0 To obtain the exact signal, an infinite number of sinusoids are required ES 240: Scientific and Engineering Computation. Introduction to Fourier Series Theoretical Concepts ( 6 ) n 1 n 1 f (t ) a0 an cos(n0t ) bn sin(n0t )
0 2 T0 Period 2 an T0 t1 T0 2 bn T0 t1 T0 Cosine terms f (t ) cos(n t )dt....n 1,2,3,... 0 t1 Sine terms f (t ) sin(n t )dt....n 1,2,3,... 0
t1 ES 240: Scientific and Engineering Computation. Introduction to Fourier Series Theoretical Concepts ( 6 ) f (t ) c0 cn cos(n0t n ) n 1 c0 a0 2 cn an bn 2 bn n tan an 1
ES 240: Scientific and Engineering Computation. Introduction to Fourier Series DC Offset What is the difference between these two functions? Average Value = 0 A -2 -1 0 1 2 Average Value ? -A A -2 -1
0 1 2 ES 240: Scientific and Engineering Computation. Introduction to Fourier Series DC Offset If the function has a DC value: 1 f (t ) a0 an cos(n0t ) bn sin( n0t ) 2 n 1 n 1 1 a0 T0 t1 T0 f (t )dt t1
ES 240: Scientific and Engineering Computation. Introduction to Fourier Series Qualitative Analysis Is it possible to have an idea of what your solution should be before actually computing it? For Sure ES 240: Scientific and Engineering Computation. Introduction to Fourier Series Properties DC Value If the function has no DC value, then a0 = ? -A DC? -1 1
2 A A -2 -1 0 DC? 1 2 ES 240: Scientific and Engineering Computation. Introduction to Fourier Series Properties Symmetry Even function f(-t) = f(t) A
Odd function 0 /2 3/2 -A f(-t) = -f(t) A 0 A /2 3/2 ES 240: Scientific and Engineering Computation.
Introduction to Fourier Series Properties Symmetry Note that the integral over a period of an odd function is? If f(t) is even: Even 2 bn T0 2 an T0 X Odd = Odd t1 T0 f (t ) sin(n0t )dt....n 1,2,3,... t1
Even X Even = Even t1 T0 f (t ) cos(n0t )dt....n 1,2,3,... t1 ES 240: Scientific and Engineering Computation. Introduction to Fourier Series Properties Symmetry Note that the integral over a period of an odd function is zero. If f(t) is odd: Odd 2 an T0
2 bn T0 X Even = Odd t1 T0 f (t ) cos(n0t )dt....n 1,2,3,... t1 Odd X Odd = Even t1 T0 f (t ) sin(n0t )dt....n 1,2,3,... t1
ES 240: Scientific and Engineering Computation. Introduction to Fourier Series 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 ES 240: Scientific and Engineering Computation. Introduction to Fourier Series Properties Symmetry If the function is half-wave symmetric, then only odd harmonics exist Half wave symmetry: f(t-T0/2) = -f(t) -A
-1 1 A 2 ES 240: Scientific and Engineering Computation. Introduction to Fourier Series 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: Which function has discontinuities? -A -1 1 2
A A -2 -1 0 1 2 Which is closer to a sinusoid? ES 240: Scientific and Engineering Computation. Introduction to Fourier Series Example Without any calculations, predict the general form of the Fourier series of: -A -1
1 2 A DC? Symmetry? No, a0 = 0; Even, bn = 0; Half wave symmetry? Discontinuities? Yes, only odd harmonics No, falls of as 1/n2 Prediction an 1/n2 for n = 1, 3, 5, ; ES 240: Scientific and Engineering Computation.
Introduction to Fourier Series Example Now perform the calculation T0 2; 0 2 2 an 2 T0 t1 T0 f (t ) cos(n0t )dt t1 1 an 2 2 At cos(nt )dt 0 an 8A 2 2 n
4A 2 2 n ...n 1,3,5... 4 T0 T0 / 2 f (t ) cos(n0t )dt 0 1 cos(n ) zero for n even ES 240: Scientific and Engineering Computation. Introduction to Fourier Series Example Now compare your calculated answer with your predicted form
DC? Symmetry? No, a0 = 0; Even, bn = 0; Half wave symmetry? Discontinuities? Yes, only odd harmonics No, falls of as 1/n2 ES 240: Scientific and Engineering Computation. Introduction to Fourier Series 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 solution A -2 -1 0 1 2
