Time-Frequency Analysis Basics of Fourier Series Philippe B. aval KSU Fall 015 Philippe B. aval (KSU) Fourier Series Fall 015 1 / 0
Introduction We first review how to derive the Fourier series of a function. We then state some important results about Fourier series. A Fourier Series of a function f is a special expansion of the form f (x) = A 0 + ( A n cos nπx + B n sin nπx ) Finding the Fourier series for a given function f (x) (if it exists) amounts to finding the coeffi cients A n for n = 0, 1,,... and B n for n = 1,, 3,... Philippe B. aval (KSU) Fourier Series Fall 015 / 0
Euler Formulas for the Coeffi cients Definition The Fourier series of a function f (x) on the interval [, ] where > 0 is given by f (x) = A 0 + ( A n cos nπx + B n sin nπx ) (1) The coeffi cients which appear in the Fourier series were known to Euler before Fourier, hence they bear his name. They are given by the following formulas. Philippe B. aval (KSU) Fourier Series Fall 015 3 / 0
Euler Formulas for the Coeffi cients To find the coeffi cients, the following formulas play an important role: 1 nπx cos dx = 0 for n = 1,,... 3 4 5 nπx sin dx = 0 for n = 1,,... nπx mπx sin cos sin nπx cos nπx mπx sin dx = mπx cos dx = dx = 0 for every m, n { = 0 if m n = if m = n { = 0 if m n = if m = n 0 Philippe B. aval (KSU) Fourier Series Fall 015 4 / 0
Euler Formulas for the Coeffi cients Theorem The coeffi cients in equation 1 are given by A n = 1 B n = 1 A 0 = 1 f (x) dx () f (x) cos nπx dx for n = 1,,... (3) f (x) sin nπx dx for n = 1,,... (4) Philippe B. aval (KSU) Fourier Series Fall 015 5 / 0
Euler Formulas for the Coeffi cients Definition For a positive integer N, we denote the N th partial sum of the Fourier series of f by S N (x). So, we have S N (x) = A 0 + N ( A n cos nπx We now illustrate what we did with some examples. + B n sin nπx ) Philippe B. aval (KSU) Fourier Series Fall 015 6 / 0
Examples Find the Fourier series of f (x) = sin x on [ π, π]. 10 5 5 10 x y 1 Figure: Graph of sin x and S (x) y 1 10 5 5 10 Figure: Graph of sin x and S Philippe B. aval (KSU) Fourier Series Fall 015 7 / 0 x (x)
Examples Find the Fourier series of f (x) = sin x on [ π, π]. We should have found A 0 = π A n = 4 π (4n 1) B n = 0 sin x = π + 4 π (4n cos nx 1) y 1 10 5 5 10 Figure: Graph of sin x and S (x) y 1 10 5 5 10 x x Figure: Graph of sin x and S Philippe B. aval (KSU) Fourier Series Fall 015 7 / 0 (x)
Examples We now look at a π-periodic function with discontinuities and derive its Fourier series using the formulas of this section (assuming it is legitimate). This function is called the sawtooth function. It is defined by g (x) = { 1 (π x) if 0 < x π g (x + π) otherwise Find the Fourier series for this function. Plot this function as well as S 1 (x), S 7 (x), S 0 (x) where S N (x) is the N th partial sum of its Fourier series. Philippe B. aval (KSU) Fourier Series Fall 015 8 / 0
Examples y 4 We should have found: A 0 = 0 A n = 0 5 5 10 4 Graph of the sawtooth function (black) and S 1 (x) (red) x B n = 1 n g (x) = sin nx n 5 5 10 Graph of the sawtooth function (black) and S 7 (x) (red) Philippe B. aval (KSU) Fourier Series Fall 015 9 / 0 y 4 4 x
Examples y 4 y 4 5 5 10 x 5 5 10 x 4 Graph of the sawtooth function (black) and S 0 (x) (red) 4 Graph of the sawtooth function (black) and S 100 (x) (red) Philippe B. aval (KSU) Fourier Series Fall 015 10 / 0
Some Remarks Several important facts are worth noticing here. 1 The Fourier series seems to agree with the function, except at the points of discontinuity. At the points of discontinuity, the series converges to 0, which is the average value of the function from the left and from the right. 3 Near the points of discontinuity, the Fourier series overshoots its limiting values. This is a well known phenomenon, known as Gibbs phenomenon. You can see an online simulation of the Gibbs phenomenon Philippe B. aval (KSU) Fourier Series Fall 015 11 / 0
Piecewise Continuous and Piecewise Smooth Functions Definition We will denote f (c ) = lim f (x) and f (c+) = lim f (x) x c x c + Remembering that a function f is continuous at c if and only if f (x) = f (c), we see that a function f is continuous at c if and only if lim x c Definition (Piecewise Continuous) f (c ) = f (c+) = f (c) A function f is said to be piecewise continuous on the interval [a, b] if the following are satisfied: 1 f (a+) and f (b ) exist. f is defined and continuous on (a, b) except possibly at a finite number of points in (a, b) where the left and right limit at these points exist. Such points are called jump discontinuities. Philippe B. aval (KSU) Fourier Series Fall 015 1 / 0
Piecewise Continuous and Piecewise Smooth Functions Definition (Piecewise Smooth) A function f, defined on [a, b] is said to be piecewise smooth if f and f are piecewise continuous on [a, b]. Definition The average of f at c is defined to be f (c ) + f (c+) Clearly, if f is continuous at c, then its average at c is f (c). Philippe B. aval (KSU) Fourier Series Fall 015 13 / 0
Convergence Theorem for Fourier Series Theorem If f is a piecewise smooth function on [, ], then, x [, ] f (x ) + f (x+) = A 0 + ( A n cos nπx + B n sin nπx ) (5) where the coeffi cients are given by equations, 3, and 4. In particular, if f is piecewise smooth and continuous at x, then f (x) = A 0 + ( A n cos nπx + B n sin nπx ) (6) Thus, at points where f is continuous, the Fourier series converges to the function. At points of discontinuity, the series converges to the average of the function at these points. This was the case in the example with the sawtooth function. Philippe B. aval (KSU) Fourier Series Fall 015 14 / 0
Fourier Series of Even and Odd Functions We finish this section by noticing that in the special cases that f is either even or odd, the series simplifies greatly. 1 If f is even, then nπx f (x) sin is odd so that B n = 0 and the series is simply a cosine series. If f is odd, then nπx f (x) cos is odd and A n = 0 and the series is simply a sine series. We summarize this in a theorem. Philippe B. aval (KSU) Fourier Series Fall 015 15 / 0
Fourier Series of Even and Odd Functions Theorem Suppose that on [, ] f has the Fourier series representation Then: f (x) = A 0 + [ A n cos nπx + B n sin nπx ] 1 If f is even then B n = 0 for all n and in this case f (x) = A 0 + A n cos nπx If f is odd then A n = 0 for all n and in this case f (x) = B n sin nπx Philippe B. aval (KSU) Fourier Series Fall 015 16 / 0
Fourier Series in Complex Form Recall Euler s identity e ±ix = cos x ± i sin x (7) The Fourier series of a function f on [, ] can be written f (x) = n= C n e i nπx (8) where C n = 1 nπx f (x) e i dx (9) Philippe B. aval (KSU) Fourier Series Fall 015 17 / 0
Some Applications One of the main uses of Fourier series is in solving some of the differential equations from mathematical physics such as the wave equation or the heat equation. Fourier developed his theory by working on the heat equation. Fourier series also have applications in music synthesis and image processing (signal processing). When we represent a signal f (t) by its [ Fourier series f (t) = A 0 + A n cos nπt + B n sin nπt ], we are finding the contribution of each frequency nπ to the signal. The value of the corresponding coeffi cients give us that contribution. The n th term of the partial sum of the Fourier series, A n cos nπt + B n sin nπt, is called the n th harmonic of f. Its amplitude is given by A n + B n. Philippe B. aval (KSU) Fourier Series Fall 015 18 / 0
Some Applications Conversely, we can create a signal by using the Fourier series [ A 0 + A n cos nπt + B n sin nπt ] for a given value of and playing with the value of the coeffi cients. Audio signals describe air pressure variations captured by our ears and perceived as sounds. We will focus here on periodic audio signals also known as tones. Such signals can be represented by Fourier series. A pure tone can be written as x (t) = a cos (ωt + φ) where a > 0 is the amplitude, ω > 0 is the frequency in radians/seconds and φ is the phase angle. An alternative way to represent the frequency is in Hertz. The frequency f in Hertz is given by f = ω π. The pitch of a pure tone is logarithmically related to the frequency. Philippe B. aval (KSU) Fourier Series Fall 015 19 / 0
Some Applications An octave is a frequency range between f and f for a given frequency f in Hertz. Tones separated by an octave are perceived by our ears to be very similar. In western music, an octave is divided into 1 notes equally spaced on the logarithmic scale. The ordering of notes in the octave beginning at the frequency 0 Hz are shown below Note A A# B C C# D D# E F Frequency (Hz) 0 33 47 6 77 94 311 330 349 A more complicated tone can be represented by a Fourier series of the form x (t) = a 1 cos (ωt + φ 1 ) + a cos (ωt + φ ) +... Philippe B. aval (KSU) Fourier Series Fall 015 0 / 0