Fourier Series and Integrals

Unit V: Fourier Series and Integrals

5.1 Introduction to Fourier Series

Fourier series decompose periodic functions into sums of sine and cosine functions. This powerful mathematical tool is used extensively in signal processing, heat conduction, vibration analysis, and electrical engineering.

The key idea: any periodic function (with certain conditions) can be represented as an infinite sum of harmonically related sinusoids.

5.2 Periodic Functions

A function f(x) is periodic with period T if f(x + T) = f(x) for all x. The fundamental frequency is ω = 2π/T.

5.3 Trigonometric Fourier Series

For a periodic function f(x) with period 2L, the Fourier series is:

f(x) = a₀/2 + ∑_n=1^∞[a_ncos(nπx/L) + b_nsin(nπx/L)]

Fourier Coefficients:

a₀ = (1/L)∫_-L^Lf(x) dx
a_n = (1/L)∫_-L^Lf(x)cos(nπx/L) dx
b_n = (1/L)∫_-L^Lf(x)sin(nπx/L) dx

5.4 Dirichlet Conditions

A function f(x) has a convergent Fourier series if:

1. f(x) is periodic
2. f(x) has a finite number of discontinuities in one period
3. f(x) has a finite number of maxima and minima in one period
4. ∫|f(x)|dx over one period is finite

At points of discontinuity, the series converges to the average of left and right limits.

5.5 Fourier Sine and Cosine Series

Even functions (f(-x) = f(x)) have only cosine terms (b_n = 0):

f(x) = a₀/2 + ∑_n=1^∞a_ncos(nπx/L)

Odd functions (f(-x) = -f(x)) have only sine terms (a_n = 0):

f(x) = ∑_n=1^∞b_nsin(nπx/L)

Half-Range Expansions: A function defined only on [0, L] can be extended as even (cosine series) or odd (sine series) to get a Fourier representation.

5.6 Complex Form of Fourier Series

Using Euler's formula e^iθ = cosθ + i sinθ, the Fourier series can be written compactly:

f(x) = ∑_n=-∞^∞c_ne^inπx/L

where c_n = (1/2L)∫_-L^Lf(x)e^-inπx/L dx

The complex coefficients relate to the real coefficients: c_n = (a_n - ib_n)/2 for n > 0.

5.7 Parseval's Identity

This relates the energy of a signal to its Fourier coefficients:

(1/L)∫_-L^L|f(x)|² dx = a₀²/2 + ∑_n=1^∞(a_n² + b_n²)

5.8 Fourier Integrals

For non-periodic functions, the Fourier series generalizes to the Fourier integral:

f(x) = (1/π)∫₀^∞[A(ω)cos(ωx) + B(ω)sin(ωx)] dω

where:

A(ω) = ∫_-∞^∞f(t)cos(ωt) dt
B(ω) = ∫_-∞^∞f(t)sin(ωt) dt

5.9 Fourier Transforms

The Fourier transform and its inverse provide a frequency-domain representation:

F(ω) = ∫_-∞^∞f(t)e^-iωt dt (Forward Transform)
f(t) = (1/2π)∫_-∞^∞F(ω)e^iωt dω (Inverse Transform)

Properties of Fourier Transform:

• Linearity: F{af + bg} = aF{f} + bF{g}
• Time Shifting: F{f(t-a)} = e^-iωaF(ω)
• Frequency Shifting: F{e^iatf(t)} = F(ω-a)
• Scaling: F{f(at)} = (1/|a|)F(ω/a)
• Convolution: F{f*g} = F(ω)G(ω)