EEEN 464 - ANALOGUE COMMUNICATION

FOURIER SERIES STUDY GUIDE/REVISION

Introduction to Fourier Series

Fourier Series is a mathematical tool that represents a periodic function as a sum of sine and cosine functions with different frequencies and amplitudes.

In communication engineering, Fourier series is fundamental because:

[Insert animation showing square wave approximation with increasing Fourier terms]

Mathematical Foundation

Periodic Functions

A function f(t) is periodic with period T if:

f(t + T) = f(t) for all t

Fourier Series Representation

The Fourier series representation of a periodic function f(t) with period T is:

f(t) = a₀ + ∑[aₙcos(nω₀t) + bₙsin(nω₀t)] (from n=1 to ∞)

where ω₀ = 2π/T is the fundamental frequency.

Fourier Coefficients

The coefficients are calculated as:

a₀ = (1/T)∫f(t)dt over one period
aₙ = (2/T)∫f(t)cos(nω₀t)dt over one period
bₙ = (2/T)∫f(t)sin(nω₀t)dt over one period

Dirichlet Conditions: For a Fourier series to exist, the function must:

  1. Be single-valued
  2. Have a finite number of maxima/minima in any finite interval
  3. Have a finite number of discontinuities in any finite interval
  4. Be absolutely integrable over one period

Different Forms of Fourier Series

1. Trigonometric Form

f(t) = a₀ + ∑[aₙcos(nω₀t) + bₙsin(nω₀t)]

2. Compact Trigonometric Form

f(t) = C₀ + ∑Cₙcos(nω₀t + θₙ)

where:

C₀ = a₀, Cₙ = √(aₙ² + bₙ²), θₙ = -tan⁻¹(bₙ/aₙ)

3. Exponential (Complex) Form

f(t) = ∑cₙe^{jnω₀t} (from n=-∞ to ∞)

where:

cₙ = (1/T)∫f(t)e^{-jnω₀t}dt over one period
Form Advantages Disadvantages
Trigonometric Easy to visualize physical meaning Two coefficients per harmonic
Compact Trigonometric Single coefficient per harmonic Phase angles complicate some calculations
Exponential Most compact form, easiest for calculations Less intuitive physical interpretation

Properties of Fourier Series

1. Linearity

If f(t) and g(t) have Fourier coefficients {aₙ, bₙ} and {cₙ, dₙ} respectively, then:

αf(t) + βg(t) has coefficients {αaₙ + βcₙ, αbₙ + βdₙ}

2. Time Shifting

If f(t) ↔ {aₙ, bₙ}, then f(t - t₀) has coefficients:

aₙ' = aₙcos(nω₀t₀) + bₙsin(nω₀t₀)
bₙ' = bₙcos(nω₀t₀) - aₙsin(nω₀t₀)

3. Time Reversal

f(-t) ↔ {aₙ, -bₙ}

4. Time Scaling

f(kt) has period T/k and frequency kω₀

5. Differentiation

df/dt ↔ {nω₀bₙ, -nω₀aₙ}

6. Integration

∫f(t)dt ↔ {-bₙ/(nω₀), aₙ/(nω₀)} (for n ≠ 0)

7. Parseval's Theorem

(1/T)∫|f(t)|²dt = a₀² + ½∑(aₙ² + bₙ²) = ∑|cₙ|²

Symmetry Considerations

Symmetry Fourier Coefficients Example
Even Function
f(-t) = f(t)		 bₙ = 0 (sine terms vanish)
Only cosine terms remain		 cos(t), t²
Odd Function
f(-t) = -f(t)		 aₙ = 0 (cosine terms vanish)
Only sine terms remain		 sin(t), t³
Half-wave Symmetry
f(t ± T/2) = -f(t)		 Only odd harmonics present
(n = 1, 3, 5, ...)		 Square wave, triangle wave

Fourier Series of Common Signals

1. Square Wave

f(t) = (4/π)[sin(ω₀t) + (1/3)sin(3ω₀t) + (1/5)sin(5ω₀t) + ...]

2. Sawtooth Wave

f(t) = (2/π)[sin(ω₀t) - (1/2)sin(2ω₀t) + (1/3)sin(3ω₀t) - ...]

3. Triangle Wave

f(t) = (8/π²)[cos(ω₀t) + (1/9)cos(3ω₀t) + (1/25)cos(5ω₀t) + ...]

4. Rectangular Pulse Train

f(t) = (τ/T) + (2τ/T)∑[sinc(nπτ/T)cos(nω₀t)]

where τ is pulse width and sinc(x) = sin(x)/x

[Insert comparison plots of these waveforms with their Fourier approximations]

Applications in Communication Engineering

1. Signal Analysis

Decomposing signals into frequency components helps analyze:

2. Modulation Techniques

Fourier series is fundamental to understanding:

3. Filter Design

Helps analyze how filters affect different frequency components

4. Signal Transmission

Understanding how periodic signals behave in transmission lines

5. Spectral Analysis

Basis for Fourier Transform used in spectrum analyzers

Practice Problems

  1. Find the Fourier series representation of the periodic function: f(t) = { -1 for -π < t < 0; 1 for 0 < t < π } with period 2π
  2. Calculate the Fourier coefficients for a half-wave rectified sine wave.
  3. Using the Fourier series of a square wave, estimate how many terms are needed to achieve less than 5% error at the discontinuities.
  4. Prove Parseval's theorem for the exponential form of Fourier series.
  5. A periodic signal has only odd cosine terms in its Fourier series. What can you deduce about its symmetry?
  6. Derive the relationship between the trigonometric and exponential Fourier coefficients.
  7. For a rectangular pulse train with duty cycle 25%, calculate the amplitude of the 3rd harmonic relative to the fundamental.
  8. Explain the Gibbs phenomenon and its implications for signal processing.
  9. Show how differentiation affects the Fourier coefficients of a triangular wave.
  10. Calculate the power in the first three harmonics of a square wave with amplitude A.

Additional Resources