EECS20N: Signals and Systems

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Week 8

• Frequencies
• Musical scale
• Phase
• Timbre
• Images
• Periodicity
• Fourier series
• Examples
• Aperiodicity
• Images
• Coefficients
• Discrete signals
• DFS
• Examples

Fourier series

We have seen that a periodic signal x:Time → Reals with period p ∈ Time is one where for all t ∈ Time

x(t) = x(t + p).

A remarkable result, due to Joseph Fourier, 1768-1830, is that such signal can (usually) be described as a constant term plus a sum of sinusoids,

x(t) = A₀ + ∑_{(k=1 to ∞)} A_k cos (kω₀t + φ _k )

Each term in the summation is a cosine with amplitude A_k and phase φ _k. The frequency ω ₀, which has units of radians per second, is called the fundamental frequency, and is related to the period by

ω ₀ = 2π /p.

In other words, a sinusoid with frequency ω ₀ has period p. The constant term A₀ is sometimes called the DC term, where "DC" stands for "direct current," a reference back to the origins of much of this theory in circuit analysis. The terms where k ≥ 2 are called harmonics.

Using the Fourier series expansion for synthesis of signals is problematic because of the infinite summation. However, for most practical signals, the coefficients A_k become very small (or even zero) for large k, so a finite summation can be used as an approximation. Even when an infinite summation is used, the expansion of a periodic waveform is not always exact. There are some technical mathematical conditions that f must satisfy for it to be exact. These conditions are beyond the scope of this course. Fortunately, these conditions are rarely an issue with practical, real-world signals.