Overview
This lecture explains how periodic signals can be represented using orthogonal basis functions, focusing on Fourier series both in trigonometric (sine/cosine) and complex exponential forms, and the computation of their coefficients.
Signal Space and Orthogonality
- Signals can be represented as vectors in "signal space" using orthogonal basis functions.
- Orthogonality for signals means the integral of their product over a period is zero.
- If we have a complete set of orthogonal basis signals, any signal in that space can be represented as their linear combination.
- Trigonometric signals (cosine and sine functions) are orthogonal over one period if their frequencies differ.
Fourier Series and Basis Functions
- The fundamental frequency for a periodic signal is (f_0 = 1/T_0), where (T_0) is the period.
- Cosine and sine functions for all harmonics ((m = 1, 2, ...)) and the DC component (constant function) are mutually orthogonal.
- All the harmonic cosines and sines, and the DC value, form a complete orthogonal basis for representing any periodic signal.
- Any periodic signal (g(t)) can be written as:
(g(t) = a_0 + \sum_{n=1}^{\infty} [a_n \cos(2\pi n f_0 t) + b_n \sin(2\pi n f_0 t)]), the classical Fourier series._
Calculation of Fourier Series Coefficients
- The coefficients are:
- (a_0 = \frac{1}{T_0} \int_0^{T_0} g(t) , dt)
- (a_n = \frac{2}{T_0} \int_0^{T_0} g(t) \cos(2\pi n f_0 t) , dt)
- (b_n = \frac{2}{T_0} \int_0^{T_0} g(t) \sin(2\pi n f_0 t) , dt)
- These coefficients tell us which harmonics (frequencies) are present in the signal and their strengths.
Frequency Domain Representation
- Any periodic signal is composed of its harmonics, each corresponding to a specific frequency component.
- Fourier series provides a way to convert a time-domain signal into its frequency-domain (spectrum) representation.
Complex Exponential Fourier Series
- Signals can also be represented using complex exponentials: (e^{j2\pi n f_0 t}), where (n) is any integer.
- These complex exponentials are mutually orthogonal for different (n).
- Any periodic signal (g(t)) can be written as:
(g(t) = \sum_{n=-\infty}^{\infty} c_n e^{j2\pi n f_0 t})
- The coefficients are:
- (c_n = \frac{1}{T_0} \int_0^{T_0} g(t) e^{-j2\pi n f_0 t} , dt)
- The choice of negative exponent in the coefficient calculation arises from the orthogonality properties of complex exponentials._
Key Terms & Definitions
- Signal Space — Mathematical space where signals are represented like vectors.
- Orthogonality — Two signals are orthogonal if the integral of their product over a period is zero.
- Fourier Series — Representation of a periodic signal as a sum of sines and cosines or complex exponentials.
- Harmonics — Integer multiples of the fundamental frequency in a periodic signal.
- Fourier Coefficient — The weight of each basis function (sine, cosine, or exponential) in the Fourier series.
- Frequency Domain — Representation of a signal in terms of its frequency components.
Action Items / Next Steps
- Review the relationship between trigonometric and complex exponential forms of the Fourier series.
- Practice calculating Fourier coefficients for given periodic signals.
- Await further discussion on frequency spectrum plotting and detailed examples.