傅里叶级数为何是正交级数？正交级数定义及通信应用问询

Great questions—these concepts are core to understanding how we analyze and transmit signals, and Couch’s Digital And Analog Communication does an excellent job grounding them in practical comms contexts. Let’s break this down step by step:

1. What does "Fourier series are a special type of orthogonal series" mean? #

First, let’s anchor this in the basics: a Fourier series is a way to decompose any periodic signal (that meets Dirichlet conditions) into a sum of sine, cosine, or complex exponential functions. Its "special orthogonal" status comes from two key traits:

• It uses a specific, universally useful orthogonal basis: The trigonometric functions (sin(nω₀t), cos(nω₀t)) or complex exponentials (e^(jnω₀t)) form an orthogonal set over any interval of length T=2π/ω₀ (the signal’s period). Orthogonal here means the inner product of any two distinct basis functions is zero:

  For example, ∫₀^T cos(mω₀t)cos(nω₀t) dt = 0 when m ≠ n, and ∫₀^T sin(mω₀t)cos(nω₀t) dt = 0 for all m,n.

• Its basis is complete: Unlike some orthogonal sets, this trigonometric basis is complete—meaning any valid periodic signal can be represented with arbitrary precision by adding enough terms of the series. No other orthogonal basis is as universally applicable to periodic signals in comms, which is why Fourier series stand out.

In short: Fourier series are a subset of orthogonal series that leverage a specific, complete orthogonal basis tailored to periodic signals.

2. Definition of orthogonal series & its uses in communication systems #

Definition of an orthogonal series #

An orthogonal series is an infinite series constructed from a set of pairwise orthogonal functions {φₙ(t)}. Mathematically, it takes the form:
f(t) = Σₙ=₀^∞ aₙφₙ(t)
Where:

• {φₙ(t)} is the orthogonal set: for any m ≠ n, the inner product <φₘ(t), φₙ(t)> = ∫φₘ(t)φₙ*(t) dt = 0 (the asterisk denotes complex conjugate)
• The coefficients aₙ are calculated as aₙ = <f(t), φₙ(t)> / <φₙ(t), φₙ(t)>, ensuring each coefficient captures the "weight" of that basis function in the signal.

Key uses in communication systems #

Orthogonal series (and Fourier series in particular) are everywhere in comms—here are the most critical applications:

• Signal analysis & spectral decomposition: We use Fourier series to break down periodic signals (like carrier waves, pulse trains) into their frequency components. This lets us analyze bandwidth requirements, detect interference, and design filters.
• Orthogonal modulation schemes: Technologies like QAM, PSK, and orthogonal FSK rely on orthogonal signals. Since orthogonal signals don’t interfere with each other (their inner product is zero), we can transmit multiple data streams simultaneously over the same channel, boosting spectral efficiency.
• Noise suppression & filtering: Orthogonal bases let us isolate specific frequency bands. For example, a low-pass filter can be implemented by zeroing out high-frequency coefficients in the Fourier series of a signal, removing noise without distorting the desired signal.
• Data compression: By retaining only the largest-magnitude coefficients in an orthogonal series (like Fourier or wavelet), we can compress audio, image, or video signals significantly. This is how MP3 or JPEG formats work—they discard negligible components that the human eye/ear won’t notice.

内容的提问来源于stack exchange，提问作者Kartikeya Gokhale