Summary of Spectral Analysis and Fourier Transforms

Spectral Analysis and Fourier Transforms: A Student's Guide

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Introduction

Discrete-time Fourier analysis studies how discrete-time signals can be represented, understood and manipulated in the frequency domain. This material focuses on spectral representations of periodic and nonperiodic discrete-time sequences, convergence considerations, and the discrete-time Fourier series (DTFS). We build intuition with examples and point out practical implications for signal processing.

Definition: A discrete-time signal (sequence) is a function $x[n]$ defined for integer time index $n\in\mathbb{Z}$.

1. Basic concepts

Time-domain vs frequency-domain

  • Time-domain describes how a sequence varies with index $n$.
  • Frequency-domain describes how the sequence energy or components are distributed across frequency.

Definition: A spectral density is a frequency-domain description that assigns a complex amplitude or power to each frequency.

Convergence considerations for infinite sums

  • Many frequency representations are defined via infinite sums. Such sums converge only under certain conditions on the sequence.
  • A common sufficient condition is finite energy: $\sum_{n=-\infty}^{\infty} |x[n]|^2 < \infty$.
  • Periodic sequences typically do not meet the finite-energy condition because their energy repeats indefinitely.

Definition: A sequence $x[n]$ has finite energy if $\sum_{n=-\infty}^{\infty} |x[n]|^2 < \infty$.

Continuity of spectral density

  • For nonperiodic sequences that yield a continuous spectral density, the frequency-domain representation is a continuous function of frequency. While mathematically convenient, continuous spectra are often impractical for discrete digital systems because they require infinite precision to represent.
💡 Věděli jste?Fun fact: Continuous spectral functions arise naturally from nonperiodic discrete-time signals, which means digital systems often rely on sampled or discrete-frequency representations to work efficiently.

2. Periodic discrete-time signals

Definition and basic property

  • A sequence $x[n]$ is periodic with period $N$ if for every integer $k$ and all $n$: $$x[n+kN] = x[n].$$
  • The smallest positive $N$ satisfying this is the fundamental period.

Definition: A periodic discrete-time signal repeats its values every $N$ samples: $x[n+N]=x[n]$.

Example

  • If $x[n]$ has period $N=10$, then values at indices $n=0,1,\dots,9$ determine the sequence for all integers $n$ because $x[n+10]=x[n]$.

Implications

  • Periodic sequences have discrete spectra composed of harmonics at frequencies that are integer multiples of the fundamental frequency $\Omega_0 = \tfrac{2\pi}{N}$.
  • The spectrum consists of a finite set of coefficients (one per harmonic within one period of the frequency index).
💡 Věděli jste?Did you know that sampling a continuous-time periodic signal does not always produce a periodic discrete-time sequence; periodicity after sampling depends on the ratio between the continuous period and the sampling interval?

3. Discrete-Time Fourier Series (DTFS)

Purpose

  • DTFS expresses a periodic discrete-time sequence as a finite sum of complex exponentials (sinusoids) at discrete harmonic frequencies.

Definition: The Discrete-Time Fourier Series represents a length-$N$ periodic sequence $x[n]$ as a sum of $N$ complex exponentials with coefficients $X[k]$.

Synthesis (series expansion)

  • A periodic sequence with period $N$ can be written as: $$x[n] = \sum_{k=0}^{N-1} X[k],e^{j,\frac{2\pi}{N}kn}.$$ Here $X[k]$ is the $k$-th complex spectral coefficient and the fundamental frequency is $\Omega_0 = \tfrac{2\pi}{N}$.

Interpretation of terms

  • The $k=0$ term is the DC component (constant term).
  • Positive $k$ correspond to harmonics at frequencies $\frac{2\pi}{N}k$.
  • Each term $X[k]e^{j\frac{2\pi}{N}kn}$ is a sinusoidal component scaled and phase-shifted by $X[k]$.

Analysis (coefficient computation)

  • The DTFS coefficients for a period-$N$ sequence are computed by projecting onto the complex exponent
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Discrete-Time Fourier Analysis

Klíčové pojmy: A discrete-time sequence is periodic if $x[n+N]=x[n]$ for all integer $n$, Finite energy condition: $\sum_{n=-\infty}^{\infty}|x[n]|^2<\infty$ ensures convergence of many frequency sums, Periodic sequences produce discrete harmonics at $\Omega_k=\tfrac{2\pi}{N}k$, DTFS synthesis: $x[n]=\sum_{k=0}^{N-1}X[k]e^{j\frac{2\pi}{N}kn}$, DTFS analysis: $X[k]=\tfrac{1}{N}\sum_{n=0}^{N-1}x[n]e^{-j\frac{2\pi}{N}kn}$, DTFS yields exactly $N$ independent coefficients describing one period, Sampling a continuous periodic signal yields a periodic discrete sequence only if $T_0/T_s$ is rational, Real-valued periodic sequences have conjugate symmetry in DTFS coefficients

## Introduction Discrete-time Fourier analysis studies how discrete-time signals can be represented, understood and manipulated in the frequency domain. This material focuses on spectral representations of periodic and nonperiodic discrete-time sequences, convergence considerations, and the discrete-time Fourier series (DTFS). We build intuition with examples and point out practical implications for signal processing. > **Definition:** A discrete-time signal (sequence) is a function $x[n]$ defined for integer time index $n\in\mathbb{Z}$. ## 1. Basic concepts ### Time-domain vs frequency-domain - Time-domain describes how a sequence varies with index $n$. - Frequency-domain describes how the sequence energy or components are distributed across frequency. > **Definition:** A spectral density is a frequency-domain description that assigns a complex amplitude or power to each frequency. ### Convergence considerations for infinite sums - Many frequency representations are defined via infinite sums. Such sums converge only under certain conditions on the sequence. - A common sufficient condition is finite energy: $\sum_{n=-\infty}^{\infty} |x[n]|^2 < \infty$. - Periodic sequences typically do not meet the finite-energy condition because their energy repeats indefinitely. > **Definition:** A sequence $x[n]$ has finite energy if $\sum_{n=-\infty}^{\infty} |x[n]|^2 < \infty$. ### Continuity of spectral density - For nonperiodic sequences that yield a continuous spectral density, the frequency-domain representation is a continuous function of frequency. While mathematically convenient, continuous spectra are often impractical for discrete digital systems because they require infinite precision to represent. Fun fact: Continuous spectral functions arise naturally from nonperiodic discrete-time signals, which means digital systems often rely on sampled or discrete-frequency representations to work efficiently. ## 2. Periodic discrete-time signals ### Definition and basic property - A sequence $x[n]$ is periodic with period $N$ if for every integer $k$ and all $n$: $$x[n+kN] = x[n].$$ - The smallest positive $N$ satisfying this is the fundamental period. > **Definition:** A periodic discrete-time signal repeats its values every $N$ samples: $x[n+N]=x[n]$. ### Example - If $x[n]$ has period $N=10$, then values at indices $n=0,1,\dots,9$ determine the sequence for all integers $n$ because $x[n+10]=x[n]$. ### Implications - Periodic sequences have discrete spectra composed of harmonics at frequencies that are integer multiples of the fundamental frequency $\Omega_0 = \tfrac{2\pi}{N}$. - The spectrum consists of a finite set of coefficients (one per harmonic within one period of the frequency index). Did you know that sampling a continuous-time periodic signal does not always produce a periodic discrete-time sequence; periodicity after sampling depends on the ratio between the continuous period and the sampling interval? ## 3. Discrete-Time Fourier Series (DTFS) ### Purpose - DTFS expresses a periodic discrete-time sequence as a finite sum of complex exponentials (sinusoids) at discrete harmonic frequencies. > **Definition:** The Discrete-Time Fourier Series represents a length-$N$ periodic sequence $x[n]$ as a sum of $N$ complex exponentials with coefficients $X[k]$. ### Synthesis (series expansion) - A periodic sequence with period $N$ can be written as: $$x[n] = \sum_{k=0}^{N-1} X[k]\,e^{j\,\frac{2\pi}{N}kn}.$$ Here $X[k]$ is the $k$-th complex spectral coefficient and the fundamental frequency is $\Omega_0 = \tfrac{2\pi}{N}$. ### Interpretation of terms - The $k=0$ term is the DC component (constant term). - Positive $k$ correspond to harmonics at frequencies $\frac{2\pi}{N}k$. - Each term $X[k]e^{j\frac{2\pi}{N}kn}$ is a sinusoidal component scaled and phase-shifted by $X[k]$. ### Analysis (coefficient computation) - The DTFS coefficients for a period-$N$ sequence are computed by projecting onto the complex exponent