Notes on Analog and Digital Signals and Fourier Analysis

Analog and Digital Data

  • Data can be analog or digital. Analogy: information that is continuous; digital: information that has discrete states.
  • Analog data take on continuous values; digital data take on discrete values.
  • Important distinction: data vs signals. Data can be transformed to electromagnetic signals for transmission.

Signals: Analog and Digital

  • Signals can be analog or digital.
  • Analog signals can have an infinite number of values in a range.
  • Digital signals can have only a limited number of values (discrete levels).

Analog and Digital Data (Summary)

  • Analog data: continuous values across a range.
  • Digital data: discrete states, e.g., binary 0/1.
  • Data transformation: to be transmitted, data must be transformed to electromagnetic signals.

Analog and Digital Signals

  • Analog signals can take an infinite number of values within a range.
  • Digital signals have a finite set of levels.
  • This distinction parallels the data types but is specific to how information is represented in time.

Periodic Analog Signals and Composite Signals

  • In data communications, we commonly use periodic analog signals and nonperiodic digital signals.
  • Simple periodic analog signal: a sine wave that cannot be decomposed into simpler signals.
  • Composite periodic signal: made of multiple sine waves.

A sine wave

  • Definition: a basic periodic waveform with a single frequency.
  • Key properties: amplitude, frequency, phase, and period.

Amplitude and Peak Amplitude

  • Amplitude: the maximum displacement of the waveform from its zero (center) position.
  • Peak amplitude: the highest absolute value reached by the wave.
  • Signals with same phase and frequency can have different amplitudes.

Frequency and Period: Inverse Relationship

  • Frequency f is the rate of repetition per unit time.
  • Period T is the time for one complete cycle.
  • Relationships:
    • f=1Tf = \frac{1}{T}
    • T=1fT = \frac{1}{f}

Units of Period and Frequency

  • Base units:
    • Seconds (s) → 1 s
    • Hertz (Hz) → 1 Hz = 1 s^{-1}
  • Common prefixes:
    • Milliseconds (ms) = $10^{-3}$ s
    • Kilohertz (kHz) = $10^{3}$ Hz
    • Microseconds (µs) = $10^{-6}$ s
    • Megahertz (MHz) = $10^{6}$ Hz
    • Nanoseconds (ns) = $10^{-9}$ s
    • Gigahertz (GHz) = $10^{9}$ Hz
    • Picoseconds (ps) = $10^{-12}$ s
    • Terahertz (THz) = $10^{12}$ Hz
  • Examples:
    • The home power frequency is typically f=60  Hzf = 60\;\text{Hz}, so the period is T=160≈0.0167s=16.7 msT = \frac{1}{60} ≈ 0.0167 \text{s} = 16.7\text{ ms}.

Example: Frequency and Period Calculations

  • Example 1: Power at home is 60 Hz.
    • Period: T=160s≈0.0167s=16.7 msT = \frac{1}{60} \text{s} \approx 0.0167 \text{s} = 16.7\text{ ms}.
  • Example 2: If the period is 100 ms, the frequency is:
    • f=1T=10.100 s=10 Hzf = \frac{1}{T} = \frac{1}{0.100\text{ s}} = 10\text{ Hz}
    • In kilohertz: f=0.01 kHzf = 0.01\text{ kHz} since 1 kHz=103 Hz1\text{ kHz} = 10^{3} \text{ Hz}.
  • Example 3: A phase corresponding to 1/6 of a cycle:
    • 1 cycle = 360° ⇒ 1/6 cycle = 60° = π3 rad\frac{\pi}{3} \text{ rad}.
  • Example 4 (not numerically specified in the transcript) would follow similar phase/frequency relationships.

Phase

  • Phase describes the position of the waveform relative to time 0.
  • Visual: three sine waves with same amplitude and frequency but different phases (0°, 90°, 180°).
    • 0° phase: reference position at t = 0.
    • 90° phase: quarter-cycle shift.
    • 180° phase: half-cycle shift.

A sine wave is offset 1/6 cycle with respect to time 0

  • If a sine is offset by 1/6 cycle, its phase is 60° (or π3\frac{\pi}{3} radians).
  • This demonstrates the relationship between phase offset and a fractional cycle shift.

Wavelength and Period

  • Wavelength relates to propagation through a transmission medium.
  • Time-domain view: a wave at time t has a certain position.
  • At time t + T (where T is period) the wave has advanced by one full cycle, and the direction of propagation remains the same.
  • Wavelength is the spatial distance corresponding to one period in space, linked to propagation speed and frequency.

Time-Domain and Frequency-Domain Plots

  • Time-domain: Amp vs Time for a sine wave (e.g., peak value 5 V, frequency 6 Hz).
  • Frequency-domain: The same sine wave is represented by a single spike at its frequency (6 Hz) with amplitude equal to the peak value.
  • Concept: a single-frequency sine wave corresponds to a single impulse (spike) in the frequency domain.

A complete sine wave in time domain can be represented by one spike in the frequency domain

  • Pure sine wave → one spectral line at its frequency f with amplitude equal to the peak value.
  • This illustrates the simplicity of a single-frequency component in the frequency domain.

The frequency domain is compact for multiple sine waves

  • When dealing with several sine waves, the time-domain signal becomes complex, but in the frequency domain it is represented by a few spikes (harmonics) at their respective frequencies and amplitudes.
  • Example: three sine waves with different amplitudes and frequencies can be represented by three spikes in the frequency domain.

Signals and the Fourier Idea

  • A single-frequency sine wave is not sufficient for data communications.
  • Real data signals are composites of many sine waves.
  • Fourier analysis states that any composite signal is a sum of sine waves with different frequencies, amplitudes, and phases.

Composite Signals and Periodicity

  • If a composite signal is periodic, its Fourier decomposition yields a series with discrete (harmonic) frequencies: f, 2f, 3f, …
  • If the composite signal is nonperiodic, the decomposition yields a continuum of frequencies (continuous spectrum).

Composite Periodic Signal (Example 5)

  • A periodic composite signal can be analyzed by decomposing into harmonics f, 3f, 9f, etc.
  • Time-domain decomposition shows harmonics at f, 3f, 9f in time.
  • Frequency-domain decomposition shows spikes at frequencies f, 3f, 9f corresponding to those harmonics.

Non-Periodic Composite Signal (Example 6)

  • A nonperiodic signal (e.g., a spoken word) does not repeat exactly with the same tone.
  • Its time-domain signal is non-repeating; the frequency-domain representation is continuous (not a discrete set of spikes).

Bandwidth and Signal Frequency

  • Bandwidth of a composite signal is the difference between the highest and lowest contained frequencies:
    • Bandwidth=f<em>high−f</em>low\text{Bandwidth} = f<em>{\text{high}} - f</em>{\text{low}}
  • Examples:
    • Periodic signal with frequencies ranging from 1 kHz to 5 kHz has bandwidth =5000 Hz−1000 Hz=4000 Hz= 5000\text{ Hz} - 1000\text{ Hz} = 4000\text{ Hz}.
    • Nonperiodic signals can also have bandwidth defined by their spectral spread.

Fourier Analysis: A Tool for Time and Frequency Domains

  • Fourier analysis converts a time-domain signal into a frequency-domain representation and vice versa.
  • It is essential for analyzing non-sinusoidal waveforms and their harmonic content.

Fourier Series

  • Every composite periodic signal can be represented as a series of sine and cosine functions.
  • The functions are integral harmonics of the fundamental frequency $f_0$ (the fundamental).
  • Decomposition allows expressing a periodic signal as a sum of harmonics.
  • General form:
    • s(t)=A<em>0+∑</em>n=1∞[A<em>ncos⁡(2πnf</em>0t)+B<em>nsin⁡(2πnf</em>0t)]s(t) = A<em>0 + \sum</em>{n=1}^{\infty} [A<em>n \cos(2\pi n f</em>0 t) + B<em>n \sin(2\pi n f</em>0 t)]
  • Coefficients:
    • A<em>0=1T∫</em>0Ts(t)  dtA<em>0 = \frac{1}{T} \int</em>{0}^{T} s(t) \; dt
    • A<em>n=2T∫</em>0Ts(t)  cos⁡(2πnf0t)  dt,n≥1A<em>n = \frac{2}{T} \int</em>{0}^{T} s(t) \; \cos(2\pi n f_0 t) \; dt, \quad n \ge 1
    • B<em>n=2T∫</em>0Ts(t)  sin⁡(2πnf0t)  dt,n≥1B<em>n = \frac{2}{T} \int</em>{0}^{T} s(t) \; \sin(2\pi n f_0 t) \; dt, \quad n \ge 1
  • Where $T$ is the period and $f_0 = \frac{1}{T}$ is the fundamental frequency.

Examples of Signals and the Fourier Series Representation

  • Time-domain example for a square wave:
    • A square wave can be represented by a sum of odd harmonics:
    • s(t)=4Aπ(11sin⁡(2πf<em>0t)+13sin⁡(2π3f</em>0t)+15sin⁡(2π5f0t)+⋯ )s(t) = \frac{4A}{\pi} \left( \frac{1}{1} \sin(2\pi f<em>0 t) + \frac{1}{3} \sin(2\pi 3 f</em>0 t) + \frac{1}{5} \sin(2\pi 5 f_0 t) + \cdots \right)
  • Sawtooth signal:
    • All harmonics are present with amplitudes decreasing as $1/n$ and alternating signs:
    • s(t)=2Aπ∑<em>n=1∞(−1)n+1nsin⁡(2πnf</em>0t)s(t) = \frac{2A}{\pi} \sum<em>{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \sin(2\pi n f</em>0 t)
  • These examples illustrate how non-sinusoidal waveforms arise from combining sinusoids.

Fourier Transform

  • Fourier Transform provides the frequency-domain representation of a nonperiodic time-domain signal:
    • S(f)=∫−∞∞s(t) e−j2πft dtS(f) = \int_{-\infty}^{\infty} s(t) \, e^{-j 2\pi f t} \, dt
  • Inverse Fourier Transform recovers the time-domain signal from its frequency-domain representation:
    • s(t)=∫−∞∞S(f) ej2πft dfs(t) = \int_{-\infty}^{\infty} S(f) \, e^{j 2\pi f t} \, df

Time-Limited and Band-Limited Signals

  • Time-limited signal: amplitude is zero outside a finite interval: s(t)=0for t∉[T<em>1,T</em>2]s(t) = 0\quad \text{for } t \notin [T<em>1, T</em>2]
  • Band-limited signal: spectrum is zero outside a finite band: S(f) = 0\quad \text{for } |f| > F0 \text{ (or outside } [F1, F_2])

Fourier Theory Overview

  • Fourier analysis is a fundamental tool in communication circuits and systems, especially for non-sinusoidal waveforms.
  • A nonsinusoidal waveform can be broken down into harmonically related sine/cosine components.
  • Classic example: a square wave decomposes into an infinite series of odd harmonics.
  • Through Fourier analysis, we can determine a signal’s bandwidth and harmonic content.

Time Domain vs Frequency Domain: Key Concepts

  • Time-domain view: variations of voltage, current, or power with respect to time.
  • Frequency-domain view: amplitude variations with respect to frequency.
  • Fourier theory provides a bridge between these views, enabling different analyses and design strategies.

Practical Instrumentation: Spectrum Analysis

  • A spectrum analyzer produces a frequency-domain display of a signal.
  • It is a key instrument in designing, analyzing, and troubleshooting communication equipment.

Notes on Notation and Conventions

  • All equations are presented in LaTeX format for clarity:
    • Fundamental relations: f=1T,T=1ff = \frac{1}{T}, \quad T = \frac{1}{f}
    • Fourier series, transforms, and characteristic equations use standard forms as shown above.

Quick Summary for Exam Preparation

  • Distinguish between data type (analog vs digital) and signal type (analog vs digital).
  • Understand the inverse relationship between frequency and period; know unit conversions for Hz, kHz, MHz, etc.
  • Recognize that a pure sine wave has a single spectral spike; composite signals require multiple harmonics.
  • Be able to write the Fourier series for a periodic signal and identify harmonic content for square and sawtooth waves.
  • Know the Fourier transform pair and the inverse transform for nonperiodic signals.
  • Understand time-limited and band-limited concepts and how bandwidth is defined.
  • Appreciate the role of Fourier analysis in communications and the practical use of spectrum analyzers.