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ELEC1010 Notes: Signals, Spectrum, and Frequency Translation

BIG IDEAS:

  • Representation of signals in time and frequency domains

  • Digitization of information

  • Coding for data compression and error protection

  • Transmission of signals

  • Cellular mobile phone and wireless communications

  • The Internet

  • No prerequisite required, but course is technical; not necessarily easy

  • Course Outline (High-Level):

    • Input signals and signal representation

    • Time-domain representations and components of signals

    • 0/1 digital representation and iPhone as a digital system

    • Digitization: converting continuous waveforms to 0/1 sequences

    • Coding for data size reduction and protection

    • Transmission and communications concepts: internet, cellular networks

Power of Information Technology in a Single Handheld Device

  • Examples of technologies enabled by IT in devices:

    • Near Field Communication (NFC) for Apple Pay

    • Facial recognition (Face ID)

    • Inductive charging (Wireless charging)

    • Speech recognition (Siri)

    • Display technologies and augmented reality (AR)

    • Gesture-based multitasking

    • TrueDepth camera and Animoji

Electronics, Information Technology, and Us

  • Big ideas from the course focus:

    • IT as a driver of productivity and economic growth

    • IT-enabled progress across scientific and engineering disciplines

    • Examples:

    • Broadband Internet enabling tele-medicine

    • Wireless sensor networks enabling natural disaster detection (earthquakes, tsunamis)

    • AI enabling autonomous vehicles

  • Job market and investment implications

Signals: Basic Concepts

  • What is a signal?

    • A pattern or variation that contains information

    • A signal can be what we see, hear, touch, smell, taste, and visualize

    • Examples: Audio, image, video signals

  • Signals are ubiquitous and can be contained in phenomena not directly sensed

  • Signals convey information and can elicit stimulations and enjoyment (sound, motion pictures, etc.)

Representation of Signals

  • Signals can be represented as variations of physical quantities over time or space

    • Time-domain representations: e.g., Hang Seng Index over one year; acoustic pressure over 1/20 ext{ s}

    • Spatial variations: altitude maps, temperature distributions

    • Images: brightness of pixels over 2D space; one-dimensional cross-sections along a line

  • Signals can vary over both space and time (e.g., videos)

  • In engineering, it is useful to view signals in the frequency domain via spectrum

  • iPhone as a digital system (illustrative example of digital processing)

Analog vs Digital Signals

  • Analog signals: vary continuously over time with continuous values

    • Examples: acoustic pressure, electrical current, etc.

  • Digital signals: defined at discrete time instances and take on finite set of values

    • Modern computer processing uses digital signals

    • Examples: digital audio, numeric data, etc.

  • Conversion between analog and digital signals (sampling):

    • Sampling converts continuous-time signal to a discrete-time sequence for storage/processing

    • Process involves approximation losses; not perfectly reversible in general

  • Signals as input/output of systems: systems map input signals to output signals

Signals and Systems: Interactions and Complexity

  • Systems relate input to output; output of one system may become input to another

  • A cell phone is a complex system with multiple sub-systems and sub-sub-systems (e.g., analog baseband, digital baseband, power management, RF front-end)

  • Summary of signal concepts to be explored: spectrum, filtering, digitalization, and the role of these concepts in modern information technology

Sound Signals: Basics

  • Sound is an audio signal produced by variations in air pressure that reach the ear

  • Atmospheric pressure \approx 100{,}000 ext{ Pa}

  • Ear’s audible range: roughly from 2 \times 10^{-5} ext{ Pa} to 120 ext{ Pa}

  • Very small pressure changes (as little as 1 Pa) can cause damage; readings to be interpreted with care

  • Sound signals can be plotted as a function of time; pitched via frequency components

Manipulation of Sound Signals

  • Playback speed affects perceived pitch: slowing down lowers pitch; speeding up raises pitch

  • Pitch perception tied to the fundamental frequency of the sound pattern in each sound bite

  • Time-domain view reveals that music is built from short, repeating sound bites with characteristic patterns

Pitch, Frequency, and Harmonics

  • Pitch is linked to the repetition frequency of the sound pattern

  • Period T is the repeating interval of a periodic signal

  • Fundamental frequency f is the reciprocal of the period: f = \frac{1}{T}

  • Human hearing range: roughly 20 \text{ Hz} \text{ to } 20{,}000 \text{ Hz}

  • Sinusoidal signal (sine wave) is a basic periodic signal and a fundamental building block

    • Sine wave: x(t) = A \sin(2\pi f t)

    • Period: T = \frac{1}{f}; Frequency: f = \frac{1}{T}

    • Frequency-domain interpretation uses harmonics at integer multiples of the fundamental frequency

Harmonics and Timbre

  • Harmonics are sine waves at integer multiples of the fundamental frequency: f_n = n f, \quad n = 1,2,3,\dots

  • Adding harmonics with different amplitudes creates more complex waveforms while preserving the same fundamental frequency

  • Timbre (tone quality) is determined by the relative amplitudes of the harmonics, not just the fundamental frequency

  • A pure tone has only the fundamental ($n=1$); richer tones have stronger higher-order harmonics

  • Examples of spectra for different instruments illustrate different harmonic distributions

Spectrum, Spectrum Analysis, and Spectrogram

  • Spectrum represents the distribution of energy across frequencies for a signal

  • A spectrum shows amplitude (or energy) vs frequency; can be represented with delta impulses for pure harmonics

    • Example: a signal with harmonics at 261 Hz, 522 Hz, 783 Hz can be represented as a sum of impulses: X(f) = A1 \delta(f-261) + A2 \delta(f-522) + A_3 \delta(f-783)

  • Frequency-domain representation vs time-domain waveform

  • Fourier Transform (and Fourier Series for periodic signals):

    • Frequency-domain description is obtained via the Fourier transform

    • For periodic signals, Fourier series expresses the signal as a sum of harmonics: x(t) = \sum{n=-\infty}^{\infty} Xn e^{j 2\pi n f0 t} where f0 = \frac{1}{T} is the fundamental frequency

  • Spectrum vs spectrogram:

    • Spectrum: cross-section of a signal's frequency content at a given time

    • Spectrogram: time-varying spectrum showing how harmonic content changes over time

  • The spectrum of a signal can be viewed as the average distribution of energy over a long time

Frequency-Domain Representation and Fourier Series

  • Two equivalent representations of signals:

    • Time-domain representation: waveform vs time

    • Frequency-domain representation: spectrum X(f) showing amplitude at frequencies

  • Example of a signal composed of multiple harmonics:

    • Time-domain: x(t) = \sum{n=1}^{N} an \sin(2\pi n f_0 t)

    • Frequency-domain: X(f) = \sum{n=1}^{N} \frac{an}{2j} [\delta(f - n f0) - \delta(f + n f0)] (illustrative form)

  • Fundamental concept: any periodic signal can be decomposed into a sum of harmonics (Fourier series)

Electromagnetic Spectrum and Light

  • The universe is filled with electromagnetic waves across a broad spectrum

  • Visible light corresponds to wavelengths roughly from 0.4 to 0.7 micrometers; frequency relates to color

  • Relationship between wavelength and frequency: \lambda f = c, where c \approx 3 \times 10^{8} \text{ m/s}

  • EM spectrum includes radio waves, microwaves, infrared, visible, ultraviolet, X-rays, gamma rays

  • The electromagnetic spectrum is the basis for all modern communication channels (radio, TV, cellular, Wi-Fi, etc.)

  • The spectrum is regulated by governments for allocation of frequencies

  • The radio spectrum has been used for about a century for communication; spectrum allocation is critical for services like 4G/5G

Spectrum, Filters, and Signal Processing

  • Systems can be viewed as filters in the frequency domain with a spectral response H(f):

    • If input spectrum is X(f), output spectrum is Y(f) = H(f) X(f)

    • Notation: Y(f) = H(f) X(f) or equivalently \hat{y}(t) = (h * x)(t) in time domain

  • Common filters (basic building blocks):

    • Lowpass: passes low frequencies up to cutoff frequency f_c; attenuates higher frequencies

    • Highpass: passes high frequencies above a cutoff; attenuates low frequencies

    • Bandpass: passes a band between fL and fH; bandwidth B = fH - fL

  • Ideal filters vs practical realizations: ideal filters cannot be perfectly realized; practical implementations approximate ideal behavior

  • Applications of filtering:

    • Telephone voice: lowpass with cutoff around 3 kHz; typical maximum voice frequency \sim 4 \text{ kHz}; helps to reduce high-frequency noise

    • Radio tuners and audio processing use bandpass filters to isolate channels

  • Spectrum visualization tools (spectrum analyzer) allow real-time or chunk-based analysis of frequency components

Frequency Translation and Modulation (AM)

  • Goals: allow multiple devices to communicate over the same space without interference; separate by frequency bands

  • AM (Amplitude Modulation): baseband signal x(t) modulates a high-frequency carrier:

    • AM signal: s(t) = x(t) \cos(2\pi f_c t)

    • Baseband signal is mixed with carrier to shift its spectrum to around the carrier frequency

  • Mixing identities (for sinusoids):

    • If you multiply two sinusoids: \sin(2\pi f1 t) \cdot \sin(2\pi f2 t) = \tfrac{1}{2} [\cos(2\pi(f1 - f2)t) - \cos(2\pi(f1 + f2)t)]

    • For cosine products: \cos(2\pi f1 t) \cos(2\pi f2 t) = \tfrac{1}{2}[\cos(2\pi(f1 - f2)t) + \cos(2\pi(f1 + f2)t)]

  • Spectrum after modulation:

    • The baseband spectrum is shifted to be centered at the carrier: around fc \pm fm for each baseband frequency component f_m

    • Modulated signal bandwidth is twice the baseband bandwidth: B{mod} \approx 2 B{base}

  • Demodulation (recovery of baseband): mix the modulated signal with the carrier again and apply a low-pass filter to remove the high-frequency components

    • Principle: multiply by the carrier and then apply a low-pass filter to recover the baseband signal

    • Envelope detection is a practical demodulation method when the envelope is positive; assumes the baseband signal is non-negative

  • RF envelope concept: the envelope of the modulated carrier carries the baseband information; if the envelope is positive, full baseband can be recovered from the envelope alone

  • Example 1: baseband sine at fm = 1 \text{ kHz}, carrier fc = 1 \text{ MHz}; after modulation, you get sidebands at fc \pm fm with a total bandwidth of 2f_m around the carrier

  • Example 2: continuous baseband signal with time-varying spectrum; after modulation, you get mirror image of the baseband spectrum around the carrier

  • Frequency Division Multiple Access (FDMA):

    • Different baseband signals occupy different carrier frequencies

    • After modulation, each occupies a separate frequency band, enabling multiple simultaneous channels

    • Demodulation is achieved by mixing with the corresponding carrier and filtering

  • Practical note: demodulation methods vary (envelope detection, coherent demodulation, etc.) depending on the modulation scheme

FDMA: Multi-Channel Examples

  • Demonstration with three carriers at 1.0 MHz, 1.04 MHz, and 1.08 MHz; baseband bandwidth 10 kHz; modulated bandwidth 20 kHz

  • Before modulation: three baseband signals share the same band; after modulation: they occupy distinct bands around their respective carriers

  • Conceptual takeaway: by shifting baseband signals to separate carrier frequencies, multiple channels can coexist without mutual interference

Frequency Translation and Modulation: Key Takeaways

  • Signals can be moved in frequency using multiplication by a carrier (mixing)

  • Modulation enables multiplexing of multiple signals in the frequency domain (FDMA)

  • Demodulation can be achieved by frequency-domain or envelope-based methods

  • Spectrum awareness is essential for designing communication systems and ensuring efficient use of the radio spectrum

Summary of Core Concepts (From Course Outline)

  • Signals: definitions, representations, and the idea that signals carry information

  • Time-domain vs frequency-domain representations; spectrum as a descriptor of signal content

  • Harmonics, Fourier series, and the role of fundamental frequency in shaping timbre and pitch

  • Spectral analysis tools: spectrum and spectrogram for time-varying signals

  • Electromagnetic spectrum and its relation to communication systems

  • Filters: lowpass, highpass, bandpass; their roles as basic building blocks in signal processing

  • AM modulation, mixing equations, and the generation of sidebands

  • Demodulation concepts and practical considerations

  • FDMA: concept of using multiple carrier frequencies to carry multiple signals

Key Formulas and Notations

  • Fundamental frequency and period:
    f = \frac{1}{T}

  • Sinusoidal signal:
    x(t) = A \sin(2\pi f t)

  • Harmonics:
    f_n = n f, \quad n=1,2,3, \dots

  • AM signal (carrier modulation):
    s(t) = x(t) \cos(2\pi f_c t)

  • Sinusoid product identities (useful for deriving sidebands):
    \cos A \cos B = \tfrac{1}{2}[\cos(A-B) + \cos(A+B)]
    \sin A \sin B = \tfrac{1}{2}[\cos(A-B) - \cos(A+B)]

  • Sidebands in AM occur at frequencies:
    fc \pm fm for each baseband component f_m

  • Modulated bandwidth around the carrier:
    B{mod} \approx 2 B{base}

  • Fourier series representation for a periodic signal (conceptual):
    x(t) = \sum{n=-\infty}^{\infty} Xn e^{j 2\pi n f0 t}, \quad f0 = \frac{1}{T}

  • Frequency-domain representation (spectrum):
    X(f) = \mathcal{F}{x(t)}, with inverse x(t) = \mathcal{F}^{-1}{X(f)}

  • Dirac delta representation (example for discrete harmonics):
    X(f) = \sum{k} ak \delta(f - f_k)

  • Relationship between wavelength and frequency in EM waves:
    \lambda f = c, \quad c \approx 3 \times 10^{8} \text{ m/s}

  • Human audible range: 20 \text{ Hz} \le f \le 20{,}000 \text{ Hz}

  • Frequency response of a filter (conceptual): output spectrum Y(f) = H(f) X(f)

  • Bandwidth definitions:

    • Bandwidth of a bandpass filter: B = fH - fL

    • Bandwidth of the modulated signal around the carrier: B{mod} \approx 2 B{base}

Practical Notes and Real-World Relevance

  • Filtering and spectrum analysis are foundational for audio, communications, and RF systems

  • Understanding harmonics and timbre explains why different instruments sound distinct even when playing the same note

  • FDMA is a core principle behind how multiple wireless services share the radio spectrum

  • Envelope detection provides a simple and robust demodulation approach when the baseband signal is non-negative

  • The electromagnetic spectrum underpins all wireless communications, sensing, and imaging technologies

Quick References

  • Carrier frequency concept: carrier frequency f_c centers the modulated spectrum

  • Baseband bandwidth: the range of frequencies in the original signal before modulation

  • Spectrum analyzer: tool to visualize X(f) or the distribution of energy across frequencies in real time

  • Spectrogram: time-varying spectrum, useful to observe changes in harmonic content over time