ELEC1010 Exam Notes

What is a Signal?

• A pattern or variation containing information.
• Can be audio, image, video, or sensed indirectly (radio waves, etc.).

Signal Representation

• Variations of physical quantities over time or space.
• Examples: Hang Seng Index, temperature distribution, brightness of pixels in an image.
• Can also vary over both space and time (videos).
• Crucially, signals can be represented as functions of frequency.

Analog Signals

• Vary continuously over time.
• Examples: Acoustic pressure, electrical current.

Digital Signals

• Defined at discrete time instances with a finite set of values.
• Processed by modern computers.
• Examples: Student attendance, Hang Seng Index at day's end, MP3 files.
• Humans process signals digitally after rounding off.

Analog to Digital Conversion

• Frequently convert between analog and digital (e.g., audio processing).
• Involves sampling, which induces approximation and losses, preventing full conversion back.

Signals and Systems

• Everything can be modeled as systems with input and output signals.
• Examples: iPhone, digital camera, bank account.
• Output signals from one system can be input to another, enabling interactions.
• Cell phones are complex systems with multiple sub-systems.

Sound Signals

• Audio signals heard through air pressure variations.
• Pitch is determined by the frequency of repetitive patterns.
• Faster repetition leads to higher pitch.

Periodic Signals

• Oscillate with a repetitive pattern.
• Period (T): Repeating interval.
• Fundamental frequency (𝑓): Reciprocal of the period ($f = \frac{1}{T}$).
• Unit of frequency: Hertz (Hz).
• Human ears distinguish frequencies from 20 to 20000 Hz.

Sinusoidal Signals

• Simplest periodic signal (sine wave).
• $A \cdot sin(2\pi \cdot f \cdot t)$; Amplitude (A), frequency (f), time (t).

Pitch and Frequency

• Different frequencies correspond to different pitches.
• Every music note has a specific pitch (fundamental frequency).

Sound Attributes

• Pitch: Perception of frequency.
• Loudness: Perception of amplitude (logarithmic scale, decibels - dB).
• Timbre: Quality of sound, determined by the details of the repetitive pattern.

Harmonics

• Sine waves at integer multiples of the fundamental frequency.
• Contribute to timbre or richness of sound.
• Adding harmonics does not affect the fundamental frequency.

Signals as Sum of Sine Waves

• Our ear basically responds to the sine waves in sound signals.
• Sinusoidal signal with frequency “f” is the “1st harmonic ”
• “f” is also known as the “fundamental frequency ”
• Sinusoidal signal with frequency “ 2f” is the “2nd harmonic”
• Sinusoidal signal with frequency “ 3f” is the “3rd harmonic,” and so forth.

Frequency Domain Representation

• Represents signals by the amplitude of sine waves at different frequencies.
• Domain conversion is done by using Fourier Transform.

Spectrum of Signals

• Represents amplitude or energy at different frequencies.
• Spectrogram: Shows how harmonics change with time.

Spectrum Properties

• Human speech has negligible energy above 4,000 Hz.
• Female speech has higher frequencies than male speech.

Signals in Frequency Domain

• Time-domain representation: plot the entire periodic amplitude variations over time
• Frequency-domain representation: plot the amplitudes of the sine waves at different frequencies

Fourier Series Decomposition

• Any periodic pattern can be created by adding up different amounts of harmonics!

Sine Waves

• Sine wave is what we see and hear.
• The tones we hear are acoustic waves at different frequencies.

Spectrum Analyzer

• We frequently use the spectrum analyzer in the laboratory to analyze the spectrum of a signal.

Electromagnetic Spectrum

• Universe filled with electromagnetic waves at different frequencies.
• Includes radio waves, light, X-rays, etc.
• $Wavelength (\lambda) \cdot frequency (f) = speed of light (c)$

Systems as Filters

• Systems process signals and can be viewed as filters in the frequency domain.
• Produce a spectral response curve H, where Output = Input * H.

Types of Filters

• Lowpass: Allows low frequencies to pass.
• Bandpass: Allows a specific frequency range to pass.
• Highpass: Allows high frequencies to pass.

Frequency Translation

• How to allow many devices to communicate over the same space without interfering each other?
• Early Analog Cellular Systems (1G)
• Operated with carrier frequencies below 1 GHz
• They were analog systems, meaning that speech signals were transmitted as analog waveforms

Amplitude Modulation (AM)

• Baseband signal: the original signal containing the information
• Carrier: electromagnetic signal at high frequency used for communication
• s(t) = x(t)sin(2\pift): AM signal = Baseband signal * Carrier frequency

Digital Signals and Systems

• George Boole invented Boolean Algebra, which for many years did not seem to have any practical use
• Advantages of Digital Communications

Lowpass Filtering

• Traditional telephone networks use lowpass filters.

Bandpass Filters

• Used in radio tuners to select specific channels.

Summary

• Concept of spectrum is applied to electromagnetic wave
• Radio, microwave, X-ray: part of the electro-magnetic spectrum
• Different parts of the radio spectrum are used by different communications systems

Logic with Bits and Bytes

• In the digital world, a bit is the most basic unit of information
• “Bit” is short for binary digit.
• It refers to a digit in the binary numeral system (base 2).
• A byte (B) = 8 bits
• Digital devices operate on binary logic
• The CPU deals with this mathematical calculations using Binary Logic

Binary Logic

• A: Input
• B: Output
• 3 basic logic operations: NOT, AND, OR.

Fundamental Forces in Physics

• Electromagnetism like gravity, is one of the four fundamental forces in physics

Analog to Digital Conversion – Sampling

• Digitization: signals into digital forms
• Analog-to-digital converter (ADC)
• Two conceptual operations in ADC
  * Sampling
  * Sampling is the process of recording values (samples) of a signal at distinct points in time or space.
  * Quantization
  * Quantization refers to putting these samples into binary format (bits), i.e., making them discrete in values.

Nyquist Sampling Theorem (1927)

• $f<em>s > 2f</em>m$
  * fs is sampling frequency
  * fm is maximum frequency is present in the signal
• To give even better quality, DVD audio allows for a sampling rate of 96 kHz – more than twice that of CD.

Nyquist Rates for Telephony and CD Audio

• Telephone network speech @ 8 kHz
• CD music @ 44.1 kHz
• Higher sampling for DVD audio @ 96 kHz

Analog to Digital Conversion – Quantization

• Refers to putting these samples into binary format (bits)
• This refers to discretizing the y-axis of the signal and representing with binary numbers
• More bits, more levels, greater precision!

Digital Pictures of Different Precisions

• Quality depends on the both resolution (sampling) and color depth (quantization)

Entropy of Coding

• In the digital world, we can send a message with finite accuracy because limits exist.
• Shannon introduced the concept that a signal value can be sent and recovered only with finite accuracy because of noise and other limit