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=T1​).
Unit of frequency: Hertz (Hz).
Human ears distinguish frequencies from 20 to 20000 Hz.
Sinusoidal Signals
Simplest periodic signal (sine wave).
A⋅sin(2π⋅f⋅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(λ)⋅frequency(f)=speedoflight(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