Analog to Digital Conversion

Digitization Overview

• Converting continually changing analog quantities into a sequence of discrete "integers"

Analog to Digital Conversion

• Focus of Lecture 10, conversion from analog to digital signals

Digitization Conceptual Operations

• Sampling is converting an analog signal (like music) to a series of numbers by measuring at discrete intervals, each number is a sample
• Represented as x(t) → x[n]
• Quantization puts samples into a binary format (bits), changing them into discrete values
• Represented as x[n] → xd[n]

Analog-to-Digital Converter (ADC)

• Conceptually performs analog to digital conversion

Sampling Rate and Bits

• Sampling rate and number of bits are key considerations in analog-to-digital conversion

Compact Disc (CD) Digitization

• Digital music on CDs results from sampling at 44,100 times per second
• Original signal is stereo and has left/right channels
• Each sample is stored as a 16-bit number.
• A 3-minute stereo song requires 254,016,000 bits
• This is around 242 Mb or 30.3 MB
• 1 MB equals 1,048,576 bytes
• Results in around 10 MB per minute.
• A 600 MB CD can store about 60 minutes of digital CD music

Sampling Process

• Records signal values at distinct points in time or space
• Examples include audio waveforms (time), images (space), and time-varying scenes (time and space)
• Samples are often taken at uniformly spaced time instants or spatial intervals

Re-creating Signals

• Samples are converted back into continuous form for human consumption

Required Sampling Rate Factors

• Assumes arbitrarily accurate storage
• Key question is how often should samples be taken to faithfully recreate the original information
• Sample spacing should be determined "how fast the signal changes"

Sampling Period and Frequency

• Sample spacing (Ts) refers to the time interval between samples, measured in seconds, also known as the sampling period
• Sampling frequency (fs) / sampling rate indicates samples taken per second, measured in Hertz
• fs = 1/Ts

Signal Change Characteristics

• Maximum frequency impacts signal change

Bandwidth and Spectrum

• Any signal can be represented by its spectrum
• Spectrum defines amplitudes of sine waves making up signal
• Signals often have max frequency (fm) and/or lower frequency (fL) beyond which component sine waves are negligible/zero
• Bandwidth is the range of frequencies for which spectrum is non-zero, calculation: fm - fL

Nyquist Sampling Theorem (1927)

• Harry Nyquist formulated theorem relating sampling rate to highest frequency of signal
• Guarantees original analog signal reconstruction w/o error, if the rate is more than twice as large as the signal frequency
• Mathematically, original can be exactly recovered by passing the sampled signal via an ideal lowpass filter
• "Nyquist rate" refers to "2 x maximum frequency in the signal"

Interpretation of Nyquist Theorem

• Theorem implies one unique curve that passes through set of sample points, smaller than fs /2

Reason Behind Nyquist Theorem

• Sine wave with frequency smaller than 1/2Ts cannot turn more than once between two sample points with the interval of Ts

"Bad" Sampling Rate (Illustration)

• Considers a sine wave sampled at frequencies 6f and 2f
• Signals can be constructed as sums of sine waves

Nyquist Rate Sampling

• The original sine wave can exactly be reconstructed with 6 samples per period
• If only 2 samples per period exist, it's not possible
• If the 2f sampling has zero values, it's not possible to distinguish original sinusoid because the signal is always 0!

Quantization Error

• Frequency f = fs/2 = 1/2Ts. providing sample set in intervals

Minimum Sampling Rate

• A rate of fs = 2f gives a minimum number of samples per second needed to determine the sine wave
• Must be greater than 2 samples per period / more than twice sinusoid frequency

Nyquist Rates (Telephony and CD audio)

• Speech signals limited at the frequency 3.5 KHz and sampled at 8 KHz in network to avoid aliasing
• CD audio has signal frequency up to 20 KHz and is sampled at 44.1 KHz
• DVD quality requires 96 kHz to improve perfection

Digitizing Signals - Sampling Rates

• Telephone = 3.5kHz, 7,000 samples/s, and 8,000 actual output
• CD Music = 20 kHz, 40,000 samples output and 44,100 actual output
• FM radio = 200kHz, 400,000 samples/s output, 500,000 actual output
• Analog TV = 6MHz, and output of 12,000,000 sample output and 14,400,000 actual output
• Digital TV, FM, and digital samples occur for digital processing

Sampling Rate Conclusion

• Signals change at a rate limited by max frequency
• Ideal signal recovery depends on larger sampling relative to max rate

Quantization Definition

• Quantization addresses how digital signals should be represented with bits
• Accuracy is the measurement
• Precision the level of details

Quantization Example

• Accuracy is the measurement
• Precision the level of details
• For the question "How old are you?" example
  • The answers vary by precision
  • An age of "20 years old" is less precise compared to "20 years, 10 days, 3 hours, 4 minutes and 5 seconds old"

2-Bit Quantization (Example)

• A binary signal between 0 V to 1 V can be separated to 4 equal parts that output 2 digital bits
• 1.00 will output bit 11 and has a level of 3.
• 0.75 will output bit 10 and has a level of 2.
• 0.50 will output bit 01 and has a level of 1
• 0.25 will output bit 00 and has a level of 0.
  • 0 V falls on 0.00 bits
• For example, 0.55 V is represented by 10, the second level binary representation

Signal Rounding After Quantization

• Two bit range of output (10 binary) converts the signal to a voltage between 0.5V to 0.75V
• Converts 10 back to 0.625V to minimize error

Bit Increase

• 2 bit quantization is crude
• Increasing the signal range reduces error
• Each bit increases precision and storage use

Quantization Error

• Signal rounding is for quantization among predefined values
• The quantization error is that level difference
• Increasing the range of values decreases the error
• The value of all sampled signals is often called quantization noise

Bits Effect

• N bits increases quantization levels (2^N)
• The reproduced signal closely reflects the original
• Increasing the bits, reduces noise and produces an accurate output

Digital Pictures & Precisions

• Original has a 8 bit precision, and declines with other pictures
• Increase in levels increases precision

Storing bits and audio signals

• Precisions depend on data
• Storage of individual pixels is 8 bits (256 levels)
• Musical CD use 16 bits (65,536 levels) because our hearing is precise
• Phones have lower high fidelity and the 8 bit signals (256 levels) are intelligible/ low fidelity

Audio Qualities

• Better CD Quality is good because of recording practices
• Storage uses higher storage and increases noise/distortion

Sampling Tool Effects

• Increasing the bit range increases performance

Lecture 11 Summary

• Precision is inherent within data amounts
• More bits increase precision
• Insufficient quantization increases noise

Claude Shannon Background

• Is considered the father of info theory
• 1937 master thesis showed binary implementation of Boolean Algebra
• Has a paper that measures information by meaning only within bits

Information Measurements

• Depends on data length
• Value (Pie)
  • Pie is an irrational # that continues

Information Limitation

• Cannot receive all exact values
  • Precision requires quantization

Information by Humans

• The data amounts reflect answer

Defining Entropy

• Entropy measure determines amount
  • Unlikely answers contain larger info
  • The question has some amount of randomness
  • Information depends on probability of the answer
• Each question requires minimum bits

Information (Boy or Girl question)

• The ratio is the data
• Information isn’t needed in deterministic situations
• Higher precision requires higher data

Genders and Storage

• Use paired coding

Channel Capacity

• Capacity depends on storage
• Bandwith is speed
• SNR is quality
• Shannon says we can communicate

Lecture #12 Conclusion

• Need precision, quantity, and rate
• Less needed for random things

Chapter 3 review

• Shannon shaped tech
• Discussing physics and data
• Universe is analog
• Digital values are precise