What Is Digital? (Part 2)

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What Is Digital?(Part 2)
CS1103 ????????

• Prof. Chung-Ta King
• Department of Computer Science
• National Tsing Hua University

(Contents from MIT EECS 6.01/6.02, Wikipedia, U.
of Indiana K361, www.arrl.org, www.sounderpro.com.
tw/Digital)
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Outline

• Continuous versus discrete
• Problems with continuous
• Going into discrete/digital
• Building digital devices
• IC to digital logic to computing devices
• Transforming analog inputs into digital
• Transforming digital to analog outputs

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?????????Sampling???

• Digital representations of analog waveforms

Continuous time Continuous values
Discrete time Discrete values
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Sampling

• The value of the analog signal is measured at
  certain intervals in time. Each measurement is
  referred to as a sample.

A series of snapshots
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Sampling in Space
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Terminologies in Sampling

• Sampling rate
• How often analog signal is measured samples per
  second, Hz, e.g. 44,100 Hz
• Sampling resolution
• Number of bits to represent each sample sample
  word length, bit depth, e.g. 16 bit

4 Samples/cycle
16 Samples/cycle
8 Samples/cycle
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Note Sampling Used Widely

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• CD???? (44.1 Khz)

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Problem with Sampling

• The analog signal is sampled at discrete
  intervals of time
• The digital output is an incomplete picture of
  the behavior of the input
• There is no way of knowing, by looking at the
  output, what the input was doing between one
  sampling instant and the next

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How Many Samples Are Enough?

• Reconstruction Is it possible to reconstruct
  the original waveform using only the discrete
  time samples?

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Shannon-Nyquist Sampling Theorem

• How many samples are necessary to preserve the
  information contained in the signal?
• If signal contains high frequency components,
  need to sample at even higher rate to avoid
  losing information in the signal ? How much
  faster?
• Sampling Theorem A signal can be exactly
  reproduced if it is sampled at a frequency that
  is greater than twice max. frequency in the signal

http//www2.egr.uh.edu/glover/applets/Sampling/Sa
mpling.html
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Sampling at Same Frequency
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Sampling at Twice Frequency
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Implications

• ??????????20 Hz 20 KHz
• ????? 40 kHz ?????? (sampling rate) ??????? 20
  kHz ?????
• ???????8kHz???
• 8kHz telephone quality
• 44.1kHz CD quality
• 48kHz radio/TV broadcast, DV, DVD video
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Aliasing

• If sampling rate not high enough
• New frequencies appear in reconstructed signals
• Need circuits to eliminate (anti-aliasing filter)

A high frequency signal
sampled at a lower rate
looks like
a lower frequency signal
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Encoding of Discrete Signals

• If we use N bits to encode the magnitude of one
  of the discrete-time samples, we can capture 2N
  possible values

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3-bit Quantization

• A 3-bit binary number has 23 8 values

Amplitude
Time measure amplitude at each tick of sample
clock
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4-bit Quantization

• A 4-bit binary number has 24 16 values

Amplitude
Time measure amplitude at each tick of sample
clock
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The Digital Audio Stream

• A series of sample numbers, to be interpreted as
  instantaneous amplitudes, one for every tick of
  the sample clock.
• This is what appears in a sound file, along with
  a header that indicates the sampling rate, bit
  depth and other things.

11 13 15 13 10 9 6 1 4 9 15 11 13 9
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Audio File Size

• CD characteristics
• Sampling rate
• 44,100 samples per second (44.1 kHz)
• Sample word length
• 16 bits (i.e., 2 bytes) per sample
• Number of channels
• 2 (stereo)
• How big is a 5-minute CD-quality sound file?

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Quantization Error

• When we quantize the scaled sample values, we may
  be off by up to 1/2 step from the true sampled
  values
• Round-off error difference between actual signal
  and quantization to integer values

Random errors sounds like low-amplitude noise
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Sampling Rate and Encoding Bits
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Resolution Trade-offs
 
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Analog to Digital Recording Chain
ADC
Microphone converts acoustic to electrical
energy. Its a transducer.
Continuously varying electrical energy is an
analog of the sound pressure wave.
ADC (Analog to Digital Converter) converts analog
to digital electrical signal.
Digital signal transmits binary numbers.
DAC (Digital to Analog Converter) converts
digital signal in computer to analog for your
headphones.
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ADC and DAC
http//www.dspguide.com/ch3/4.htm
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ADC in Digital Camera
http//www.tasi.ac.uk/advice/creating/camera.html
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Converting Analog into Digital

• The simplest form of ADCuses a resistance
  ladderto switch in the appropriatenumber of
  resistors in seriesto create the desiredvoltage
  that is compared to the input(unknown) voltage

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Converting Analog into Digital

• The resistance ladder changes its configuration
  systematically and the output voltage is compared
  to the analog voltage in a comparator
• When there is a match, the digital equivalent
  (switch configuration) is captured

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How? Binary Search

• Initial conditions
• Expected high 5 volts
• Expected low 0 volts
• 5 volts ? 256
• 0 volts ? 0
• Voltage to be converted
• 3.42 volts
• Equates to 175

Analog
Digital
256
5-volts
Unknown (175)
3.42-volts
2.5-volts
128
0
0-volts
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What to Do with Digital Signals?

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Summary

• Digitization of analog signals by sampling
• Sampling rate and resolution
• Problems with sampling aliasing, quantization
  errors
• Analog to digital conversion
• Analysis of digital signals FFT