From Information to Numbers

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From Information to Numbers

• The world is full of information
• Digital devices store numbers as bits
• How do we turn signals into numbers?
• Answer Digitization
• Sampling
• Quantization

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Sampling

• Sampling Taking signal values at
  regularly-spaced intervals
• Formula
• sn s(nTs)
• s(t) original signal
• sn sampled signal
• Ts sampling period
• fs 1/Ts is the sampling rate.

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Sampling Involves a Tradeoff

• Too small a Ts makes too many samples
• Too large a Ts ruins the sampled signal
• How do we design Ts?
• Answer The Nyquist Rate

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The Nyquist Rate

• We already know Any signal can be made from a
  sum of sinusoids.
• Suppose that a particular signal is made up of
  sinusoids whose frequencies are between fhighest
  and flowest.
• (fhighest - flowest) is called the bandwidth.
• This signal can be exactly reconstructed from its
  samples if fs gt 2 (fhighest).
• The value of 2 (fhighest) is called the
  Nyquist Rate.

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Sampling Above and Below the Nyquist Rate

• fs gt 2 (fhighest - flowest)

• fs lt 2 (fhighest - flowest)

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Sampling Rates for Some Important Signals

• Designers use these sampling rates to design CD
  players, DVD players, cell phones, car radios,
  and satellite TV.

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Sampling Below Nyquist Causes Aliasing

• Example A 720Hz sinusoid sampled at a rate of
  660Hz

looks like a 60Hz sinusoid!

• Filters are used before sampling to prevent this.

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Representing Text in Binary ASCII

• ASCII American Standard Code for Information
  Interchange
• A 7-bit code (8-bit representation) to represent
  128 symbols
• Capital and small letters (a-z, A-Z)
• Numbers (0-9)
• Punctuation (e.g. !_at_()_)
• Other control codes (line feed, end of file)

Why would a standard code be so important?
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Partial ASCII Code Listing
What is 1001001 1001110 1000110
1001001 1001110 1001001 1010100 1011001 ?
Answer INFINITY
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Storing Samples Using Bits Quantization

• Digital devices must use a limited number of bits
  to store each sample of a signal.
• The errors caused by quantization are seen and
  heard as noise.

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More Bits Mean Smaller Errors

• Example
• Top 3 bits per sample
• Middle 4 bits per sample
• Bottom 16 bits per sample
• More bits mean higher accuracy, but more storage
  and effort

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How Many Bits Per Sample?

• We would like a measure of signal quality as a
  function of number of bits
• The answer Signal-to-Noise Ratio
• Used in almost every multimedia system design
• Definition (linear scale) SNR
  max(ltsignalgt)
• max(ltnoisegt)
• Definition (dB scale) SNR in dB 20
  log10 max(ltsignalgt)
• max(ltnoisegt)

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Decibel Scale

• Named after Alexander Graham Bell, inventor of
  the telephone
• Convenient for very large and vary small SNRs

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Problem SNR

• What is the SNR for the signals on the left?
• What is the SNR in dB?

• Answer 0.8/0.06 13.3
• 20log10(80/0.6)22.5dB

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Example Problem SNR
Fact SNR can be used to measure the quality of
many signals
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SNR for Quantized Signals

• For a B-bit quantized signal
• SNR (2B-1)/2-1 2B
• Simple dB Rule SNR 20 log10(2B) 6.02 B
• Each bit adds 6 dB to the SNR
• Example CDs use 16 bits
• SNRCD 96 dB