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1.3 Entropy coding

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In this module, the concepts of entropy and variable-length coding are introduced, motivating the description of the Huffman encoder.
  • Binary Scalar Encoding: Previously we have focused on the memoryless scalar quantizer y = Q ( x ) , where y takes a value from a set of L reconstruction levels. By coding each quantizer output in binary format, we transmit(store) the information at a rate (cost) of
    R = ⌈ log 2 L ⌉ bits/sample .
    If, for example, L = 8 , then we transmit at 3 bits/sample. Say we can tolerate a bit more quantization error, e.g., as resultsfrom L = 5 . We hope that this reduction in fidelity reduces our transmissionrequirements, but with this simple binary encoding scheme we still require R = 3 bits/sample!
  • Idea—Block Coding: Let's assign a symbol to each block of 3 consecutive quantizer outputs.We need a symbol alphabet of size ≥ 5 3 = 125 , which is adequately represented by a 7-bit word ( 2 7 = 128 ). Transmitting these words requires only 7 / 3 = 2 . 33 bits/sample!
  • Idea—Variable Length Coding: Assume some of the quantizer outputs occur more frequently than others.Could we come up with an alphabet consisting of short words for representing frequent outputs and longer words for infrequent outputsthat would have a lower average transmission rate?

    Variable length coding)

    Consider the quantizer with L = 4 and output probabilities indicated in [link] . Straightforward 2-bit encoding requires average bit rate of 2bits/sample, while the variable length code in [link] gives average R = ∑ k P k n k = 0 . 6 · 1 + 0 . 25 · 2 + 0 . 1 · 3 + 0 . 05 · 3 = 1 . 55 bits/sample.

    output P k code
    y 1 0.60 0
    y 2 0.25 01
    y 3 0.10 011
    y 4 0.05 111
  • (Just enough information about) Entropy:

    Given an arbitrarily complex coding scheme, what is the minimum bits/sample required to transmit (store) the sequence { y ( n ) } ?

    When random process { y ( n ) } is i.i.d., the minimum average bit rate is

    R min = H y + ϵ ,
    where H y is the entropy of random variable y ( n ) in bits:
    H y = - ∑ k = 1 L P k log 2 P k ,
    and ϵ is an arbitrarily small positive constant (see textbooks by Berger and by Cover&Thomas).

  • Huffman Encoding: Given quantizer outputs y k or fixed-length blocks of outputs ( y j y k y ℓ ) , the Huffman procedure constructs variable length codes that are optimal in certain respects (see Cover&Thomas). For example, when the probabilities of { P k } are powers of 1/2 (and { y ( n ) } is i.i.d.), the entropy rate of a Huffman encoded output attains R min .

    Huffman procedure (binary case)

    1. Arrange ouput probabilities P k in decreasing order and consider them as leaf nodes of a tree.
    2. While there exists more than one node:
      • Merge the two nodes with smallest probability to form a new node whose probability equals the sum of the twomerged nodes.
      • Arbitrarily assign 1 and 0 to the two branches of the merging pair.
    3. The code assigned to each output is obtained by reading the branch bits sequentially from root note to leaf node.

    Huffman encoder attaining r min

    In [link] , a Huffman code was constructed for the output probabilities listed below.Here H y = - 0 . 5 log 2 0 . 5 + 0 . 25 log 2 0 . 25 + 2 · 0 . 125 log 2 0 . 125 = 1 . 75 bits, so that R min = 1 . 75 bits/sample (with the i.i.d. assumption). Since the average bit rate for the Huffman code is also R = 0 . 5 · 1 + 0 . 25 · 2 + 0 . 125 · 3 + 0 . 125 · 3 = 1 . 75 bits/sample, Huffman encoding attains R min for this output distribution.

    output P k code
    y 1 0.5 0
    y 2 0.25 01
    y 3 0.125 011
    y 4 0.125 111

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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cm
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A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
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Can you compute that for me. Ty
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2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
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you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Source:  OpenStax, An introduction to source-coding: quantization, dpcm, transform coding, and sub-band coding. OpenStax CNX. Sep 25, 2009 Download for free at http://cnx.org/content/col11121/1.2
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