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LNS Arithmetic for MPEG Encoding Using a Fast DCT. Jie Ruan and Mark G. Arnold. LNS (Logarithmic Number System). Represents a number by a sign bit and an exponent to a certain base (b). S. Exponent (n-1 bits). F (Precision). Properties of LNS. Large dynamic range

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lns logarithmic number system
LNS (Logarithmic Number System)
  • Represents a number by a sign bit and an exponent to a certain base (b)


Exponent (n-1 bits)

F (Precision)

properties of lns
Properties of LNS
  • Large dynamic range
  • Easy for multiplications, divisions and exponentiations
  • Additions are not linear operations for LNS
  • Cost of adders is exponential to word length
  • Has advantages at low precisions.
lns arithmetic units


The cost is a fixed-point adder


More complex process than multiplication

E.g., when calculating logb(X+Y), (x=logbX, y=logbY)

Calculate z=x-y

Table-lookup sb(z)=logb(1+2z)


LNS Arithmetic Units
major parts of mpeg encoding
Major parts of MPEG Encoding
  • Motion Estimation
  • Discrete Cosine Transform (DCT)
  • Variable Length Coding
2 d discrete cosine transform
2-D Discrete Cosine Transform

2 Dimensional 8x8 DCT

berkeley mpeg encoder
Berkeley MPEG Encoder
  • Berkeley MPEG Tool is a software to perform MPEG encoding and decoding, written in C language.
  • The DCT function is modified to accommodate LNS features.
visual effect of mpeg encoding with lns dct
Visual Effect of MPEG Encoding with LNS DCT
  • At F=4, a visually acceptable precision

10-bit LNS

16-bit Fixed-point

64-bit Floating-point

hardware implementation for dct
Hardware implementation for DCT
  • NxN 2 Dimensional DCT
    • Usually performed through 2 rounds of 1-D N-point DCT, with an NxN buffer
    • Less hardware and simpler data input arrangement than direct 2-D implementations
  • Chen’s 1-D DCT algorithm
    • Directly factorizes the DCT matrix
    • 16 multiplications
    • 26 additions
compares lns dct with fixed point dct
Compares LNS DCT with Fixed-point DCT

Comparison Hardware cost for Chen’s algorithm, at F=4

  • At low precisions LNS is a good way for DCT in MPEG, compared to Fixed-point
    • shorter word length
    • less area
    • Comparable visual result
  • LNS is a good alternative for MPEG encoding