Quantization - PowerPoint PPT Presentation

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Quantization

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Signal x(t) is quantized in a finite number of levels ... One noise source e(n) is fed into a linear filter and the output is f(n) Mean: Power Spectrum: ... – PowerPoint PPT presentation

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Title: Quantization


1
  • Quantization

2
Quantization
  • Signal x(t) is quantized in a finite number of
    levels
  • Assume that x is in the dynamic range -1,1 and
    we quantize it with b1 bits -gt 2b1 levels
  • Quantization introduces an error e Qx - x

3
Rounding versus truncation
Truncation -2-b lt Qx x lt 0 Example 010111
-gt 0101
Rounding -2-b/2 lt Qx x lt 2-b/2 Example
010111 -gt0110
4
Statistical model
  • Additive error model The quantized signal is the
    sum of the original signal and a quantization
    noise signal e(t)

5
Fixed point representation
  • Twos complement fractional representation
  • Dynamic range
  • Advantage of fractional
  • Multiplier can not overflow (except for
    (-1).(-1))

6
Twos complement fractional
7/8 0111
6/8 0110
5/8 0101
4/8 0100
3/8 0011
2/8 0010
1/8 0001
0 0000
-1/8 1111
-2/8 1110
-3/8 1101
-4/8 1100
-5/8 1011
-6/8 1010
-7/8 1001
-1 1000
  • Example b3

7
Twos complement arithmetic (1)
  • Addition
  • Addition can lead to an overflow. Need to scale
    the input.
  • If no overflow xy can be represented exactly
    with b1 bits. Example
  • In a filter check the scaling (max/min signal
    values) at the output of each adder.
  • An adder does not inject quantization noise

- 6/8 1010
1/8 0001
- 5/8 1011
8
Twos complement arithmetic's (2)
  • Multiplication
  • If coefficient is not 1, multiplier can not
    overflow
  • x,y represented with b1 bits -gt x.y is exactly
    represented with 2b1 bits
  • Example
  • If the multiplier output is forced into a
    register of lenght lt 2b1 it introduces a
    quantization noise whose max amplitude is 1 lsb
    (truncation) or - ½ lsb (rounding).

7/8 0111
x 1/8 0001
7/64 0000111
9
Multiplier model
  • If b3 ltb1b2-1 the truncation or rounding of the
    multiplier output is modeled as an additive noise
    e(n)

10
First-order IIR example
  • Feedback path -gt register size at multiplier
    output can not be increased at each iteration!
    Problem with IIR filters
  • Noise e2(n) is amplified by the feedback loop (a
    is close to 1). We will choose b2 gtb1
  • Filter design compute the PSD at the filter
    output due to each quantization noise. Statistics
    on e(n) needed

11
First order statistics (1)
  • e(n) is a random process signal that can, for
    each time index n, be described as a random
    variable e
  • Probability Density Function pe(u)
  • pe(u).De probability that the random
    variable e be in the interval u,uDu
  • Uniform PDF pe(u) is constant over a range W and
    zero elsewhere. Recall that the total area
    underneath pe(u) must integrate to 1

Truncation
Rounding
12
First order statistics (2)
  • The hypothesis that the quantization error is
    uniform does not hold if the input signal covers
    less than 1 lsb! It is assumed valid if the
    signals in all registers of the filter cover a
    significant portion of the register dynamic
    range.
  • Mean
  • DC component of the noise
  • Variance
  • total AC power of the noise

13
Second order statistics
  • Power Spectral Density of a random process e(n)
  • See(ej?) d ? power in the band d ?
    centered at ?
  • In z domain, power spectral density PSD when
    there is only one noise input is defined as
  • PSD, when there are K noise inputs, is

14
Quantization noise
  • One noise source e(n) is fed into a linear filter
    and the output is f(n)
  • Mean
  • Power Spectrum

15
Quantization noise
  • Variance
  • Special case if e(n) is white

We will assume that the quantization noise e(n)
is white
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