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Quantization

Spring 2012

© Ammar Abu-Hudrouss- Islamic University Gaza

Slide 2Digital Signal Processing

QuantizationSampling converts the analogue signal into discrete value of

samples.

The values of theses samples depends on the sampling instants.

We need to encode each sample value in order to store it in b bits memory location.

But as b is limited, we have to consider a finite values of samples.

For example If b = 2 , we can have 2b=4 different possible sample values.If b = 4, we can have 2b=16 different possible sample values...

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Slide 3Digital Signal Processing

Quantization

Example: let x (t) = 0.9n.

If we sample at every 1 s, X (n) = {1, 0.9, 0.81, 0.229, 0.6561, 0.59049,….}

The sample values are infinite set of numbers between 0 and 1

If b = 4, then we have 16 possible codeword's {0000, 0001, ..,1111}

Each code word can be assigned to one sample value. So we can have maximum 16 possible values of the samples.

Slide 4Digital Signal Processing

Quantization

Eq(n)= xq(n)-x(n)Xq(n)X(n)=x(nT)

0.01.010

0.00.90.91

-0.010.80.812

-0.0290.70.7293

0.04390.70.65614

0.009510.60.590495

-0.0314410.50.5314416

0.02170310.50.47829697

-0.030467210.40.430467218

0.0125795110.40.3874204899

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Slide 5Digital Signal Processing

Before quantization

0 1 2 3 4 5 6 7 8 90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11

The value of x(n) = x(nTs) can take any value from 0 1 (continuous values)

Slide 6Digital Signal Processing

After quantization

0 1 2 3 4 5 6 7 8 90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11

The value of xq(n) can take only discrete values from the set {0, 0.1, 0.2, …, 1}

00010010001101000101

0110

0000

0111100010011010

∆

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Slide 7Digital Signal Processing

Quantization error

The quantization error eq (n) is limited to the range -∆/2 to ∆/2 that is

- ∆/2 eq (t ) ∆/2

If xmin and xmax represent the maximum and the minimum values of x(n ) and L is the number of quantization levels, then

∆ =(xmax - xmin)/(L-1)

Quantization noise can be reduced by increasing L.

Quantization is an irreversible process

Slide 8Digital Signal Processing

Quantization of sinusoidal signal

Consider the following sinusoidal signal

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Slide 9Digital Signal Processing

Quantization of sinusoidal signal

The mean-square error power pq is

But as eq(t )= ∆t/(2) - t

If the quantizer has b bits accuracy and covers the entire range 2A. Then the quantization step is ∆=2A/2b (we assume here that L = 2b is large )

Pq = A2/ (3*22b)

dtteP qq

2

21

12221 2

22

dttPq

Slide 10Digital Signal Processing

Quantization of sinusoidal signal

The average power of the sinusoidal signal is Ps = A2/2

Then the signal to quantization noise ratio is SQNR = Ps /Pq = (3/2)22b

Expressed in dB

SQNR = 1.76 + 6.02b dB

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Slide 11Digital Signal Processing

Coding of quantized samples

Each level is assigned a unique codeword

Number of required bits ‘b’ is equal or greater than log2 L ; where L is the number of levels

Digital to analogue Conversion Sample and hold

Interpolation between samples

Slide 12Digital Signal Processing

Homework

Students are encouraged to solve the following questions from the textbook

1.7, 1.8, 1.9 and 1.10.