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Quantization Error Analysis

Author: Author: Anil PothireddyAnil Pothireddy

12/10/200212/10/2002

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Organization of the Presentation

Introduction to Quantization.Introduction to Quantization. Quantization Error Analysis.Quantization Error Analysis. Quantization Error Reduction Techniques.Quantization Error Reduction Techniques.

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QUANTIZATIONQUANTIZATION

Definition :Definition : The transformation of a signal x[n] The transformation of a signal x[n] into one of a set of prescribed values.into one of a set of prescribed values.

Quantization converts a Discrete-Time Signal to Quantization converts a Discrete-Time Signal to a Digital Signal.a Digital Signal.

MATHEMATICAL REPRESENTATION:MATHEMATICAL REPRESENTATION:

xxqq[n] = Q( x[n] )[n] = Q( x[n] )

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EXAMPLE OF QUANTIZATIONEXAMPLE OF QUANTIZATION

(a) Unquantized samples of x[n] = 0.99cos(n/10). (b) with a 3-bit quantizer.(a) Unquantized samples of x[n] = 0.99cos(n/10). (b) with a 3-bit quantizer.

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QUANTIZERQUANTIZER

Quantizers can be defined with either uniformly Quantizers can be defined with either uniformly or non uniformly spaced quantization levels.or non uniformly spaced quantization levels.

Quantizers can also be customized to work on Quantizers can also be customized to work on either uni-polar or bipolar signals.either uni-polar or bipolar signals.

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TYPICAL QUANTIZERTYPICAL QUANTIZER

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QUANTIZATION LEVELSQUANTIZATION LEVELS

In the previous figure, the 8-quantization levels, In the previous figure, the 8-quantization levels, can be labeled using a binary code of 3–bits.can be labeled using a binary code of 3–bits.

In general, to represent B-quantization levels we In general, to represent B-quantization levels we need logneed log22B(rounded to next highest integer) bits. B(rounded to next highest integer) bits.

The step size of the quantizer will be:The step size of the quantizer will be:

∆ ∆ = 2X= 2Xm m // 2 2BB

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ADVANTAGES OF QUANTIZATIONADVANTAGES OF QUANTIZATION

The quantized signal, which is an approximation The quantized signal, which is an approximation of the original signal, can be more efficiently of the original signal, can be more efficiently separated from ADDITIVE NOISE. (by using separated from ADDITIVE NOISE. (by using repeaters).repeaters).

Transmission bandwidth can be controlled by Transmission bandwidth can be controlled by using an appropriate number of quantization using an appropriate number of quantization levels (and hence the bits to represent them).levels (and hence the bits to represent them).

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QUANTIZATION ERRORQUANTIZATION ERROR

The quantized sample will generally differ from The quantized sample will generally differ from the original signal. The difference between them the original signal. The difference between them is called the quantization error.is called the quantization error.

e[n] = xe[n] = xqq[n] - x[n][n] - x[n]

For a 3-bit Quantizer, if For a 3-bit Quantizer, if ∆/2 < x[n] =< 3 ∆/2, then ∆/2 < x[n] =< 3 ∆/2, then xxqq[n] = [n] = ∆, and it follows that: ∆, and it follows that:

-∆/2 < e[n] =< ∆/2-∆/2 < e[n] =< ∆/2

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QUANTIZER MODELQUANTIZER MODEL

In this model, the quantization error samples are In this model, the quantization error samples are thought of as an ADDITIVE NOISE SIGNAL. (The thought of as an ADDITIVE NOISE SIGNAL. (The model is exactly equivalent to a Quantizer if e[n] is model is exactly equivalent to a Quantizer if e[n] is exactly known).exactly known).

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STATISTICAL REPRESENTATION OF STATISTICAL REPRESENTATION OF QUANTIZATION ERRORSQUANTIZATION ERRORS

ASSUMPTIONSASSUMPTIONS e[n] is a sample sequence of a stationary random e[n] is a sample sequence of a stationary random

process.process. e[n] is uncorrelated with the sequence x[n].e[n] is uncorrelated with the sequence x[n]. The random variables of the error process are The random variables of the error process are

uncorrelated.uncorrelated. The probability distribution of the error process is The probability distribution of the error process is

uniform over the range of quantization error.uniform over the range of quantization error.

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QUANTIZATION ERROR (3-BIT & 8-BIT)QUANTIZATION ERROR (3-BIT & 8-BIT)

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STATISTICAL REPRESENTATION OF STATISTICAL REPRESENTATION OF QUANTIZATION ERRORS (2)QUANTIZATION ERRORS (2)

We know that : -∆/2 < e[n] =< ∆/2We know that : -∆/2 < e[n] =< ∆/2 For small ∆,it is reasonable to assume that e[n] For small ∆,it is reasonable to assume that e[n]

is a Random variable uniformly distributed from is a Random variable uniformly distributed from --∆/2 to ∆/2.--∆/2 to ∆/2.

Thus e[n] is a uniformly distributed white-noise Thus e[n] is a uniformly distributed white-noise sequencesequence

The mean value of e[n] = 0.The mean value of e[n] = 0.

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STATISTICAL REPRESENTATION OF STATISTICAL REPRESENTATION OF QUANTIZATION ERRORS (3)QUANTIZATION ERRORS (3)

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STATISTICAL REPRESENTATION OF STATISTICAL REPRESENTATION OF QUANTIZATION ERRORS (4)QUANTIZATION ERRORS (4)

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OBSERVATIONSOBSERVATIONS

We see that the signal-to-noise ratio increases We see that the signal-to-noise ratio increases approximately 6dB for each bit added to the approximately 6dB for each bit added to the word length of the Quantized samples.word length of the Quantized samples.

If σIf σxx = X = Xm m / 4 then:/ 4 then: SNR SNR (approx) = 6B – 1.25dB.(approx) = 6B – 1.25dB.

Obtaining a 90-96dB SNR for use in High-Obtaining a 90-96dB SNR for use in High-Quality audio requires a 16-bit Quantization.Quality audio requires a 16-bit Quantization.

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QUANTIZATION ERROR REDUCTION QUANTIZATION ERROR REDUCTION TECHNIQUES.TECHNIQUES.

INCREASING THE SAMPLING RATE.INCREASING THE SAMPLING RATE.

DIFFERENTIAL QUANTIZATION.DIFFERENTIAL QUANTIZATION.

NON UNIFORM QUANTIZATIONNON UNIFORM QUANTIZATION

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INCREASING THE SAMPLING RATEINCREASING THE SAMPLING RATE

It has been proved that: for every doubling of the It has been proved that: for every doubling of the oversampling ratio M, we need ½ bit less to oversampling ratio M, we need ½ bit less to achieve a given Signal-to-Quantization-Noise achieve a given Signal-to-Quantization-Noise ratio.ratio.

If we oversample by a factor M=4, we need one If we oversample by a factor M=4, we need one less bit to achieve a desired accuracy in less bit to achieve a desired accuracy in representing a signal. (i.e. M = 4representing a signal. (i.e. M = 4 (no of bits reduced)(no of bits reduced)).).

This technique is of little practical importance, as This technique is of little practical importance, as it involves a rather high overheadit involves a rather high overhead

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DIFFERENTIAL QUANTIZATIONDIFFERENTIAL QUANTIZATION

In many practical situations, due to the statistical In many practical situations, due to the statistical nature of the message signal, the sequence x[n] nature of the message signal, the sequence x[n] will consist of samples that are correlated with will consist of samples that are correlated with each other.each other.

For a given number of levels per sample, For a given number of levels per sample, differential quantization schemes yield a lower differential quantization schemes yield a lower value of quantizing noise value than direct value of quantizing noise value than direct quantizing schemes.quantizing schemes.

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DIFFERENTIAL QUANTIZING SCHEMEDIFFERENTIAL QUANTIZING SCHEME

The error reduction is possible as long as the sample The error reduction is possible as long as the sample to sample correlation is non-zero.to sample correlation is non-zero.

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EXAMPLE PROBLEMEXAMPLE PROBLEM

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SOLUTIONSOLUTION

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NON UNIFORM QUANTIZATIONNON UNIFORM QUANTIZATION The qunatization error (noise) depends on the The qunatization error (noise) depends on the

step size step size ∆. Hence if the steps are uniform in ∆. Hence if the steps are uniform in size, small-amplitude signals will have a poor size, small-amplitude signals will have a poor Signal-to-Quantization-Noise ratio.Signal-to-Quantization-Noise ratio.

To illustrate this effect, assume a full scale voltage of 10V To illustrate this effect, assume a full scale voltage of 10V and that the actual resolution is +/- 4mV (i.e. and that the actual resolution is +/- 4mV (i.e. ∆ = 8mV).∆ = 8mV).When the signal is close to 10V, the peak quantization When the signal is close to 10V, the peak quantization error is in the neighborhood of (4mV / 10V ) * 100% = error is in the neighborhood of (4mV / 10V ) * 100% = 0.04%. 0.04%. When the signal level hovers around 10mV , the error is in When the signal level hovers around 10mV , the error is in the vicinity of (4mV / 10mV) * 100% = 40% !!!the vicinity of (4mV / 10mV) * 100% = 40% !!!

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NON UNIFORM QUANTIZATION (2)NON UNIFORM QUANTIZATION (2)

The severity of this problem depends on the The severity of this problem depends on the dynamic range of the signal and the number of dynamic range of the signal and the number of bits used in encoding (quantizing).bits used in encoding (quantizing).

In theory, a sufficient number of bits could be In theory, a sufficient number of bits could be added to decrease the peak quantization error to added to decrease the peak quantization error to a more tolerable level, but this is an inefficient a more tolerable level, but this is an inefficient and often impractical process.and often impractical process.

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COMPANDINGCOMPANDING To correct this situation within the constraint of To correct this situation within the constraint of

fixed number of levels, it is advantageous to fixed number of levels, it is advantageous to taper the step size so that the steps are close taper the step size so that the steps are close together at low signal amplitudes and further together at low signal amplitudes and further apart at large amplitudesapart at large amplitudes

This leads to the SNR improvement for small This leads to the SNR improvement for small signals, but the strong signals will be impaired.signals, but the strong signals will be impaired.

However the Inst. speech signal amplitude < ¼ However the Inst. speech signal amplitude < ¼ rms signal value, for more than 50% of the time.rms signal value, for more than 50% of the time.

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COMPANDER (2)COMPANDER (2)

While it is possible to build a quantizer with While it is possible to build a quantizer with tapered steps, it is more feasible/practical to tapered steps, it is more feasible/practical to achieve an equivalent effect by distorting the achieve an equivalent effect by distorting the signal before quantizing.signal before quantizing.

An inverse distortion is introduced at the An inverse distortion is introduced at the receiving end so that the overall transmission is receiving end so that the overall transmission is distortionless. distortionless.

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COMPANDER (3)COMPANDER (3)

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COMPANDER (4)COMPANDER (4)

At low amplitudes the slope is larger than at high At low amplitudes the slope is larger than at high amplitudes. amplitudes.

Consequently a given signal change at low Consequently a given signal change at low amplitude will carry the quantizer through more amplitude will carry the quantizer through more steps than will be the case at large amplitudes.steps than will be the case at large amplitudes.

This network is called a COMPRESSOR. The This network is called a COMPRESSOR. The inverse operation is performed by a inverse operation is performed by a EXPANDER. The combination is called a EXPANDER. The combination is called a COMPANDER.COMPANDER.

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REFERENCESREFERENCES Discrete Time Signal Processing.Discrete Time Signal Processing.

Oppenheim and Schaffer,Prentice Hall.Oppenheim and Schaffer,Prentice Hall. Digital and Analog Communication Systems.Digital and Analog Communication Systems.

K. Shanmugam, John Wiley.K. Shanmugam, John Wiley. Principles of Communication Systems.Principles of Communication Systems.

Taub and Shilling, McGraw-Hill.Taub and Shilling, McGraw-Hill.

Web Sources:Web Sources: www.dspguru.comwww.dspguru.com www.ece.utexas.eduwww.ece.utexas.edu