Chapter 6

Crosscorrelations between Quantization Noise, Quantizer

Input, and Quantizer Output

Quantization noise is deterministically related to the quantizer input signal. If this noise were independent of the quantizer input, which it is not, then knowledge of the statistics of the noise would be sufficient to determine input–output relations for the quantizer. The actual situation is somewhat more complicated.

Of interest are the crosscorrelation between quantization noise and the quantizer input, between quantization noise and the quantizer output, and between the quantizer input and output. General expressions of these crosscorrelations will be obtained, but we will begin this chapter by finding simple relations that are valid only when certain of the Quantizing Theorems are satisfied.

6.1 CROSSCORRELATIONS WHEN QUANTIZING THEOREM II IS SATISFIED

6.1.1 Crosscorrelation between Quantization Noise and the Quantizer Input

Fig. 1.1(a) shows actual quantization noise ν being added to the quantizer input x to make the sum

x′D x C ν .  (6.1)

Fig. 1.1(b) shows an independent pseudo quantization noise n being added to the quantizer input x to make the sum x C n. This figure contrasts a quantization model with a related pseudo quantization noise model (PQN).

PSfrag replacements

x x C ν D x′ ν n C x C n (a)

(b) 6

Figure 6.1 Comparison between a quantization model and a pseudo quantization noise model: (a) quantization; (b) PQN.

Referring to Fig. 1.1(a), the square of the actual quantizer output can be expressed as

▷x C ν◁^{2} D x^{2} C ν^{2} C 2xν .  (6.2)

Taking expectation of both sides gives

E{▷x C ν◁^{2}} D E{x^{2}} C E{ν^{2}} C 2E{xν} .  (6.3)

Referring now to Fig. 1.1(b), the square of the sum of x and the pseudo noise n can be written as

▷x C n◁^{2} D x^{2} C n^{2} C 2xn .  (6.4)

The expected value of both sides is

E{▷x C n◁^{2}} D E{x^{2}} C E{n^{2}} C 2E{xn} .  (6.5)

Regarding the actual quantization process, assume that the conditions for Quantizing Theorem II are met. From Eqs. (

E{▷x C ν◁^{2}} D E{▷x C n◁^{2}},  and E{ν^{2}} D E{n^{2}} . (6.6)

Combining (1.6) with (1.5) and (1.3), we conclude that

E{xν} D E{xn} .  (6.7)

Since n has zero mean and is independent of x, E{xn} D 0, and therefore

E{xν} D 0 .  (6.8)

So, if Quantizing Theorem II is satisfied, the crosscorrelation between the quantization noise and the quantizer input is zero. Since the mean of ν is zero, the covariance of ν and x is also zero. This is a remarkable result, considering that ν is deterministically related to x. If Quantizing Theorem II is only approximately satisfied, then the quantization noise and the quantizer input will only be approximately uncorrelated. Examples will show how this works.

6.1.2 Crosscorrelation between Quantization Noise and the Quantizer Output

Referring once again to Fig. 1.1(a), the crosscorrelation between quantization noise and the quantizer output can be seen to be

E{νx′} D E{ν▷x C ν◁} D E{xν} C E{ν^{2}} .  (6.9)

From Fig. 1.1(b), a similar relation for the PQN model is

E{n▷x C n◁} D E{xn} C E{n^{2}} .  (6.10)

Assuming conditions for Quantizing Theorem II are met, then

E{ν^{2}} D E{n^{2}},  and E{xν} D E{xn} .  (6.11)

Combining (1.11) with (1.10) and (1.9), we obtain

E{νx′} D E{ν▷x C ν◁} D E{n▷x C n◁} .  (6.12)

Since n has zero mean and is independent of x,

E\{νx′\} D E\{ν▷x C ν◁\} D E\{n^{2}\} D \frac{1}{12}q^{2} .  (6.13)

Therefore when the conditions for Quantizing Theorem II are met, the crosscorrelation between quantization noise ν and the quantizer output x′equals the mean square of the quantization noise, q^{2}/12. Since the mean of ν is zero, the covariance of ν and x′is equal to the crosscorrelation. The covariance of n and x C n is also

cov\{n, x C n\} D cov\{ν, x C ν\} D \frac{q^{2}}{12} .  (6.14)

6.1.3 Crosscorrelation between the Quantizer Input and the Quantizer Output

Regarding Fig. 1.1(a), the crosscorrelation between the quantizer input and output is

E{xx′} D E{x▷x C ν◁} D E{x^{2}} C E{xν} .  (6.15)

For the PQN model of Fig. 1.1(b), a similar relation can be written as

E{x▷x C n◁} D E{x^{2}} C E{xn} .  (6.16)

Assuming that the conditions for QT II are met, (1.11) is fulfilled. Combining this with (1.16) and (1.15), we have

E{xx′} D E{x▷x C ν◁} D E{x▷x C n◁} .  (6.17)

Since n has zero mean and is independent of x, we may conclude that

E{xx′} D E{x▷x C ν◁} D E{x^{2}} .  (6.18)

Therefore, when QT II is satisfied, the crosscorrelation between the quantizer input and output equals the mean square of the input. Since the mean of x′is the same as the mean of x, the covariance of x and x′is equal to their crosscorrelation minus the square of the mean, i.e.

cov{x, x′} D E{x^{2}} − ▷E{x}◁^{2} D var{x} .  (6.19)

A similar result is obtained for the PQN model,

cov{x, ▷x C n◁} D E{x^{2}} − ▷E{x}◁^{2} D var{x} .  (6.20)

6.2 GENERAL EXPRESSIONS OF CROSSCORRELATIONS

6.2.1 Crosscorrelation between Quantization Noise and the Quantizer Input

In Chapter Referring to Eq. (

Substituting this into Eq. (

E\{▷x′◁^{2}\}DE\{x^{2}\} C E\{ν^{2}\} C 2\frac{q}{2π} ∑_{{lD−∞ \atop lD̸0}}^{∞} P 8_{x}▷l9◁\frac{▷−1◁^{lC1}}{l}

DE\{x^{2}\} C E\{ν^{2}\} C 2\frac{q}{π} ∑_{lD1}^{∞} Re\{ P 8_{x}▷l9◁\} \frac{▷−1◁^{lC1}}{l} . (6.21)

From (1.3), we have

E{▷x′◁^{2}} D E{x^{2}} C E{ν^{2}} C 2E{xν} .  (6.22)

Combining (1.22) and (1.21), we conclude that

E\{xν\}D \frac{q}{2π} ∑_{{lD−∞ \atop lD̸0}}^{∞} P 8_{x}▷l9◁\frac{▷−1◁^{lC1}}{l}

D \frac{q}{π} ∑_{lD1}^{∞} Re\{ P 8_{x}▷l9◁\} \frac{▷−1◁^{lC1}}{l} .  (6.23)

This is a general expression of the crosscorrelation between the quantization noise ν and the quantizer input signal x. Knowing 8 _{x}▷u◁, a number of terms in the series could be evaluated and summed to approximate E{xν}.

Since the mean of ν is not necessarily zero, and the mean of x is not necessarily zero, the covariance of x and ν is not necessarily equal to the crosscorrelation of x and ν. The covariance can be generally written as

cov{x, ν} D E{xν} − E{x}E{ν} .  (6.24)

By using (1.23) and (

cov\{x, ν\}D \frac{q}{2π} ∑_{{lD−∞ \atop lD̸0}}^{∞} P 8_{x}▷l9◁\frac{▷−1◁^{lC1}}{l} − E\{x\}\frac{q}{2π} ∑_{{lD−∞ \atop lD̸0}}^{∞} \frac{8 _{x}▷l9◁}{j} \frac{▷−1◁^{l}}{l}

D \frac{q}{π} ∑_{lD1}^{∞} Re\{ P 8_{x}▷l9◁\} \frac{▷−1◁^{lC1}}{l} − E\{x\}\frac{q}{π} ∑_{lD1}^{∞} Im\{8 _{x}▷l9◁\} \frac{▷−1◁^{l}}{l} .(6.25)

If QT II is satisfied, Eq. (1.23) indicates that E{xν} D 0, and (1.25) also indicates that cov{x, ν} D 0. The reason is that when 8 _{x}▷u◁ is bandlimited, so is P 8_{x}▷u◁. The replicas contained in the sums (1.23) and (1.25) do not overlap at the origin u D 0, and the replicas at u D 0 are not present in the sums.

The correlation and the covariance between x and ν can also be expressed in terms of the mean value and the characteristic function of the zero-mean variable, Q xD x − μ (see Eq. (

E\{xν\}D \frac{q}{2π} ∑_{{lD−∞ \atop lD̸0}}^{∞} e^{jl9μ} (jμ 8 Q x▷l9◁ C P 8Q x▷l9◁) \frac{▷−1◁^{lC1}}{l}

D \frac{q}{π} ∑_{lD1}^{∞} Re\{e^{jl9μ} (jμ 8 Q x▷l9◁ C P 8Q x▷l9◁) \} \frac{▷−1◁^{lC1}}{l}

Dμ\frac{q}{π} ∑_{lD1}^{∞} Im\{e^{jl9μ} 8 Q x▷l9◁\} \frac{▷−1◁^{l}}{l}

C \frac{q}{π} ∑_{lD1}^{∞} Re\{e^{jl9μ} P 8Q x▷l9◁\} \frac{▷−1◁^{lC1}}{l} ,  (6.26)

and

cov\{x, ν\}D \frac{q}{2π} ∑_{{lD−∞ \atop lD̸0}}^{∞} P 8Q x▷l9◁\frac{▷−1◁^{lC1}}{l} e^{jl9μ}

D \frac{q}{π} ∑_{lD1}^{∞} Re\{ P 8Q x▷l9◁\frac{▷−1◁^{lC1}}{l} e^{jl9μ} \} .  (6.27)

By using (

E{xν} D μE{ν} C cov{x, ν} .  (6.28)

This equation can also be obtained from the definition of Q x:

E{xν} D E{▷μ C Q x◁ν} D μE{ν} C E{ Q xν} D μE{ν} C cov{x, ν} . (6.29)

When the PDF of the quantizer input is symmetric about its mean value, these expressions can be further simplified:

E\{xν\}Dμ\frac{q}{π} ∑_{lD1}^{∞} 8 Q x▷l9◁ sin▷l9μ◁ \frac{▷−1◁^{l}}{l}

C \frac{q}{π} ∑_{lD1}^{∞} P 8Q x▷l9◁\frac{▷−1◁^{lC1}}{l} cos▷l9μ◁ .  (6.30)

cov\{x, ν\} D \frac{q}{π} ∑_{lD1}^{∞} P 8Q x▷l9◁\frac{▷−1◁^{lC1}}{l} cos▷l9μ◁ .  (6.31)

Equations (1.23), (1.25), (1.26), (1.27), (1.30), and (1.31) are exact expressions of the crosscorrelation and covariance of ν and x, and they apply whether QT II is satisfied or not. If the conditions for QT II are met, then the crosscorrelation and the covariance will both be zero. If QT II is approximately satisfied, then these moments will be small, close to zero. This will be demonstrated later in this chapter with the quantization of Gaussian signals.

We have learned in Chapter

6.2.2 Crosscorrelation between Quantization Noise and the Quantizer Output Signal

In this section, we will obtain a general expression of the crosscorrelation between the quantization noise and the quantizer output. From this crosscorrelation, we will derive the corresponding covariance. To get the general expression, we will not assume that conditions for any of the quantizing theorems are being met.

The crosscorrelation between the quantization noise and the quantizer output signal is

E{νx′} D E{ν▷x C ν◁} D E{νx} C E{ν^{2}} .  (6.32)

In Subsection 1.2.1, it was established that

E\{νx\} D \frac{q}{2π} ∑_{{lD−∞ \atop lD̸0}}^{∞} P 8_{x}▷l9◁\frac{▷−1◁^{lC1}}{l} .  (1.23)

In Section Therefore,

E\{νx′\}D \frac{q^{2}}{12} C \frac{q}{2π} ∑_{{lD−∞ \atop lD̸0}}^{∞} P 8_{x}▷l9◁\frac{▷−1◁^{lC1}}{l} C \frac{q^{2}}{2π^{2}} ∑_{{lD−∞ \atop lD̸0}}^{∞} 8 _{x}▷l9◁\frac{▷−1◁^{l}}{l^{2}}

D \frac{q^{2}}{12} C \frac{q}{π} ∑_{lD1}^{∞} Re\{ P 8_{x}▷l9◁\} \frac{▷−1◁^{lC1}}{l} C \frac{q^{2}}{π^{2}} ∑_{lD1}^{∞} Re\{8 _{x}▷l9◁\}\frac{▷−1◁^{l}}{l^{2}} .(6.33)

This is the general expression. With knowledge of 8 _{x}▷u◁, it could be used to evaluate E{νx′}.

If the PDF of x is symmetric about its mean value, this can be written as:

E\{νx′\}D \frac{q^{2}}{12}

C \frac{q}{π} ∑_{lD1}^{∞} μ 8 Q x▷l9◁\frac{▷−1◁^{l}}{l} sin▷l9μ◁

C \frac{q}{π} ∑_{lD1}^{∞} P 8Q x▷l9◁\frac{▷−1◁^{lC1}}{l} cos▷l9μ◁

C \frac{q^{2}}{π^{2}} ∑_{lD1}^{∞} 8 Q x▷l9◁\frac{▷−1◁^{l}}{l^{2}} cos▷l9μ◁ .  (6.34)

The covariance of ν and x′may be obtained in the following manner,

cov{ν, x′}DE{▷ν − E{ν}◁▷x′− E{x′}◁}
DE{νx′} − E{ν}E{x′} − E{x′}E{ν} C E{ν}E{x′}
DE{νx′} − E{ν}E{x′} .  (6.35)

From Eq. (1.33), we have E{νx′}. From (

cov\{ν, x′\}D \frac{q^{2}}{12}

C \frac{q}{2π} ∑_{{lD−∞ \atop lD̸0}}^{∞} P 8_{x}▷l9◁\frac{▷−1◁^{lC1}}{l}

C \frac{q^{2}}{2π^{2}} ∑_{{lD−∞ \atop lD̸0}}^{∞} 8 _{x}▷l9◁\frac{▷−1◁^{l}}{l^{2}}

− ( \frac{q}{2π} ∑_{{lD−∞ \atop lD̸0}}^{∞} \frac{8 _{x}▷l9◁}{j} \frac{▷−1◁^{l}}{l} ) ( E\{x\} C \frac{q}{2π} ∑_{{lD−∞ \atop lD̸0}}^{∞} \frac{8 _{x}▷l9◁}{j} \frac{▷−1◁^{l}}{l} )

D \frac{q^{2}}{12}

C \frac{q}{π} ∑_{lD1}^{∞} Re\{ P 8_{x}▷l9◁\}\frac{▷−1◁^{lC1}}{l}

C \frac{q^{2}}{π^{2}} ∑_{lD1}^{∞} Re\{8 _{x}▷l9◁\}\frac{▷−1◁^{l}}{l^{2}}

− ( \frac{q}{π} ∑_{lD1}^{∞} Im\{8 _{x}▷l9◁\}\frac{▷−1◁^{l}}{l} ) ( E\{x\} C \frac{q}{π} ∑_{lD1}^{∞} Im\{8 _{x}▷l9◁\}\frac{▷−1◁^{l}}{l} ).

(6.36)

This is a general expression of the covariance. If QT II were satisfied, cov{ν, x′} D q^{2}/12 would be true.

If the PDF of x is symmetrical about its mean value, this can be written as:

cov\{ν, x′\}D \frac{q^{2}}{12}

C \frac{q}{π} ∑_{lD1}^{∞} ( − μ 8 Q x▷l9◁ sin▷l9μ◁ C P 8Q x▷l9◁ cos▷l9μ◁ ) \frac{▷−1◁^{lC1}}{l}

C \frac{q^{2}}{π^{2}} ∑_{lD1}^{∞} 8 Q x▷l9◁ cos▷l9μ◁ \frac{▷−1◁^{l}}{l^{2}}

− ( \frac{q}{π} ∑_{lD1}^{∞} 8 Q x▷l9◁ sin▷l9μ◁ \frac{▷−1◁^{l}}{l} )

× ( E\{x\} C \frac{q}{π} ∑_{lD1}^{∞} 8 Q x▷l9◁ sin▷l9μ◁ \frac{▷−1◁^{l}}{l} )

D \frac{q^{2}}{12}

C \frac{q}{π} ∑_{lD1}^{∞} P 8Q x▷l9◁ cos▷l9μ◁ \frac{▷−1◁^{lC1}}{l}

C \frac{q^{2}}{π^{2}} ∑_{lD1}^{∞} 8 Q x▷l9◁ cos▷l9μ◁ \frac{▷−1◁^{l}}{l^{2}}

− ( \frac{q}{π} ∑_{lD1}^{∞} 8 Q x▷l9◁ sin▷l9μ◁ \frac{▷−1◁^{l}}{l} ) ^{2} .  (6.37)

6.2.3 Crosscorrelation between the Quantizer Input and Output Signals

In this section, we will obtain a general expression of the crosscorrelation between the quantizer input signal and the quantizer output signal. From this, we will obtain the corresponding covariance. For the general expression, we will not assume that conditions for any of the quantizing theorems are met. Accordingly,

E{xx′} D E{x▷x C ν◁} D E{x^{2}} C E{xν} .  (6.38)

This can be written (see (1.23)) as

E\{xx′\}DE\{x^{2}\} C \frac{q}{2π} ∑_{{lD−∞ \atop lD̸0}}^{∞} P 8_{x}▷l9◁\frac{▷−1◁^{lC1}}{l}

DE\{x^{2}\} C \frac{q}{π} ∑_{lD1}^{∞} Re\{ P 8_{x}▷l9◁\}\frac{▷−1◁^{lC1}}{l} .  (6.39)

This is the general expression. If QT II is satisfied, then E{xx′} D E{x^{2}}. If the PDF of x is symmetrical about its mean value, this can be written as:

E\{xx′\} D E\{x^{2}\} C \frac{q}{π} ∑_{lD1}^{∞} ( − μ 8 Q x▷l9◁ sin▷l9μ◁ C P 8Q x▷l9◁ cos▷l9μ◁ ) \frac{▷−1◁^{lC1}}{l} .

(6.40)

The covariance of x and x′can be expressed as follows.

cov{x, x′} D E{▷x − E{x}◁▷x′− E{x′}◁} D E{xx′} − E{x}E{x′} . (6.41)

Equation (1.39) gives E{xx′}, and Eq. (

cov{x, x′}DE{x^{2}}

C \frac{q}{2π} ∑_{{lD−∞ \atop lD̸0}}^{∞} P 8_{x}▷l9◁\frac{▷−1◁^{lC1}}{l}

− E\{x\} ( E\{x\} C \frac{q}{2π} ∑_{{lD−∞ \atop lD̸0}}^{∞} \frac{8 _{x}▷l9◁}{j} \frac{▷−1◁^{l}}{l} )

DE{x^{2}} − ▷E{x}◁^{2}

C \frac{q}{2π} ∑_{{lD−∞ \atop lD̸0}}^{∞} P 8_{x}▷l9◁\frac{▷−1◁^{lC1}}{l} − E\{x\}\frac{q}{2π} ∑_{{lD−∞ \atop lD̸0}}^{∞} \frac{8 _{x}▷l9◁}{j} \frac{▷−1◁^{l}}{l}

Dvar\{x\} C \frac{q}{2π} ∑_{{lD−∞ \atop lD̸0}}^{∞} P 8_{x}▷l9◁\frac{▷−1◁^{lC1}}{l} − E\{x\}\frac{q}{2π} ∑_{{lD−∞ \atop lD̸0}}^{∞} 8 _{x}▷l9◁\frac{▷−1◁^{l}}{l}

Dvar\{x\} C \frac{q}{π} ∑_{lD1}^{∞} Re\{ P 8_{x}▷l9◁\}\frac{▷−1◁^{lC1}}{l} − E\{x\}\frac{q}{π} ∑_{lD1}^{∞} Im\{8 _{x}▷l9◁\}\frac{▷−1◁^{l}}{l} .

(6.42)

This is the general expression of the covariance of x and x′. If conditions for QT II are met, cov{x, x′} D var{x}.

If the PDF of x is symmetrical about its mean value, this can be written as:

cov\{x, x′\} D var\{x\} C \frac{q}{π} ∑_{lD1}^{∞} P 8Q x▷l9◁ cos▷l9μ◁ \frac{▷−1◁^{lC1}}{l} .  (6.43)

The first term corresponds to the PQN model. The second term vanishes when conditions for QT II are fulfilled.

6.3 CORRELATION AND COVARIANCE BETWEEN GAUSSIAN QUANTIZER INPUT AND ITS QUANTIZATION NOISE

The Gaussian CF is not bandlimited and does not perfectly satisfy QT II. However, conditions for QT II are approximately met for a wide range of values of q, from zero up to 1.5σ_{x} or more.

If QT II were perfectly satisfied, the PQN model yields the following moments: E{ν} D 0, E{ν^{2}} D q^{2}/12, E{xν} D 0, cov{x, ν} D 0, E{νx′} D q^{2}/12, cov{ν, x′} D q^{2}/12, E{xx′} D E{x^{2}} , and cov{x, x′} D E{ Q x^{2}} . With a Gaussian quantizer input, these moments will depart somewhat from the PQN moments. Exact expressions from the previous sections were used to calculate these moments.

Table 1.1 lists calculated moments for the case of a zero-mean Gaussian input. Tables 1.2, and 1.3, and 1.4 list calculated moments for the case of a Gaussian input with a mean of q/4. The calculations were made by making use of Eqs. (1.30), (1.31), (1.34), (1.37), (1.40), and (1.43).

124  6 Crosscorrelations

TABLE 6.1 Quantization correlations, Gaussian input with μ D 0.

\frac{E\{xν\}}{\sqrt{E\{x^{2}\}E\{ν^{2}\}}}(D ρ_{x,ν})  E\{x′ν\} = cov\{x′, ν\}  E\{xx′\} = cov\{x, x′\}
q D 2σ  −2.50 · 10^{−2}  (1 − 5.19 · 10^{−2} ) \frac{q^{2}}{12} (1 − 1.44 · 10^{−2} ) E\{x^{2}\}
q D 1.5σ  −7.15 · 10^{−4}  (1 − 1.84 · 10^{−3} ) \frac{q^{2}}{12} (1 − 3.10 · 10^{−4} ) E\{x^{2}\}
q D σ  −1.85 · 10^{−8}  (1 − 6.75 · 10^{−8} ) \frac{q^{2}}{12} (1 − 5.35 · 10^{−9} ) E\{x^{2}\}
q D 0.5σ  −7.10 · 10^{−34}  (1 − 4.98 · 10^{−33} ) \frac{q^{2}}{12} (1 − 1.02 · 10^{−34} ) E\{x^{2}\}

TABLE 6.2 Normalized quantization correlations, Gaussian input with μ D q/4.

\frac{E\{xν\}}{\sqrt{E\{x^{2}\}E\{ν^{2}\}}} \frac{cov\{x, ν\}}{\sqrt{var\{x\} var\{ν\}}}D ρ_{x,ν}
q D 2σ  −3.55 · 10^{−3}  −9.27 · 10^{−9}
q D 1.5σ −6.00 · 10^{−5}  −2.66 · 10^{−15}
q D σ  −7.15 · 10^{−10} −1.13 · 10^{−24}
q D 0.5σ −7.01 · 10^{−34} −4.35 · 10^{−50}

TABLE 6.3 Quantization output-noise correlations, Gaussian input with μ D q/4.

E{x′ν}  cov{x′, ν}
q D 2σ  (1 − 6.87 · 10^{−3} ) \frac{q^{2}}{12} (1 − 6.29 · 10^{−5} ) \frac{q^{2}}{12}
q D 1.5σ (1 − 1.48 · 10^{−4} ) \frac{q^{2}}{12} (1 − 2.92 · 10^{−8} ) \frac{q^{2}}{12}
q D σ  (1 − 2.55 · 10^{−9} ) \frac{q^{2}}{12} (1 − 8.70 · 10^{−18} ) \frac{q^{2}}{12}
q D 0.5σ (1 − 4.89 · 10^{−35} ) \frac{q^{2}}{12} (1 − 3.04 · 10^{−49} ) \frac{q^{2}}{12}

TABLE 6.4 Quantization input–output correlations, Gaussian input with μ D q/4.

E{xx′}  cov{x, x′}
q D 2σ  (1 − 6.86 · 10^{−2} ) E{x^{2}} (1 − 1.02 · 10^{−2} ) E{ Q x^{2}}
q D 1.5σ (1 − 1.61 · 10^{−4} ) E{x^{2}} (1 − 1.55 · 10^{−4} ) E{ Q x^{2}}
q D σ  (1 − 2.00 · 10^{−10} ) E{x^{2}} (1 − 3.10 · 10^{−25} ) E{ Q x^{2}}
q D 0.5σ (1 − 9.64 · 10^{−35} ) E{x^{2}} (1 − 1.02 · 10^{−49} ) E{ Q x^{2}}

The moments listed in Tables 1.1, 1.2, 1.3, and 1.4 are presented in a way that it brings into evidence the differences between them and the corresponding moments of the PQN model. The first entry of Table 1.1 is the correlation coefficient between x and ν.

ρ_{x,ν} D \frac{cov\{x, ν\}}{\sqrt{var\{x\} var\{ν\}}}.  (6.44)

This correlation coefficient is a normalized form of the correlation, and shows a value of 2.5% when q D 2σ, a value of 0.0715% when q D 1.5σ, and even more negligible values for finer quantization. The PQN model gives this correlation a value of zero. What we see is that when quantizing a Gaussian signal, the quantization noise is essentially uncorrelated with the quantizer input, even when the quantization is extremely rough, as rough as q D 2σ. The same idea is evident from inspection of Table 1.2, where the Gaussian input is biased by q/4.

Also shown in Table 1.1 is the crosscorrelation between ν and x′. The PQN model gives this a value of q^{2}/12. The table shows this correlation to differ from that of PQN by 5.19% when q D 2σ. Table 1.2 shows this correlation to differ from that of PQN by 3.19% when the input is biased by q/4 and q D 2σ. Once again, the PQN model works very well even with very rough quantization.

Table 1.1 shows the crosscorrelation between x and x′. The PQN model that works when the conditions for QT II are met shows this crosscorrelation to be equal to E{x^{2}}. According to the table, this correlation differs from that of the PQN model by 1.44% when q D 2σ. Table 1.4 shows this difference to be 1.03% when the input is biased by q/4. Again the deviations from the PQN model’s moments are slight even with very rough quantization.

6.4 CONDITIONS OF ORTHOGONALITY OF x AND ν:

QUANTIZING THEOREM III/B

It is possible for E{xν} to equal zero without the conditions of QT II being met. In this section we will formulate and prove the necessary and sufficient conditions for x D Q xC μ to be orthogonal to ν, for any value of μ.

Quantizing Theorem III/B^{1}

The input variable x D Q xCμ and the quantization noise ν are orthogonal for any value of μ if and only if the value and the first derivative of the CF of x is zero at integer multiples of 9:

8 _{x}▷l9◁ D 0, P 8_{x}▷l9◁ D 0 for l D ±1, ±2 …  (6.45)

Moreover, if and only if (1.45) is true,

E{xν^{t}} D E{xn^{t}} for t D 1, 2, …  (6.46)

holds, independently of the value of μ.

We will prove here the significantly simpler statement (x and ν are orthogonal, that is, E{xν} D E{xn} D 0, for the given value of μ, if P 8_{x}▷l9◁ D 0 for l D ±1, ±2 …), and will develop a general proof in Appendix

We establish the following arguments. Sufficient conditions of orthogonality for a given value of μ come directly from Eq. (1.23), as follows:

E{xν} D 0  (6.47)

when

P 8_{x}▷l9◁ D 0 , for l D ±1, ±2, …  (6.48)

However, if condition (1.48) is satisfied, a small change in the mean value of x may change the value of the derivative, so condition (1.48) may not be satisfied. Therefore, condition (1.48) is not very useful by itself.

Let us require that the correlation of x and ν be zero independently of the signal mean value μ.

Considering (1.26), it is clear that (1.45) is a sufficient condition for the correlation to be zero, for x and ν to be orthogonal, independently of the mean value of the input signal.

We will prove that this is also a necessary condition. If (1.26) is zero, independently of the value of μ, it remains zero when the mean value is changed by an arbitrary value μ_{m} to μ C μ_{m}. The mean value enters the expressions implicitly,

1Although it is natural to discuss QT III/B here, this turns out to be a special case of a more general theorem, QT III, yet to be introduced in Section

through 8 _{x}▷u◁. By changing the mean value to μ C μ_{m}, the characteristic function is changed from 8 _{x}▷u◁ to e^{juμ_{m}} 8 _{x}▷u◁. Substituting this into the first line of ▷1.26◁, and by choosing for μ_{m} the arbitrary values μ_{0} and μ_{0} C q, the equation becomes

0D \frac{q}{2π} ∑_{{lD−∞ \atop lD̸0}}^{∞} e^{jl9μ_{0}} (jμ_{0} 8 _{x}▷l9◁ C P 8_{x}▷l9◁) \frac{▷−1◁^{lC1}}{l}

0D \frac{q}{2π} ∑_{{lD−∞ \atop lD̸0}}^{∞} e^{jl9μ_{0}} (j ▷μ_{0} C q◁ 8 _{x}▷l9◁ C P 8_{x}▷l9◁) \frac{▷−1◁^{lC1}}{l} . (6.49)

By taking the difference,

\frac{q}{2π} ∑_{{lD−∞ \atop lD̸0}}^{∞} jqe^{jl9μ_{0}} 8 _{x}▷l9◁\frac{▷−1◁^{lC1}}{l} D 0 .  (6.50)

This is a Fourier series as a function of μ_{0}. Since the Fourier series representation of a function is unique, it can be equal to zero for all values of μ_{0} only if all coefficients are equal to zero, that is,

8 _{x}▷l9◁ D 0 for l D ±1, ±2 ….  (6.51)

Substituting this back into the first line of (1.49), the remaining terms form again a Fourier series, so the derivatives of 8 _{x}▷u◁ must also be all zero:

P 8_{x}▷l9◁ D 0 , l D ±1, ±2 ….  (6.52)

Consequently, (1.45) is a necessary and sufficient condition for x and ν to be orthogonal, independent of the mean value of x.

6.5 CONDITIONS OF UNCORRELATEDNESS BETWEEN x AND ν: QUANTIZING THEOREM IV/B

It is possible for cov{x, ν} to equal zero without the conditions of QT II being met. This gives rise to another quantizing theorem, QT IV/B. This quantizing theorem gives conditions on the derivative of 8 Q x▷u◁ for which ν and x will be mutually uncorrelated, that is, their covariance will be zero.

It is sufficient to have P 8Q x▷l9◁ equal to zero for all nonzero integer values of l to assure that Eq. (1.27) is zero. However, this is not necessary, because the covariance can also be zero for specific combinations of P 8Q x▷l9◁ and μ. But if we seek the conditions of uncorrelatedness of x D ▷ Q xCμ◁ and ν, independently of the value of μ, a necessary and sufficient condition can be formulated. This is the weakest condition that would make the input-noise covariance be equal to zero, for all possible values of μ.

Equation (1.27) is a Fourier series and is a function of μ. It can be zero for all values of μ only if all the coefficients of the series are zero. As a consequence, we can state the following quantizing theorem:

Quantizing Theorem IV/B^{2}

The input variable x D Q xC μ and the quantization noise ν are uncorrelated for any value of μ if and only if the first derivative of the CF of Q xis zero at integer multiples of 9:

P 8Q x▷l9◁ D 0 for l D ±1, ±2 …  (6.53)

Moreover, if and only if (1.53) is true,

E{ Q xν^{t}} D E{ Q xn^{t}} for t D 1, 2, …  (6.54)

holds, independently of the value of μ.^{3}

This is QT IV/B. The conditions of QT IV/B are similar to those of QT III/A, introduced in Chapter

Satisfaction of QT III/A is not sufficient for cov{ν, x} D 0, but does give ν a uniform PDF between ±q/2. Satisfaction of QT IV/B makes cov{ν, x} D 0, but does not give ν a uniform PDF. It is also possible for both QT III/A and QT IV/B to be satisfied without having QT II satisfied. Satisfaction of QT III/A and QT IV/B satisfies QT III/B, causing ν to be uniformly distributed and uncorrelated with x. Examples of PDFs that meet these various conditions are given

6.6 SUMMARY

The means, mean squares, and crosscorrelations comprise the first- and second-order moments that are most important to statistical signal processing and control systems. That the PQN model works so well with these moments when a Gaussian signal is quantized explains why this model is so widely accepted and used. What we have developed are measures of accuracy of the PQN model that apply not only to the Gaussian case but to any quantizer input.

2Like QT III/A is a special case of QT III, QT IV/B is a special case of the more general theorem QT IV that will be developed in Section

3Validity of (1.54) is equivalent to cov{ Q x, ν^{t}} D cov{ Q x, n^{t}}, since E{ Q x} D 0.

Several expressions for means and crosscorrelations are the following: E{ν} comes from Eq. (

It is useful to note that when either QT II or both QT III/A and QT IV/B are satisfied, all of the above crosscorrelations and covariances correspond exactly when one compares actual quantization with the addition of pseudo quantization noise (PQN).

Regarding these moments, the PQN model is precise when conditions for QT II are met. The conditions of QT II are sufficient for this to be the case. There are however less stringent conditions that would allow orthogonality between x and ν (E{xν} D 0), and other conditions that would allow uncorrelatedness between x and ν (cov{x, ν} D 0). This subject was discussed in Sections 1.4 and 1.5. The ideas expressed in these sections will find important applications in Chapter

6.7 EXERCISES

6.1 Check the numerical calculations for the entries in Tables 1.1, 1.3, 1.4 (starting in page 12) for q D σ.

6.2 The derivations in Chapter 1 analyze the mid-tread quantizer, with a “dead zone” around zero.

(a) Generalize the expressions of the crosscorrelation (1.23) and of the covariance (1.25) between x and ν to the case when the quantizer is shifted by s both horizontally and vertically (that is, the characteristic is shifted along the 45^{∘} line).

(b) Investigate the results for the case of the mid-riser quantizer (s D q/2).

(c) Show how the same result as in (a) can be obtained from (1.23) and (1.25) by one of the transformations illustrated in Fig. 1.2 (page 5).

(d) Are these cases different from the case of input offset? Why?

6.3 Consider the probability density function of Example

(a) Determine the probability density function and the characteristic function of the quantizer output for the quantizer characteristics as defined in Fig. 1.2 (page 5), but let the shift of the quantizer characteristic be s, instead of q/2.

(b) Determine the probability density function and the characteristic function of the quantization noise for the same case.

(c) Prove that the covariance between the input and the noise is equal to zero, independently of the input mean.

(d) Determine and plot the correlation between the quantizer input and the quantization noise as a function of the input mean value μ in the interval 0 ≤ μ ≤ 2q, for s D q/2.

(e) Determine the values of μ and s so that the correlation between the input and the noise is equal to zero.

6.4 Repeat the questions in Exercise 1.3 (except for the CF in item (b), and except for (c)) for the PDF given in Example

6.5 Prove Eq. (1.42) using Eq. (1.25).

6.6 It is obvious that for a uniform input distribution of width kq, where k is a small positive integer, neither of Sheppard’s corrections nor the conditions of the PQN model are fulfilled. However, it has been shown in Chapter

6.7 One would like to measure the correlation coefficient between the quantization noise and the input. The input distribution is one of the distributions described in Appendix