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# 信号代写|数字信号处理作业代写digital signal process代考|Signal Quantization

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## 信号代写|数字信号处理作业代写digital signal process代考|Classical Quantization Model

Quantization is described by Widrow’s quantization theorem [Wid61]. This says that a quantizer can be modeled (see Fig. 2.2) as the addition of a uniform distributed random signal $e(n)$ to the original signal $x(n)$ (see Fig. 2.2, [Wid61]). This additive model,
$$x_{Q}(n)=x(n)+e(n),$$
is based on the difference between quantized output and input according to the error signal
$$e(n)=x_{Q}(n)-x(n) .$$

## 信号代写|数字信号处理作业代写digital signal process代考|Quantization Theorem

The statement of the quantization theorem for amplitude sampling (digitizing the amplitude) of signals was given by Widrow [Wid61]. The analogy for digitizing the time axis is the sampling theorem due to Shannon [Sha48]. Figure $2.6$ shows the amplitude quantization and the time quantization. First of all, the PDF of the output signal of a quantizer is determined in terms of the PDF of the input signal. Both probability density functions are shown in Fig. 2.7. The respective characteristic functions (Fourier transform of a PDF) of the input and output signals form the basis for Widrow’s quantization theorem.

Quantization of a continuous-amplitude signal $x$ with PDF $p_{X}(x)$ leads to a discreteamplitude signal $y$ with PDF $p_{Y}(y)$ (see Fig. 2.8). The continuous PDF of the input is sampled by integrating over all quantization intervals (zone sampling). This leads to a discrete PDF of the output.

In the quantization intervals, the discrete PDF of the output is determined by the probability
$$W[k Q]=W\left[-\frac{Q}{2}+k Q \leq x<\frac{Q}{2}+k Q\right]=\int_{-Q / 2+k Q}^{Q / 2+k Q} p_{X}(x) d x$$

## 信号代写|数字信号处理作业代写digital signal process代考|Statistics of Quantization Error

The PDF of the quantization error depends on the PDF of the input and is dealt with in the following. The quantization error $e=x_{Q}-x$ is restricted to the interval $\left[-\frac{Q}{2}, \frac{Q}{2}\right]$. It depends linearly on the input (see Fig. 2.13). If the input value lies in the interval $\left[-\frac{Q}{2}, \frac{Q}{2}\right]$ then the error is $e=0-x$. For the PDF we obtain $p_{E}(e)=p_{X}(e)$. If the input value lies in the interval $\left[-\frac{Q}{2}+Q, \frac{Q}{2}+Q\right]$ then the quantization error is $e=Q\left\lfloor Q^{-1} x+0.5\right\rfloor-x$ and is again restricted to $\left[-\frac{Q}{2}, \frac{Q}{2}\right]$. The PDF of the quantization error is consequently $p_{E}(e)=p_{X}(e+Q)$ and is added to the first term. For the sum over all intervals we can write
$$p_{E}(e)= \begin{cases}\sum_{k=-\infty}^{\infty} p_{X}(e-k Q), & -\frac{Q}{2} \leq e<\frac{Q}{2}, \ 0, & \text { elsewhere. }\end{cases}$$

Because of the restricted values of the variable of the PDF, we can write
\begin{aligned} p_{E}(e) &=\operatorname{rect}\left(\frac{e}{Q}\right) \sum_{k=-\infty}^{\infty} p_{X}(e-k Q) \ &=\operatorname{rect}\left(\frac{e}{Q}\right)\left[p_{X}(e) * \delta_{Q}(e)\right] \end{aligned}

## 信号代写|数字信号处理作业代写DIGITAL SIGNAL PROCESS代考|CLASSICAL QUANTIZATION MODEL

X问(n)=X(n)+和(n),

$$e(n)=x_{Q}(n)-x(n) .$$

## 信号代写|数字信号处理作业代写DIGITAL SIGNAL PROCESS代考|QUANTIZATION THEOREM

$$W[k Q]=W\left[-\frac{Q}{2}+k Q \leq x<\frac{Q}{2}+k Q\right]=\int_{-Q / 2+k Q}^{Q / 2+k Q} p_{X}(x) d x$$

## 信号代写|数字信号处理作业代写DIGITAL SIGNAL PROCESS代考|STATISTICS OF QUANTIZATION ERROR

p和(和)={∑ķ=−∞∞pX(和−ķ问),−问2≤和<问2, 0, 别处。

\begin{aligned} p_{E}(e) &=\operatorname{rect}\left(\frac{e}{Q}\right) \sum_{k=-\infty}^{\infty} p_{X}(e-k Q) \ &=\operatorname{rect}\left(\frac{e}{Q}\right)\left[p_{X}(e) * \delta_{Q}(e)\right] \end{aligned}