C3 - Digitization

Sampling

Converting an analog signal into a sequence of numbers at discrete time or space

$$x(t)=x[n]$$

Nyquist Sampling Theorem

A sampled signal can be converted to its original analog signal without any error if

$$f_s>2f_{max}$$

$f_s$ is the Nyquist rate

Telephone network filter away frequency > 4 kHz to avoid aliasing (under sampling)

Quantization

Putting samples into binary format

$$x[n]=x_d[n]$$

Level

$$\text{Quantization Levels}=2^n$$

The range of number is $0\to 2^n-1$

Precision

$$\begin{array}{rl}& \frac{\text{Signal Range}}{\text{Levels}}=\text{Level Range}\\ & \\ & \text{Precision}=\pm \frac{\text{Level Range}}{2}\\ & \end{array}$$

Reproduced analog value is the average of a level’s boundaries

Error

$$\text{Quantization Error}=|\text{Actual Analog Value}-\text{Reproduced Value}|$$

This will never be $>\frac{1}{2}$ of a level’s range

Quantization noise refers to the average of these errors, the more quantization levels, the less quantization noise

Information

$$\begin{array}{rl}\text{Information}=\text{Entropy}=H(X)& =\sum _{i=1}^{n}p(x_i){\log }_2\frac{1}{p(x_i)}\\ & =-\sum _{i=1}^{n}p(x_i){\log }_{2}p(x_i)\end{array}$$

The unit for information is $\mathrm{bits}$

In 1948, Claude Shannon revolutionized the concept of information - information is meaningfully measured only in bits

Information Capacity

$$\begin{array}{rl}& \text{Information Capacity}=\text{Bandwidth}\cdot {\log }_2(1+\mathrm{SNR})\\ & \\ & {\mathrm{SNR}}_{\mathrm{db}}=10\cdot {\log }_{10}(\mathrm{SNR})\end{array}$$

The unit for information capacity is $bits/s$