# Exercise 1.2Z: Puls Code Modulation

PCM components

All modern communication systems are digital.  The principle of digital transmission of speech signals goes back to  Alec Reeves,  who invented the so-called  "Puls Code Modulation"  $\rm (PCM)$  as early as 1938.

On the right you see the (simplified) block diagram of the PCM transmitter with three functional units:

• The band-limited speech signal  ${q(t)}$  is sampled, where the  Sampling Theorem  is observed, and yields the sampled signal  $q_{\rm A}(t)$.
• Each sample  $q_{\rm A}(t)$  is mapped to one of  $M = 2^N$  results in the quantized signal  $q_{\rm Q}(t)$.
• Each individual quantized value is represented by a code sequence of  $N$  binary symbols and results in the coded signal  $q_{\rm C}(t)$.

In this task only the different signals of the PCM transmitter are to be classified.
Later tasks will deal with other properties of the puls code modulation.

Note:   This task belongs to the chapter  Signal classification.

### Questions

1

Which of the statements are true for the source signal  ${q(t)}$ ?

 In normal operation  ${q(t)}$  is a stochastic signal. A deterministic source signal is only useful in test operation or for theoretical investigations. ${q(t)}$  is a time-discrete signal. ${q(t)}$  is a continuous-valued signal.

2

Which of the statements apply to the sampled signal  $q_{\rm A}(t)$ ?

 $q_{\rm A}(t)$  is a discrete-valued signal. $q_{\rm A}(t)$  is a time-discrete signal. The higher the maximum frequency of the source signal, the higher the sampling rate must be selected.

3

Which statements are true for the quantized signal  $q_{\rm Q}(t)$  if  $N = 8$  is taken as a base?

 $q_{\rm Q}(t)$  is a time-discrete signal. $q_{\rm Q}(t)$  is a discrete-valued signal with  $M = 8$  possible values. $q_{\rm Q}(t)$  is a discrete-valued signal with  $M = 256$  possible values. $q_{\rm Q}(t)$  is a binary signal.

4

Which statements are true for the coded signal  $q_{\rm C}(t)$  if  $N = 8$  is taken as a base?

 $q_{\rm C}(t)$  is a time-discrete signal. $q_{\rm C}(t)$  is a discrete-valued signal with  $M = 8$  possible values. $q_{\rm C}(t)$  is a binary signal. When sampling at distance  $T_{\rm A}$  the bit duration is  $T_{\rm B} = T_{\rm A}$. When sampling at distance  $T_{\rm A}$  the bit duration is  $T_{\rm B} = T_{\rm A}/8$.

### Solution

#### Solution

(1)  Correct are the solutions 1, 2 and 4:

• The source signal  ${q(t)}$  is analog, i.e. "continuous in time and value".
• In general, it makes no sense to transmit a deterministic signal.
• For the mathematical description, a deterministic source signal – such as a periodic signal – is better suited than a random signal.
• Deterministic signals are also used for testing in order to be able to reconstruct detected errors.

(2)  Correct are the solution suggestions 2 and 3:

• After sampling, the signal  $q_{\rm A}(t)$  is still  value-continuous, but now also time-discrete.
• The sampling frequency  $f_{\rm A}$  is given by the so-called  "Sampling Theorem".
• The greater the maximum frequency  $f_{\rm N,\,max}$  of the source signal, the greater must  $f_{\rm A} ≥ 2 \cdot f_{\rm N,\,max}$  be selected.

(3)  Correct are the solution suggestions 1 and 3:

• The quantized signal  $q_{\rm Q}(t)$  is discrete in time and value, where the number of levels are  $M = 2^8 = 256$ .
• A binary signal, on the other hand, is a value–discrete signal with the level number  $M = 2$.

(4)  Correct are the solutions 1, 3 and 5:

• The coded signal  $q_{\rm C}(t)$  is binary  $($level number  $M = 2)$  with bit duration  $T_{\rm B} = T_{\rm A}/8$.