Exercise 1.2Z: Puls Code Modulation

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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

(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  $T_{\rm B} = T_{\rm A}/8$.