Exercise 1.2Z: Puls Code Modulation

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Components of pulse code modulation

All modern communication systems are digital. The principle of digital transmission of speech signals goes back to  Alec Reeves , who invented the so-called  Pulscodemodulation  (PCM)  already at 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  Abtasttheorem  is observed, and yields the sampled signal  $q_{\rm A}(t)$.
• Each sample  $q_{\rm A}(t)$  is mapped to one of  $M = 2^N$  and 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 pulse code modulation.

Notes:   This task belongs to the chapter  Klassifizierung von Signalen.

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

	$q_{\rm C}(t)$  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  $T_{\rm B} = T_{\rm A}$.
	For sampling at distance  $T_{\rm A}$  the bit duration is  $T_{\rm B} = T_{\rm A}/8$.

Solutions

Solution

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

• The source signal  ${q(t)}$  is analog, i.e. time- and value-continuous.
• Im Allgemeinen macht es keinen Sinn, ein deterministisches Signal zu übertragen.
• Für die mathematische Beschreibung eignet sich ein deterministisches Quellensignal – wie zum Beispiel ein periodisches Signal – besser als ein Zufallssignal.
• Deterministische Signale werden auch für den Testbetrieb herangezogen, um erkannte Fehlfunktionen rekonstruieren zu können.

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

• Das Signal  $q_{\rm A}(t)$  nach der Abtastung ist weiterhin  wertkontinuierlich, aber nun  zeitdiskret.
• Die Abtastfrequenz  $f_{\rm A}$  ist dabei durch das so genannte  Abtasttheorem  vorgegeben.
• Je größer die maximale Frequenz  $f_{\rm N,\,max}$  des Nachrichtensignals ist, desto größer muss  $f_{\rm A} ≥ 2 \cdot f_{\rm N,\,max}$  gewählt werden.

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

• Das quantisierte Signal  $q_{\rm Q}(t)$  ist zeit- und wertdiskret, wobei die Stufenzahl  $M = 2^8 = 256$  beträgt.
• A binary signal, on the other hand, is a discrete value signal with the number of steps  $M = 2$.

(4)  Richtig sind hier die Lösungsvorschläge 1, 3 und 5:

• Das codierte Signal  $q_{\rm C}(t)$  ist binär  $($Stufenzahl  $M = 2)$  mit Bitdauer  $T_{\rm B} = T_{\rm A}/8$.