Pulse Code Modulation on Mac

14. Pulse Code Modulation We have seen that sampling a bandlimited signal at or above the Nyquist sampling rate does not destroy any information content and fully characterises the bandlimited signal. A system transmitting these sampled values of the bandlimited signal is called a sampleddata or pulse modulation system. In modern communication systems, these sampled signals are often quantised and coded before transmission. We have pulse code modulation (PCM). Pulse code modulation is very popular because of the many advantages it offers. These include: 1. 2.

3. 4. 5.

Inexpensive digital circuitry may be used in the system. All-digital transmission. PCM signals derived from analogue signals may be timedivision multiplexed with data from digital computers and transmitted over a common high-speed channel. Further digital signal processing such as encryption is possible. Errors may be minimised by appropriate coding of the signals. Signals may be regularly reshaped or regenerated using repeaters at appropriate intervals.

Figure 14.1 shows a single-channel PCM system. Figure 14.1 A single-channel PCM transmission system. An analogue message m(t) is first sampled at or above the Nyquist sampling rate. These sampled signals are then converted into a finite number of discrete amplitude levels. The conversion process is called quantisation. Figure 14.2 shows how an analogue message is converted into 8 amplitude levels with equal spacing by an 8-level quantiser. Figure 14.2 Message and quantised signal. Quantisation obviously reduces the degree of accuracy of representation of the sampled signal and introduces some error in the reproduction of the signal at the receiver. Error introduced by the quantiser is called quantisation error or quantisation noise. To reduce the quantisation error, we simply increase the total number of amplitude levels (decreasing the spacing between adjacent levels). What is the minimum number of quantisation levels for speech? 8 ~ 16 levels are sufficient. In practical digital telephone systems, 256 = 28 levels are used to keep the quantisation error to a tolerable level. 65,536 = 216 levels are used for the CD digital system.

14.1

Pulse Code Modulation on Mac

If the quantised samples are transmitted directly over a channel, we have a quantised PAM system. If, instead, we code each quantised sample into a block of digits for transmission, , we have a PCM system. The decimal-to-binary conversion can be done in various ways. Table 15.1 shows two possible coding rules (binary and gray coding) for converting a 16-level sample into 4 binary digits. ________________________________________________________________________ Binary Code Gray Code ________________ _______________ Digit [b1 b2 b3 b4] [g1 g2 g3 g4] ________________________________________________________________________ 0 0000 0000 1 0001 0001 2 0010 0011 3 0011 0010 4 0100 0110 5 0101 0111 6 0110 0101 7 0111 0100 8 1000 1100 9 1001 1101 10 1010 1111 11 1011 1110 12 1100 1010 13 1101 1011 14 1110 1001 15 1111 1000 ________________________________________________________________________ Table 14.1 Decimal-to-binary conversion. The elements bk and gk are related to each other by the following equations: g1 = b1, gk = bk ⊕ bk-1, k > 2

(14.1)

b1 = g1, bk = gk ⊕ bk-1, k > 2

(14.2)

where ⊕ represents the modulo-2 operation. It can be seen from Table 14.1 that, in changing from one decimal digit to an adjacent digit, the binary code may change by more than one binary digit. This makes the binary code highly susceptible to error in recording 14.2

Pulse Code Modulation on Mac

the analogue-to-digital conversion. One would prefer a code in which only one binary digit at a time changed as the corresponding input digit changed by one level. Gray code has the above property and is the preferred coding method. Figure 14.3 shows 3 quantised samples and their corresponding coded bit sequences. Figure 14.3 Binary and Gray coding of samples. In Figure 14.4, we show a complete 10-channel PCM system and its associated signal shapes at various transmitting points. Clearly, the bandwidth required at the output of the binary encoder is three times the bandwidth required at the input and the output of the quantiser. Thus, a binary PCM system requires more transmission bandwidth than the PAM and the quantised PAM systems. Figure 14.4

Ten-channel PCM system. (a) Transmitter. (b) Receiver. (c) Signal shapes.

Bandwidth Reduction Technique Binary coding is just one special case of a coding method in a PCM system. In general, we can code a quantised sample into a group of m pulses every T seconds, each pulse with a duration of τ = T/m seconds and n possible amplitude levels. Clearly, the total number of amplitude levels that a quantised signal can have is M = nm. The ability to choose n and m gives us some freedom to reduce the transmission bandwidth. Figure 14.5 shows the bandwidth reduction effects when we vary n and m. If n is fixed, we can reduce the transmission bandwidth by reducing the value of m. This is shown in Figure 14.5 (a). If M is fixed, we can reduce the transmission bandwidth by increasing the value of n and reducing the value of m. This is shown in Figure 14.5 (b). The collapsing of successive pulses onto one much wider pulse reduces the transmission bandwidth. However, there is one major drawback for the fixed M case. If the spacing between adjacent levels is fixed, the required peak power goes up as n increases. On the other hand, if the peak power or amplitude swing is fixed, adjacent levels get closer to each other. This makes easier for noise to obscure adjacent levels. Not a very good bandwidth reduction technique! The technique is only useful for very-low-noise environments. n = 2 is the most noise-immune choice. As we are only dealing with on-off signalling, the exact magnitude is not important. Reshaping of signals by repeaters facilitates the signal decision process at the receiver. Figure 14.5 Bandwidth reduction technique. (a) Fixed n, (b) Fixed M.

14.3

Pulse Code Modulation on Mac

References [1]

H. P. Hsu, Analog and Digital Communications, McGraw-Hill, 1993.

[2]

M. Schwartz, Information Transmission, Modulation, and Noise, 4/e, McGraw-Hill, 1990.

[3]

L. W. Couch II, Analog and Digital Communication Systems, 6/e, Prentice Hall, 2001.

14.4

Pulse Code Modulation on Mac

Message

m (t )

Quantiser

Sampler

Encoder ~ ~

Low-pass filter

m^ ( t )

Decoder

Figure 14.1 A single-channel PCM transmission system.

Volts

message m ( t )

7 6 5 4 3 2 1

a

A0

0

2

4

6

10

8

12

Seconds

Figure 14.2 Message and quantised signal.

7

6

5

T

T

T

7

6

5

t

Quantised samples

1 1 1 1 1 0 1 0 1

T

T

t

T

Binary code

1 0 0 1 0 1 1 1 1

T

T

t

T

Gray code

Figure 14.3 Binary and Gray coding of samples.

14.5

Pulse Code Modulation on Mac

30-3200 Hz m 1( t )

8000 cycles/s

m 2( t ) : m

10

1

8-level quantiser

2

Binary 3 encoder

(t ) (a)

m (t ) 1 Filter m 2( t ) Filter : m 10 ( t ) Filter 30-3200 Hz

8000 cycles/s Decoder

(b) 5.3 1. TDM PAM

2.7

4.8

1 2 3

2. Quantised

3 1 2 3

12.5 µ s 3. Binary coded 4.2 µ s

...

125 µ s

12.5 µ s 5

5.5 2.4 4 10 1 2 3

5

...

4.2 V 10

125 µ s 6 2 4 10 1 2 3

...

125 µ s

... 125 µ s

1 0 1 0 1 1 1 0 1 1 3

4V 10

t

t

t

12.5 µ s (c) Figure 14.4

Ten-channel PCM system. (a) Transmitter. (b) Receiver. (c) Signal shapes.

14.6

Pulse Code Modulation on Mac

M = n m Total no. of amplitude levels m = T / τ - No. of pulses per sample - No. of possible amplitude levels per pulse n Fixed n ( = 2): (Β = m / T ) Quantised sample

∝m T t

τ = T /3 m = 3 pulses per sample

τ = T /2

m = 2 pulses per sample

n=2 Β = 3/ T t n=2 Β = 2/ T t

(a) Fixed M (= 16):

n

m

B

T Quantised sample

τ = T /4

t n =2 Β = 4/ T t

m = 4 pulses per sample

1

a

τ = T /2 m = 2 pulses per sample

Fixed peak power swing

Fixed spacing

n=4 Β = 2/ T V t

a

1 0

3 2 1 0

V

a 0

V a

(b) Figure 14.5 Bandwidth reduction technique. (a) Fixed n, (b) Fixed M.

14.7

3 2 1 0