Communication Theory IILecture 18: Pulse Code Modulation

Ahmed Elnakib, PhDAssistant Professor, Mansoura University, Egypt

April 19th, 2015 1

Lecture OutlinesoPulse Code Modulation (PCM)Sampling and Pulse Amplitude ModulationQuantization

• Uniform quantization vs non uniform quantization• Compressor and expander (compander)

EncodingBandwidth of PCM

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Pulse Code Modulation (PCM)o Method of converting an analog into digital signal (A/D conversion)o Composed of three components: sampling, quantizing, and encoding

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Block Diagram of PCM System

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oThere are 3 sampling methods: Ideal - an impulse at each sampling instant Natural - a pulse of short width with varying amplitude Flattop - sample and hold, like natural but with single amplitude value

Sampling Methods

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Pulse Amplitude Modulation (PAM)

o PAM is a linear modulation process where the amplitudes of regularly spacedpulses are varied in proportion to the corresponding sample values of a continuousmessage signal.

o In contrast to ideal sampling, it involve two operation: Instantaneous sampling of the message signal m(t) every Ts seconds, where the

sampling rate fs = 1/Ts is chosen in accordance with the sampling theorem Lengthening the duration of each sample so obtained to some constant value T

Flat-top samples

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Quantizationo Rounding the analog signal value to

one of the closest permissible numbers (quantization levels)

o Number of levels: L=8 levelsL-ary digital signal

o Interval width: ∆v= (max - min)/L

o If m(t) lies between mp,-mp∆v=2mp/L Amplitude of m(t) beyond

± 𝑚𝑚𝑚𝑚 is chopped

o Each sample falling in a zone (subinterval) is then approximated to the value of the midpoint

o Maximum quantization error ± ∆v/2

-12.5

Quantization Error 1.4 0 -1.3 2.3 -1.5 2 1.2 -1.9 1.5

-17.5

-7.5-2.5

2.57.5

12.5

mp = 2017.5

mp = 20

∆v=2mp/L=2.5

L=8

sampled signal 7.5 16.2 19.7 11 -5.5 -11.3 -9.4 -6

Quantized signal -7.5 7.5 17.5 17.5 12.5 -7.5 -6.1 -7.5 -7.5

QuantizationoSampling results in a series of pulses of varying amplitude values ranging

between two limits: a min and a max.oThe amplitude values are infinite between the two limits.oWe need to map the infinite amplitude values onto a finite set of known

values.oThis is achieved by dividing the distance between min and max into L zones,

each of height ∆ v .∆v= (max - min)/L

Quantization Levels

oThe midpoint of each zone is assigned a value from 0 to L-1 (resulting in L values)

oEach sample falling in a zone is then approximated to the value of the midpoint.

Quantization ZonesoAssume we have a voltage signal with amplitutes Vmin=-20V and Vmax=+20V.oWe want to use L=8 quantization levels.oZone width ∆ = (20 - -20)/8 = 5oThe 8 zones are: -20 to -15, -15 to -10, -10 to -5, -5 to 0, 0 to +5, +5 to +10,

+10 to +15, +15 to +20oThe midpoints are: -17.5, -12.5, -7.5, -2.5, 2.5, 7.5, 12.5, 17.5

Two types of uniform quantization: (a) midtread and (b) midriseBoth are uniform quantization and symmetric about the origin

Types of Uniform Quantization

To have a zero output level

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Assigning Codes to Zones (subintervals)oEach zone is then assigned a binary code.oThe number of bits required to encode the zones, or the number of bits per

sample as it is commonly referred to, is obtained as follows: n = log2 L

oGiven our example, n= 3oThe 8 zone (or level) codes are therefore: 000, 001, 010, 011, 100, 101,

110, and 111oAssigning codes to zones: 000 will refer to zone -20 to -15 001 to zone -15 to -10, etc.

Quantization noise

Illustration of the quantization process.13

Quantization noise

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Random Uniformly distributed Quantization Noise

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Signal to Noise Ratio

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Problems of Uniform Quantization: S/N depends on the message signal power

Solution 1: using a non uniform quantizerwith non uniform levels Solution 2:

using a compressor with logarithmic characteristics followed by a uniform quantizer17

(a) Nonuniform quantization of the message signal in the transmitter (b) Uniform quantization of the original message signal in the receiver

Non-uniform Quantization

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Example 1: Sampling of telephone voice signalo An audio signal bandwidth is about 15 kHZ Subjective tests show that he signal articulation is not affected if all the

components above 3.4 kHz are suppressed An anti-aliasing filter suppress all the components above 3.4 kHz before

samplingo Typical sampling rate used is fs=8kHz (higher than the Nyquist rate to avoid

unrealizable filters for signal reconstruction) Cut-off frequency 3.4 kHz Transition band 6.8 to 8kHz

o Each sample is finally quantized into 256 levels (L=256=28), which require 8 bits toencode each sample

o The telephone signal require 8x8000=64k pulses per second19

Example 2: Sampling of voice signal on a Compact Disk (CD)

o An audio signal bandwidth is about 15 kHZ An anti-aliasing filter suppress all the components above 15 kHz before

samplingo Typical sampling rate used is fs=44.1kHz (higher than the Nyquist rate to avoid

unrealizable filters for signal reconstruction)o Each sample is finally quantized into 65,536 levels (L=216), which require 16 bits to

encode each sampleo The audio signal on a typical CD require 16x44.1k=705.6k pulses per second

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Questions

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