Pulse Code Modulation

Pulse Code ModulationPCM is a method of converting an analog

signal into a digital signal. (A/D conversion)The amplitude of Analog signal can take any

value over a continuous range i.e. it can take on an infinite values.

Digital signal amplitude can take on finite values.

Analog signal can be converted into digital by sampling and quantizing.

Cont.

Cont.The amplitude of analog signal m(t) lie in the

range (-mp, mp) and is partitioned into L sub-intervals each of magnitude 2mp/L

Binary pulse codes

QuantizingDigital signals come from variety of sources

e.g. computerSome sources are analog but are converted

into digital form by variety of techniques such as PCM and DM

For quantizing , we limit the amplitude of m(t) to a range(-mp, mp) as shown in the previous slides

This amplitude is uniformly divided into L subintervals and each interval is ,

Cont.A sample value is approximated by the mid

point of the intervalThe quantized samples are coded and

transmitted as binary pulsesAt the receiver some pulses will be detected

incorrectlyThere are two types of errors

Quantization errorPulse detection error

Cont.In almost all practical schemes, the pulse

detection error is very small compared to the quantization error and can be ignored

Now we analyze the quantization error

Cont.

Cont.

Cont.The integral of the cross product terms is zero

and we obtain,

Because the sampling rate is 2B, hence the total number of samples over the averaging interval is 2BT

This is called the mean of the quantization error

Cont.The quantized levels are separated by 2mp/LSince sample value is approximated by the

midpoint of the subinterval in which the sample falls

The maximum quantization error is The mean square quantizing error is

Cont.

Cont.

Non-uniform quantizationSNR is an indication of the quality of the received

signalIdeally we would like to have constant SNRUnfortunately, the SNR is directly proportional to

the signal power, which varies from talker to talkerThe signal power can also vary because of the

connecting circuitsSNR vary even for the same talker, when the

person speaks softlySmaller amplitudes pre-dominate in speech and

larger amplitude much less frequent.This means the SNR will be low most of the time

Cont.The root of this difficulty is that the

quantization steps are of uniform valueThe quantization noise is directly

proportional to the square of the step size.The problem can be solved by using smaller

steps for smaller amplitudes as shown in fig. on the next slide

Cont.

Cont.The same result can be obtained by first

compressing a signal and then using uniform quantization

The input-output characteristics of compressor are shown in fig.

Cont.The horizontal axis is normalized input signal

and the vertical axis is the output signal y.The compressor maps the input signal into

larger increments Hence the interval delta(m) contains large

number of steps when m is smallThe quantization noise is small for smaller input

signalThus loud talker and stronger signals are

penalized with higher noise steps in order to compensate the soft talker and weak signals

Compression LawsThere are two laws regarding compressions(1)This law is used in North America and Japan

(2) A-LawThis law is used in Europe and the rest of the

word

Cont.The compressed samples are restored to their

original values at receiver by using an expanderThe compressor and expander together are

called compandor.Compression of a signal increases its bandwidth

but in PCM, we are not compressing the signal but its samples the number of samples does not change, therefore bandwidth does not rise

When meu-law compandor is used then output SNR is

Transmission BW and output SNRFor binary PCM, we assign distinct group of

n binary digits to each of the L quantization levels

Each quantized level is encoded into n-bitsMinimum channel BW isThis is the theoretical minimum transmission

bandwidth required to transmit the PCM signal

Example 6.2A signal m(t) band-limited to 3kHz is sampled

at a rate 33.33% higher than Nyquist rate, a maximum acceptable error in the sample amplitude is 0.5% of the peak amplitude. The quantized samples are binary coded. Find the minimum channel BW required to transmit the coded signal. If 24 such channels are time-division multiplexed, determine the minimum transmission BW required to transmit the multiplexed signal

Solution

Exponential Increase of output SNR

SNR in decibel scale

Cont.

Example 6.3

Comments on Logarithmic UnitsVery small and very large values are

expressed in logarithmic units

T1 carrier systemA schematic of T1-system is shown in fig.

Cont.

Cont.

Differential Pulse Code Modulation

Taylor's series

Cont.

Analysis of DPCM

Cont.

Delta Modulation

Cont.

Cont.

Delta Modulator

Delta Demodulator

Delta Modulator output

Working of DM