PAM, PPM,PCM, and Delta Modulation

KEEE343 Communication Theory

Lecture #24, June 7, 2011Prof. Young-Chai Kokoyc@korea.ac.kr

Pulse-Amplitude Modulation

• The amplitude of regularly spaced pulses are varied in proportion to the corresponding sample values of a continuous message signal.

• Two operations involved in the generation of PAM signal

Sampling

Holding: Lengthening the duration of each sample, so that it occupies some finite value T

=> Sample-and-Hold filter

[Ref: Haykin & Moher, Textbook]

Sample-and-Hold Filter : Analysis The PAM signal is

The h(t) is a standard rectangular pulse of unit amplitude and duration

The instantaneously sampled version of m(t) is

h(t) = rect

t� T

2

T

!=

8<

:

1, 0 < t < T12 , t = 0, t = T0, otherwise

m�(t) =1X

n=�1m(nTs)�(t� nTs)

s(t) =X

m(nTs)h(t� nTs)

• To modify so as to assume the same form as the PAM

• The PAM signal s(t) is mathematically equivalent to the convolution of and the pulse h(t).

m�(t)

m�(t) ⇤ h(t) =

Z 1

�1m�(t)h(t� ⇥) d⇥

=

Z 1

�1

1X

n=�1m(nTs)�(⇥ � nTs)h(t� ⇥) d⇥

=1X

n=�1m(nTs)

Z 1

�1�(⇥ � nTs)h(t� ⇥) d⇥

m�(t)

s(t) = m�(t) ⇤ h(t)

• Fourier Transform

s(t) = m�(t) ⇤ h(t), M�(f) = fs

1X

k=�1M(f � kfs)

S(f) = M�(f)H(f), S(f) = fs

1X

k=�1M(f � kfs)H(f)

[Ref: Haykin & Moher, Textbook]

Aperture Effect and Its Equalization

• Aperture effect

• The distortion caused by the use of pulse-amplitude modulation to transmit an analog information-bearing signal

• Equalization filter

1

|H(f)| =1

T sinc(fT)=

�f

sin(�fT )

[Ref: Haykin & Moher, Textbook]

Pulse-Position Modulation

• PDM (Pulse-Duration Modulation)

• Pulse-width or Pulse-length modulation

• The samples of the message signal are used to vary the duration of the individual pulses.

• PDM is wasteful of power

• PPM (Pulse-position modulation)

• The position of a pulse relative to its unmodulated time of occurrence is varied in accordance with the message signal

s(t) =1X

n=�1g(t� nTs � kpm(nTs))

g(t) = 0, |t| > (Ts/2)� kp|m(t)|max

, ⇥ kp|m(t)|max

< T2

/2

[Ref: Haykin & Moher, Textbook]

Completing the Transition from Analog to Digital

The advantages offered by digital pulse modulation Performance

Digital pulse modulation permits the use of regenerative repeaters, when placed along the transmission path at short enough distances, can practically eliminate the degrading effects of channel noise and signal distortion.

Ruggedness A digital communication system can be designed to withstand the effects of

channel noise and signal distortion Reliability

Can be made highly reliable by exploiting powerful error-control coding techniques.

Security Can be made highly secure by exploiting powerful encryption algorithms

Efficiency Inherently more efficient than analog communication system in the tradeoff

between transmission bandwidth and signal-to-noise ratio System integration

To integrate digitized analog signals with digital computer data

Quantization Process

Amplitude quantization The process of transforming the sample amplitude m(nTs) of a baseband signal

m(t) at time t=nTs into a discrete amplitude v(nTs) taken from a finite set of possible levels.

Representation level (or Reconstruction level) The amplitudes vk , k=1,2,3,……,L

Quantum (or step-size) The spacing between two adjacent representation levels

Ik : {mk < m � mk+1}, k = 1, 2, ..., L

v = g(m)

[Ref: Haykin & Moher, Textbook]

[Ref: Haykin & Moher, Textbook]

Pulse-Code Modulation

• Operation in the Transmitter

Sampling: Sampling faster than Nyquist rate

Nonuniform Quantization

Encoding

• Operation in the Receiver

Decoding and Expanding

Reconstruction

Non-Uniform Quantization

• Pass the message signal through the compressor

• Two different compressor laws

-law

A-law

µ

|v| = log(1 + µ|m|)log(1 + µ)

|v| =(

A|m|1+logA , 0 � m � 1

A1+log(A|m|)

1+logA , 1

A � |m| � 1

Encoding

• Sampled value is digitized using the binary digits (bits).

000001

010011

100101

110

111R=3 bits/samplecodeword

codeword length= 2R

Regeneration of the PCM

• Operation in the Receiver

Decoding and Expanding

Decoding: regenerating a pulse whose amplitude is the linear sum of all pulses in the codeword

Expander: a subsystem in the receive with a characteristic complementary to the compressor

Reconstruction

Recover the message signal: passing the expander output through a low-pass reconstruction filter.

Delta Modulation

• An incoming message signal is oversampled to purposely increase the correlation between adjacent samples of the signal.

• The difference between the input signal and its approximation is quantized into only two levels - corresponding to positive and negative differences

e(nTs) = m(nTs)�mq(nTs � Ts)

eq(nTs) = �sgn[e(nTs)]

mq(nTs) = mq(nTs � Ts) + eq(nTs)

• System details

Comparator

Computes the difference between its two inputs

Quantizer

Consider of a hard limiter with an input-output characteristic that is a scaled version of the signum function

Accumulator

Operates on the quantizer output so as to produce an approximation to the message signal

mq(nTs) = mq(nTs � Ts) + eq(nTs)

= mq(nTs � 2Ts) + eq(nTs � Ts) + eq(nTs)

...

=nX

i=1

eq(iTs)