Advance Communication System Lectures Part 3

8/2/2019 Advance Communication System Lectures Part 3

1/23

Chapter 6

Sampling & PCM


2/23

Advantages of Digital Communications

Rugged: Can withstand channel noise and

distortion much better.

Use of repeaters (travels as far as needed).

Use of TDM

Can be encrypted (Security and Privacy)

Can be encoded for error correction (reliability). Easy to process, store and search.


3/23

Analog to Digital Conversion (A/D)

In converting an analog signal to an equivalentsequence of 0s and 1s, we go through threeprocesses:

Sampling:o converting continuoustime analog signals to discretetime

analog signals.

Quantization

o converting discretetime analog signals to discretetime

digital signals (finite set of amplitude levels). Coding

o Map each amplitude level to a binary sequence.


4/23

[1] Sampling: Mathematical Representation

One sample of g(t) can be obtained from

If we want to sample g(t) periodically every Ts

sec then we

can repeat this process periodically

0 0 0( ) ( ) ( ) ( ) ( )sg t g t t t g t t t

( ) ( ) ( )

( ) ( )

( ) ( ).

s

n

s

n

s s

n

g t g t t nT

g t t nT

g nT t nT


5/23

Sampling: Time-Domain Plot

s s s s s ss s s s s s

g(t)

s

s

s

s

s

s

s

s

s

s

s

s

g(t)


6/23

Sampling: Frequency-Domain Analysis (1/2)

0 1 2 3

1 2 3

( ) ( )

cos( ) cos(2 ) cos(3 )

sin( ) sin(2 ) sin(3 )

sT s

n

s s s

s s s

t t nT

a a t a t a t

b t b t b t

2

s

sT

2 2

0

2 2

1 1 1( ) ( )

s s

s

s s

T T

T

T Ts s s

a t dt t dt T T T

2 2 2

0

2 2 2

2 2 2 2( ) cos( ) ( )cos( ) ( )cos(0)

s s s

s

s s s

T T T

T s s

T T Ts s s s

a t t dt t t dt t dt T T T T

2 2 2

0

2 2 2

2 2 2( ) sin( ) ( )sin( ) ( )sin(0) 0

s s s

s

s s s

T T T

T s s

T T Ts s s

b t t dt t t dt t dt T T T

)()()( ttgtgsT

an

bn


7/23

Sampling: Frequency-Domain Analysis (2/2)

( ) ( ) ( )

1 2 2 2( ) ( )cos( ) ( )cos(2 ) ( )cos(3 ) .

s

n

s s s

s s s s

g t g t t nT

g t g t t g t t g t t T T T T

1 1 1( ) ( ) ( ) ( ) ( 2 ) ( 2 )

1

( 3 ) ( 3 ) .

1( )

s s s s

s s s

s ss

s

ns

G G G G G GT T T

G GT

G nT


8/23

Spectrum of Sampled Function

G()

+2B

2B s

s

s

s s

s

A

G()

+2B

2B s

s

s

s s

s

A/Ts

......

s+2B

s2B

s+2B

s2B


9/23

Recovering the Continuous-Time Signal

G()

+2B

2B s

s

s

s s

s

A/Ts

......

s+2B

s2B

s+2B

s2B

LPF for reconstructing the origianl

signal from the sampled signal

Reconstructed Signal

+2B2B s

s

s

s s

s

A/Ts

Ts

Magnitude of LPF should be Ts to cancel

the scaling factor caused by sampling

s> 2(2B) No interference between Images


10/23

Sampling Theorem

A baseband signal whose spectrum is band-

limited to B Hz can be reconstructed exactly

(without any error) from its samples taken

uniformly at a ratefs 2B.

fs 2B is called Nyquist Criterion of sampling.

fs = 2B is called the Nyquist rate of sampling.

Does Sampling Theorem Make Sense?


11/23

Aliasing

Sampling a signal at a rate less that the Nyquist rate results inAliasing.

In aliasing, the higher frequency components take the identity oflower frequencies.

Real life Example: Sampling a rotating wheel.

G()

s

s

s

s

s

A/Ts

......

LPF for reconstructing the origianl

signal from the sampled signal

Reconstructed Signal

A/Ts

Ts

s < 2(2B) Interference between imageswill occur

s

s

s

Damaged part of the signal

s

s

s

s

s

s

s

s


12/23

Time Division Multiplexing

(TDM)

Multiplexing: The process ofsending two or more signalstogether

FDM: Sending them together at thesame time over different bandsusing carrier modulation (AM &FM broadcasting)

TDM: Sending them together overthe same band by sampling thesignals and sending the samples atdifferent time instants(interleaved).

Ts

g1(t)

g2(t)

g3(t)

Ts

Ts

gTDM

(t)

Ts

Ts/3


13/23

How to Transmit the Samples?

Pulse Modulation:

Use the samples to modulate a carrier of pulses

Pulse Amplitude Modulation (PAM)

Pulse Width Modulation (PWM)

Pulse Position Modulation (PPM)

Pulse Code Modulation (PCM)

Quantization of samples

Coding


14/23

Pulse Amplitude Modulation (PAM)

Ts

gPAM(t)

t


15/23

Pulse Width Modulation (PWM)

Ts

gPWM(t)

t


16/23

Pulse Position Modulation (PPM)

Ts

gPPM(t)

t


17/23

[2] Quantization

Analog samples with an amplitude that may take valuein a specific range are converted to a digital sampleswith an amplitude that takes one of a specific predefined set of values.

The range of possible values of the analog samples isdivide intoL levels. L is usually taken to be a power of2 (L = 2n).

The center value of each level is assigned to anysample that falls in that quantization interval.

For almost all samples, the quantized samples willdiffer from the original samples by a small amount,called the quantization error.


18/23

Quantization: Illustration

t4TsTs 3Ts 5Ts2Ts0

mp

mp

L = 2n

L levels

n bits0

v

Qu antizer Ou tput Sam plesq

x

Qu antizer Input Samp les x

A quantization interval Corresponding quantization value

2 pmvL


19/23

Input-Output Characteristics of Quantizer

v v v vvvvv

v/2

v/2

v/2

v/2

v/2

v/2

v/2

v/2

Quantizer

Inputx

Quantizer

Output xq

qx

x

mp


20/23

[3] Coding

4TsTs 3Ts 5Ts2Ts0

mp

mp

L = 2n

L levels

n bits

000

001

010

111

PCM Code

n bits/sample

0

v

Quantizer Output Samples qx

Quantizer Input Samples x

A quantization interval Corresponding quantization value

011

100

101

110

001 011 100 110 110 110 100 010 010 010 100 101 101


21/23

We want to scan and send a black-and-white image ofheight 11 inches and width 8.5 inches (Letter size paper).The resolution of the scanner is 600600 dots per inchsquare. The picture will be quantized using 256 levels.Find the size of the scanned image and the time it takesto transmit the image using a modem of speed 56 kbps.

Size of image =11(in)8.5(in)600600(samples/in2)8bits/sample

= 269280000 bits = 269 Mbits Time to transmit = 269280000 / 56,000 = 4808 sec = 80 min


22/23


23/23

Nyquist Theorem for Transmission

Note that the larger the transmission rate

(pulses/sec) the narrower the pulse, the wider

its spectrum, the higher the channel bandwidth

required for transmission.

The minimum theoretical bandwidth

required to transmit R pulses/sec is R/2 Hz.

(To be demonstrated later)

Advance Communication System Lectures Part 3

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