review of probability theory - guceee.guc.edu.eg/courses/communications/comm502... ·...

Dr. Ahmed El-Mahdy COM 502: Comm. Theory

Comm. 502: Communication Theory

Lecture 3

Transmission of Baseband Signals

- Pulse Code Modulation.


: Transmission of Baseband SignalsRemember

Baseband channel: Coaxial cable or

pair of wires Conversion from a bit stream to a

Sequence of pulse waveform

PAM PCM


Pulse Code Modulation (PCM)

PCM consists of three steps to digitize an analog signal:

1. Sampling: Convert the message signal to samples.

2. Quantization: Converting of samples to a set of discrete amplitudes.

3. Binary encoding: Translating the discrete set of samples into digital data.


. Quantization2 • It is not necessary to transmit the exact amplitude of the sampled

signal

• The original continuous signal may be approximated by a signal constructed of discrete amplitudes selected on a minimum error basis from an available set. (i.e quantization)

Definition:

• Quantization is the process of transforming the amplitude of a sampled signal into a discrete amplitude taken from a finite set.

sampled signal Quantized signal

)( snTX )( snTXq

Quantizer

XqnTXq s )()( snTX


Uniform Quantizer

im

1im

2im

iiiq

q

xXxifmX

toaccordinggenerated

isXoutputquantizedThe

1

:

1ix

ix

1ix

2ix

X

Lixx

m iii ...,,2,1,

2

1

The distance between min and max of the signal is divided into L levels, with

step size

= (max - min)/L L is the number of levels.

Each sample falling in a zone is then approximated to the value of the level

in this zone.

Levels Levels Zone


Example: (Uniform Quantizer)

• Assume we have a voltage signal with amplitudes Vmax=+40V and Vmin=-40V. • We want to use L=16 quantization levels. • Step Size = [40 – (-40)]/16 = 5

• The 16 zones are: -40 to -35 0 to 5 -35 to -30 5 to 10 -30 to -25 10 to 15 -25 to -20 15 to 20 -20 to -15 20 to 25 -15 to -10 25 to 30 -10 to -5 30 to 35 -5 to 0 35 to 40


Uniform Quantization of Sinusoidal signal

Sampling and quantization of a signal.

Quantization A/D Conversion

16L


Effect of changing the number of levels


Uniform Quantizers

0 2 4 -4 -2

2

4

-4

-2

0 2 4 -4 -2

2

4

-4

-2

Mid-Tread Uniform Quantizer Mid-Rise Uniform Quantizer

Used for odd number of levels Used for even number of levels. The

levels do not include the value of zero

Xq Xq

Xs Xs

8 Levels 7 Levels


. Coding3

• Each level is then assigned a binary code.

• The number of bits required to encode the levels, or the number of bits per sample as it is commonly referred to, is obtained as follows:

n = log2 L L is the number of levels

• In our example, n = log2 16= 4

• The 16 level codes are therefore: 0000, 0001,0010,…….. and 1111

PCM


. Coding (Cont.)3

1111

0001

0000

0111

bit4n

Coding of a Quantized signal


Quantization Noise (Error) of the Uniform Quantizer • When a signal is quantized, this introduces an error - the quantized signal is

an approximation of the actual amplitude value.

• The difference between quantized value (midpoint) and actual value is referred to as the quantization error q:

• The quantization error is uniformly distributed with pdf:

• The more levels, the smaller which results in smaller errors.

• BUT, the more levels, the more bits required to encode the samples.

Δ Δq

2 2

)1(

0

22

1

)(

otherwise

qqfQ

0 q

;)( XXqerroronQuantizati q


• Consider an input signal of continuous amplitude in the range and we use a uniform quantizer, then we define the step size of the quantizer as:

Since the mean of the quantization error

is zero, its variance (power or mean square value)

is derived as:

12

|3

1

1

)(

][

2

2/2/

3

2/

2/

2

2/

2/

2

22

q

dqq

dqqfq

QE

Q

Q

Variance (power) of Quantization error

0

Quantization Noise Ratio of a -to-Derivation of Output SignalUniform Quantizer

),( maxmax VV

L

Vmax2

2Q

Using equation (1)

/1

q


• Let n denotes the number of bits per sample used in the construction of binary code. We may write:

Using these equations, the step size can be written as:

and

Then the variance of the quantization error (noise power) becomes:

• The peak power of the analog signal can be expressed as:

• The output signal-to-quantization noise ratio of a uniform quantizer is given by:

nL 2 Ln 2log

n

V

2

2 max

n

Q V 22

max

22 2

3

1

12

L

Vmax2

Quantization Noise Ratio of a Uniform Quantizer-to-Derivation of Output Signal

2max

LV

4

222

max

LVP

n

Q

LLP

SQNR 22

2

22

2233

12/

4/


Quantization Noise Ratio of a Uniform -to-Output SignalQuantizer

• The output signal-to-quantization noise ratio of a uniform quantizer is given by:

This equation shows that the output signal to quantization noise

ratio of the quantizer increases exponentially with increasing

the number of bits per sample n. Any increase in n leads

to increase in the transmission bandwidth

n

Q

LLP

SQNR 22

2

22

2233

12/

4/


Quantization & Coding


Example: Statistics of Speech Amplitudes

•Very low speech

amplitudes

predominante 50%

of the time.

•Large amplitude

values are

relatively rare.

•Uniform

quantization would

be wasteful for

speech signals. (normalized to its rms values))


Uniform Quantizers-Non

• A nonuniform quantizer uses a variable step size.

• Basic Concept of Non-Uniform Quantizers

– Concentrate quantization levels within the range where the pdf of the message signal is large

1 iii xx

x

fX(x)

Representation Levels

Li ...,,2,1

1ixix

i 1 i

1ix

pdf


Uniform Quantizers (Cont.)-Non

• Non-uniform quantization can provide small quantization errors of the weak signals and increase quantization errors of strong signals.

• The effect is to improve the overall SQNR by reducing the noise for the predominant weak signals at the expense of an increase in noise for the rarely occurring strong signals.


Uniform Quantizers (Cont.)-Non

• The most commonly used compander (Compressor/Expander) uses a logarithmic compression Y=log X where the levels are crowded near the origin and spaced farther apart near the peak values of X.

• Two commonly used logarithmic compression laws are so called and A-compression laws defined as:


Example:

Original Signal

After Compression

The dynamic range of a signal is compressed before

transmission and is expanded to the original value

at the receiver

Dynamic range

describes the ratio

between the largest and

smallest amplitude of the

signal.

http://upload.wikimedia.org/wikipedia/commons/9/94/Waveformoriginal.png

http://upload.wikimedia.org/wikipedia/commons/2/2e/Waveformcompanded.png


Law Quantizer-μ • Used in North America and Japan

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

m

m'

=5 =100=0

ln 1+μ mm' =

ln 1+μ

where m' is the compressed signal

Normalized to

maximum

value of the

amplitude of

the signal input

Output

Output

signal

input

signal


Law Quantizer-A

• Used in Europe

A m 10 m

1+ln A Am' =

1+ln A m 1m 1

1+ln A A

where m' is the compressed signal

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

m

m'

A=1 A=100A=2

input

output

The µ-law algorithm

provides a slightly

larger dynamic range

than the A-law at the

cost of worse

proportional distortion

for small signals


Companding (Compressor/Expander)

• It is often much cheaper to build ADCs which implement uniform quantization. • In the Transmitter: Compress the signal first, then apply uniform PCM. • In the Receiver: Demodulate the uniform PCM then expand the signal.

Compressor

Analog-to-Digital

Converter (ADC)

Input

Signal Transmitter

Expander

Output

Signal Receiver

Digital-to-Analog

Converter (DAC)

Channel


Bandwidth of PCM signals • PCM codeword length is: n bits

• Sample rate: fs samples /second

• Bit rate R: n fs bits/second

• First null BW (with rectangular pulse shaping) the first null bandwidth is: n fs Hz

Bit rate ss nffSample

bitsR

#

)(log2 LffnRB ssPCM

Without Proof


Bandwidth of PCM signals

We have:

The bandwidth of a PCM signal is at least 2n times larger

than the maximum frequency of the analog signal.

max2: ffIf s

max2: ffIf s

LfnffnRB sPCM 2maxmax log22

LfnfnfB sPCM 2maxmax log22

lowest bandwidth of PCM


Digital Telephone

Voice channel: 4 kHz

8000 samples per second required

8 bits per sample

==> 64 kbps bit rate )( sfnR


SNR of PCM

• The peak signal power to the average noise power is given by:

• The average signal power to the average noise power is given by:

• Where is the channel bit error rate due to noise.

e

outpkPL

L

)1(41

3SNR

2

2

e

outPL

L

)1(41SNR

2

2

eP

Proof : see the

reference book


• Neglecting : the peak SNR resulting only from quantizing error is given by:

• The average SNR due to quantizing error only is given by:

• Since , the previous equations in dB can be written as:

• (6 dB law)

• for peak SNR and for average SNR

eP

23SNR Loutpk

2SNR Lout

nL 2

ndB 02.6SNR

77.4 0

Proof next page


Proof of:

dBn

n

LSNR

LSince

n

n

dB

n

n

02.677.4

301.0277.4

]2log3[log10

)23log(10SNR

233

2

2

2

22

ndB 02.6SNR


SNR of PCM with Compression

• The 6-dB law (output SNR) in case of using compression becomes:

ndB 02.6SNR

)compandinglaw(,)]1[ln(log2077.4

)compandinglaw(,)]ln1[log2077.4 AA

levelsinputlargelysufficientFor

or

Proof : see the

reference book


Quantization & Coding

Remember

word

review of probability theory - guceee.guc.edu.eg/courses/communications/comm502... ·...

Documents