Digital Coding of Analog Signal Ut1

8/8/2019 Digital Coding of Analog Signal Ut1

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Digital Coding ofAnalog Signal

Prepared By:

Amit Degada

Teaching Assistant

Electronics Engineering Department,

Sardar Vallabhbhai National Institute of Technology,

Surat-395007.


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Outline Analog To Digital Converter Review of sampling

Nyquist sampling theory: frequency and time domain Alliasing Bandpass sampling theory

Natural Sampling Aperture Effect

Quantization Quantization. Quantization Error. Companding.

Two optimal rules A law/u law

Coding Differential PCM


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Claude Elwood Shannon, Harry Nyquist


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Sampling Theory In many applications it is useful to represent a

signal in terms of sample values taken atappropriately spaced intervals.

The signal can be reconstructed from thesampled waveform by passing it through an ideallow pass filter.

In order to ensure a faithful reconstruction, theoriginal signal must be sampled at an appropriaterate as described in the sampling theorem. A real-valued band-limited signal having no spectral

components above a frequency of FM Hz is determineduniquely by its values at uniform intervals spaced nogreater than (1/2FM) seconds apart.


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Sampling Block Diagram

Consider a band-limited signal f(t) having nospectral component above B Hz.

Let each rectangular sampling pulse have unit

amplitudes, seconds in width and occurring atinterval of T seconds.

A/D

conversionf(t)

T

fs(t)

Sampling


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Sampling

Sampled waveform

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Sampled waveform

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Sampled waveform

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Signal waveform

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1 201

Impulse sampler

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Impulse Samplingwith increasing sampling time T

Sampled waveform

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1 201

Sampled waveform

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1 20

Sampled waveform

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1 201

Sampled waveform

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1 20


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EE 541/451 Fall 2006

Introduction

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Interpolation FormulaInterpolation Formula

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Interpolation

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If the sampling is at exactly the Nyquist rate, then


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Under Sampling, Aliasing


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Avoid Aliasing

Band-limiting signals (by filtering)before sampling.

Sampling at a rate that is greaterthan the Nyquist rate.

A/D

conversionf(t)

T

fs(t)

Sampling

Anti-aliasing

filter


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Practical InterpolationSinc-function interpolation is theoretically perfect but it cannever be done in practice because it requires samples from

the signal for all time. Therefore real interpolation must

make some compromises. Probably the simplest realizable

interpolation technique is hat a DAC does.


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Natural sampling(Sampling with rectangular waveform)

S al wa efor

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1 201 401 601 801 1001 1201 1401 1601 1801 200

Natural sampler

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Bandpass Sampling A signal of bandwidth B, occupying the frequency

range between fL and fL + B, can be uniquelyreconstructed from the samples if sampled at arate fS :

fS >= 2 * (f2-f1)(1+M/N)where M=f2/(f2-f1))-N and N = floor(f2/(f2-f1)),

B= f2-f1, f2=NB+MB.


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Entire spectrum is allocated for a channel (user) for a limitedtime.

The user must not transmit until its

next turn.

Used in 2nd generation

Advantages:

Only one carrier in the medium at any given time

High throughput even for many users

Common TX component design, only one power amplifier

Flexible allocation of resources (multiple time slots).

f

t

c

k2 k3 k4 k5 k6k1

Frequency

Time

Time Division Multiplexing


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Quantization Scalar Quantizer Block Diagram


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Quantization Procedure


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Quantization Error


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Quantization Type

Mid-tread Mid-rise


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Quantization Noise


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Quantization Noise

What happens if no. of representationlevel increases?

>64 distortion is significant

Quantization error is uniformlydistributed in interval (-/2 to /2).

The Avg. Power of Quantizing error qe


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0 V

K

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Sample of

AmplitudeK+ qe

Pq

Math


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Example

A sinusoidal Signal of amplitude Amuses all Representation levelsprovided for Quantization in the caseof full load condition. CalculateSignal to Noise ratio in db assumingthe number of quantization levels to

be 512. ANS: 55.8 db.


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Example SNR for varying number of representation

levels for sinusoidal modulation 1.8+6 XdB

Number ofrepresentation level L

Number ofBits perSample, R

SNR (dB)

32 5 31.864 6 37.8

128 7 43.8

256 8 49.8


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Companding

Process of uniform Quantization is notpossible.

Example: Speech, Video.

The variation in power from weak signal topowerful signal is 40 db.

So Ratio 1000:1

Excursion in Large amplitude occurs less

frequently. This Scenario is cared by Non- Uniform

Quantization.


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Non-uniform Quantizer

x^

ExampleF: y=log(x) F-1: x=exp(x)

F: nonlinear compressing function

F-1: nonlinear expanding function

F and F-1: nonlinear compander

We will study nonuniform quantization by PCM example next

A la and Q la

y^

yX F Q F-1XXX Q

y^y^y^y^y^


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Input-Output characteristicof Compressor


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Q Law/A Law The Q-law algorithm (-law) is a companding algorithm,

primarily used in the digital telecommunication systems ofNorth America and Japan. Its purpose is to reduce thedynamic range of an audio signal. In the analog domain,

this can increase the signal to noise ratio achieved duringtransmission, and in the digital domain, it can reduce thequantization error (hence increasing signal to quantizationnoise ratio).

A-law algorithm used in the rest of worlds.

A-law algorithm provides a slightly larger dynamic rangethan the mu-law at the cost of worse proportionaldistortion for small signals. By convention, A-law is used foran international connection if at least one country uses it.


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Q Law


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EE 541/451 Fall 2006

A Law


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Implementation ofCompander

Diode equation

Piece-wise linear Approach


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Coding

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Digital Coding of Analog Signal Ut1

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