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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
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Sampling
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Sampling
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Impulse Samplingwith increasing sampling time T
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EE 541/451 Fall 2006
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Interpolation
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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)
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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).
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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
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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