Meixia Tao @ SJTU

Principles of Communications

Chapter 4: Analog-to-Digital Conversion

Selected from: Chapter 7.1 – 7.4 of Fundamentals of Communications Systems, Pearson Prentice Hall

2005, by Proakis & Salehi

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Digital Communication Systems

Source A/D converter

Source encoder

Channelencoder Modulator

Channel

DetectorChannel decoder

Source decoder

D/A converter

User

Transmitter

Receiver

Absent if source is

digitalNoise

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Topics to be Covered

Sampling

Quantization

Encoding

Pulse Code Modulation

Analog signal

Discrete-time continuous-valued signal

Discrete-time discrete-valued signal

Bit mapping

Digital transmission

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Harry Nyquist (1928)

Continuous-time Discrete-time

Sampling Theorem

"Certain topics in telegraph transmission theory", Trans. AIEE, vol. 47, pp. 617–644, Apr. 1928

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Sampling Process

( ) ( ) ( ) ( ) ( )s s sn n

x t x nT t nT x t t nTδ δ δ∞ ∞

=−∞ =−∞

= − = −∑ ∑

The sampling process can be regarded a modulation process with carrier given by periodic impulses. It’s also called pulse modulation

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If or

The minimum sampling rate is known

as Nyquist sampling rate

1 1( ) ( )n ns s s s

n nX f X f f X fT T T Tδ δ

∞ ∞

=−∞ =−∞

= ∗ − = −

∑ ∑

W-W f

( )X f

f1/Ts-1/Ts

( )X fδ

12sTW

>

aliasing error, reconstruction is not possible,

1 2ss

f WT

= <

Sampling Process

f1/Ts-1/Ts

( )X fδ

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Reconstruction LPF with frequency response

Ideal LPF

where

( ) 10

s

s

T f WH f

f WT

<= ≥ −

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Reconstruction With this choice, we have

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Meixia Tao @ SJTU

Topics to be Covered

Sampling

Quantization

Encoding

Pulse Code Modulation

Analog signal

Discrete-time continuous-valued signal

Discrete-time discrete-valued signal

Bit mapping

Digital transmission

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Quantization

quantization (rounding)

sampling

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Quantization Region Quantization regions:

Quantization level for each :

Quantization: ( ) , for all , 1,...,k kQ x x x k N= ∈ =R

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Performance Measure of Quantization

Quantization error:

Average (mean square error) distortion:

Signal-to-quantization noise ratio (SQNR):

( , ( )) ( )e x Q x x Q x= −

[ ]( )2( )D E X Q x= −

[ ]( )2

2

( )( )

E XSQNRE X Q x

=−

( )

/2 2

/2

/2 2

/2

1lim ( )

1lim ( ) ( ( ))

T

TT

T

TT

x t dtTSQNR

x t Q x t dtT

−→∞

−→∞

=−

∫

∫

For random variable X

For signal x(t)

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Example The source X(t) is a stationary Gaussian source with mean

zero and power spectral density

It is sampled at the Nyquist rate and each sample is quantized using the 8-level quantizer with

What is the resulting distortion ?

What is the SQNR?

Are the sampled signals independent to each other?

2 100( )

0 otherwisexf Hz

S f <

=

0 1 2 3 4 5 6 7, 60, 40, 20, 0, 20, 40, 60a a a a a a a a= −∞ = − = − = − = = = =

1 2 3 4 5 6 7 870, 50, 30, 10, 10, 30, 50, 70x x x x x x x x= − = − = − = − = = = =

Sol: 33.38

Sol: 10.78dB

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Uniform Quantizer Range of the input samples = [-a, a]

Number of quantization levels = .

Length of each quantization region

Quantized value are the midpoints of quantization regions.

Quantization distortion (assume quantization error uniformly distributed on )

1

22v

a aN −∆ = =

( , )2 2∆ ∆

−/2 2 2 2

2 22

/2

1[ ]12 2 3 4v

a aE e x dxN

∆

−∆

∆= = = =

∆ ⋅∫

102 2 2

3 4 10log 6 4.8[ ]

v dBX X XP P PSQNR v

E e a a⋅

= = = + +One extra bit increases the SQNR by 6 dB!

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Example Find the SQNR for a signal uniformly distributed on

and quantized by a uniform quantizer with 256

levels.

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Nonuniform Quantizer By relaxing the condition that the quantization regions be of

equal length, we can minimize the distortion with less constraints

The resulting quantizer will perform better than a uniform quantizer

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Optimal Quantizer Lloyd-Max Conditions (criteria for optimal quantizer)

The boundaries of the quantization regions are the midpoints of the corresponding quantized values

The quantized values are the centroids of the quantization regions.

( )112i i ia x x += +

1

1

( )

( )

i

i

i

i

a

ai a

a

xf x dxx

f x dx−

−

=∫∫

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Meixia Tao @ SJTU

Example How would the results of the example in slide#13 change if

an optimal non-uniform quantizer with the same number of levels?

Sol: 14.6dB (i.e. 3.84dB better)

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Vector Quantization Scalar quantization

Each sample is quantized individually

Vector quantization: Take blocks of source outputs of length n, and design the quantizer

in the n-dim Euclidean space, rather than doing the quantization based on single samples in an one-dim space

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An Illustration

4-level quantizer 4-level quantizer applied to 2 samples

Vector quantization in 2-D

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Optimal Vector Quantization Region Ri is the set of all points in the n-dim space that are

closer to xi than any other xj, for all j\= i; i.e.

xi is the centroid of the region Ri, i.e.

{ }: ,ni i jR R j i= ∈ − < − ∀ ≠x x x x x

1 ( )( )

i

ii R

f dP R

=∈ ∫ xx x x x

x

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Application of Quantizer Application of quantizer:

ADC in digital communication Channel feedback in FDD cellular networks

More reading: Allen Gersho, "Quantization", IEEE Communications Society

Magazine, pp. 16–28, Sept. 1977.

Meixia Tao @ SJTU

Topics to be Covered

Sampling

Quantization

Encoding

Pulse Code Modulation

Analog signal

Discrete-time continuous-valued signal

Discrete-time discrete-valued signal

Bit mapping

Digital transmission

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Encoding The encoding process is to assign v bits to

quantization levels. Since there are v bits for each sample and fs

samples/second, we have a bit rate of

Natural binary coding Assign the values of 0 to N-1 to different quantization levels

in order of increasing level value.

Gray coding Adjacent levels differ only in one bit

bits/secondsR vf=

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Examples Natural binary code (NBC), folded binary code (FBC), 2-

complement code (2-C), 1-complement code (1-C), and Gray code

Level no NBC FBC 2-C Gray code Amplitude level7 111 011 011 100 3.56 110 010 010 101 2.55 101 001 001 111 1.54 100 000 000 110 0.53 011 100 100 010 -0.52 010 101 111 011 -1.51 001 110 110 001 -2.50 000 111 101 000 -3.5

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Most widely used method for A/D conversion of audio source Block diagram of a PCM system

Uniform PCM: the quantizer is a uniform quantizer Bit rate

If a signal has a bandwidth of W and is sampled at and v bits are used for each sampled signal, then

Bandwidth requirement: With binary transmission

Pulse Code Modulation (PCM)

Sampler Quantizer Encoder( )x t { }nx ˆ{ }nx { 0110 }

(to be discussed in later chapters)

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Non-uniform PCM For speech waveform, there exists a higher probability for smaller

amplitudes and a lower probability for larger amplitudes. It makes sense to design a quantizer with more quantization regions at

lower amplitudes and fewer quantization regions at large amplitudes. The usual method:

First pass the samples through a nonlinear filter that compress the large amplitudes and then perform a uniform quantization

Apply the inverse (expansion) of this nonlinear operation at the receiver

=> Companding (compressing-expanding)

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Compander (speech coding) μ law compander

The standard PCM system in US & Canada employs a compressor with followed by a uniform quantizer with 8 bits/sample

( )log 1 | |( ) sgn( ), | | 1

log(1 )x

g x x xµµ

+= ≤

+

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Compander A law compander

1 log | |( ) sgn( ), | | 11 log

A xg x x xA

+= ≤

+

81

8486

1

x

y

21

41

81

1

6.87=A

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Differential PCM (DPCM) For a bandlimited random process, the sampled values are usually

correlated random variables This correlation can be exploited to improve the performance Differential PCM: quantize the difference between two adjacent

samples. As the difference has small variation, to achieve a certain level of

performance, fewer bits are required

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DPCM

Quantizer

delay

delay

encoder

decoder

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Delta Modulation (DM) DM is a simplified version of DPCM having a two-level quantizer with

magnitude In DM, only 1-bit per symbol is employed. So adjacent samples must

have high correlation.

±∆

×××)( 1−

′itm )( itm′

)( itm)(tm

)(tm′

∆The step size is critical in designing a DM system.

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Step Size

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Suggested Reading Chapter 7.1 – 7.4 of Fundamentals of Communications

Systems, Pearson Prentice Hall 2005, by Proakis & Salehi

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