Sampling Dan Pcm

7/31/2019 Sampling Dan Pcm

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Sampling and Pulse CodeModulation

Chapter 6

Dr. Yun Q. Shi

Dept of Electrical & Computer Engineering

New Jersey Institute of [email protected]

Dr. Shi Digital Communications 2

Sampling Theorem A Signal is said to be band-limited if

g (t), its spectrum (FT) G()

G () = 0 as | | > 2B

Sampling Theorem:The signal can be reconstructed from its samples takenuniformly at a rate

R > 2B.

That is, the minimum sampling frequency is

fs = 2B [Ts = 1/2B] (Ts: sampling interval)fs: Nyguist rate for g(t)Ts: Nyguist interval for g(t)

Ts = 1/fs


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Signal Reconstruction:

Interpolation1. Zero-order hold circuit (LPF)

Figure 6.2 ! " Simple interpolation using

zero-order hold circuit (LPF)

Frequency response of the LPF:

(6.8)

Interpolation filter impulse response h(t):

=

=

Bc

B

TcTH ss

4sin

2

1

2sin)(

)()(*)( tgthtg =



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Signal Reconstruction:

Interpolation

2. Ideal interpolation (sinc function)

Figure 6.3 ! " Ideal interpolation

10.6.

)]2[sin)(

)](2[sin)(

)()()(

Eq

kBtckTg

kTtBckTg

kTthkTgtg

ks

k

ss

k

ss

=

=

=



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Maximum Information Rate:

Statement

Two pieces of information per second per

Hertz bandwidth.

That is, a maximum of 2B independent pieces

of information per second can be transmitted

error-freely, over a noise-less channel of

bandwidth B Hz.


Maximum Information Rate:

Justification

Noise free

A channel of bandwidth B Hz can transmit a signal

of bandwidth B Hz error-freely.

Sampling Theorem:

A signal of bandwidth B can be reconstructed from

its Nyguist samples at a rate of 2B Hz.

The signal can be reconstructed by 2B independent

pieces of information per second (lower bound).


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Pulse Modulation PAM Pulse Amplitude Modulation

PWM Pulse Width Modulation

PPM Pulse Position Modulation

PCM Pulse Code Modulation

By far, the most popular among

pulse modulation.

Figure 6.8 shown next



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Time-Division Multiplexing (TDM)

One advantage of using pulse modulation:

It permits the simultaneous transmission of

several signals on a time-sharing basis.

Interweaving several signals along time

domain.

Figure 6.9


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Quantization:

Discretization of magnitude of a signal

Input-output characteristic of a uniform

midtread quantizer (Figure 2.4, Shi & Sun)

Input-output characteristic of a uniform midrise

quantizer (Figure 2.5, Shi & Sun, 1999)

Yi = Q(x) if x belongs to (di , d i+1) di : decision levels

i : index of intervals

yi : reconstruction level (quantizing level)

: step size=+ ii dd 1


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Quantization

1. Except for possibly the right-most and left-most intervals, all intervals (hence, decisionlevels) along the x-axis are uniformly spaced(same interval length).

2. Except for possibly the outer intervals, thereconstruction levels are also uniformlyspaced. Each inner reconstruction level is the

arithmetic average of the two decision levelsof the corresponding interval alongx-axis.



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Quantization Distortion

Quant. noise, quant. error,

Mean-square quant. error

N: number of intervals

fx (x): pdf .

qe)(xQxeq =

=

+

=N

i

d

dxq

i

i

dxxfxQxMSE1

21 )())((

( ) 22 :)( qexQx



Assume thatfx (x) is uniformly distributed

Variance of inputx:

=

=

q

d

dq

MSE

dxN

xQxNMSEi

12

1))((

2

22

2

qqMSE

=( )

12

2

2 =N

x


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Signal-to-noise ratio, SNRms

If

then

2

102

2

10 log10log10 NSNRq

xms ==

nN 2=

ndbnSNR nms 02.62log202log20 1010 ===


Quantization Distortion Meaning:

If we use the Natural Binary Code to code the

reconstruction levels of a uniform quantizer with a

uniformly distributed input source, then every

increased bit in the coding brings out a 6.02 dB

increase in the SNR ms .

That is, whenever the step size of the uniform

quantizer decreases by a half, the decreases

four times.qMSE


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