Sampling PAM- Pulse Amplitude Modulation (continued)

1

SamplingPAM- Pulse Amplitude Modulation

(continued)

EELE445-14Lecture 14

Sampling

Properties will be looking at for:

•Impulse Sampling•Natural Sampling•Rectangular Sampling

2

Figure 2–18 Impulse sampling.

Impulse Sampling

sT2

( )

( ) [ ]L++++=

====

++⇒−= ∑∑∞

=

∞

−∞=

)3cos(2)2cos(2)cos(211

2120

)cos()(

0

10

tttT

t

TD

TD

T

tnDDnTtt

ssss

T

sn

sssn

nnsn

nsT

s

s

ωωωδ

πωϕ

ϕωδδ

3

Impulse Sampling

( ) )(∑∞

−∞=

−=n

sT nTtts

δδ

)(tws

( ) [ ]

[ ] )(1)()(

)3cos(2)2cos(2)cos(21)()()(

∑∞

−∞=

−==

++++==

ns

sss

ssss

Ts

nffWT

twFfW

tttT

twttwtws

Lωωωδ

Impulse Sampling- text

∑

∑

∑∑

∞

−∞=

∞

−∞=

∞

−∞=

∞

−∞=

−=

=

−−=

−=

n ss

s

tjnn

ss

n ss

n ss

nffWT

fW

eT

twtw

gsubsitutin

eqnTtnTw

nTttwtw

s

)(1)(

1)()(

,

1712)()(

)()()(

ω

δ

δ

4

Impulse SamplingThe spectrum of the impuse sampled signal is the spectrum of the unsampled signal that is repeated every fs Hz, where fs is the sampling frequency or rate (samples/sec). This is one of the basic principles of digital signal processing.

Note:This technique of impulse sampling is often used to

translate the spectrum of a signal to another frequency band that is centered on a harmonic of the sampling frequency, fs.

If fs>=2B, (see fig 2-18), the replicated spectra around each harmonic of fs do not overlap, and the original spectrum can be regenerated with an ideal LPF with a cutoff of fs/2.

Impulse SamplingUndersampling and aliasing.

5

Natural SamplingGeneration of PAM with natural sampling (gating).

Natural Sampling

Duty cycle =1/3

6

Natural Sampling

ss f

df 3= at Null

PAM and PCM

• PAM- Pulse Amplitude Modulation:– The pulse may take any real voltage value that is

proportional to the value of the original waveform. No information is lost, but the energy is redistributed in the frequency domain.

• PCM- Pulse Code Modulation:– The original waveform amplitude is quantized with

a resulting loss of information

7

Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0

Figure 3–4 Demodulation of a PAM signal (naturally sampled).

nth Nyquist region recovery

Comb Oscillatorn=1,2,3….


Figure 3–5 PAM signal with flat-top sampling.Impulse sample and hold

8


Figure 3–6 Spectrum of a PAM waveform with flat-top sampling.


Figure 3–7 PCM trasmission system.

9

PCM Transmission

• Negative: The transmission bandwidth of the PCM signal is much larger than the bandwidth of the original signal

• Positive: The transmission range of a PCM signal may be extended with the use of a regenerative repeater.


Figure 3–8 Illustration of waveforms in a PCM system.

10


Figure 3–8 Illustration of waveforms in a PCM system.


Illustration of waveforms in a PCM system.

11


Illustration of waveforms in a PCM system.

•The error signal, (quantization noise), is inversely proportional to the number of quantization levels usedto approximate the waveform x(t)

•Companding is used to improve the SQNR (signal toquantization noise ratio) for small amplitude x(T) signals


Figure 3–9 Compression characteristics (first quadrant shown).

12


Figure 3–9 Compression characteristics (first quadrant shown).


Figure 3–10 Output SNR of 8-bit PCM systems with and without companding.

( )[ ]

( )[ ] compandinglawAAnNS

compandinglawnNS

quantionuniformx

Vn

NS

dB

dB

rms

rangepeak

dB

−+−+=⎥⎦⎤

⎢⎣⎡

−+−+=⎥⎦⎤

⎢⎣⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−+=⎥⎦

⎤⎢⎣⎡

ln1log2002.677.4

1lnlog2002.677.4

log2002.677.4

μμ

13

26

Quantization Noise,Analog to Digital Converter-A/D

EELE445Lecture 16


Figure 3–7 PCM trasmission system.

q(x)

14

Quantization

V

-V

Δ

x(t)

Q(x)

n

n

M

VMV

2

22

1

=

==Δ −

)(ˆ xQxn =

Quantization – Results in a Loss of Informationx(t)

Q(x)

2Δ

2Δ

−

Lost Information

After sampling, x(t)=xi

R∈ix

After Quantization:

R∈= x,x̂Q(x)

122

−==Δ nV

MV

15

Quantization Noise

Quantization function:

Define the mean square distortion:

2)(

~))(()( 22

Δ≤−

=−=

xQx

andxxQxxq

Quantization

( ) noiseonquantizatithePMV

dqqq

nq==

Δ=

Δ= ∫

Δ

Δ−

2

2

2

2

2

22

3

12

1

where M=2n, V is ½ the A/D input range,and n is the number of bits

n

n

M

VMV

2

22

1

=

==Δ −

16

Quantization Noise from the expectation operator:

)~(xf

2Δ

−2Δ

Δ1

x~

( )[ ] [ ]

)(~

))((ˆ,

~ˆ

2

XQXX

XQXEXXdED

XandXX

−=

−==

:ddistribute uniformly is error the of pdf The

:as n)(distortio error squared mean the define can weSo are so variable, random a is Since

⎪⎩

⎪⎨⎧ Δ

≤≤Δ

−Δ

=otherwise

xxf

02

~22)~(

SQNR – Signal to Quantization Noise Ratio

17


Px may be found using:


The distortion, or “noise”, is therefore:

Where Px is the power of the input signal

18

SQNR – Linear Quantization[ ]

12max

2max

2

<

≤

xP

xXE

x

The SQNR decreases asThe input dynamic rangeincreases

U-Law Nonuniform PCMused to increase SQNR for given Px , xmax, and n

U=255 U.S

19

A-Law Nonuniform PCM

a=87.56 U.S

u-Law v.s. Linear Quantization8 bitPx is signal powerRelative to full scale

Px

20

Pulse Code Modulation, PCM, Advantagecompared with analog systems

• b is the number of bits

• γ is (S/N)basebandRelative to full scale

• PPM is pulse position modulation

Documents

Sampling PAM- Pulse Amplitude Modulation (continued)