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EECS 247 Lecture 8: Sampling 2002 B. Boser 1A/DDSP

Sampling and Aliasing

Any continuous time signal can be sampled andprocessed in the sampled-data domain A/D converters look at signals at only discrete points in time

Computer models of continuous systems must operate atdiscrete time intervals

Even analog filters can use continuous or sampled timesignal representations

Multiple continuous time signals can yield exactly thesame sampled data signal aliasing Lets look at samples taken at 1sec intervals of several

sinusoidal waveforms


Sampling Sinewaves

timevoltage

v(t) = sin [2(101000)t]

T = 1s

fs = 1/T = 1MHz

fx = 101kHz

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Sampling Sinewaves

timevoltage

v(t) = - sin [2(899000)t]

T = 1s

fs = 1MHz

fx = 899kHz


Sampling Sinewaves

timevoltage

v(t) = sin [2(1101000)t]

T = 1s

fs = 1MHz

fx = 1101kHz

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Aliasing

Multiple continuous time signals can produceidentical series of sampled voltages

The translation of signals from NfSfIN downto fIN is called aliasing

Sampling theorem: fs > 2fB

If aliasing occurs, no signal processingoperation downstream of the samplingprocess can recover the original continuoustime signal

If you dont like it, sample faster!


Time vs Frequency Domain

Time Frequency

multiply convolve

T fS=1/T

fB

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Nomenclature

Continuous time signal x(t)

Sampling interval T

Sampling frequency f s = 1/T

Sampled signal x(kT) = x(k)

time

x(kT) x(k)

T

x(t)


Sampled Signal Properties

Its obvious from the preceding slides that the

frequency content of a continuous signal can

be grossly distorted by sampling

What do we know about the relationships

between x(t) and x(k)? Well assume that x(t) is stationary; that is, itsmean is constant and its autocorrelation R(t1,t2)

depends only on the time difference t1-t2

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Sampled Signal Properties

If x(t) is stationary, then(Athanasios Papoulis, Signal Analysis, 1977, Section 9.4.):

x(k) is also stationary

Sampling doesnt change the mean: E{x(k)}=E{x(t)}

E{} is the expectation operator

Sampling doesnt change energy:

E{[x(k)]2}= E{[x(t)]2}

Continuous time energy will show up someplace in

the frequency domain after sampling

Aliasing may move continuous signal frequency

components to wrong frequencies


Anti-Aliasing Filter

time time

fs

frequency time

frequency time

Anti-Aliasing Filter

Aliasing

fs

fs

Filtered Spectrum

fB

fB

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Sampled Signals

Sampled data signals are valid only atsampling instances

In an actual circuit, the signal transitions tothe new value between sampling instances

The value between sampling instances is

insignificant and often erroneous (e.g.amplifier slewing distortion)


Zero-Order Hold

Reconstructs CT signal

from SD signal

Frequency response?

0 0.5 1 1.5 2 2.5 3 3.5

x 10-5

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time

Amplitude

sampled dataafter ZOH

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Zero-Order Hold

( ) d

d

d

fTj

d

d

d

sT

d

sT

d

efT

fT

sT

e

sT

e

sT

=

=

2

2sin

1

1

Step response Laplace transform

Td

timeTd0


0 0.5 1 1.5 2 2.5 3

x 106

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency

Magnitude

sampled signalsinczoh signal

Spectrum of Reconstructed Signal

Reconstructed

signal

SD signal:

periodic spectrum

Sinc

Td = T = 1/fs

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0 0.5 1 1.5 2 2.5 3

x 106

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency

Magnitude

sampled signalsinczoh signal

Spectrum of Reconstructed Signal

Reconstructed

signal

SD signal:

periodic spectrum

Sinc

Td

= 0.5T


Summary

Quantization in Time:Continuous time (CT) vs Sampled Data (SD)

Sampling theorem: fs > 2fB

IF sampling theorem is met, CT signal can berecovered from SD signal without loss ofinformation

ZOH reconstructs CT from SD signal

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