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SAMPLING & ALIASING

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OVERVI

EW

Periodic sampling, the process of

representing a continuous signal with a

sequence of discrete data values, pervades

the field of digital signal processing.

In practice, sampling is performed by

applying a continuous signal to an analog-to-digital (A/D) converter whose output is a

series of digital values.

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OVERVI

EW(cont.)

With regard to sampling, the primary

concern is how fast must the given

continuous signal be sampled in order to

preserve its information content.

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ALIASING There is a frequency-domain ambiguity

associated with the discrete-time signal

samples that is absent in the continuoussignal world.

eg. Suppose you are given the

following sequence of values,x(0) = 0

x(1) = 0.866

x(2) = 0.866

x(3)= 0

x(4) = -0.866

x(5) = -0.866

x(6) = 0

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and were told that they represented instantaneous

values of a sinewave. Next you were told to

draw that sinewave. You would be able to draw a

sinewave that passes through the points

representing the original sequence.

yn

n

0 2 4 61

0

1

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x1i

0

x2i

ti

However, someone else might draw another

sine wave with a different frequency through

the same points.

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As an example, consider two sine waves that

differ only in frequency.O

ne has a frequency of500 Hz, and the frequency of the subsequent sine

wave is 8500 Hz.

x1i

0

x2i

ti

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Consider the continuous time domain sinusoidal

signal

This a garden variety sinewave with a frequency

fo Hz. If we sample at a rate offs samples per

second, where Ts = 1/fs. If we start sampling at

t = 0, we will obtain samples at 0 ts, 1 ts, 2 ts,

and so on.

x t fto( ) sin( )! 2T

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The final equation defines the value of the nth

sample of ourx(n) sequence to be equal to the

original sine wave at the time instant nTs.

!

!

!

!

0

0

0

0

2sin)(

22sin)2(

2sin)(

02sin)0(

T

nTnTx

T

TTx

T

TTx

Tx

ss

ss

ss

T

T

T

T

/

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Because two values of a sine wave are

identical if theyre separated by an integermultiple of 2T radians, i.e.,

where m is any integer. Now

)2sin()sin( mTJJ !

)22sin()2sin()( mntfntfnx soso TTT !!

-

! s

s

o ntnt

mfnx T2sin)(

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If we let mbe an integer multiple ofn, m=kn,

we can replace the m/n ratio with kso that

The fo and (fo + kfs) factors are equal. It means

that anx(n) sequence of digital sample values,

representing a sine wave offo Hz, also exactlyrepresents sine waves at other frequencies,

namely fo + kfs . This is one of the most

important relationships in the field of DSP.

-

! s

s

o ntt

kfnx T2sin)(

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When sampling at a rate offs samples/sec., ifkis any positive or negative integer, we cannot

distinguish between the sampled values of a

sine wave offo Hz and a sine wave of(fo + kfs)

Hz.

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In many textbooks you will find sampling described as a

multiplication of the input analog waveform with a

periodic delta, Dirac, or impulse function.

In an idealized system our sampling waveform would

consist of a train of impulse functions spaced evenly by

a period Ts.We can describe our idealized sampling function,z(t), as

the sum of all the individual impulse functions:

g g!

tTtTttTtTtttz

ss

ss

HHHHHHH

-

-

22)(

g!

g!

!n

n

snTttz H)(

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If we multiply these by our analog input signal,f(t), we

obtain a train of pulses whose amplitudes are equal to the

amplitude off(t) at that moment in time.

Mathematically, the output sampled waveform,y(t), is

just the multiplication ofz(t) with the input analog signal

f(t):

g!

g!

!n

n

snTttfty H)()(

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Oversampling

If the original waveform does not vary much

over the duration ofp(t), then we will also

obtain a good construction. Oversampling, i.e.,

using a sampling rate that is much greater than

the Nyquist rate, can ensure this.