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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.