SAMPLING & ALIASING
OVERVIEW
• 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 analogto-digital (A/D) converter whose output is a
series of digital values.
OVERVIEW (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.
ALIASING
• There is a frequency-domain ambiguity
associated with the discrete-time signal
samples that is absent in the continuous
signal 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
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.
1
y
n
0
1
0
2
4
n
6
However, someone else might draw another
sine wave with a different frequency through
the same points.
x1
i
0
x2
i
t
i
As an example, consider two sine waves that
differ only in frequency. One has a frequency of
500 Hz, and the frequency of the subsequent sine
wave is 8500 Hz.
x1
i
0
x2
i
t
i
Consider the continuous time domain sinusoidal
signal
x (t )  sin(2f o t )
This a garden variety sinewave with a frequency
fo Hz. If we sample at a rate of fs 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.

0 

x(0)  sin  2

T
0 

 Ts 

x(Ts )  sin  2

T
0 

 2Ts
x( 2Ts )  sin  2
T0






 nTs 

x( nTs )  sin  2

T
0 

The final equation defines the value of the nth
sample of our x(n) sequence to be equal to the
original sine wave at the time instant nTs.
Because two values of a sine wave are
identical if they’re separated by an integer
multiple of 2 radians, i.e.,
sin(  )  sin(   2m)
where m is any integer. Now
x(n)  sin( 2 f o nt s )  sin( 2 f o nt s  2m)
 
m  
nt s 
x(n)  sin 2  f o 
nt s  
 
If we let m be an integer multiple of n, m=kn,
we can replace the m/n ratio with k so that
 
k 
x(n)  sin 2  f o  nt s 
ts  
 
The fo and (fo + kfs ) factors are equal. It means
that an x(n) sequence of digital sample values,
representing a sine wave of fo Hz, also exactly
represents sine waves at other frequencies,
namely fo + kfs . This is one of the most
important relationships in the field of DSP.
When sampling at a rate of fs samples/sec., if k
is any positive or negative integer, we cannot
distinguish between the sampled values of a
sine wave of fo Hz and a sine wave of (fo + kfs)
Hz.
IMPULSE FUNCTION
The impulse function is represented by d(t). Its domain is all t, and
the range contains only two values, 0 and .
0,
d (t )  
,
t0
t 0
The range is zero for all values of time except when t = 0, when
the time function has the value of infinity (i.e. blows up).
The graphical representation of the delta function is given below
d(t)
t
To obtain some insight into the d function, we can consider the
function D(t) as shown below
D
D
- t 
2
2
otherwise
1
 ,
Dt    D
0,
The function has a
height of 1/D and a
width of D, resulting in
an area of 1.
D
2
1
1  D  D 
Dt dt  D dt   -  -   1
D  2  2 
-
- D
2



D(t)
-D/2
D/2
t
The d function is obtained from D(t) by the limiting process
d (t )  lim D(t )
D 0
The width of function goes to zero as its amplitude goes to infinity
in a manner that causes the area under the curve to remain constant
at unity.
D(t)
t
As D decreases in value, the function becomes
narrower and higher. In the limit, as D  0,
Dt  d(t).
The integral of D(t) is unity for any value of D. It
follows that the integral of the d function is also unity
provided that the origin is included in the limits of
integration. Otherwise, it is zero.
1,
d (t )dt  
a
0,

b
a0b
otherwise
If we multiply the d function by a constant, k, and then
integrate the product over all time, we get




kd (t )dt  k d (t )dt  k
-
-
Multiplication of d(t) by a constant is equivalent to
choosing D(t) with a height of k/D. This results in an area
equal to k.
TIME-SHIFTED FUNCTIONS
If the argument of the d function is replaced with t-t, we
see that the impulse still occurs when the argument takes
on the value of zero (i.e., when t-t = 0 or t = t).
0,
d (t - t )  
,
t -t  0
t -t  0
TIME-SIFTING PROPERTY
If we integrate the product of the d function and any time
function, we can observe the sifting property of d
function.


f (t )d (t )dt 
-

e
f (t )d (t )dt 
-
where e is arbitrarily small.

e
f (t )d (t )dt 
-e


f (t )d (t )dt
-e
The first and last integrals on the right-hand side of the
previous equation are zero, since the limits of integration
do not include the origin. If f(t) is reasonably well
behaved at the origin (i.e., no jumps or discontinuities), as
e  0, f(t) can be approximate by f(0). Since f(0) is a
constant, it can be taken outside the integral.
lim
e 0

e
f (t )d (t )dt 
-e

e

e
f (0)d (t )dt  f (0) d (t )dt  f (0)
-e
-e
So all the values of f(t) are sifted out except the value at
the origin, f(0).


f (t )d (t )dt  f (0)
-
If we use the time-shifted d function, we obtain
f (t ) 

 f (t )d (t - t )dt
-
Note that
(t - t )  0  t  t  0  (t - t )
Since expressions equal to the same expression are
equal to each other,
(t - t )  (t - t )
and
d (t - t )  d (t - t )
Because d (t - t )  d (t - t ) , we can interchange t
witht to obtain
f (t ) 


f (t )d (t - t )dt
-
Evaluate the following integrals.
a.
c.

4
-1
d (t ) dt
10

-10
3d (t ) dt
-8
b.  d (t ) dt
- 28
50
d.  7d (t ) dt
40
In many DSP 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:
z (t )  d t -      d t - 2Ts   d t - Ts   d t 
d t  Ts   d t  2Ts     d t   
n  -
z (t ) 
 d t - nT 
s
n 
If we multiply these by our analog input signal, f(t), we
obtain a train of pulses whose amplitudes are equal to the
amplitude of f(t) at that moment in time.
Mathematically, the output sampled waveform, y(t), is
just the multiplication of z(t) with the input analog signal
f(t):
n  -
y (t ) 
 f (t )d t - nT 
s
n 
Interpolation
A digital-to-analog converter (DAC) converts
digital signals to continuous (analog) signals.
A general formula that describes a broad class of
DACs is
y (t ) 

 y[n] p(t - nTs)
n -
where p(t) is the characteristic pulse shape of the
converter.
The equation state that the output signal is
produced by adding together many pulses, each
shifted in time. That is, at each sample time
tn = nTs, a pulse p(t-nTs) is emitted with an
amplitude
The simplest pulse shape is a symmetric square
pulse:
R
1
|
p( t )  S
|T0
1
1
- Ts  t  Ts
2
2
otherwise
The figure below shows a pulse is Ts = 5ms
and a amplitude of 1.
1
p( t )
0
0.005
0
t
0.005
The reconstructed waveform for the square pulse is
a poor approximation of the original sine wave.
1
w( t )
0
1
0.005
0
0.005
0.01
t
0.015
0.02
0.025
Although this a poor approximation, this is a
useful model since many physically realizable
DACs produce outputs that look exactly like
this.
Since a constant is a polynomial of zero order,
and since the effect of the flat pulse is to hold or
replicate each sample for Ts seconds, the use of
a flat pulse is called a zero-order hold
reconstruction.
The triangular pulse is defined as a pulse
consisting of the first-order polynomial (straightline segments:
Rt
1|
p(t )  S Ts
|T0
- Ts  t  Ts
otherwise
1
0.5
p( t )
0
0.005
0
t
0.005
1
w( t )
0
1
0.005
0
0.005
0.01
t
0.015
0.02
0.025
1
w( t )
0
1
0.005
0
0.005
0.01
t
0.015
0.02
0.025
Oversampling
If the original waveform does not vary much
over the duration of p(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.