1 Copyright ? 2001,S,K,Mitra
Discrete-Time Signals,
Time-Domain Representation
? Signals are represented as sequences of
numbers,called samples
? Sample value of a typical signal or sequence
denoted as x[n] with n being an integer in
the range
? x[n] defined only for integer values of n and
undefined for non-integer values of n
? Discrete-time signal represented by {x[n]}
????? n
2 Copyright ? 2001,S,K,Mitra
Discrete-Time Signals,
Time-Domain Representation
? Discrete-time signal may also be written as
a sequence of numbers inside braces,
? In the above,
etc,
? The arrow is placed under the sample at
time index n = 0
},9.2,7.3,2.0,1.1,2.2,2.0,{]}[{ ?? ???
?
nx
,2.0]1[ ???x,2.]0[ ?x,1.1]1[ ?x
3 Copyright ? 2001,S,K,Mitra
Discrete-Time Signals,
Time-Domain Representation
? Graphical representation of a discrete-time
signal with real-valued samples
4 Copyright ? 2001,S,K,Mitra
Discrete-Time Signals,
Time-Domain Representation
? In some applications,a discrete-time
sequence {x[n]} may be generated by
periodically sampling a continuous-time
signal at uniform intervals of time
)(txa
5 Copyright ? 2001,S,K,Mitra
? Block-diagram representation of the
sampling process
()axt,,
Discrete-Time Signals,
Time-Domain Representation
[ ] ( ) ( )aat n Tx n x t x n T???
??,1,0,1,2,???n
6 Copyright ? 2001,S,K,Mitra
Discrete-Time Signals,
Time-Domain Representation
? Here,the n-th sample is given by
? The spacing T between two consecutive
samples is called the sampling interval or
sampling period
? Reciprocal of sampling interval T,denoted
as,is called the sampling frequency,
),()(][ nTxtxnx anTta ?? ? ??,1,0,1,2,???n
TF
TFT
1?
7 Copyright ? 2001,S,K,Mitra
Discrete-Time Signals,
Time-Domain Representation
? Unit of sampling frequency is cycles per
second,or Hertz (Hz),if T is in seconds
? Whether or not the sequence {x[n]} has
been obtained by sampling,the quantity
x[n] is called the n-th sample of the
sequence
? {x[n]} is a real sequence,if the n-th sample
x[n] is real for all values of n
? Otherwise,{x[n]} is a complex sequence
8 Copyright ? 2001,S,K,Mitra
Discrete-Time Signals,
Time-Domain Representation
? A complex sequence {x[n]} can be written
as where
and are the real and imaginary
parts of x[n]
? The complex conjugate sequence of {x[n]}
is given by
? Often the braces are ignored to denote a
sequence if there is no ambiguity
][nxre ][nxim
]}[{]}[{]}[{ nxjnxnx imre ??
]}[{]}[{]}[*{ nxjnxnx imre ??
9 Copyright ? 2001,S,K,Mitra
Discrete-Time Signals,
Time-Domain Representation
? Example - is a real
sequence
? is a complex sequence
? We can write
where
}.{ c o s]}[{ nnx 250?
}{]}[{, njeny 30?
}.s in.{ c o s]}[{ njnny 3030 ??
}.{ s in}.{ c o s njn 3030 ??
}.{ c o s]}[{ nny re 30?
}.{ s in]}[{ nny im 30?
10 Copyright ? 2001,S,K,Mitra
Discrete-Time Signals,
Time-Domain Representation
? Example -
is the complex conjugate sequence of {y[n]}
? That is,
}{}.{ s i n}.{ c o s]}[{, njenjnnw 303030 ????
]}[*{]}[{ nynw ?
11 Copyright ? 2001,S,K,Mitra
Discrete-Time Signals,
Time-Domain Representation
? Two types of discrete-time signals,
- Sampled-data signals in which samples
are continuous-valued
- Digital signals in which samples are
discrete-valued
? Signals in a practical digital signal
processing system are digital signals
obtained by quantizing the sample values
either by rounding or truncation
12 Copyright ? 2001,S,K,Mitra
Discrete-Time Signals,
Time-Domain Representation
? Example -
Sampled-data signal Digital signal
Am
pli
tude
t,time A
mpl
itude
t,time
13 Copyright ? 2001,S,K,Mitra
Discrete-Time Signals,
Time-Domain Representation
? A discrete-time signal may be a finite-
length or an infinite-length sequence
? Finite-length (also called finite-duration or
finite-extent) sequence is defined only for a
finite time interval,
where and with
? Length or duration of the above finite-
length sequence is
21 NnN ??
1N??? ??2N 21 NN ?
112 ??? NNN
14 Copyright ? 2001,S,K,Mitra
Discrete-Time Signals,
Time-Domain Representation
? Example - is a finite-
length sequence of length
? Example - is an infinite-
length sequence
432 ???? nnnx,][
8134 ???? )(
nny 40,c o s][ ?
15 Copyright ? 2001,S,K,Mitra
Discrete-Time Signals,
Time-Domain Representation
? A length-N sequence is often referred to as
an N-point sequence
? The length of a finite-length sequence can
be increased by zero-padding,i.e.,by
appending it with zeros
16 Copyright ? 2001,S,K,Mitra
Discrete-Time Signals,
Time-Domain Representation
? Example -
is a finite-length sequence of length 12
obtained by zero-padding
with 4 zero-valued samples
??
?
??
????
850
432
n
nnnx
e,
,][
432 ???? nnnx,][
17 Copyright ? 2001,S,K,Mitra
Discrete-Time Signals,
Time-Domain Representation
? A right-sided sequence x[n] has zero-
valued samples for
? If a right-sided sequence is called a
causal sequence
,01 ?N
1Nn ?
A right-sided sequence
18 Copyright ? 2001,S,K,Mitra
Discrete-Time Signals,
Time-Domain Representation
? A left-sided sequence x[n] has zero-valued
samples for
? If a left-sided sequence is called an
anti-causal sequence
2Nn ?
,02 ?N
A left-sided sequence
19 Copyright ? 2001,S,K,Mitra
Operations on Sequences
? A single-input,single-output (SISO)
discrete-time system operates on a
sequence,called the input sequence,
according some prescribed rules and
develops another sequence,called the
output sequence,with more desirable
properties
x[n] y[n]
Input sequence Output sequence
Discrete-time
system
20 Copyright ? 2001,S,K,Mitra
Operations on Sequences
? For example,the input may be a signal
corrupted with additive noise
? Discrete-time system is designed to
generate an output by removing the noise
component from the input
? In most cases,the operation defining a
particular discrete-time system is composed
of some basic operations
21 Copyright ? 2001,S,K,Mitra
Basic Operations
? Product (modulation) operation,
– Modulator
? An application is in forming a finite-length
sequence from an infinite-length sequence
by multiplying the latter with a finite-length
sequence called a window sequence
? Process called windowing
?x[n] y[n]
w[n] ][][][ nwnxny ??
22 Copyright ? 2001,S,K,Mitra
Basic Operations
? Addition operation,
– Adder
? Multiplication operation
– Multiplier
][][][ nwnxny ??
A
x[n] y[n] ][][ nxAny ??
x[n] y[n]
w[n]
?
23 Copyright ? 2001,S,K,Mitra
Basic Operations
? Time-shifting operation,
where N is an integer
? If N > 0,it is delaying operation
– Unit delay
? If N < 0,it is an advance operation
– Unit advance
][][ Nnxny ??
y[n] x[n] z
1?z y[n] x[n] ][][ 1?? nxny
][][ 1?? nxny
24 Copyright ? 2001,S,K,Mitra
Basic Operations
? Time-reversal (folding) operation,
? Branching operation,Used to provide
multiple copies of a sequence
][][ nxny ??
x[n] x[n]
x[n]
25 Copyright ? 2001,S,K,Mitra
Basic Operations
? Example - Consider the two following
sequences of length 5 defined for,
? New sequences generated from the above
two sequences by applying the basic
operations are as follows,
40 ?? n
}{]}[{ 09643 ??na
}{]}[{ 35412 ???nb
26 Copyright ? 2001,S,K,Mitra
Basic Operations
? Operations on two or more sequences can
be carried out if all sequences involved are
of same length and defined for the same
range of the time index n
}{]}[][{]}[{ 0452446 ????? nbnanc
}{]}[][{]}[{ 341035 ????? nbnand
}..{]}[{]}[{ 0513965423 ??? nane
27 Copyright ? 2001,S,K,Mitra
Basic Operations
? However if the sequences are not of same
length,in some situations,this problem can
be circumvented by appending zero-valued
samples to the sequence(s) of smaller
lengths to make all sequences have the same
range of the time index
? Example - Consider the sequence of length
3 defined for, }{]}[{ 312 ???nf20 ?? n
28 Copyright ? 2001,S,K,Mitra
Basic Operations
? We cannot add the length-3 sequence
to the length-5 sequence {a[n]} defined
earlier
? We therefore first append with 2
zero-valued samples resulting in a length-5
sequence
? Then
]}[{ nf
]}[{ nf
{ [ ] } { 2 1 3 00 }efn ? ? ?
}09351{]}[{]}[{]}[{ ???? nfnang e
29 Copyright ? 2001,S,K,Mitra
Combinations of Basic
Operations
? Example -
]3[]2[]1[][][ 4321 ??????? nxnxnxnxny ????
30 Copyright ? 2001,S,K,Mitra
Sampling Rate Alteration
? Employed to generate a new sequence y[n]
with a sampling rate higher or lower
than that of the sampling rate of a given
sequence x[n]
? Sampling rate alteration ratio is
? If R > 1,the process called interpolation
? If R < 1,the process called decimation
TF
'TF
T
T
F
FR '?
31 Copyright ? 2001,S,K,Mitra
Sampling Rate Alteration
? In up-sampling by an integer factor L > 1,
equidistant zero-valued samples are
inserted by the up-sampler between each
two consecutive samples of the input
sequence x[n],
1?L
?
?
? ????
o th e r w is e,0
,2,,0],/[][ ?LLnLnxnx
u
L ][nx ][nxu
32 Copyright ? 2001,S,K,Mitra
Sampling Rate Alteration
? An example of the up-sampling operation
0 10 20 30 40 50
-1
- 0.5
0
0.5
1
O ut put s e que nc e up- s a m pl e d by 3
T i m e i nde x n
A
m
pl
i
t
ude
0 10 20 30 40 50
-1
- 0.5
0
0.5
1
I nput S e que nc e
T i m e i nde x n
A
m
pl
i
t
ude
33 Copyright ? 2001,S,K,Mitra
Sampling Rate Alteration
? In down-sampling by an integer factor
M > 1,one every M samples of the input
sequence is kept and in-between
samples are removed,
1?M
][][ nMxny ?
][nx ][nyM
34 Copyright ? 2001,S,K,Mitra
Sampling Rate Alteration
? An example of the down-sampling
operation
0 10 20 30 40 50
-1
- 0.5
0
0.5
1
O ut put s e que nc e dow n- s a m pl e d by 3
A
m
pl
i
t
ude
T i m e i nde x n
0 10 20 30 40 50
-1
- 0.5
0
0.5
1
I nput S e que nc e
T i m e i nde x n
A
m
pl
i
t
ude
35 Copyright ? 2001,S,K,Mitra
Classification of Sequences
Based on Symmetry
? Conjugate-symmetric sequence,
If x[n] is real,then it is an even sequence
][*][ nxnx ??
An even sequence
36 Copyright ? 2001,S,K,Mitra
Classification of Sequences
Based on Symmetry
? Conjugate-antisymmetric sequence,
If x[n] is real,then it is an odd sequence
][*][ nxnx ???
An odd sequence
37 Copyright ? 2001,S,K,Mitra
Classification of Sequences
Based on Symmetry
? It follows from the definition that for a
conjugate-symmetric sequence {x[n]},x[0]
must be a real number
? Likewise,it follows from the definition that
for a conjugate anti-symmetric sequence
{y[n]},y[0] must be an imaginary number
? From the above,it also follows that for an
odd sequence {w[n]},w[0] = 0
38 Copyright ? 2001,S,K,Mitra
Classification of Sequences
Based on Symmetry
? Any complex sequence can be expressed as
a sum of its conjugate-symmetric part and
its conjugate-antisymmetric part,
where
][][][ nxnxnx cacs ??
? ?][*][][ 21 nxnxnx cs ???
? ?][*][][ 21 nxnxnx ca ???
39 Copyright ? 2001,S,K,Mitra
Classification of Sequences
Based on Symmetry
? Example - Consider the length-7 sequence
defined for,
? Its conjugate sequence is then given by
? The time-reversed version of the above is
},,,,,,{]}[{ 32652432410 jjjjjng ????????
},,,,,,{]}[*{ 32652432410 jjjjjng ???????
},,,,,,{]}[*{ 04132246523 jjjjjng ????????
?
?
?
33 ??? n
40 Copyright ? 2001,S,K,Mitra
Classification of Sequences
Based on Symmetry
? Therefore
? Likewise
? It can be easily verified that
and
]}[*][{]}[{ ngngng cs ??? 21
]}[*][{]}[{ ngngng ca ??? 21
}.,.,..,,..,.,.{ 51505151251515051 jjjjj ?????????
}.,.,..,,..,.,.{ 5135054534545335051 jjjj ???????
?
? ][*][ n
csgncsg ?? ][*][ n
cagncag ???
41 Copyright ? 2001,S,K,Mitra
Classification of Sequences
Based on Symmetry
? Any real sequence can be expressed as a
sum of its even part and its odd part,
where
][][][ nxnxnx odev ??
? ?][][][ 21 nxnxnx ev ???
? ?][][][ 21 nxnxnx od ???
42 Copyright ? 2001,S,K,Mitra
Classification of Sequences
Based on Symmetry
? A length-N sequence x[n],
can be expressed as
where
is the periodic conjugate-symmetric part and
is the periodic conjugate-antisymmetric part
,10 ??? Nn
][][][ nxnxnx p c ap c s ??
? ?,][*][][ Np c a nxnxnx ????? 21,10 ??? Nn
? ?,][*][][ Np c s nxnxnx ????? 21,10 ??? Nn
43 Copyright ? 2001,S,K,Mitra
Classification of Sequences
Based on Symmetry
? For a real sequence,the periodic conjugate-
symmetric part,is a real sequence and is
called the periodic even part
? For a real sequence,the periodic conjugate-
antisymmetric part,is a real sequence and is
called the periodic odd part
][nx pe
][nx po
44 Copyright ? 2001,S,K,Mitra
Classification of Sequences
Based on Symmetry
? A length-N sequence x[n] is called a
periodic conjugate-symmetric sequence if
and is called a periodic conjugate-
antisymmetric sequence if
][*][*][ nNxnxnx N ??????
][*][*][ nNxnxnx N ????????
45 Copyright ? 2001,S,K,Mitra
Classification of Sequences
Based on Symmetry
? A finite-length real periodic conjugate-
symmetric sequence is called a symmetric
sequence
? A finite-length real periodic conjugate-
antisymmetric sequence is called an
antisymmetric sequence
46 Copyright ? 2001,S,K,Mitra
Classification of Sequences
Based on Symmetry
? Example - Consider the length-4 sequence
defined for,
? Its conjugate sequence is given by
? To determine the modulo-4 time-reversed
version observe the following,
30 ?? n
},,,{]}[{ 65243241 jjjjnu ???????
},,,{]}[*{ 65243241 jjjjnu ???????
]}[*{ 4??? nu
47 Copyright ? 2001,S,K,Mitra
Classification of Sequences
Based on Symmetry
? Hence
4100 4 juu ?????? ][*][* 6531
4 juu ??????? ][*][* 2422
4 juu ?????? ][*][* 3213
4 juu ??????? ][*][*
},,,{]}[*{ 322465414 jjjjnu ??????????
48 Copyright ? 2001,S,K,Mitra
Classification of Sequences
Based on Symmetry
? Therefore
? Likewise
]}[*][{]}[{ 421 ????? nununu p c s
]}[*][{]}[{ 421 ????? nununu p c a
}..,,..,{ 5453454531 jj ?????
}..,,..,{ 5151251514 jjj ?????
49 Copyright ? 2001,S,K,Mitra
Classification of Sequences
Based on Periodicity
? A sequence satisfying
is called a periodic sequence with a period N
where N is a positive integer and k is any
integer
? The smallest value of N satisfying
is called the fundamental
period
][~ nx ][~][~ kNnxnx ??
][~][~ kNnxnx ??
50 Copyright ? 2001,S,K,Mitra
Classification of Sequences
Based on Periodicity
? Example -
? A sequence not satisfying the periodicity
condition is called an aperiodic sequence
7N ?6 44 7 4 48
51 Copyright ? 2001,S,K,Mitra
Classification of Sequences,
Energy and Power Signals
? Total energy of a sequence x[n] is defined by
? An infinite length sequence with finite sample
values may or may not have finite energy
? A finite length sequence with finite sample
values has finite energy
??
?
???n
nx 2x ][?
52 Copyright ? 2001,S,K,Mitra
Classification of Sequences,
Energy and Power Signals
? The average power of an aperiodic
sequence is defined by
? Define the energy of a sequence x[n] over a
finite interval as
??
?????
K
KnKK
nxP 212 1x ][lim
KnK ???
??
??
K
KnKx
nx 2,][?
53 Copyright ? 2001,S,K,Mitra
Classification of Sequences,
Energy and Power Signals
? Then
? The average power of a periodic sequence
with a period N is given by
? The average power of an infinite-length
sequence may be finite or infinite
KxKKxP,12
1lim ?
????
?
?
?
?
1
0
21 N
nN
x nxP ][
~
][~ nx
54 Copyright ? 2001,S,K,Mitra
Classification of Sequences,
Energy and Power Signals
? Example - Consider the causal sequence
defined by
? Note,x[n] has infinite energy
? Its average power is given by
5.412 )1(9lim1912 1lim
0
?????
?
??
?
? ?
?? ????? K
K
KP K
K
nK
x
??
?
?
???
00
013
n
nnx n
,
,)(][
55 Copyright ? 2001,S,K,Mitra
Classification of Sequences,
Energy and Power Signals
? An infinite-energy signal with finite average
power is called a power signal
Example - A periodic sequence which has a
finite average power but infinite energy
? A finite-energy signal with zero average
power is called an energy signal
Example - A finite-length sequence which has
finite energy but zero average power
56 Copyright ? 2001,S,K,Mitra
Other Types of Classifications
? A sequence x[n] is said to be bounded if
? Example - The sequence is a
bounded sequence as
??? xBnx ][
[ ] c o s ( 0, 3 )x n n??
[ ] c o s ( 0, 3 ) 1x n n???
57 Copyright ? 2001,S,K,Mitra
Other Types of Classifications
? A sequence x[n] is said to be absolutely
summable if
? Example - The sequence
is an absolutely summable sequence as
? ???
???n
nx ][
??
?
?
??
00
030
n
nny n
,
,.][
???
?
??
?
?
4 2 8 5 71
301
130
0
.
.
.
n
n
58 Copyright ? 2001,S,K,Mitra
Other Types of Classifications
? A sequence x[n] is said to be square-
summable if
? Example - The sequence
is square-summable but not absolutely
summable
? ???
???n
nx 2][
n nnh ?? 4.0s i n][
59 Copyright ? 2001,S,K,Mitra
Basic Sequences
? Unit sample sequence -
? Unit step sequence -
?
?
?
?
??
0,0
0,1][
n
nn?
?
?
?
?
??
0,0
0,1][
n
nn?
60 Copyright ? 2001,S,K,Mitra
? The unit sample and unit step sequences
relate as follows,
Basic Sequences [ ] [ ]
[ ] [ ] [ 1 ]
n
k
nk
n n n
??
? ? ?
? ??
?
? ? ?
?
61 Copyright ? 2001,S,K,Mitra
Basic Sequences
? Real sinusoidal sequence -
where A is the amplitude,is the angular
frequency,and is the phase of x[n]
Example -
)c o s (][ ???? nAnx o
o?
?
62 Copyright ? 2001,S,K,Mitra
Basic Sequences
? Exponential sequence -
where A and are real or complex numbers
? If we write
then we can express
where
,][ nAnx ?? ????? n
?
,)( oo je ?????,?? jeAA
],[][][ )( nxjnxeeAnx imrenjj oo ??? ????
),c o s(][ ???? ? neAnx onre o
)sin (][ ???? ? neAnx onim o
63 Copyright ? 2001,S,K,Mitra
Basic Sequences
? and of a complex exponential
sequence are real sinusoidal sequences with
constant,growing,and
decaying amplitudes for n > 0
][nxre ][nxim
? ?0??o ? ?0??o
? ?0?? o
njnx )e x p (][ 6121 ????
0 10 20 30 40
-1
- 0.5
0
0.5
1
T i m e i nde x n
A
m
pl
i
t
ude
R e a l pa r t
0 10 20 30 40
-1
- 0.5
0
0.5
1
T i m e i nde x n
A
m
pl
i
t
ude
I m a gi na r y pa r t
64 Copyright ? 2001,S,K,Mitra
Basic Sequences
? Real exponential sequence -
where A and ? are real numbers
,][ nAnx ?? ????? n
0 5 10 15 20 25 30
0
10
20
30
40
50
T i m e i nde x n
A
m
pl
i
t
ude
? = 1.2
0 5 10 15 20 25 30
0
5
10
15
20
T i m e i nde x n
A
m
pl
i
t
ude
? = 0.9
65 Copyright ? 2001,S,K,Mitra
Basic Sequences
? Sinusoidal sequence and
complex exponential sequence
are periodic sequences of period N if
where N and r are positive integers
? Smallest value of N satisfying
is the fundamental period of the sequence
? To verify the above fact,consider
)c o s ( ??? nA o
)e x p ( njB o?
rNo ??? 2
rNo ??? 2
)c o s (][1 ???? nnx o
))(c o s (][2 ????? Nnnx o
66 Copyright ? 2001,S,K,Mitra
Basic Sequences
? Now
which will be equal to
only if
and
? These two conditions are met if and only if
or
))(c o s (][2 ????? Nnnx o
NnNn oooo ?????????? s in)s in (c o s)c o s (
][)c o s ( 1 nxno ????
0s i n ?? No 1c o s ?? No
rNo ??? 2 rNo ???2
67 Copyright ? 2001,S,K,Mitra
Basic Sequences
? If is a noninteger rational number,then
the period will be a multiple of
? Otherwise,the sequence is aperiodic
? Example - is an aperiodic
sequence
o??/2
o??/2
)3s in (][ ??? nnx
68 Copyright ? 2001,S,K,Mitra
0 10 20 30 40
0
0.5
1
1.5
2
T i m e i nde x n
A
m
pl
i
t
ude
?
0
= 0
? Here
? Equation is satisfied for
r = 0 and any positive integer N; N=1 is
the smallest
Basic Sequences
0??o
02o Nr????
69 Copyright ? 2001,S,K,Mitra
Basic Sequences
? Here
? Hence for r = 1
??? 1.0o
201.02 ?? ?? rN
0 10 20 30 40
-2
-1
0
1
2
T i m e i nde x n
A
m
pl
i
t
ude
?
0
= 0.1 ?
70 Copyright ? 2001,S,K,Mitra
Basic Sequences
? Property 1 - Consider and
with and
where k is any positive
integer
? If then x[n] = y[n]
? Thus,x[n] and y[n] are indistinguishable
)e x p (][ 1 njnx ??
)e x p (][ 2 njny ?? ???? 10
)1(22 2 ?????? kk
,212 k?????
71 Copyright ? 2001,S,K,Mitra
Basic Sequences
? Property 2 - The frequency of oscillation of
increases as increases from 0
to ?,and then decreases as increases from
to
? Thus,frequencies in the neighborhood of
are called low frequencies,whereas,
frequencies in the neighborhood of are
called high frequencies
)c o s ( nA o? o?
o?? ?2
???
0??
72 Copyright ? 2001,S,K,Mitra
Basic Sequences
? Because of Property 1,a frequency in
the neighborhood of ? = 2? k is
indistinguishable from a frequency
in the neighborhood of ? = 0
and a frequency in the neighborhood of
is indistinguishable from a
frequency in the
neighborhood of ? = ?
o?
ko ??? 2
o?
)12( ???? ko
)12( ???? k
73 Copyright ? 2001,S,K,Mitra
Basic Sequences
? Frequencies in the neighborhood of ? = 2? k
are usually called low frequencies
? Frequencies in the neighborhood of
? = ? (2k+1) are usually called high
frequencies
? is a low-
frequency signal
? is a high-
frequency signal
)9.1c o s ()1.0c o s (][1 nnnv ????
)2.1c o s ()8.0c o s (][2 nnnv ????
74 Copyright ? 2001,S,K,Mitra
Basic Sequences
? An arbitrary sequence can be represented in
the time-domain as a weighted sum of some
basic sequence and its delayed (advanced)
versions
]2[]1[5.1]2[5.0][ ?????? nnnnx ???
]6[75.0]4[ ???? nn ??
75 Copyright ? 2001,S,K,Mitra
The Sampling Process
? Often,a discrete-time sequence x[n] is
developed by uniformly sampling a
continuous-time signal as indicated
below
? The relation between the two signals is
)(txa
),()(][ nTxtxnx anTta ?? ? ??,2,1,0,1,2,???n
76 Copyright ? 2001,S,K,Mitra
The Sampling Process
? Time variable t of is related to the time
variable n of x[n] only at discrete-time
instants given by
with denoting the sampling
frequency and
denoting the sampling angular
frequency
)(txa
TT
n
n
F
nnTt
?
???? 2
nt
TF T /1?
TT F??? 2
77 Copyright ? 2001,S,K,Mitra
The Sampling Process
? Consider the continuous-time signal
? The corresponding discrete-time signal is
where
is the normalized digital angular frequency
of x[n]
)c o s ()2c o s ()( ???????? tAtfAtx oo
2[ ] c o s ( ) c o s o
o
T
x n A n T A n??? ?? ?? ? ? ? ????
??
)c o s ( ???? nA o
ToToo ??????? /2
78 Copyright ? 2001,S,K,Mitra
The Sampling Process
? If the unit of sampling period T is in
seconds
? The unit of normalized digital angular
frequency is radians/sample
? The unit of normalized analog angular
frequency is radians/second
? The unit of analog frequency is hertz
(Hz)
o?
o?
of
79 Copyright ? 2001,S,K,Mitra
The Sampling Process
? The three continuous-time signals
of frequencies 3 Hz,7 Hz,and 13 Hz,are
sampled at a sampling rate of 10 Hz,i.e,
with T = 0.1 sec,generating the three
sequences
)6c o s ()(1 ttg ??
)14c o s ()(2 ttg ??
)26c o s ()(3 ttg ??
)6.2c o s (][3 nng ??
)6.0c o s (][1 nng ?? )4.1c o s (][2 nng ??
80 Copyright ? 2001,S,K,Mitra
The Sampling Process
? Plots of these sequences (shown with circles)
and their parent time functions are shown
below,
? Note that each sequence has exactly the same
sample value for any given n
81 Copyright ? 2001,S,K,Mitra
The Sampling Process
? This fact can also be verified by observing that
? As a result,all three sequences are identical
and it is impossible to associate a unique
continuous-time function with each of these
sequences
? ? )6.0c o s ()6.02(c o s)4.1c o s (][2 nnnng ????????
? ? )6.0c o s ()6.02(c o s)6.2c o s (][3 nnnng ????????
82 Copyright ? 2001,S,K,Mitra
The Sampling Process
? The above phenomenon of a continuous-
time signal of higher frequency acquiring
the identity of a sinusoidal sequence of
lower frequency after sampling is called
aliasing
83 Copyright ? 2001,S,K,Mitra
The Sampling Process
? Since there are an infinite number of
continuous-time signals that can lead to the
same sequence when sampled periodically,
additional conditions need to imposed so
that the sequence can
uniquely represent the parent continuous-
time signal
? Under these conditions,can be fully
and uniquely recovered from {x[n]}
)}({]}[{ nTxnx a?
)(txa
)(txa
84 Copyright ? 2001,S,K,Mitra
The Sampling Process
? Example - Determine the discrete-time signal v[n]
obtained by uniformly sampling at a sampling rate
of 200 Hz the continuous-time signal
? Note,is composed of 5 sinusoidal signals of
frequencies 30 Hz,150 Hz,170 Hz,250 Hz and
330 Hz
)340c o s (2)300s in (3)60c o s (6)( ttttv a ??????
)660s i n (10)500c o s (4 tt ????
)(tva
85 Copyright ? 2001,S,K,Mitra
The Sampling Process
? The sampling period is
? The generated discrete-time signal v[n] is
thus given by
se c0 0 5.02 0 01 ??T
)7.1c o s (2)5.1s i n (3)3.0c o s (6][ nnnnv ??????
)()( )3.02(c o s2)5.02(s in3)3.0c o s (6 nnn ??????????
)3.3s i n (10)5.2c o s (4 nn ????
)()( )7.04(s i n10)5.02(c o s4 nn ????????
86 Copyright ? 2001,S,K,Mitra
The Sampling Process
? Note,v[n] is composed of 3 discrete-time
sinusoidal signals of normalized angular
frequencies,0.3?,0.5?,and 0.7?
)5.0c o s (4)3.0c o s (2)5.0s i n (3)3.0c o s (6 nnnn ????????
)7.0s i n (10 n??
)7.0s i n (10)6 4 3 5.05.0c o s (5)3.0c o s (8 nnn ???????
87 Copyright ? 2001,S,K,Mitra
The Sampling Process
? Note,An identical discrete-time signal is
also generated by uniformly sampling at a
200-Hz sampling rate the following
continuous-time signals,
)140s in (10)6435.0100c o s (5)60c o s (8)( ttttaw ???????
)2 6 0s in (10)1 0 0c o s (4)60c o s (2)( ttttag ??????
)7 0 0s i n (3)4 6 0c o s (6 tt ????
88 Copyright ? 2001,S,K,Mitra
The Sampling Process
? Recall
? Thus if,then the corresponding
normalized digital angular frequency of
the discrete-time signal obtained by
sampling the parent continuous-time
sinusoidal signal will be in the range
? Thus,no aliasing occurs
T
o
o ?
???? 2
o?
oT ??? 2
??????
89 Copyright ? 2001,S,K,Mitra
The Sampling Process
? On the other hand,if,the
normalized digital angular frequency will
fold into a lower digital frequency
in the range
because of aliasing
? Hence,to prevent aliasing,the sampling
frequency should be greater than 2
times the frequency of the sinusoidal
signal being sampled
oT ??? 2
?????? ???????? 2/2 Too
T?
o?
90 Copyright ? 2001,S,K,Mitra
The Sampling Process
? Generalization,Consider an arbitrary
continuous-time signal composed of a
weighted sum of a number of sinusoidal
signals
? can be represented uniquely by its
sampled version {x[n]} if the sampling
frequency is chosen to be greater than 2
times the highest frequency contained in
)(txa
)(txa
T?
)(txa
91 Copyright ? 2001,S,K,Mitra
The Sampling Process
? The condition to be satisfied by the
sampling frequency to prevent aliasing is
called the sampling theorem (Shannon)
Discrete-Time Signals,
Time-Domain Representation
? Signals are represented as sequences of
numbers,called samples
? Sample value of a typical signal or sequence
denoted as x[n] with n being an integer in
the range
? x[n] defined only for integer values of n and
undefined for non-integer values of n
? Discrete-time signal represented by {x[n]}
????? n
2 Copyright ? 2001,S,K,Mitra
Discrete-Time Signals,
Time-Domain Representation
? Discrete-time signal may also be written as
a sequence of numbers inside braces,
? In the above,
etc,
? The arrow is placed under the sample at
time index n = 0
},9.2,7.3,2.0,1.1,2.2,2.0,{]}[{ ?? ???
?
nx
,2.0]1[ ???x,2.]0[ ?x,1.1]1[ ?x
3 Copyright ? 2001,S,K,Mitra
Discrete-Time Signals,
Time-Domain Representation
? Graphical representation of a discrete-time
signal with real-valued samples
4 Copyright ? 2001,S,K,Mitra
Discrete-Time Signals,
Time-Domain Representation
? In some applications,a discrete-time
sequence {x[n]} may be generated by
periodically sampling a continuous-time
signal at uniform intervals of time
)(txa
5 Copyright ? 2001,S,K,Mitra
? Block-diagram representation of the
sampling process
()axt,,
Discrete-Time Signals,
Time-Domain Representation
[ ] ( ) ( )aat n Tx n x t x n T???
??,1,0,1,2,???n
6 Copyright ? 2001,S,K,Mitra
Discrete-Time Signals,
Time-Domain Representation
? Here,the n-th sample is given by
? The spacing T between two consecutive
samples is called the sampling interval or
sampling period
? Reciprocal of sampling interval T,denoted
as,is called the sampling frequency,
),()(][ nTxtxnx anTta ?? ? ??,1,0,1,2,???n
TF
TFT
1?
7 Copyright ? 2001,S,K,Mitra
Discrete-Time Signals,
Time-Domain Representation
? Unit of sampling frequency is cycles per
second,or Hertz (Hz),if T is in seconds
? Whether or not the sequence {x[n]} has
been obtained by sampling,the quantity
x[n] is called the n-th sample of the
sequence
? {x[n]} is a real sequence,if the n-th sample
x[n] is real for all values of n
? Otherwise,{x[n]} is a complex sequence
8 Copyright ? 2001,S,K,Mitra
Discrete-Time Signals,
Time-Domain Representation
? A complex sequence {x[n]} can be written
as where
and are the real and imaginary
parts of x[n]
? The complex conjugate sequence of {x[n]}
is given by
? Often the braces are ignored to denote a
sequence if there is no ambiguity
][nxre ][nxim
]}[{]}[{]}[{ nxjnxnx imre ??
]}[{]}[{]}[*{ nxjnxnx imre ??
9 Copyright ? 2001,S,K,Mitra
Discrete-Time Signals,
Time-Domain Representation
? Example - is a real
sequence
? is a complex sequence
? We can write
where
}.{ c o s]}[{ nnx 250?
}{]}[{, njeny 30?
}.s in.{ c o s]}[{ njnny 3030 ??
}.{ s in}.{ c o s njn 3030 ??
}.{ c o s]}[{ nny re 30?
}.{ s in]}[{ nny im 30?
10 Copyright ? 2001,S,K,Mitra
Discrete-Time Signals,
Time-Domain Representation
? Example -
is the complex conjugate sequence of {y[n]}
? That is,
}{}.{ s i n}.{ c o s]}[{, njenjnnw 303030 ????
]}[*{]}[{ nynw ?
11 Copyright ? 2001,S,K,Mitra
Discrete-Time Signals,
Time-Domain Representation
? Two types of discrete-time signals,
- Sampled-data signals in which samples
are continuous-valued
- Digital signals in which samples are
discrete-valued
? Signals in a practical digital signal
processing system are digital signals
obtained by quantizing the sample values
either by rounding or truncation
12 Copyright ? 2001,S,K,Mitra
Discrete-Time Signals,
Time-Domain Representation
? Example -
Sampled-data signal Digital signal
Am
pli
tude
t,time A
mpl
itude
t,time
13 Copyright ? 2001,S,K,Mitra
Discrete-Time Signals,
Time-Domain Representation
? A discrete-time signal may be a finite-
length or an infinite-length sequence
? Finite-length (also called finite-duration or
finite-extent) sequence is defined only for a
finite time interval,
where and with
? Length or duration of the above finite-
length sequence is
21 NnN ??
1N??? ??2N 21 NN ?
112 ??? NNN
14 Copyright ? 2001,S,K,Mitra
Discrete-Time Signals,
Time-Domain Representation
? Example - is a finite-
length sequence of length
? Example - is an infinite-
length sequence
432 ???? nnnx,][
8134 ???? )(
nny 40,c o s][ ?
15 Copyright ? 2001,S,K,Mitra
Discrete-Time Signals,
Time-Domain Representation
? A length-N sequence is often referred to as
an N-point sequence
? The length of a finite-length sequence can
be increased by zero-padding,i.e.,by
appending it with zeros
16 Copyright ? 2001,S,K,Mitra
Discrete-Time Signals,
Time-Domain Representation
? Example -
is a finite-length sequence of length 12
obtained by zero-padding
with 4 zero-valued samples
??
?
??
????
850
432
n
nnnx
e,
,][
432 ???? nnnx,][
17 Copyright ? 2001,S,K,Mitra
Discrete-Time Signals,
Time-Domain Representation
? A right-sided sequence x[n] has zero-
valued samples for
? If a right-sided sequence is called a
causal sequence
,01 ?N
1Nn ?
A right-sided sequence
18 Copyright ? 2001,S,K,Mitra
Discrete-Time Signals,
Time-Domain Representation
? A left-sided sequence x[n] has zero-valued
samples for
? If a left-sided sequence is called an
anti-causal sequence
2Nn ?
,02 ?N
A left-sided sequence
19 Copyright ? 2001,S,K,Mitra
Operations on Sequences
? A single-input,single-output (SISO)
discrete-time system operates on a
sequence,called the input sequence,
according some prescribed rules and
develops another sequence,called the
output sequence,with more desirable
properties
x[n] y[n]
Input sequence Output sequence
Discrete-time
system
20 Copyright ? 2001,S,K,Mitra
Operations on Sequences
? For example,the input may be a signal
corrupted with additive noise
? Discrete-time system is designed to
generate an output by removing the noise
component from the input
? In most cases,the operation defining a
particular discrete-time system is composed
of some basic operations
21 Copyright ? 2001,S,K,Mitra
Basic Operations
? Product (modulation) operation,
– Modulator
? An application is in forming a finite-length
sequence from an infinite-length sequence
by multiplying the latter with a finite-length
sequence called a window sequence
? Process called windowing
?x[n] y[n]
w[n] ][][][ nwnxny ??
22 Copyright ? 2001,S,K,Mitra
Basic Operations
? Addition operation,
– Adder
? Multiplication operation
– Multiplier
][][][ nwnxny ??
A
x[n] y[n] ][][ nxAny ??
x[n] y[n]
w[n]
?
23 Copyright ? 2001,S,K,Mitra
Basic Operations
? Time-shifting operation,
where N is an integer
? If N > 0,it is delaying operation
– Unit delay
? If N < 0,it is an advance operation
– Unit advance
][][ Nnxny ??
y[n] x[n] z
1?z y[n] x[n] ][][ 1?? nxny
][][ 1?? nxny
24 Copyright ? 2001,S,K,Mitra
Basic Operations
? Time-reversal (folding) operation,
? Branching operation,Used to provide
multiple copies of a sequence
][][ nxny ??
x[n] x[n]
x[n]
25 Copyright ? 2001,S,K,Mitra
Basic Operations
? Example - Consider the two following
sequences of length 5 defined for,
? New sequences generated from the above
two sequences by applying the basic
operations are as follows,
40 ?? n
}{]}[{ 09643 ??na
}{]}[{ 35412 ???nb
26 Copyright ? 2001,S,K,Mitra
Basic Operations
? Operations on two or more sequences can
be carried out if all sequences involved are
of same length and defined for the same
range of the time index n
}{]}[][{]}[{ 0452446 ????? nbnanc
}{]}[][{]}[{ 341035 ????? nbnand
}..{]}[{]}[{ 0513965423 ??? nane
27 Copyright ? 2001,S,K,Mitra
Basic Operations
? However if the sequences are not of same
length,in some situations,this problem can
be circumvented by appending zero-valued
samples to the sequence(s) of smaller
lengths to make all sequences have the same
range of the time index
? Example - Consider the sequence of length
3 defined for, }{]}[{ 312 ???nf20 ?? n
28 Copyright ? 2001,S,K,Mitra
Basic Operations
? We cannot add the length-3 sequence
to the length-5 sequence {a[n]} defined
earlier
? We therefore first append with 2
zero-valued samples resulting in a length-5
sequence
? Then
]}[{ nf
]}[{ nf
{ [ ] } { 2 1 3 00 }efn ? ? ?
}09351{]}[{]}[{]}[{ ???? nfnang e
29 Copyright ? 2001,S,K,Mitra
Combinations of Basic
Operations
? Example -
]3[]2[]1[][][ 4321 ??????? nxnxnxnxny ????
30 Copyright ? 2001,S,K,Mitra
Sampling Rate Alteration
? Employed to generate a new sequence y[n]
with a sampling rate higher or lower
than that of the sampling rate of a given
sequence x[n]
? Sampling rate alteration ratio is
? If R > 1,the process called interpolation
? If R < 1,the process called decimation
TF
'TF
T
T
F
FR '?
31 Copyright ? 2001,S,K,Mitra
Sampling Rate Alteration
? In up-sampling by an integer factor L > 1,
equidistant zero-valued samples are
inserted by the up-sampler between each
two consecutive samples of the input
sequence x[n],
1?L
?
?
? ????
o th e r w is e,0
,2,,0],/[][ ?LLnLnxnx
u
L ][nx ][nxu
32 Copyright ? 2001,S,K,Mitra
Sampling Rate Alteration
? An example of the up-sampling operation
0 10 20 30 40 50
-1
- 0.5
0
0.5
1
O ut put s e que nc e up- s a m pl e d by 3
T i m e i nde x n
A
m
pl
i
t
ude
0 10 20 30 40 50
-1
- 0.5
0
0.5
1
I nput S e que nc e
T i m e i nde x n
A
m
pl
i
t
ude
33 Copyright ? 2001,S,K,Mitra
Sampling Rate Alteration
? In down-sampling by an integer factor
M > 1,one every M samples of the input
sequence is kept and in-between
samples are removed,
1?M
][][ nMxny ?
][nx ][nyM
34 Copyright ? 2001,S,K,Mitra
Sampling Rate Alteration
? An example of the down-sampling
operation
0 10 20 30 40 50
-1
- 0.5
0
0.5
1
O ut put s e que nc e dow n- s a m pl e d by 3
A
m
pl
i
t
ude
T i m e i nde x n
0 10 20 30 40 50
-1
- 0.5
0
0.5
1
I nput S e que nc e
T i m e i nde x n
A
m
pl
i
t
ude
35 Copyright ? 2001,S,K,Mitra
Classification of Sequences
Based on Symmetry
? Conjugate-symmetric sequence,
If x[n] is real,then it is an even sequence
][*][ nxnx ??
An even sequence
36 Copyright ? 2001,S,K,Mitra
Classification of Sequences
Based on Symmetry
? Conjugate-antisymmetric sequence,
If x[n] is real,then it is an odd sequence
][*][ nxnx ???
An odd sequence
37 Copyright ? 2001,S,K,Mitra
Classification of Sequences
Based on Symmetry
? It follows from the definition that for a
conjugate-symmetric sequence {x[n]},x[0]
must be a real number
? Likewise,it follows from the definition that
for a conjugate anti-symmetric sequence
{y[n]},y[0] must be an imaginary number
? From the above,it also follows that for an
odd sequence {w[n]},w[0] = 0
38 Copyright ? 2001,S,K,Mitra
Classification of Sequences
Based on Symmetry
? Any complex sequence can be expressed as
a sum of its conjugate-symmetric part and
its conjugate-antisymmetric part,
where
][][][ nxnxnx cacs ??
? ?][*][][ 21 nxnxnx cs ???
? ?][*][][ 21 nxnxnx ca ???
39 Copyright ? 2001,S,K,Mitra
Classification of Sequences
Based on Symmetry
? Example - Consider the length-7 sequence
defined for,
? Its conjugate sequence is then given by
? The time-reversed version of the above is
},,,,,,{]}[{ 32652432410 jjjjjng ????????
},,,,,,{]}[*{ 32652432410 jjjjjng ???????
},,,,,,{]}[*{ 04132246523 jjjjjng ????????
?
?
?
33 ??? n
40 Copyright ? 2001,S,K,Mitra
Classification of Sequences
Based on Symmetry
? Therefore
? Likewise
? It can be easily verified that
and
]}[*][{]}[{ ngngng cs ??? 21
]}[*][{]}[{ ngngng ca ??? 21
}.,.,..,,..,.,.{ 51505151251515051 jjjjj ?????????
}.,.,..,,..,.,.{ 5135054534545335051 jjjj ???????
?
? ][*][ n
csgncsg ?? ][*][ n
cagncag ???
41 Copyright ? 2001,S,K,Mitra
Classification of Sequences
Based on Symmetry
? Any real sequence can be expressed as a
sum of its even part and its odd part,
where
][][][ nxnxnx odev ??
? ?][][][ 21 nxnxnx ev ???
? ?][][][ 21 nxnxnx od ???
42 Copyright ? 2001,S,K,Mitra
Classification of Sequences
Based on Symmetry
? A length-N sequence x[n],
can be expressed as
where
is the periodic conjugate-symmetric part and
is the periodic conjugate-antisymmetric part
,10 ??? Nn
][][][ nxnxnx p c ap c s ??
? ?,][*][][ Np c a nxnxnx ????? 21,10 ??? Nn
? ?,][*][][ Np c s nxnxnx ????? 21,10 ??? Nn
43 Copyright ? 2001,S,K,Mitra
Classification of Sequences
Based on Symmetry
? For a real sequence,the periodic conjugate-
symmetric part,is a real sequence and is
called the periodic even part
? For a real sequence,the periodic conjugate-
antisymmetric part,is a real sequence and is
called the periodic odd part
][nx pe
][nx po
44 Copyright ? 2001,S,K,Mitra
Classification of Sequences
Based on Symmetry
? A length-N sequence x[n] is called a
periodic conjugate-symmetric sequence if
and is called a periodic conjugate-
antisymmetric sequence if
][*][*][ nNxnxnx N ??????
][*][*][ nNxnxnx N ????????
45 Copyright ? 2001,S,K,Mitra
Classification of Sequences
Based on Symmetry
? A finite-length real periodic conjugate-
symmetric sequence is called a symmetric
sequence
? A finite-length real periodic conjugate-
antisymmetric sequence is called an
antisymmetric sequence
46 Copyright ? 2001,S,K,Mitra
Classification of Sequences
Based on Symmetry
? Example - Consider the length-4 sequence
defined for,
? Its conjugate sequence is given by
? To determine the modulo-4 time-reversed
version observe the following,
30 ?? n
},,,{]}[{ 65243241 jjjjnu ???????
},,,{]}[*{ 65243241 jjjjnu ???????
]}[*{ 4??? nu
47 Copyright ? 2001,S,K,Mitra
Classification of Sequences
Based on Symmetry
? Hence
4100 4 juu ?????? ][*][* 6531
4 juu ??????? ][*][* 2422
4 juu ?????? ][*][* 3213
4 juu ??????? ][*][*
},,,{]}[*{ 322465414 jjjjnu ??????????
48 Copyright ? 2001,S,K,Mitra
Classification of Sequences
Based on Symmetry
? Therefore
? Likewise
]}[*][{]}[{ 421 ????? nununu p c s
]}[*][{]}[{ 421 ????? nununu p c a
}..,,..,{ 5453454531 jj ?????
}..,,..,{ 5151251514 jjj ?????
49 Copyright ? 2001,S,K,Mitra
Classification of Sequences
Based on Periodicity
? A sequence satisfying
is called a periodic sequence with a period N
where N is a positive integer and k is any
integer
? The smallest value of N satisfying
is called the fundamental
period
][~ nx ][~][~ kNnxnx ??
][~][~ kNnxnx ??
50 Copyright ? 2001,S,K,Mitra
Classification of Sequences
Based on Periodicity
? Example -
? A sequence not satisfying the periodicity
condition is called an aperiodic sequence
7N ?6 44 7 4 48
51 Copyright ? 2001,S,K,Mitra
Classification of Sequences,
Energy and Power Signals
? Total energy of a sequence x[n] is defined by
? An infinite length sequence with finite sample
values may or may not have finite energy
? A finite length sequence with finite sample
values has finite energy
??
?
???n
nx 2x ][?
52 Copyright ? 2001,S,K,Mitra
Classification of Sequences,
Energy and Power Signals
? The average power of an aperiodic
sequence is defined by
? Define the energy of a sequence x[n] over a
finite interval as
??
?????
K
KnKK
nxP 212 1x ][lim
KnK ???
??
??
K
KnKx
nx 2,][?
53 Copyright ? 2001,S,K,Mitra
Classification of Sequences,
Energy and Power Signals
? Then
? The average power of a periodic sequence
with a period N is given by
? The average power of an infinite-length
sequence may be finite or infinite
KxKKxP,12
1lim ?
????
?
?
?
?
1
0
21 N
nN
x nxP ][
~
][~ nx
54 Copyright ? 2001,S,K,Mitra
Classification of Sequences,
Energy and Power Signals
? Example - Consider the causal sequence
defined by
? Note,x[n] has infinite energy
? Its average power is given by
5.412 )1(9lim1912 1lim
0
?????
?
??
?
? ?
?? ????? K
K
KP K
K
nK
x
??
?
?
???
00
013
n
nnx n
,
,)(][
55 Copyright ? 2001,S,K,Mitra
Classification of Sequences,
Energy and Power Signals
? An infinite-energy signal with finite average
power is called a power signal
Example - A periodic sequence which has a
finite average power but infinite energy
? A finite-energy signal with zero average
power is called an energy signal
Example - A finite-length sequence which has
finite energy but zero average power
56 Copyright ? 2001,S,K,Mitra
Other Types of Classifications
? A sequence x[n] is said to be bounded if
? Example - The sequence is a
bounded sequence as
??? xBnx ][
[ ] c o s ( 0, 3 )x n n??
[ ] c o s ( 0, 3 ) 1x n n???
57 Copyright ? 2001,S,K,Mitra
Other Types of Classifications
? A sequence x[n] is said to be absolutely
summable if
? Example - The sequence
is an absolutely summable sequence as
? ???
???n
nx ][
??
?
?
??
00
030
n
nny n
,
,.][
???
?
??
?
?
4 2 8 5 71
301
130
0
.
.
.
n
n
58 Copyright ? 2001,S,K,Mitra
Other Types of Classifications
? A sequence x[n] is said to be square-
summable if
? Example - The sequence
is square-summable but not absolutely
summable
? ???
???n
nx 2][
n nnh ?? 4.0s i n][
59 Copyright ? 2001,S,K,Mitra
Basic Sequences
? Unit sample sequence -
? Unit step sequence -
?
?
?
?
??
0,0
0,1][
n
nn?
?
?
?
?
??
0,0
0,1][
n
nn?
60 Copyright ? 2001,S,K,Mitra
? The unit sample and unit step sequences
relate as follows,
Basic Sequences [ ] [ ]
[ ] [ ] [ 1 ]
n
k
nk
n n n
??
? ? ?
? ??
?
? ? ?
?
61 Copyright ? 2001,S,K,Mitra
Basic Sequences
? Real sinusoidal sequence -
where A is the amplitude,is the angular
frequency,and is the phase of x[n]
Example -
)c o s (][ ???? nAnx o
o?
?
62 Copyright ? 2001,S,K,Mitra
Basic Sequences
? Exponential sequence -
where A and are real or complex numbers
? If we write
then we can express
where
,][ nAnx ?? ????? n
?
,)( oo je ?????,?? jeAA
],[][][ )( nxjnxeeAnx imrenjj oo ??? ????
),c o s(][ ???? ? neAnx onre o
)sin (][ ???? ? neAnx onim o
63 Copyright ? 2001,S,K,Mitra
Basic Sequences
? and of a complex exponential
sequence are real sinusoidal sequences with
constant,growing,and
decaying amplitudes for n > 0
][nxre ][nxim
? ?0??o ? ?0??o
? ?0?? o
njnx )e x p (][ 6121 ????
0 10 20 30 40
-1
- 0.5
0
0.5
1
T i m e i nde x n
A
m
pl
i
t
ude
R e a l pa r t
0 10 20 30 40
-1
- 0.5
0
0.5
1
T i m e i nde x n
A
m
pl
i
t
ude
I m a gi na r y pa r t
64 Copyright ? 2001,S,K,Mitra
Basic Sequences
? Real exponential sequence -
where A and ? are real numbers
,][ nAnx ?? ????? n
0 5 10 15 20 25 30
0
10
20
30
40
50
T i m e i nde x n
A
m
pl
i
t
ude
? = 1.2
0 5 10 15 20 25 30
0
5
10
15
20
T i m e i nde x n
A
m
pl
i
t
ude
? = 0.9
65 Copyright ? 2001,S,K,Mitra
Basic Sequences
? Sinusoidal sequence and
complex exponential sequence
are periodic sequences of period N if
where N and r are positive integers
? Smallest value of N satisfying
is the fundamental period of the sequence
? To verify the above fact,consider
)c o s ( ??? nA o
)e x p ( njB o?
rNo ??? 2
rNo ??? 2
)c o s (][1 ???? nnx o
))(c o s (][2 ????? Nnnx o
66 Copyright ? 2001,S,K,Mitra
Basic Sequences
? Now
which will be equal to
only if
and
? These two conditions are met if and only if
or
))(c o s (][2 ????? Nnnx o
NnNn oooo ?????????? s in)s in (c o s)c o s (
][)c o s ( 1 nxno ????
0s i n ?? No 1c o s ?? No
rNo ??? 2 rNo ???2
67 Copyright ? 2001,S,K,Mitra
Basic Sequences
? If is a noninteger rational number,then
the period will be a multiple of
? Otherwise,the sequence is aperiodic
? Example - is an aperiodic
sequence
o??/2
o??/2
)3s in (][ ??? nnx
68 Copyright ? 2001,S,K,Mitra
0 10 20 30 40
0
0.5
1
1.5
2
T i m e i nde x n
A
m
pl
i
t
ude
?
0
= 0
? Here
? Equation is satisfied for
r = 0 and any positive integer N; N=1 is
the smallest
Basic Sequences
0??o
02o Nr????
69 Copyright ? 2001,S,K,Mitra
Basic Sequences
? Here
? Hence for r = 1
??? 1.0o
201.02 ?? ?? rN
0 10 20 30 40
-2
-1
0
1
2
T i m e i nde x n
A
m
pl
i
t
ude
?
0
= 0.1 ?
70 Copyright ? 2001,S,K,Mitra
Basic Sequences
? Property 1 - Consider and
with and
where k is any positive
integer
? If then x[n] = y[n]
? Thus,x[n] and y[n] are indistinguishable
)e x p (][ 1 njnx ??
)e x p (][ 2 njny ?? ???? 10
)1(22 2 ?????? kk
,212 k?????
71 Copyright ? 2001,S,K,Mitra
Basic Sequences
? Property 2 - The frequency of oscillation of
increases as increases from 0
to ?,and then decreases as increases from
to
? Thus,frequencies in the neighborhood of
are called low frequencies,whereas,
frequencies in the neighborhood of are
called high frequencies
)c o s ( nA o? o?
o?? ?2
???
0??
72 Copyright ? 2001,S,K,Mitra
Basic Sequences
? Because of Property 1,a frequency in
the neighborhood of ? = 2? k is
indistinguishable from a frequency
in the neighborhood of ? = 0
and a frequency in the neighborhood of
is indistinguishable from a
frequency in the
neighborhood of ? = ?
o?
ko ??? 2
o?
)12( ???? ko
)12( ???? k
73 Copyright ? 2001,S,K,Mitra
Basic Sequences
? Frequencies in the neighborhood of ? = 2? k
are usually called low frequencies
? Frequencies in the neighborhood of
? = ? (2k+1) are usually called high
frequencies
? is a low-
frequency signal
? is a high-
frequency signal
)9.1c o s ()1.0c o s (][1 nnnv ????
)2.1c o s ()8.0c o s (][2 nnnv ????
74 Copyright ? 2001,S,K,Mitra
Basic Sequences
? An arbitrary sequence can be represented in
the time-domain as a weighted sum of some
basic sequence and its delayed (advanced)
versions
]2[]1[5.1]2[5.0][ ?????? nnnnx ???
]6[75.0]4[ ???? nn ??
75 Copyright ? 2001,S,K,Mitra
The Sampling Process
? Often,a discrete-time sequence x[n] is
developed by uniformly sampling a
continuous-time signal as indicated
below
? The relation between the two signals is
)(txa
),()(][ nTxtxnx anTta ?? ? ??,2,1,0,1,2,???n
76 Copyright ? 2001,S,K,Mitra
The Sampling Process
? Time variable t of is related to the time
variable n of x[n] only at discrete-time
instants given by
with denoting the sampling
frequency and
denoting the sampling angular
frequency
)(txa
TT
n
n
F
nnTt
?
???? 2
nt
TF T /1?
TT F??? 2
77 Copyright ? 2001,S,K,Mitra
The Sampling Process
? Consider the continuous-time signal
? The corresponding discrete-time signal is
where
is the normalized digital angular frequency
of x[n]
)c o s ()2c o s ()( ???????? tAtfAtx oo
2[ ] c o s ( ) c o s o
o
T
x n A n T A n??? ?? ?? ? ? ? ????
??
)c o s ( ???? nA o
ToToo ??????? /2
78 Copyright ? 2001,S,K,Mitra
The Sampling Process
? If the unit of sampling period T is in
seconds
? The unit of normalized digital angular
frequency is radians/sample
? The unit of normalized analog angular
frequency is radians/second
? The unit of analog frequency is hertz
(Hz)
o?
o?
of
79 Copyright ? 2001,S,K,Mitra
The Sampling Process
? The three continuous-time signals
of frequencies 3 Hz,7 Hz,and 13 Hz,are
sampled at a sampling rate of 10 Hz,i.e,
with T = 0.1 sec,generating the three
sequences
)6c o s ()(1 ttg ??
)14c o s ()(2 ttg ??
)26c o s ()(3 ttg ??
)6.2c o s (][3 nng ??
)6.0c o s (][1 nng ?? )4.1c o s (][2 nng ??
80 Copyright ? 2001,S,K,Mitra
The Sampling Process
? Plots of these sequences (shown with circles)
and their parent time functions are shown
below,
? Note that each sequence has exactly the same
sample value for any given n
81 Copyright ? 2001,S,K,Mitra
The Sampling Process
? This fact can also be verified by observing that
? As a result,all three sequences are identical
and it is impossible to associate a unique
continuous-time function with each of these
sequences
? ? )6.0c o s ()6.02(c o s)4.1c o s (][2 nnnng ????????
? ? )6.0c o s ()6.02(c o s)6.2c o s (][3 nnnng ????????
82 Copyright ? 2001,S,K,Mitra
The Sampling Process
? The above phenomenon of a continuous-
time signal of higher frequency acquiring
the identity of a sinusoidal sequence of
lower frequency after sampling is called
aliasing
83 Copyright ? 2001,S,K,Mitra
The Sampling Process
? Since there are an infinite number of
continuous-time signals that can lead to the
same sequence when sampled periodically,
additional conditions need to imposed so
that the sequence can
uniquely represent the parent continuous-
time signal
? Under these conditions,can be fully
and uniquely recovered from {x[n]}
)}({]}[{ nTxnx a?
)(txa
)(txa
84 Copyright ? 2001,S,K,Mitra
The Sampling Process
? Example - Determine the discrete-time signal v[n]
obtained by uniformly sampling at a sampling rate
of 200 Hz the continuous-time signal
? Note,is composed of 5 sinusoidal signals of
frequencies 30 Hz,150 Hz,170 Hz,250 Hz and
330 Hz
)340c o s (2)300s in (3)60c o s (6)( ttttv a ??????
)660s i n (10)500c o s (4 tt ????
)(tva
85 Copyright ? 2001,S,K,Mitra
The Sampling Process
? The sampling period is
? The generated discrete-time signal v[n] is
thus given by
se c0 0 5.02 0 01 ??T
)7.1c o s (2)5.1s i n (3)3.0c o s (6][ nnnnv ??????
)()( )3.02(c o s2)5.02(s in3)3.0c o s (6 nnn ??????????
)3.3s i n (10)5.2c o s (4 nn ????
)()( )7.04(s i n10)5.02(c o s4 nn ????????
86 Copyright ? 2001,S,K,Mitra
The Sampling Process
? Note,v[n] is composed of 3 discrete-time
sinusoidal signals of normalized angular
frequencies,0.3?,0.5?,and 0.7?
)5.0c o s (4)3.0c o s (2)5.0s i n (3)3.0c o s (6 nnnn ????????
)7.0s i n (10 n??
)7.0s i n (10)6 4 3 5.05.0c o s (5)3.0c o s (8 nnn ???????
87 Copyright ? 2001,S,K,Mitra
The Sampling Process
? Note,An identical discrete-time signal is
also generated by uniformly sampling at a
200-Hz sampling rate the following
continuous-time signals,
)140s in (10)6435.0100c o s (5)60c o s (8)( ttttaw ???????
)2 6 0s in (10)1 0 0c o s (4)60c o s (2)( ttttag ??????
)7 0 0s i n (3)4 6 0c o s (6 tt ????
88 Copyright ? 2001,S,K,Mitra
The Sampling Process
? Recall
? Thus if,then the corresponding
normalized digital angular frequency of
the discrete-time signal obtained by
sampling the parent continuous-time
sinusoidal signal will be in the range
? Thus,no aliasing occurs
T
o
o ?
???? 2
o?
oT ??? 2
??????
89 Copyright ? 2001,S,K,Mitra
The Sampling Process
? On the other hand,if,the
normalized digital angular frequency will
fold into a lower digital frequency
in the range
because of aliasing
? Hence,to prevent aliasing,the sampling
frequency should be greater than 2
times the frequency of the sinusoidal
signal being sampled
oT ??? 2
?????? ???????? 2/2 Too
T?
o?
90 Copyright ? 2001,S,K,Mitra
The Sampling Process
? Generalization,Consider an arbitrary
continuous-time signal composed of a
weighted sum of a number of sinusoidal
signals
? can be represented uniquely by its
sampled version {x[n]} if the sampling
frequency is chosen to be greater than 2
times the highest frequency contained in
)(txa
)(txa
T?
)(txa
91 Copyright ? 2001,S,K,Mitra
The Sampling Process
? The condition to be satisfied by the
sampling frequency to prevent aliasing is
called the sampling theorem (Shannon)
