Discrete-Time Signals:
Time-Domain Representation
Signals 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 n
x[n] defined only for integer values of n and
undefined for noninteger values of n
Discrete-time signal represented by {x[n]}
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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:
{x[n]} = {K, 0.2, 2.2,1.1, 0.2, 3.7, 2.9,K}
In the above, x[1] = 0.2, x[0] = 2.2, x[1] = 1.1,
etc.
The arrow is placed under the sample at
time index n = 0
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Copyright 2001, S. K. Mitra
Discrete-Time Signals:
Time-Domain Representation
Graphical representation of a discrete-time
signal with real-valued samples is as shown
below:
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 xa (t ) at uniform intervals of time
Copyright 2001, S. K. Mitra
Discrete-Time Signals:
Time-Domain Representation
Here, n-th sample is given by
x[n] = xa (t ) t = nT = xa (nT ), n = K, 2, 1,0,1,K
The spacing T between two consecutive
samples is called the sampling interval or
sampling period
Reciprocal of sampling interval T, denoted
as FT , is called the sampling frequency:
1
FT =
T
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
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Copyright 2001, S. K. Mitra
Discrete-Time Signals:
Time-Domain Representation
A complex sequence {x[n]} can be written
as {x[n]} = {xre [ n]} + j{xim [n]} where
xre [n] and xim [n] are the real and imaginary
parts of x[n]
The complex conjugate sequence of {x[n]}
is given by {x * [n]} = {xre [n]} j{xim [n]}
Often the braces are ignored to denote a
sequence if there is no ambiguity
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Copyright 2001, S. K. Mitra
Discrete-Time Signals:
Time-Domain Representation
Example - {x[n]} = {cos 0.25n} is a real
sequence
j 0.3n
} is a complex sequence
{y[n]} = {e
We can write
{y[n]} = {cos 0.3n + j sin 0.3n}
= {cos 0.3n} + j{sin 0.3n}
where {yre [n]} = {cos 0.3n}
{yim [n]} = {sin 0.3n}
Copyright 2001, S. K. Mitra
Discrete-Time Signals:
Time-Domain Representation
Example j 0.3n
{w[n]} = {cos 0.3n} j{sin 0.3n} = {e
}
is the complex conjugate sequence of {y[n]}
That is,
{w[n]} = {y * [n]}
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
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Copyright 2001, S. K. Mitra
Discrete-Time Signals:
Time-Domain Representation
Amplitude
Amplitude
Example -
time, t
time, t
Boxedcar signal
11
Digital signal
Copyright 2001, S. K. Mitra
Discrete-Time Signals:
Time-Domain Representation
A discrete-time signal may be a finitelength or an infinite-length sequence
Finite-length (also called finite-duration or
finite-extent) sequence is defined only for a
finite time interval: N1 n N 2
where < N1 and N 2 < with N1 N 2
Length or duration of the above finitelength sequence is N = N 2 N1 + 1
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Copyright 2001, S. K. Mitra
Discrete-Time Signals:
Time-Domain Representation
2
x
[
n
]
=
n
, 3 n 4 is a finite Example length sequence of length 4 (3) + 1 = 8
y[n] = cos 0.4n is an infinite-length sequence
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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
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Copyright 2001, S. K. Mitra
Discrete-Time Signals:
Time-Domain Representation
Example n 2 , 3 n 4
xe [n] =
0, 5 n 8
is a finite-length sequence of length 12
obtained by zero-padding x[n] = n 2 , 3 n 4
with 4 zero-valued samples
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Copyright 2001, S. K. Mitra
Discrete-Time Signals:
Time-Domain Representation
A right-sided sequence x[n] has zerovalued samples for n < N1
N1
A right-sided sequence
If N1 0, a right-sided sequence is called a
causal sequence
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Copyright 2001, S. K. Mitra
Discrete-Time Signals:
Time-Domain Representation
A left-sided sequence x[n] has zero-valued
samples for n > N 2
N2
A left-sided sequence
If N 2 0, a left-sided sequence is called a
anti-causal sequence
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Copyright 2001, S. K. Mitra
Operations on Sequences
A single-input, single-output 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]
Input sequence
18
Discrete-time
system
y[n]
Output sequence
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
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Copyright 2001, S. K. Mitra
Basic Operations
Product (modulation) operation:
x[n]
Modulator
y[n]
y[n] = x[n] w[n]
w[n]
An application is in forming a finite-length
sequence from an infinite-length sequence
by multiplying the latter with a finite-length
sequence called an window sequence
Process called windowing
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Copyright 2001, S. K. Mitra
Basic Operations
Addition operation:
x[n]
y[n]
Adder
y[n] = x[n] + w[n]
w[n]
Multiplication operation
A
Multiplier
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x[n]
y[n]
y[n] = A x[n]
Copyright 2001, S. K. Mitra
Basic Operations
Time-shifting operation: y[n] = x[n N ]
where N is an integer
If N > 0, it is delaying operation
Unit delay
x[n]
z 1
y[n]
y[n] = x[n 1]
If N < 0, it is an advance operation
x[n]
Unit advance
22
y[n]
y[n] = x[n + 1]
Copyright 2001, S. K. Mitra
Basic Operations
Time-reversal (folding) operation:
y[n] = x[ n]
Branching operation: Used to provide
multiple copies of a sequence
x[n]
x[n]
x[n]
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Copyright 2001, S. K. Mitra
Basic Operations
Example - Consider the two following
sequences of length 5 defined for 0 n 4 :
{a[n]} = {3 4 6 9 0}
{b[n]} = {2 1 4 5 3}
New sequences generated from the above
two sequences by applying the basic
operations are as follows:
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Copyright 2001, S. K. Mitra
Basic Operations
{c[n]} = {a[n] b[n]} = {6 4 24 45 0}
{d [n]} = {a[n] + b[n]} = {5 3 10 4 3}
{e[n]} = 3 {a[n]} = {4.5 6 9 13.5 0}
2
As pointed out by the above example,
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
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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 0 n 2: { f [n]} = { 2 1 3}
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Copyright 2001, S. K. Mitra
Basic Operations
We cannot add the length-3 sequence { f [n]}
to the length-5 sequence {a[n]} defined
earlier
We therefore first append { f [n]} with 2
zero-valued samples resulting in a length-5
sequence { f e [n]} = { 2 1 3 0 0}
Then
{g[n]} = {a[ n]} + { f e [n]} = {1 5 3 9 0}
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Copyright 2001, S. K. Mitra
Combinations of Basic
Operations
Example -
y[n] = 1x[n] + 2 x[n 1] + 3 x[n 2] + 4 x[n 3]
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Copyright 2001, S. K. Mitra
Sampling Rate Alteration
Employed to generate a new sequence y[n]
'
with a sampling rate FT higher or lower
than that of the sampling rate FT of a given
sequence x[n]
FT'
Sampling rate alteration ratio is R =
FT
If R > 1, the process called interpolation
If R < 1, the process called decimation
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Copyright 2001, S. K. Mitra
Sampling Rate Alteration
In up-sampling by an integer factor L > 1,
L 1 equidistant zero-valued samples are
inserted by the up-sampler between each
two consecutive samples of the input
sequence x[n]:
x[n / L], n = 0, L, 2 L,L
xu [n] =
otherwise
0,
x[n]
30
xu [n]
Copyright 2001, S. K. Mitra
Sampling Rate Alteration
An example of the up-sampling operation
Output sequence up-sampled by 3
1
0.5
0.5
Amplitude
Amplitude
Input Sequence
1
-0.5
-1
31
-0.5
10
20
30
Time index n
40
50
-1
10
20
30
Time index n
40
50
Copyright 2001, S. K. Mitra
Sampling Rate Alteration
In down-sampling by an integer factor
M > 1, every M-th samples of the input
sequence are kept and M 1 in-between
samples are removed:
y[n] = x[nM ]
x[n]
32
y[n]
Copyright 2001, S. K. Mitra
Sampling Rate Alteration
An example of the down-sampling
operation
Output sequence down-sampled by 3
1
0.5
0.5
Amplitude
Amplitude
Input Sequence
1
-0.5
-1
33
-0.5
10
20
30
Time index n
40
50
-1
10
20
30
Time index n
40
Copyright 2001, S. K. Mitra
50
Classification of Sequences
Based on Symmetry
Conjugate-symmetric sequence:
x[n] = x * [ n]
If x[n] is real, then it is an even sequence
An even sequence
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Copyright 2001, S. K. Mitra
Classification of Sequences
Based on Symmetry
Conjugate-antisymmetric sequence:
x[n] = x * [ n]
If x[n] is real, then it is an odd sequence
An odd sequence
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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
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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:
x[n] = xcs [n] + xca [n]
where
xcs [n] = 12 ( x[n] + x * [ n])
xca [n] = 12 ( x[n] x * [n])
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Copyright 2001, S. K. Mitra
Classification of Sequences
Based on Symmetry
Example - Consider the length-7 sequence
defined for 3 n 3 :
{g [ n ]} = {0, 1+ j 4, 2 + j 3, 4 j 2, 5 j 6, j 2, 3}
Its conjugate sequence is then given by
{g * [ n ]} = {0, 1 j 4, 2 j 3, 4 + j 2, 5+ j 6, j 2, 3}
The time-reversed version of the above is
{g * [ n ]} = {3, j 2, 5+ j 6, 4 + j 2, 2 j 3, 1 j 4, 0}
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Copyright 2001, S. K. Mitra
Classification of Sequences
Based on Symmetry
Therefore {g cs [n]} = 12 {g [n] + g * [ n]}
= {1.5, 0.5+ j 3, 3.5+ j 4.5, 4, 3.5 j 4.5, 0.5 j 3, 1.5}
Likewise {g ca [n]} = 1 {g [n] g * [ n]}
2
= {1.5, 0.5+ j , 1.5 j1.5, j 2, 1.5 j1.5, 0.5 j , 1.5}
* [ n]
It can be easily verified that gcs [n] = gcs
* [ n]
and gca [n] = gca
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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:
x[n] = xev [n] + xod [n]
where
xev [n] = 12 ( x[n] + x[ n])
xod [n] = 12 ( x[n] x[ n])
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Copyright 2001, S. K. Mitra
Classification of Sequences
Based on Symmetry
A length-N sequence x[n], 0 n N 1,
can be expressed as x[n] = x pcs [n] + x pca [n]
where
x pcs [n] = 12 ( x[n] + x * [ n N ]), 0 n N 1,
is the periodic conjugate-symmetric part
and
x pca [n] = 12 ( x[n] x * [ n N ]), 0 n N 1,
41
is the periodic conjugate-antisymmetric
part
Copyright 2001, S. K. Mitra
Classification of Sequences
Based on Symmetry
For a real sequence, the periodic conjugatesymmetric part, is a real sequence and is
called the periodic even part x pe [n]
For a real sequence, the periodic conjugateantisymmetric part, is a real sequence and is
called the periodic odd part x po [n]
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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
x[n] = x * [ n N ] = x * [ N n]
and is called a periodic conjugateantisymmetric sequence if
x[n] = x * [ n N ] = x * [ N n]
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Copyright 2001, S. K. Mitra
Classification of Sequences
Based on Symmetry
A finite-length real periodic conjugatesymmetric sequence is called a symmetric
sequence
A finite-length real periodic conjugateantisymmetric sequence is called a
antisymmetric sequence
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Copyright 2001, S. K. Mitra
Classification of Sequences
Based on Symmetry
Example - Consider the length-4 sequence
defined for 0 n 3 :
{u[n]} = {1 + j 4, 2 + j 3, 4 j 2, 5 j 6}
Its conjugate sequence is given by
{u * [n]} = {1 j 4, 2 j 3, 4 + j 2, 5 + j 6}
To determine the modulo-4 time-reversed
version {u * [n 4 ]} observe the following:
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Copyright 2001, S. K. Mitra
Classification of Sequences
Based on Symmetry
u * [0 4 ] = u * [0] = 1 j 4
u * [1 4 ] = u * [3] = 5 + j 6
u * [2 4 ] = u * [2] = 4 + j 2
u * [3 4 ] = u * [1] = 2 j 3
Hence
{u * [ n 4 ]} = {1 j 4, 5 + j 6, 4 + j 2, 2 j 3}
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Copyright 2001, S. K. Mitra
Classification of Sequences
Based on Symmetry
Therefore
{u pcs [n]} = 1 {u[n] + u * [ n 4 ]}
2
= {1, 3.5 + j 4.5, 4, 3.5 j 4.5}
Likewise
{u pca [n]} = 1 {u[n] u * [ n 4 ]}
2
= { j 4, 1.5 j1.5, 2, 1.5 j1.5}
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Copyright 2001, S. K. Mitra
Classification of Sequences
Based on Periodicity
x [n] satisfying ~
A sequence ~
x [n ] = ~
x [n + kN ]
is called a periodic sequence with a period N
where N is a positive integer and k is any
integer
Smallest value of N satisfying ~
x [n ] = ~
x [n + kN ]
is called the fundamental period
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Copyright 2001, S. K. Mitra
Classification of Sequences
Based on Periodicity
Example -
A sequence not satisfying the periodicity
condition is called an aperiodic sequence
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Copyright 2001, S. K. Mitra
Classification of Sequences:
Energy and Power Signals
Total energy of a sequence x[n] is defined by
x =
x[n]
n =
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
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Copyright 2001, S. K. Mitra
Classification of Sequences:
Energy and Power Signals
The average power of an aperiodic
sequence is defined by
Px =
1
lim 2 K +1
K
x[n]
n= K
Define the energy of a sequence x[n] over a
finite interval K n K as
51
x,K
= x[n]
n= K
Copyright 2001, S. K. Mitra
Classification of Sequences:
Energy and Power Signals
Then
Px = lim
1
K 2 K +1
x. K
The average power of a periodic sequence
~
x [n] with a period N is given by
Px =
1
N
N 1
2
~
x [n ]
n =0
The average power of an infinite-length
sequence may be finite or infinite
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Copyright 2001, S. K. Mitra
Classification of Sequences:
Energy and Power Signals
Example - Consider the causal sequence
defined by
3(1)n , n 0
x[n] =
n<0
0,
Note: x[n] has infinite energy
Its average power is given by
1 K
9( K + 1)
= 4.5
Px = lim
9 1 = lim
K 2 K + 1 n = 0 K 2 K + 1
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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
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Copyright 2001, S. K. Mitra
Other Types of Classifications
A sequence x[n] is said to be bounded if
x[n] Bx <
Example - The sequence x[n] = cos 0.3n is a
bounded sequence as
x[n] = cos 0.3n 1
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Copyright 2001, S. K. Mitra
Other Types of Classifications
A sequence x[n] is said to be absolutely
summable if
x[n] <
n =
Example - The sequence
0.3n , n 0
y[n] =
0, n < 0
is an absolutely summable sequence as
1
n
0.3 = 1 0.3 = 1.42857 <
n =0
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Copyright 2001, S. K. Mitra
Other Types of Classifications
A sequence x[n] is said to be squaresummable if
2
x[n] <
n =
Example - The sequence
sin 0.4 n
h[n] = n
is square-summable but not absolutely
summable
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Copyright 2001, S. K. Mitra
Basic Sequences
1, n = 0
Unit sample sequence - [n] =
0, n 0
1
n
4
1, n 0
[ n] =
0, n < 0
Unit step sequence 1
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Copyright 2001, S. K. Mitra
Basic Sequences
Real sinusoidal sequence x[n] = A cos(o n + )
where A is the amplitude, o is the angular
frequency, and is the phase of x[n]
Example = 0.1
o
Amplitude
1
0
-1
-2
0
59
10
20
Time index n
30
40
Copyright 2001, S. K. Mitra
Basic Sequences
Exponential sequence n
x[n] = A , < n <
where A and are real or complex numbers
j
( o + jo )
A
=
A
e
,
If we write = e
,
then we can express
x[n] =
where
j ( o + jo ) n
Ae e
o n
cos(o n + ),
o n
sin(o n + )
xre [n] = A e
60
xim [n] = A e
= xre [n] + j xim [n],
Copyright 2001, S. K. Mitra
Basic Sequences
xre [n] and xim [n] of a complex exponential
sequence are real sinusoidal sequences with
constant (o = 0 ), growing (o > 0 ), and
decaying (o < 0 ) amplitudes for n > 0
Imaginary part
0.5
0.5
Amplitude
Amplitude
Real part
0
-0.5
-1
0
-0.5
61
10
20
Time index n
30
x[n] =
40
-1
1
exp( 12
10
20
Time index n
+ j 6 )n
30
Copyright 2001, S. K. Mitra
40
Basic Sequences
Real exponential sequence x[n] = A n , < n <
where A and are real numbers
= 0.9
= 1.2
20
50
15
Amplitude
Amplitude
40
30
20
10
0
0
62
10
10
15
20
Time index n
25
30
0
0
10
15
20
Time index n
25
Copyright 2001, S. K. Mitra
30
Basic Sequences
Sinusoidal sequence A cos(o n + ) and
complex exponential sequence B exp( jo n)
are periodic sequences of period N if o N = 2r
where N and r are positive integers
Smallest value of N satisfying o N = 2r
is the fundamental period of the sequence
To verify the above fact, consider
x1[n] = cos(o n + )
x2 [n] = cos(o (n + N ) + )
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Copyright 2001, S. K. Mitra
Basic Sequences
Now x2 [n] = cos(o (n + N ) + )
= cos(o n + ) cos o N sin(o n + ) sin o N
which will be equal to cos(o n + ) = x1[n]
only if
sin o N = 0 and cos o N = 1
These two conditions are met if and only if
o N = 2 r or 2 = N
o r
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Copyright 2001, S. K. Mitra
Basic Sequences
If 2/o is a noninteger rational number, then
the period will be a multiple of 2/o
Otherwise, the sequence is aperiodic
Example - x[n] = sin( 3n + ) is an aperiodic
sequence
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Copyright 2001, S. K. Mitra
Basic Sequences
=0
0
Amplitude
1.5
1
0.5
0
0
10
20
Time index n
30
40
Here o = 0
2 r
Hence period N =
= 1 for r = 0
0
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Copyright 2001, S. K. Mitra
Basic Sequences
= 0.1
0
Amplitude
1
0
-1
-2
0
10
20
Time index n
30
40
Here o = 0.1
2 r
Hence N =
= 20 for r = 1
0.1
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Copyright 2001, S. K. Mitra
Basic Sequences
Property 1 - Consider x[n] = exp( j1n) and
y[n] = exp( j2 n) with 0 1 < and
2k 2 < 2(k + 1) where k is any positive
integer
If 2 = 1 + 2k , then x[n] = y[n]
Thus, x[n] and y[n] are indistinguishable
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Copyright 2001, S. K. Mitra
Basic Sequences
Property 2 - The frequency of oscillation of
A cos(o n) increases as o increases from 0
to , and then decreases as o increases from
to 2
Thus, frequencies in the neighborhood of
= 0 are called low frequencies, whereas,
frequencies in the neighborhood of = are
called high frequencies
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Copyright 2001, S. K. Mitra
Basic Sequences
Because of Property 1, a frequency o in
the neighborhood of = 2 k is
indistinguishable from a frequency o 2 k
in the neighborhood of = 0
and a frequency o in the neighborhood of
= (2 k + 1) is indistinguishable from a
frequency o (2 k + 1) in the
neighborhood of =
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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
v1[n] = cos(0.1 n) = cos(1.9 n) is a lowfrequency signal
v2 [n] = cos(0.8 n) = cos(1.2 n) is a highfrequency signal
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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
x[n] = 0.5 [n + 2] + 1.5 [n 1] [n 2]
+ [n 4] + 0.75 [n 6]
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Copyright 2001, S. K. Mitra
The Sampling Process
Often, a discrete-time sequence x[n] is
developed by uniformly sampling a
continuous-time signal xa (t ) as indicated
below
The relation between the two signals is
x[n] = xa (t ) t =nT = xa (nT ), n = K, 2, 1, 0,1, 2,K
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Copyright 2001, S. K. Mitra
The Sampling Process
Time variable t of xa (t ) is related to the time
variable n of x[n] only at discrete-time
instants tn given by
tn = nT = n = 2 n
FT T
with FT = 1 / T denoting the sampling
frequency and
T = 2 FT denoting the sampling angular
frequency
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Copyright 2001, S. K. Mitra
The Sampling Process
Consider the continuous-time signal
x(t ) = A cos(2 fot + ) = A cos(ot + )
The corresponding discrete-time signal is
2 o
x[n] = A cos(o nT + ) = A cos(
n + )
T
= A cos(o n + )
where o = 2 o / T = oT
is the normalized digital angular frequency
of x[n]
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If the unit of sampling period T is in
seconds
The unit of normalized digital angular
frequency o is radians/sample
The unit of normalized analog angular
frequency o is radians/second
The unit of analog frequency f o is hertz
(Hz)
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The three continuous-time signals
g1 (t ) = cos(6 t)
g 2 (t ) = cos(14 t)
g3 (t ) = cos(26 t)
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
g1[n] = cos(0.6 n) g 2 [n] = cos(1.4 n)
g3[n] = cos(2.6 n)
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Plots of these sequences (shown with circles)
and their parent time functions are shown
below:
1
Amplitude
0.5
-0.5
-1
0.2
0.4
0.6
0.8
time
Note that each sequence has exactly the same
sample value for any given n
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This fact can also be verified by observing that
g 2 [n] = cos(1.4 n) = cos((2 0.6)n ) = cos(0.6 n)
g3[n] = cos(2.6 n) = cos((2 + 0.6)n ) = cos(0.6 n)
As a result, all three sequences are identical
and it is difficult to associate a unique
continuous-time function with each of these
sequences
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The above phenomenon of a continuoustime signal of higher frequency acquiring
the identity of a sinusoidal sequence of
lower frequency after sampling is called
aliasing
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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 {x[n]} = {xa (nT )} can
uniquely represent the parent continuoustime signal xa (t )
In this case, xa (t ) can be fully recovered
from {x[n]}
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Example - Determine the discrete-time
signal v[n] obtained by uniformly sampling
at a sampling rate of 200 Hz the continuoustime signal
va (t ) = 6 cos(60 t) + 3 sin(300 t) + 2 cos(340 t)
+ 4 cos(500t ) + 10 sin(660t )
Note: va (t ) is composed of 5 sinusoidal
signals of frequencies 30 Hz, 150 Hz, 170
Hz, 250 Hz and 330 Hz
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The sampling period is T =
= 0.005 sec
The generated discrete-time signal v[n] is
thus given by
1
200
v[ n ] = 6 cos( 0.3 n) + 3 sin(1.5 n) + 2 cos(1.7 n)
+ 4 cos( 2.5 n) + 10 sin( 3.3 n)
= 6 cos( 0.3n ) + 3 sin(( 2 0.5) n ) + 2 cos(( 2 0.3) n )
+ 4 cos(( 2 + 0.5) n) + 10 sin(( 4 0.7 ) n)
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= 6 cos( 0.3 n) 3 sin( 0.5 n) + 2 cos( 0.3 n) + 4 cos( 0.5 n)
10 sin( 0.7 n)
= 8 cos( 0.3 n) + 5 cos( 0.5 n + 0.6435) 10 sin( 0.7 n)
Note: v[n] is composed of 3 discrete-time
sinusoidal signals of normalized angular
frequencies: 0.3, 0.5, and 0.7
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Note: An identical discrete-time signal is
also generated by uniformly sampling at a
200-Hz sampling rate the following
continuous-time signals:
wa (t ) = 8 cos( 60 t) + 5 cos(100 t + 0.6435) 10 sin(140 t)
g a (t ) = 2 cos( 60 t) + 4 cos(100 t) + 10 sin( 260 t)
+ 6 cos( 460 t) + 3 sin( 700 t)
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Recall
2o
o =
T
Thus if T > 2o , then the corresponding
normalized digital angular frequency o of
the discrete-time signal obtained by
sampling the parent continuous-time
sinusoidal signal will be in the range < <
No aliasing
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On the other hand, if T < 2o , the
normalized digital angular frequency will
foldover into a lower digital frequency
o = 2o / T 2 in the range < <
because of aliasing
Hence, to prevent aliasing, the sampling
frequency T should be greater than 2
times the frequency o of the sinusoidal
signal being sampled
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Generalization: Consider an arbitrary
continuous-time signal xa (t ) composed of a
weighted sum of a number of sinusoidal
signals
xa (t ) can be represented uniquely by its
sampled version {x[n]} if the sampling
frequency T is chosen to be greater than 2
times the highest frequency contained in
xa (t )
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The condition to be satisfied by the
sampling frequency to prevent aliasing is
called the sampling theorem
A formal proof of this theorem will be
presented later
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