Simple operations and properties of sequences

2.1 Simple operations on signals
In analyzing discrete-time systems, operations on sequences occur frequently.
Some operations are discussed below.
2.1.1 Sequence addition:
Let {x[n]} and {y[n]} be two sequences. The sequence addition is defined as
term by term addition. Let {z[n]} be the resulting sequence
{z[n]} = {x[n]} + {y[n]} where each term z[n] = x[n] + y[n]
We will use the following notation
{x[n]} + {y[n]} = {x[n] + y[n]}
2.1.2 Scalar multiplication:
Let a be a scalar. We will take a to be real if we consider only the real valued
signals, and take a to be a complex number if we are considering complex
valued sequence. Unless otherwise stated we will consider complex valued
sequences. Let the resulting sequence be denoted by w[n]
{w[n]} = ax[n]
is defined by w[n] = ax[n],
each term is multiplied by a
We will use the notation aw[n] = aw[n]
Note: If we take the set of sequences and define these two operators as
addition and scalar multiplication they satisfy all the properties of a linear
vector space.
2.1.3 Sequence multiplication:
Let {x[n]} and {y[n]} be two sequences, and {z[n]} be resulting sequence
{z[n]} = {x[n]}{y[n]}
1
where z[n] = x[n]y[n]
The notation used for this will be {x[n]}{y[n]} = {x[n]y[n]} Now we consider some operations based on independent variable n.
2.1.4 Shifting
This is also known as translation. Let us shift a sequence {x[n]} by n0 units,
and the resulting sequence by {y[n]}
{y[n]} = z−n0({x[n]})
where z−n0()is the operation of shifting the sequence right by n0 unit. The
terms are defined by y[n] = x[n−n)]. We will use short notation {x[n−n0]} to denote shift by n0.
{x[n]}
1
n
.....
-3 -2 -1 0 1 2 3
{x[n − 2]}
n
.....
{x[n + 1]}
n
.....
-1 0 1 2 3
Figure above show some examples of shifting. A negative value of n0 means
shift towards right.
2
2.1.5 Reflection:
Let {x[n]} be the original sequence, and {y[n]} be reflected sequence, then
y[n] is defined by
y[n] = x[−n]
{x[n]}
n
.....
-3 -2 -1 0 1 2 3
-3 -2 -1 0 1 2 3
.....
.....
.....
1
2
3
4
1
2
3
4
We will denote this by {x[n]}.
When we have complex valued signals, sometimes we reflect and do the complex
conjugation, ie, y[n] is defined by y[n] = x∗[−n], where * denotes complex
conjugation. This sequence will be denoted by {x∗[−n]}.
We will learn about more complex operations later on. Some of these operations
commute, ie. if we apply two operations we can interchange their order
and some do not commute. For example scalar multiplication and reflection
commute.
{x[n]}
Scalar multiplication by a{y[n]}
reflection {z[n]}

{x[n]}
reflection {w[n]}
Scalar multiplication {
v[n]}
3
Then v[n] = z[n] for all n. Shifting and scaling do not commute.
{x[n]} {y[n]} − {x[n − 1]} {z[n]} = {y[−n]}
{x[n]}
shift by n0 {y[n]}
reflection
{z[n]}
reflection shift by n0
{x[n]}
{w[n]
}
{v[n]}
n
.....
n
.....
n
.....
0 1 2 3 -1 0 1 2
.....
{x[n]} {w[n]} = {x[−n]} {v[n]} = {w[n − 1]}
n
.....
n
.....
n
.....
0 1 2 3 0 1 2
.....
We can combine many of these operations in one step, for example {y[n]} may be defined as y[n] = 2x[3 − n].
2.2 Some properties of signals:
2.2.1 Energy of a Signal:
The total enery of a signal {x[n]} is defined by
Ex =
∞

n=−∞
x[n]2
4
A signal is reffered to as an energy signal, if and only if the total energy of
the signal Ex is finite.
2.2.2 Power of a signal:
If {x[n]} is a signal whose energy is not finite, we define power of the signal
as
Px = lim
N→∞
1
(2N + 1)
N


n=−N
x[n]2
A signal is referred to as a power signal if the power Px satisfies the condition
0 < Px < ∞
An energy signal has a zero power and a power signal has infinite energy.
There are signals which are neither energy signals nor power signals. For
example {x[n]} defined by x[n] = n does not have finite power or energy.
2.2.3 Periodic Signals:
An important class of signals that we encounter frequently is the class of
periodic signals. We say that a signal {x[n]} is periodic period N, where N
is a positive integer, if the signal is unchanged by the time shift of N ie.,
{x[n]} = {x[n + N]}
or x[n] = x[n + N for all n.
Since {x[n]} is same as {x[n + N]}, it is also periodic so we get
{x[n]} = {x[n + N]} = {x[n + N + N]} = {x[n + 2N]}
Generalizing this we get {x[n]} = {x[n+kN]}, where k is a positive integer.
From this we see that {x[n]} is periodic with 2N, 3N, ..... The fundamental
period N0 is the smallest positive value N for which the signal is periodic.
The signal illustrated below is periodic with fundamental period N0 = 4
{x[n]} By change of variable we can write {x[n]} = {x[n +N]} as
{x[m − N]} = {x[m]} and then .........as before, we see that
{x[n]} = {x[n + kN]},
for all integer values of k, positive, negative or zero. By definition, period of
a signal is always a positive integer n.
Except for a all zero signal all periodic signals have infinite energy. They may
5
have finite power. Let {x[n]} be periodic with period N, then the power Px
is given by
P − x = lim
M→∞
1
(2M + 1)
M


n=−M
x[n]2
= lim
M→∞
1
2M+1[
N−1
n=0 x[n]2 +
2N−1
n=N x[n]2 + ...
+
kN−1
n=(k−1)N−1 x[n]2 +
M
n=kN x[n]2 +
−1
n=−N x[n]2 + ...
+
−(k−1)N−1
n=−kN x[n]2 +
−kN−1
n=−M x[n]2]
where k is largest integer such that kN −1 ≤ M. Since the signal is periodic,
sum over one period will be same for all terms. We see that k is approximately
equal to M/N(it is integer part of this) and for large M we get 2M/N terms
and limit 2M/(2M + 1) as M goes to infinite is one we get
Px =
1
N
N−1


n=0
x[n]2
2.2.4 Even and odd signals:
A real valued signal {x[n]} is referred as an even signal if it is identical to its
time reversed counterpart ie, if
{x[n]} = {x[−