Discrete - Time Signal
Definition
Notation
Classification
Manipulation

Discrete Time Signal
As discussed before a signal is a parameter that depends on an
independent parameter
Definition
: A Discrete-time Signal is a function of an independent
variable that is an integer and is formally denoted by x = {x(n)} , -
∞ < n < ∞
This means that the independent parameter of the Discrete-time
signal has to be represented by integers.
For eg: if we sample an analog signal X(t) at time instants 0, T
s
, 2T
s
,
….
i.e at t=nT
s
, then the discrete time signal obtained is denoted by x(n)
and x(n) = X(nT
s
)
So here the independent parameter is time and n=0,1,2,3,…
represent t=0, T
s
, 2T
s
, ….
A discrete time signal is basically just a sequence of numbers, this
will become clear as we discuss representation of discrete time
signals

Representation of Discrete Time Signals
Graphical Representation
A Discrete time signal may be
represented graphically as shown
in the figure. This is known as the
lollipop representation.
Note that the signal is represented only at integers, -9,-8,….,0,…. This is because
discrete
time signal domain is only integers. In between integers the signal is not equal to
zero , In fact
at these points (for eg between 1 and 2) , the signal does not exist
!!!
Functional Representation
Another kind of representation is by denoting the discrete time signal
as a function
Eg: 1) x(n) = Asin(ω
0
n+θ)
3n-5, if -9≤n<0
2) x(n) = 4n+2, if 0 ≤n<5
0, otherwise

Sequential Representation
A discrete time signal may also be represented as a sequence of
numbers.
For example x = {2,3,4,9,3} is a signal. What this means is that n=0 is at
the
arrow, i.e. x(0)=3, and therefore x(-1)=2; x(1)=4; x(2)=9; x(3)=3.
Therefore we see that any discrete time signal is basically just a
sequence of
numbers. It is for this reason we often refer to discrete time signals as
sequences .
Finite Duration Sequence: If a sequence is non-zero only over a
finite period of time, e.g. x={1,4,1}; This sequence has 3 samples,
so it is
called a 3 point sequence.
Infinite Duration Sequence: If a sequence is non-zero over an
infinite duration of time, so it may range over (- ∞ <n< ∞) or (-
∞<n<a) or
(a<n< ∞ ). Eg: {…….0,4,3,-2,1,6,7,……….}
Representation of Discrete Time Signals
contd.

Some Basic/Important Sequences/Signals
Finite duration sequences:
Unit Sample Sequence:
δ(n) = 1 , n=0
0 , otherwise
Infinite duration sequences:
Unit Step Sequence:
u(n) = 1 , n=0
0 , otherwise
Unit Ramp Sequence:
u
r
(n) = n , n≥0
0 , otherwise
Exponential Sequence:
x(n)= a
n
, for all n
Sinusoidal Sequence:
x(n)= Asin(ω
0
n+θ)
0<a<1

Classification of Discrete Time Signals
Cannot predict
value of the
signal, varies
randomly, hence
also called random
signal
All past, present,
future values of
the signal are
known. Can
predict the value
of the signal
Non-
Deterministic
Deterministic
Energy of the signal is
infinite, power is
finite, E ∞ and
0<P<∞
Eg: All periodic signals
Energy of the
signal
is finite, 0<E<∞,
All finite duration
signals are
energy signals
Power Signal
Energy Signal
Does not
have any N
for which
condition is
satisfied
If signal satisfies the

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