•
For a C.T. ramp signal is designated by r(t) and
mathematically defined as
0
0
0
)
(
t
t
t
t
r
•
Graphical representation
•
We can also express r(t)in
terms of u(t) as
r(t)=t·u(t).
d
u
t
r
t
)
(
)
(
)
(
)
(
t
r
dt
d
t
u
DT Unit ramp signal
•
Similarly the DT ramp signal can be defined as
•
Graphical representation
•
Representation of DT ramp
in terms of u[n]
0
0
0
)
(
n
n
n
n
r
•
And DT unit step in terms of ramp can written
as
n
k
k
u
n
r
)
(
)
(
)
(
)
1
(
)
(
n
r
n
r
n
u
Unit impulse (sample) signal
•
An Impulse function is having important role in
signal analysis.
•
Continuous time unit impulse function also
known as Dirac delta function was first defined
by Dirac as
(t)=1 for t
0 and
1
)
(
dt
t
An impulse function
(t) is a pulse defined about
t=0, which covers 1 unit area about the origin.
•
It is a delicate concept to represent impulse
graphically see the figures in next slide where the
impulse function is represented graphically.
Unit impulse (sample) signal
•
Graphical representation
•
Therefore the shape of the signal doesn’t matter, if
area covered by pulse about the origin is 1 then it
is called as impulse function
(t).
•
Generally it is represented by arrow at t=0 for CT
signal as shown in fig 1.30 (c) with strength equals
to 1.
Unit impulse (sample) signal
•
If width of the impulse gradually approaches to
zero then amplitude of impulse will go on
increasing and at width equal to 0 the amplitude
will be infinite.
•
That is impulse at 0 has infinite amplitude.
•
The impulse function is a derivative of unit step
function or we can say unit step function is a
integration of unit impulse function.
d
t
u
t
)
(
)
(
)
(
)
(
t
u
dt
d
t
Properties of impulse signal
•
The reflection of impulse is impulse itself.
•
This property of impulse response is termed as
reflection property of impulse response.
•
Time scaling property
)
(
)
(
t
t
)
(
)
(
at
a
t
•
Time shift property: Shall be covered after
studying basic operations on signals
DT Unit impulse (sample) signal
•
Discrete version of unit impulse signal is easier to
understand which can be defined as
•
Graphical representation
0
0
0
1
)
(
n
n
n
•
Every signal shall be expressed in terms of unit
sample sequence(signal)
Rectangular pulse
•
Definition
•
Graphical representation
otherwise
t
t
0
1
)
(
2
1
Triangular Pulse
•
Definition
•
Graphical representation
a
t
a
t
t
a
t
a
0
1
)
(
x
(t)
t
a

a
1
Signum Function
•
Definition
•
Graphical representation
a
t
a
t
t
a
t
a
0
1
)
(
x
(t)
t
1
1
Sinc Pulse
•
Definition
•
Graphical representation
t
t
t
Sin
t
Sinc
)
(
)
(
•
The Sinc function oscillates with period 2
and
decays with increasing t. It’s value is zero at n
.
Revise the elementary signals studied
•
Exponential signal
•
Sinusoidal signal
•
Unit step signal
•
Unit ramp signal
•
Unit impulse (sample) signal
•
Rectangular pulse
•
Triangular pulse
•
Signum function
•
Sinc function
Basic operations on signals
•
Whenever we need to perform any
operation on the signal either amplitude or
time variable of the signal will be modified.
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 Summer '20
 Meenakshi Patil
 Signal Processing, DT Signals, 2t