04 elementary signals.pdf
TRANSCRIPT
-
11
Introduction to Communication Engineering(H61ICM)
2
Signals representationSinusoidal signalsExponential Signal & Complex Exponential SignalUnit impulse and unit step (Discrete and Continuous)Sinc & Rect function
-
23
SinusoidsMathematically represented as:
a cost, a sint, a cos(t+), a sin(t+)
a cost = a cos2pifta, amplitude (volts, current, etc)
, angular frequency (radians/sec)f, frequency in Hertz (oscillations/sec)
=2pif where (radians or degrees, where pi radians=180 degrees)
A(t)time or radians or degreesT
a
T, 2pi radians, 360o0
4
Adding a Phase shiftThe start point of the wave can be changed by altering the starting phase, For the previous signal, a cos(t+), =0Consider a cosine wave with =-pi/2
i.e. A(t)=a cos(t-pi/2)
the cosine wave has been shifted along the time axis by 1/4 of a cycle [=a sin (t) ]
A(t)time or radians or degreesT
a
T, 2pi radians, 360o0
-
35
Signals representationSinusoidal signalsExponential Signal & Complex Exponential SignalUnit impulse and unit step (Discrete and Continuous)Sinc function
6
The continuous-time complex exponential signal is of the form
or
where a is generally complex numbers. Depending on the values of this parameter, the complex exponential exhibit several different characteristics
atCetx =)( [ ] nCrnx =
Exponential Signals
r = e
-
47
Exponential SignalsConsider
when a is real and a>0,
x(t) grows exponentially as time increases
atAetx =)(
c
x(t)
t
8
when a is real and a
-
59
When a is purely imaginary, i.e.
An important property of this signal is that it is periodic.
tjAetx 0)( =
Complex Exponential Signals
10
When a is complex, i.e.
The signal is oscillating and bounded by the curves
tjteAetx 0)( =0 ja +=
tAe
Complex Exponential Signals
-
613
Signals representationSinusoidal signalsExponential Signal & Complex Exponential SignalUnit impulse and unit step (Discrete and Continuous)Sinc function
14
Unit impulse (Discrete-time)In discrete-time, unit impulse is defined as
=
=
0,10,0][
n
nn
n0
1 [n]
Discrete Time Unit Impulses & Step Sequences
-
715
Unit step (Discrete-time)Definition
Unit step function is denoted by u[n] and is defined as
It is made up of a sequence of discrete unit impulse