01Introduction_Lecture2SigSize.pdf girisha

A strict definition of a signal is a time-varying occurrence that conveys information, and a strict definition of system is a collection of modules which take in signals and generate some sort of response. As you can see, we will overshoot the line every time

1. Lecture 2 Introduction to signals and systems Continued
 Even and Odd components of the signal:
 A real function xe(t) is said to be an even function of t if
xe(t) = xe(-t) i.e symmetrical about vertical axis at t= 0
 A real function xo(t) is said to be an odd function of t if
xo(t) = -xo(-t) i.e anti -symmetrical about vertical axis at t= 0
OR
OR
o Some properties:
 Even function Χ odd function = odd function
 Odd function Χ odd function = Even function
 Even function Χ Even function = Even function
 Area ∫??????
??????
??????
−??????
(??????)????????????=2∫??????
??????
??????
0
(??????)???????????? and ∫??????
0
??????
−??????
(??????)????????????=0
Given any arbitrary signal x(t), odd and even parts can be found as
 Even part of the signal xe(t) =
1
2
[x(t)+x(-t)]
 Odd part of the signal xo(t) =
1
2
[x(t)-x(-t)]

2. NOTE: To find the ODD components OR EVEN components of any signal one can use the same equations either using theoretically OR by
computation (using OCTAVE/MATLAB).
Example:

Original signal EVEN ODD
 Typical MATLAB / OCTAVE Implementation example
t = -4:0.01:5; creating array of time
x = exp(-t).*cos(2*pi*t).*((t>=-2)&(t<1)); defining the signal
plot(t,x) sketching the signal
Ex = sum(x.^2*0.01) computing energy (discussed in following section)
xr = exp(t).*cos(2*pi*(-t)).*((t>=-1)&(t<2)); creating time reversal of signal
subplot(3,1,1); plot(t,x) sketch using subplots (3 plots)
subplot(3,1,2); plot(t, 0.5*(x+xr))
subplot(3,1,3); plot(t, 0.5*(x-xr))
 There is another method using ‘inline’ function demonstrated in the manual
 You may compute the size of these signals using the method discussed as follows

3. Size of a signal:
--------> It indicates the largeness or strength of the signal
Eg: human size -----> is it volume ? ; not the height only OR is it some thing else? Like knowledge, popularity etc.
 There are two ways in which one can see the size of a signal
SIGNAL ENERGY OR SIGNAL POWER OR NONE

 Signal energy: Ex = ∫??????
2
∞
−∞
(??????)???????????? for real signal OR Ex = ∫|??????(??????)|
2
∞
−∞
???????????? for complex signal
o Signal size is the area under square of the signal x(t), i.e. x
2
(t)
o Signal energy should be finite and non zero for it to be a meaningful measure of signal size
o Necessary condition is that amplitude of signal x(t) ----> 0 as |??????| ---> ∞
Examples ???
x(t) = 2 : -1 ≤ t ≤ 0
2 e
-t/2
: t ≥ 0
Energy of the signal is 8.0018 units

4.  Signal power:
o If the amplitude of the signal x(t) is not ----> 0 as |??????| ---> ∞,

o signal energy is infinite.
o Then, a more meaningful measure of signal size in such a case would be Time average of the energy, with some condition
o Px = lim
??????→∞
1
??????
∫??????
2
(??????)????????????
??????
2
−
??????
2
for real signal OR Px = lim
??????→∞
1
??????
∫|??????(??????)|
2
????????????
??????
2
−
??????
2
for complex, where, T is the period of the periodic signal.
o Signal power should be finite for it to be a meaningful measure of signal size.

5. Example:
Power of the signal is 0.3333 units
 √??????
?????? = RMS Value of the signal, generally applicable for periodic signal.

Note: Generally mean of an entity averaged over a large time interval approaching infinity, exists if the entity is either periodic or has a
statistical regularity. If such a condition is not satisfied, the average may not exists.
 Power is the time average of energy
 A signal cannot be both an energy signal and power signal
 A ramp signal is neither energy nor power signal



NOTE:
For caluculating power and OR energy using MATLAB / OCTAVE, one may use the above equations over a range accordingly
( for energy : entire time range for which the signal is non-zero and
for power : entire time range for one time period)

o For convenience many times we may use a causal signal but periodic is also referred as power signal (in real sense it is not true).

6. TASK: a derivation example :
determine the power and RMS value of
X(t) = C cos (ωot +θ) and
X(t) = C1 cos (ω1t +θ1) + C2 cos (ω2t +θ2) with ω1 ≠ ω2
........................................



Px =
??????
1
2
2
+
??????
2
2
2
OR Px =
1
2
∑????????????
2
∞
??????=1