![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 ∫ 𝑥𝑒
𝑎
−𝑎
(𝑡)𝑑𝑡 = 2 ∫ 𝑥𝑒
𝑎
0
(𝑡)𝑑𝑡 and ∫ 𝑥0
𝑎
−𝑎
(𝑡)𝑑𝑡 =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)]](https://image.slidesharecdn.com/01introductionlecture2sigsize-240822113409-3f102881/85/01Introduction_Lecture2SigSize-pdf-girisha-1-320.jpg)
01Introduction_Lecture2SigSize.pdf girisha
- 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 ∫ 𝑥𝑒 𝑎 −𝑎 (𝑡)𝑑𝑡 = 2 ∫ 𝑥𝑒 𝑎 0 (𝑡)𝑑𝑡 and ∫ 𝑥0 𝑎 −𝑎 (𝑡)𝑑𝑡 =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 = ∫ 𝑥2 ∞ −∞ (𝑡)𝑑𝑡 for real signal OR Ex = ∫ |𝑥(𝑡)|2 ∞ −∞ 𝑑𝑡 for complex signal o Signal size is the area under square of the signal x(t), i.e. x2 (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 |𝑡| ---> ∞ 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 |𝑡| ---> ∞, 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 = lim𝑇→∞ 1 𝑇 ∫ 𝑥2(𝑡)𝑑𝑡 𝑇 2 − 𝑇 2 for real signal OR Px = lim𝑇→∞ 1 𝑇 ∫ |𝑥(𝑡)|2 𝑑𝑡 𝑇 2 − 𝑇 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 √𝑃 𝑥 = 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 = 𝐶1 2 2 + 𝐶2 2 2 OR Px = 1 2 ∑ 𝐶𝑛 2 ∞ 𝑛=1