TA Seminar
Leo Lam 2010-2011Signals and SystemsEE235
Oh beerAn infinite amount of mathematicians walk into a bar. The
first one orders a beer. The second one orders half a beer. The
third one orders one quarter beer.
The fourth one starts to order, but the bartender interrupts
"Here's two beers; you lot can figure the rest out yourself."Leo
Lam 2010-2011
Leo Lam 2010-2011Todays menuLab 1 started today!A few little
clarificationsFrom yesterday: Describing Common SignalsIntroduced
the three building blocksGeneral description for sinusoidal
signalsToday: PeriodicityPeriodic signalsDefinition: x(t) is
periodic if there exists a T (time period) such that:
The minimum T is the periodFundamental frequency f0=1/TLeo Lam
2010-2011
For all integers n
Periods unit is in seconds, and so the frequency must be counted
in Hz.4Periodic signals: examplesSinusoidsComplex exponential
(non-decaying or increasing)
Infinite sum of shifted signals v(t) (more later)Leo Lam
2010-2011x(t)=A cos(w0 t+f)
T0Periodicity of the sum of periodic signalsQuestion:If x1(t) is
periodic with period T1 andx2(t) is periodic with period T2What is
the period of x1(t)+x2(t)?Can we rephrase this using our language
in math?Leo Lam 2010-2011Use 27 and 90 as example3 x 3 x 32 x 3 x 3
x 56Rephrasing in mathLeo Lam 2010-2011
Goal: find T such that
Rephrasing in mathLeo Lam 2010-2011
Goal: find T such that
Need:
T=LCM(T1,T2)Solve it for r=1, true for all rPeriodic sum
exampleIf x1(t) has T1=2 and x2(t) has T2=3, what is the period of
their sum, z(t)?LCM (2,3) is 6And you can see it, too.Leo Lam
2010-2011-1 0 1 2 3 4 5 6 7 8 9 10 11 121211T1T2Your turn!Find the
period of:
Find the period of:Leo Lam 2010-2011
No LCM exists! Why?No LCM because the period of the second term
is NOT rational. It is NOT periodic.10A few moreLeo Lam
2010-2011
Not rational, so not periodic
Decaying term means pattern does not repeat exactly, so not
periodicSummaryDescription of common signalsPeriodicityLeo Lam
2010-2011Playing with signalsOperations with signalsadd, subtract,
multiply, divide signals pointwisetime delay, scaling,
reversalProperties of signals (cont.)even and oddLeo Lam
2010-2011Adding signalsLeo Lam 2010-201111123tt+=
??x(t)y(t)1t123x(t)+y(t)Delay signalsLeo Lam 2010-2011unit pulse
signal(memorize)t011What does y(t)=p(t-3) look like?
P(t)034Multiply signalsLeo Lam 2010-2011
Scaling timeLeo Lam 2010-2011Speed-up: y(t)=x(2t) is x(t) sped
up by a factor of 2
t011t01.5y(t)=x(2t)How could you slow x(t) down by a factor of
2?y(t)=x(t)
Scaling timeLeo Lam 2010-2011 y(t)=x(t/2) is x(t) slowed down by
a factor of 2t01-1t01-1y(t)=x(t/2)2-2y(t)=x(t)
Playing with signalsLeo Lam 2010-2011What is y(t) in terms of
the unit pulse p(t)?t835t011Need:Wider (x-axis) factor of 2Taller
(y-axis) factor of 8Delayed (x-axis) 3 secondsNot to scale! But the
need should be clear, the pulse needs to be wider (factor of 2) and
taller (factor of 8), and then shifted/delayed to the right by 3
seconds.19Playing with signalsLeo Lam 2010-2011t835
in terms of unit pulse p(t)t82first step:
35t8
second step: 20Playing with signalsLeo Lam 2010-2011t835
in terms of unit pulse p(t)t82first step:
35t8
second step: replace t by t-3:
Is it correct?21
Playing with signalsLeo Lam 2010-201135t8
Double-check:
pulse starts:
pulse ends:
22Do it in reverseLeo Lam 2010-2011tSketch
123Do it in reverseLeo Lam 2010-2011t
Let
then
that is, y(t) is a delayed pulse p(t-3) sped up by 3.11 4/313
4
Double-check
pulse starts:3t-3 = 0pulse ends:3t-3=124Order mattersLeo Lam
2010-2011With time operations, order mattersy(t)=x(at+b) can be
found by:
Shift by b then scale by a (delay signal by b, then speed it up
by a)w(t)=x(t+b) y(t)=w(at)=x(at+b)
Scale by a then shift by b/aw(t)=x(at)
y(t)=w(t+b/a)=x(a(t+b/a))=x(at+b)25Playing with timeLeo Lam
2010-2011t1
What does
look like?2
1-2Time reverse of speech:
Also a form of time scaling, only with a negative
number26Playing with timeLeo Lam 2010-2011t1
2Describe z(t) in terms of w(t)1-21
3t27
