Chapter 2
Updated 6/30/2016
Time reversal:
X(t) Time
Reversal
Y=X(-t)
Time scaling
Example: Given x(t), find y(t) = x(2t). This
SPEEDS UP x(t) (the graph is shrinking)
The period decreases!
What happens to the period T?
X(t)
Time
Scaling
The period of x(t) is 2 and the period of y(t) is 1,
a>1  Speeds up  Smaller period  Graph shrinks!
a<1  slows down  Larger period  Graph expands
Y=X(at)
Example of Time Scaling (tt/a)
http://tintoretto.ucsd.edu/jorge/teaching/mae143a/lectures/1analogsignals.pdf
Time scaling
• Given y(t),
– find w(t) = y(3t)
– v(t) = y(t/3).
Time Shifting
• The original signal g(t) is
shifted by an amount t0 .
Time Shift: y(t)=g(t-to)
• g(t)g(t-to) // to>0 Signal
Delayed Shift to the right
X(t)
Time
Shifting
Y=X(t-to)
Delay
Time Shifting
X(t)
• The original signal x(t) is
shifted by an amount t0 .
Time Shift: y(t)=x(t-to)
• X(t)X(t-to) // to>0
Signal Delayed Shift
to the right
• X(t)X(t+to) // to<0
Signal Advanced Shift
to the left
Time
Shifting
Y=X(t-to)
Time Shifting
Example
• Given x(t) = u(t+2) -u(t-2),
– find
• x(t-t0)=
• x(t+t0)=
Answer:
• x(t-t0)= u(t-to+2) -u(t-to-2),
• x(t+t0)= u(t+to+2) -u(t+to-2),
But how can we draw this function?
Note: Unit Step Function
a discontinuous continuous-time signal
Draw
• x(t) = u(t+1)- u(t-2)
u(t+1)- u(t-2)
t=0
t
Time Shifting
t=0
•
Determine x(t) + x(2-t) , where x(t) = u(t+1)- u(t-2)
– Which is x(t):
u(t+1)- u(t-2)
– find x(2-t): Reverse and then advance in time
–  First find y(t)=x[-(t-2)];
First we reverse x(t)
Then, we delay it by 2 unites
as shown below:
Add the two functions: x(t) + x(2-t)
Summary
Shifting to the right; increasing in time  Delaying the signal!
Delayed/
Moved right
shift to the left of t=0 by
two units!
Advanced/
Moved left
Reversed &
Delayed
Or rewrite as: X[-(t+1)]
Hence, reverse the signal in time.
Then shift to the left of t=0
by one unit!
See
Notes
This is really:
X(-(t+1))
Or rewrite as: X[-(t-2)]
Hence, reverse the signal in time.
Then shift to the right of t=0
by two units!
Amplitude Operations
In general:
y(t)=Ax(t)+B
Reversal
B>0  Shift up
B<0  Shift down
|A|>1 Gain
|A|<1 Attenuation
Scaling
A>0NO reversal
A<0 reversal
Scaling
Amplitude Operations
Given x2(t), find 1 - x2(t).
Remember:
This is y(t) =1
Ans.
Multiplication of two signals:x2(t)u(t)
Step unit function
Ans.
Signals can be added or multiplied
Amplitude Operations
Note: You can also think of it as X2(t)
being amplitude revered and then
shifted by 1.
Given x2(t), find 1 - x2(t).
Remember: This is y(t) =1
Multiplication of two signals:x2(t)u(t)
Step unit function
Signals can be added or multiplied  e.g., we can filter parts of a signal!
Signal Characteristics
•
Even and odd signals
1. X(t) = Xe(t) + Xo(t)
2. X(-t) = X(t)  Even
3. X(-t) = -X(t)  Odd
•
Represent Xo(t) in terms of X(t)
only!
Represent Xe(t) in terms of X(t)
only!
Properties
Xe + Ye = Ze
Xo + Yo = Zo
Xe + Yo = Ze + Zo
Xe * Ye = Ze
Xo * Yo = Ze
Xe * Yo = Zo
Signal Characteristics
•
Even and odd signals
1. X(t) = Xe(t) + Xo(t)
2. X(-t) = X(t)  Even
3. X(-t) = -X(t)  Odd
•
Properties
Xe + Ye = Ze
Xo + Yo = Zo
Xe + Yo = Ze + Zo
Know
These!
Xe * Ye = Ze
Xo * Yo = Ze
Xe * Yo = Zo
Proof Examples
• Prove that product of two even
signals is even.
Change t -t
x(t )  x1 (t )  x2 (t ) 
x(t )  x1 (t )  x2 (t ) 
x1 (t )  x2 (t )  x(t )
• Prove that product of two odd
signals is odd.
• What is the product of an even
signal and an odd signal?
Prove it!
x(t )  x1 (t )  x2 (t ) 
x(t )  x1 (t )  x2 (t ) 
x1 (t )   x2 (t )   x(t ) 
x(t )  Even
Signal Characteristics:
Find uo(t) and ue(t)
Remember:
Given:
Signal Characteristics
Symmetric across the vertical axis
Anti-symmetric
across the vertical axis
Example
• Given x(t) find xe(t) and xo(t)
4___
5
2___
2___
5
5
Example
• Given x(t) find xe(t) and xo(t)
4___
5
2___
2___
-5
5
-5
5
Example
• Given x(t) find xe(t) and xo(t)
4___
4e-0.5t
2___
2___
2___
2e-0.5t
5
-2___
2e+0.5t
2___
5 2e-0.5t
Example
• Given x(t) find xe(t) and xo(t)
4___
4e-0.5t
2___
2___
2___
2e-0.5t
5
-2e+0.5t
-2___
2e+0.5t
2___
5 2e-0.5t
Periodic and Aperiodic Signals
• Given x(t) is a continuous-time signal
• X (t) is periodic iff X(t) = x(t+nT) for any T
and any integer n
• Example
– Is X(t) = A cos(wt) periodic?
• X(t+nT) = A cos(w(t+Tn)) =
• A cos(wt+w2np)= A cos(wt)
– Note: f0=1/T0; wo2p/To <= Angular freq.
– T0 is fundamental period; T0 is the minimum
value of T that satisfies X(t) = x(t+T)
Periodic signals
Examples:
• =>Show that sin(t) is in fact a periodic
signal.
• Use a graph
• Show it mathematically
– What is the period?
– Is this an even or odd signal?
• =>Is tesin(t) periodic?
– X(t) = x(t+T)?
Sum of periodic Signals
•
•
•
•
X(t) = x1(t) + X2(t)
X(t+nT) = x1(t+m1T1) + X2(t+m2T2)
m1T1=m2T2 = To = Fundamental period
Example:
– cos(tp/3)+sin(tp/4)
– T1=(2p)/(p/3)=6; T2 =(2p)/(p/4)=8;
– T1/T2=6/8 = ¾ = (rational number) = m2/m1
– m1T1=m2T2  Find m1 and m2
– 6.4 = 3.8 = 24 = To (n=1)
Sum of periodic Signals
•
•
•
•
Note that T1/T2 must be
RATIONAL (ratio of integers)
X(t) = x1(t) + X2(t)
X(t+nT) = x1(t+m1T1) + X2(t+m
An irrational2T
number
2) is any real
number which cannot be
m1T1=m2T2 = To=Fundamental
expressed asperiod
a fraction a/b,
where a and b are integers, with
b non-zero.
Example:
– cos(tp/3)+sin(tp/4)
– T1=(2p)/(p/3)=6; T2 =(2p)/(p/4)=8;
– T1/T2=6/8 = ¾
– m1T1=m2T2  T1/T2= m2/m16/8=3/46.4 =
3.8  24 = To
Read about Irrational Numbers: http://www.mathsisfun.com/irrational-numbers.html
Product of periodic Signals
X(t) = xa(t) * Xb(t) =
= 2sin[t(7p/24)]* cos[t(p/24)];
find the period of x(t)
• We know: = 2sin[t(7p/12)/2]* cos[t(p/12)/2];
– Using Trig. Itentity:
• x(t) = sin(tp/3)+sin(tp/4)
• Thus, To=24 , as before!
http://www.sosmath.com/trig/Trig5/trig5/pdf/pdf.html
Sum of periodic Signals – may not
always be periodic!
x(t )  x1 (t )  x2 (t )  cos t  sin 2t
–
–
–
–
T1=(2p)/(1)= 2p;
T2 =(2p)/(sqrt(2));
T1/T2= sqrt(2);
Note: T1/T2 = sqrt(2)
is an irrational number
– X(t) is aperiodic
Note that T1/T2 is NOT
RATIONAL (ratio of integers) In
this case!
Sum of periodic Several Signals
• Example
– X1(t) = cos(3.5t)
– X2(t) = sin(2t)
– X3(t) = 2cos(t7/6)
– Is v(t) = x1(t) + x2(t) + x3(t) periodic?
– What is the fundamental period of v(t)?
– Find even and odd parts of v(t).
Do it on your own!
Sum of periodic Signals (cont.)
–
–
–
–
–
–
X1(t) = cos(3.5t)  f1 = 3.5/2p  T1 = 2p /3.5
X2(t) = sin(2t)  f2 = 2/2p  T2 = 2p /2
X3(t) = 2cos(t7/6)  f3 = (7/6)/2p  T3 = 2p /(7/6)
 T1/T2 = 4/7 Ratio or two integers
 T1/T3 = 1/3 Ratio or two integers
 Summation is periodic
T1
2p/3.5
T2
2p/2
T3
12p/7
m1
7.3=21
m2
2.6=12
m3
m1xT1
12p
m2xT2
12p
m3xT3
12p
– m1T1 = m2T2 = m3T3 = To ; Hence we find To
– The question is how to choose m1, m2, m3 such that the above
relationship holds
– We know: 7(T1) = 4(T2) & 3(T1) = 1(T3) ;  m1(T1)=m2(T2)
– Hence:
21(T1) = 12(T2)= 7(T3);
Thus, fundamental period: To = 21(T1) = 21(2p /3.5)=12(T2)=12p
Find even and odd parts of v(t).
7.1=7
Important Engineering Signals
• Euler’s Formula (polar form or complex exponential form)
– Remember: ejF = 1|_F and arg [ ejF ] = F (can you prove these?)
notes
• Unit Step Function (Singularity Function)
notes
What is sin(A+B)? Or cos(A+B)?
• Use Unit Step Function to express a block function (window)
-T/t
T/t
notes
Important Engineering Signals
• Euler’s Formula
notes
• Unit Step Function (Singularity Function)
– Can you draw x(t) = cos(t)[u(t) – u(t-2p)]?
notes
• Use Unit Step Function to express a block function (window)
Next
-T/2
T/2
Unit Step Function Properties
u(at  t0 )  u (t  t0 / a), a  0
u (t  t0 )  [u (t  t0 )]2  [u (t  t0 )]k
u(-t) =1- u(t) ® u(-(t - t0 )) =1- u(t - t0 )
Examples:
Note:
U(-t+3)=1-u(t-3)
Unit Step Function Applications:
Creating Block Function (window)
1
•
•
rect(t/T)
Note:
Period is T; & symmetric
-T/2
T/2
1
Can be expressed as u(T/2-t)-u(-T/2-t)
– Draw u(t+T/2) first; then reverse it!
-T/2
T/2
1
•
Can be expressed as u(t+T/2)-u(t-T/2)
•
Can be expressed as u(t+T/2).u(T/2-t)
-T/2
T/2
-T/2
T/2
Unit Step Function Applications:
Creating Unit Ramp Function
Unit ramp function can be achieved by:
notes
t
t
0
t0
1
f (t )   u (  t0 )d   d (t  t0 )u (t  t0 )
to
Non-zero only
for t>t0
Example: Using Mathematica:
(t-2)*unitstep[t-2] – click here:
http://www.wolframalpha.com/input/?i=%28t-2%29*unitstep%5Bt-2%5D
to+1
Unit Step Function Applications:
Example
f (t )  3u (t )  tu(t )  [t  1]u (t  1)  5u (t  2)
Unit Step Function Applications:
Example
• Plot
f (t )  3[t  2]u (t  2)  6[t  1]u (t  1)  3[t  1]u (t  1)  3u (t  3)
•
•
•
•
•
t<-2
-2<t<-1
-1<t<1
1<t<3
3<t<
 f(t)=0
 f(t)=3[t+2]
 f(t)=-3t
 f(t)=-3
 f(t)=0
Using Mathematica - click here:
http://www.wolframalpha.com/input/?i=3*%28t%2B2%29*UnitStep%28t%2B2%296*%28t%2B1%29*UnitStep%28t%2B1%29%2B3*%28t-1%29*UnitStep%28t-1%29%2B3*UnitStep%28t3%29
Unit Impulse Function d(t)
•
•
Not real (does not exist in nature – similar to i=sqrt(-1)
Also known as Dirac delta function
–
•
•
Generalized function or testing function
The Dirac delta can be loosely thought of as a
function of the real line which is zero everywhere
except at the origin, where it is infinite
Note that impulse function is not a true function – it is
not defined for all values
–
= f(0)
It is a generalized function
Mathematical definition
Mathematical definition
d(t)
0
d(t-to)
0
to
Unit Impulse Function d(t)
• Also note that
• Also
Unit Impulse Properties
1. Scaling
– Kd(t) Area (or weight) under = K
2. Multiplication
– X(t) d(t) X(0) d(t) = Area (or weight) under
3. Time Shift
– X(t) d(t-to) X(to) d(t-to)
• Example: Draw 3x(t-1) d(t-3/2) where
x(t)=sin(t)
– Using x(t) d(t-to) X(to) d(t-to) ;
– 3x(3/2-1) d(t-3/2)=3sin(1/2) d(t-3/2)
Unit Impulse Properties
• Integration of a test function
– Example
• Other properties:
Make sure you can understand why!
Unit Impulse Properties
• Example: Verify
Schaum’s p38
• Evaluate the following
1
 (3t
2
 1)d (t )  ?
1
Schaum’s p40
Remember:
Continuous-Time Systems
• A system is an operation for which causeand-effect relationship exists
– Can be described by block diagrams
– Denoted using transformation T[.]
• System behavior described by
mathematical model
X(t)
T [.]
y(t)
System - Example
• Consider an RL series circuit
– Using a first order equation:
VL (t )  L
R
di (t )
dt
di (t )
V (t )  VR  VL (t )  i (t )  R  L
dt
V(t)
L
Interconnected Systems
• Parallel
• Serial (cascaded)
• Feedback
notes
R
V(t)
R
L
L
Interconnected System Example
• Consider the following systems with 4 subsystem
• Each subsystem transforms it input signal
• The result will be:
– y3(t)=y1(t)+y2(t)=T1[x(t)]+T2[x(t)]
– y4(t)=T3[y3(t)]= T3(T1[x(t)]+T2[x(t)])
– y(t)= y4(t)* y5(t)= T3(T1[x(t)]+T2[x(t)])* T4[x(t)]
Feedback System
•
•
Used in automatic control
Example: The following system has 3 subsystems. Express the equation
denoting interconnection for this system - (mathematical model will depend
on each individual subsystem)
–
–
–
–
e(t)=x(t)-y3(t)= x(t)-T3[y(t)]=
y(t)= T2[m(t)]=T2(T1[e(t)])
 y(t)=T2(T1[x(t)-y3(t)])= T2(T1( [x(t)] - T3[y(t)] ) ) =
=T2(T1([x(t)] –T3[y(t)]))
Find this
first
Then,
calculate
this
System Properties
Continuous-Time Systems - Properties
• Systems with Memory:
• Examples – Memoryless/Memory?
Has memory if output depends on inputs other than the one defined at current time
Continuous-Time Systems - Properties
Inverse of a System (invertibility)
Examples
Noninvertible Systems
Thermostat
Example!
(notes)
Each distinct input  distinct output
Continuous-Time Systems - Invertible
• If a system is invertible it has an Inverse System
x(t)
System
y(t)
Inverse
System
x(t)
• Example: y(t)=2x(t)
– System is invertible: For any x(t) we get a distinct output y(t)
– Thus, the system must have an Inverse
• x(t)=1/2 y(t)=z(t)
x(t)
System
(multiplier)
y(t)=2x(t)
Inverse
System
(divider)
x(t)
If the system is not invertible it does not have an INVERSE!
Continuous-Time Systems - Causality
Causality (non-anticipatory system)
- A System can be causal with non-causal components!
Examples
Remember: Reverse is not TRUE!
??
?? What it t<0?
Depends on cause-and-effect  no future dependency
Continuous-Time Systems - Stability
• Many different definitions
• Bounded-input-bounded-output
• Example:
– An ideal amplifier y(t) = 10 x(t)  B2=10 B1
– Square system: y(t)=x^2  B2=B1^2
• A system can be unstable or marginally stable
• More examples
Continuous-Time Systems – Time
Invariance
Testing for Time Invariance
To test: y(t)|t-to  y(t)|x(t-to)
Example:
y(t) = e^x(t):
y(t)|t=t-to  e^x(t-to)
y(t)|x(t-to)  e^x(t-to)
 Time Invariance
What if the system is time reversal? (next slide)
Time-shift in input results in time-shift in output  system always acts the same way (Fix System)
Continuous-Time Systems –
Time Invariance
Example of a system:
U(1-t)
Draw the First output!
Draw the Second output!
Are they the same?
Continuous-Time Systems –
Time Invariance
Example of a system:
U(1-t)
Pay attention!
Due to time - reversal
Time reversal operation is NOT time invariant!
Continuous-Time Systems – Time
Invariance
• Examples:
– Do these yourself!
notes
Continuous-Time Systems – Linearity
• A linear system must satisfy superposition
condition (additive and homogeneity hoh-muh-juh-nee-i-tee)
We need to prove: aT[x1]+bT[x2]=T[a.x1+b.x2]
• Note: for a linear system a zero input always
generates an output zero
• Example
notes
More…
• Example
• Summary
notes
Practice Problems
• Schaum’s Outlines:
– 1.1(p19), 1.5(p23). 1.6-1.7(p24)1.16(p30),
1.22(p35)
– There will be more….(check back)
• Textbook: Chapter 2- Solved Problems
– 1, 3, 5-10, 13, 14, 16, 17, 21, 22, 24, 25-30,
33-35, 48, 51, 53-61