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EE 312 Signals and Systems
Handout #1
Signals: Concepts & Properties
Prof. Mohamed Zribi
Updated 7 September 2015
EE312 Signals and Systems Dr. Mohamed Zribi 1
Outline
I. Definition of Signals
II. Classifications of Signals
III. Important CT Signals
IV. Important DT Signals
V. Operations on signals
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I. Definition of Signals
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o A signal is a function of time representing a physical
variable, e.g. voltage, current, spring displacement, share
market prices, number of student asleep in the Lab, cash in
the bank account.
o Typically we will use a mathematical function such as f(t),
u(t) or y(t) to describe a signal which is a continuous
function of time.
EE312 Signals and Systems Dr. Mohamed Zribi
What is a Signal?
4
Remarks about Signals
o Usually, a signal is a function of an independent variableExample 1: Daily high temperature measured over a month
o Continuous-time signals are functions of a real argumentx(t) where t can take any real valuex(t) may be 0 for a given range of values of t
o Discrete-time signals are functions of an argument that takes values from a discrete setx[n] where n {...-3,-2,-1,0,1,2,3...}Often, we use “index” instead of “time” for discrete-time
signals
o Values for x may be real or complex
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II. Classification of Signals
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Classification of Signals
1. One-dimensional and Multi-dimensional Signals2. Continuous-time and discrete-time Signals3. Analog and digital Signals4. Deterministic and Random Signals5. Periodic and Aperiodic Signals6. Causal and Anti-causal vs. Non-causal Signals7. Even and Odd Signals8. Finite and Infinite Length Signals9. Energy of a Signal 10. Power of a Signal11. Energy signals and power signals
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1.One-dimensional vs. Multi-dimensional
A signal can be a function of a single variable or a function of multiple variables.
Example 2:
Speech varies as a function of time one-dimensional
Image intensity varies as a function of (x , y) coordinates
multi-dimensional
In this course, we will focus on one-dimensional
signals.
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A signal is continuous time if it is defined for all
time, x(t).
A signal is discrete time if it is defined only at
discrete instants of time, x[n].
A discrete time signal can be derived from a
continuous time signal through sampling, i.e.:
s[ ] ( ), T is the sampling periodd c sx n x nT
2. Continuous-time vs. Discrete-time
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Continuous-Time (CT) Signals
Most signals in the real world are continuous time.
E.g. voltage, velocity,
A CT signal is denoted by x (t), where the time interval
may be bounded (finite) or infinite
EE312 Signals and Systems Dr. Mohamed Zribi
x(t)
t
10
Discrete-Time (DT) Signals
Some real world and many digital signals are discrete time, as
they are sampled. For example, pixels, daily stock price
(anything that a digital computer processes)
A DT signal is denoted by x[n], where n is an integer value that
varies discretely.
EE312 Signals and Systems Dr. Mohamed Zribi
x[n]
n
11
Continuous-Time vs. Discrete-Time
Graphically,
It is meaningless to say 1.5 sample of a DT signal because it is
not defined.
t
)(tx
0 4-3 1 3
][nx
0
]2[x ]1[x]2[x ...
2 ...-1-2...
]0[x
]3[x
]1[x
...
n
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Continuous time signal
xa(t)
Discrete time signal (sequence)
x[n]
x[n] = xa(nT ) T : sampling periodfs = 1/T : sampling rate
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Examples of Signals
Graphical Representation of a DT is shown below:
Example 3:
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Example 4:Graphical Representation of a DT is shown below:
15
A discrete-time signal is usually (not always) obtained from sampling acontinuous signal at a regular time period known as the sampling period,which we will represent by the parameter T.
Thus the discrete-time signal is equal in amplitude to the continuoussignal at the sampling instants.
A discrete-time signal is essentially a sequence of numbers (0,1,2, ...)where each of those numbers represents the amplitude of a continuous-time signal at a time equal to kT.
Note that often we neglect to put in the sampling period and write thesignal simple as a function of the sample number k.
EE312 Signals and Systems Dr. Mohamed Zribi
Discrete-time Signals
kTk ff cd
16
Sampling of an analog signal is shown below:
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Discrete-time Signals
17
Converting between a continuous and discrete signal.
Consider the continuous-time signal such as:
The sampling interval T = 0.1 seconds. Start sampling at t = 0
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Example 5:
0
1*
2*
3*
0.1**
[0] 1.0000
[1] 0.9048
[2] 0.8187
[3] 0.7408
[ ]
dT
dT
dT
d
kk Td
x ex ex ex e
x k e e
tcx t e
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We can also write:
Either method gives
i.e. a sequence of number, known as the discrete-time signal.
Note that the values of this sequence depend on the sampling period.
If the sampling interval T were doubled then :
Thus a single continuous function can yield an infinite number of sampled sequences, depending on the value chosen for T.
EE312 Signals and Systems Dr. Mohamed Zribi
* 0.9048k Td
k kTx k e e
1.0000, 0.9048, 0.8187, 0.7408,dx k
1.0000, 0.8187,dx k
19
Continuous-Time vs. Discrete Time
To distinguish CT and DT signals, t is used to denote CT and the
independent variable in (.). Also, n is used to denote DT
independent variable in [.]
Continuous x(t), t is real
Discrete x[n], n is integer Signals can be represented in mathematical form. For example, x(t) = et, x[n] = n/2
Discrete signals can also be represented as sequences: {y[n]} = {…,1,0,1,0,1,0,1,0,1,0,…}
550
)( 2 ,t,t
tty
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3. Analog vs. Digital Signals
The difference between analog and
digital signals is with respect to the
value of the function (y-axis).
An analog signal corresponds to a
continuous y-axis, while a digital
signal corresponds to a discrete y-
axis.
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A signal whose amplitude can take on any value in a
continuous range is an analog signal.
A digital signal is one whose amplitude can take on only a
finite number of values. For example, Binary signals are
digital signals.
An analog signal can be converted into a digital signal
through quantization.
Analog vs. Digital Signals
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1
-1
Analog vs. Digital Signals
The amplitude of an analog signal can take any real or
complex value at each time/sample
Analog
The amplitude of a digital signal takes values from a discrete
set
Digital
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Digital vs. Analog Signals
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Digital vs. Analog Signals
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Digital vs. Analog Signals
Examples of analog technology photocopiers telephones audio tapes televisions (intensity and color info per scan line) VCRs (same as TV)
Examples of digital technology Digital computers!
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Analog and Digital Signals
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Deterministic signal is a signal in which each value of the
signal is fixed and can be determined by a mathematical
expression, rule, or table. Because of this the future values of the
signal can be calculated from past values with complete
confidence.
Random signal has a lot of uncertainty about its behavior. The
future values of a random signal cannot be accurately predicted
and can usually only be guessed based on the averages of sets of
signals
4. Deterministic vs. Random Signals
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A deterministic signal is known for all time and can bepredicted in advance exactly, e.g. a sine-wave with knownphase.
A random signal cannot be predicted exactly, e.g. weathertemperatures.
Random signals are usually dealt with by statistical rather thananalytical techniques.
Noise is simply a signal we don't want. Sometime it iscompletely random, but on occasions it can be someone else'ssignal, such as cross-talk on a telephone line. So, since we don'twant it we call it noise.
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In other words,
29
Example 6: Deterministic vs. Random Signals
Deterministic
Random
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Deterministic signal (An example is sin wave, square wave)
Stochastic signal (An example is noise signal or human voice)
Example 7: Deterministic vs. Random Signals
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Examples of Signals
DeterministicSignal
.
.
.
.RandomSignal
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Signal with noise
Deterministic Signal
Signal with noise
Examples of Signals
A periodic signal x(t) is a function of time that satisfies
The smallest T, that satisfies this relationship is called the
fundamental period. is called the frequency of the signal (Hz).
Angular frequency, (radians/sec).
A signal is either periodic or aperiodic.
A periodic signal must continue forever.
A non-periodic signal is called aperiodic.
Tf 1
Tf 22
5. Periodic and Aperiodic Signals
0, )()( TtTtxtx
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Example 8: The voltage at an AC power source is periodic.
Remark:0 0
0
( ) ( ) ( )a T b T
a b T
v t dt v t dt v t dt
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Example 9: Periodic and Aperiodic signals
Periodic
Aperiodic
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Periodic
Periodic
Example 10: Periodic and Aperiodic signals
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Example 11: An example of a CT periodic signal is:
If x(t) is periodic with T then
Thus, x(t) is also periodic with 2T, 3T, 4T, ...
The fundamental period T0 of x(t) is the smallest value of T for which holds.
)( tx
0 TTT2 T2
ZnnTtxtx for )()(
EE312 Signals and Systems Dr. Mohamed Zribi
Periodic
38
cos (t +2) = cos (t)
sin (t +2) = sin (t)
These two signals are both periodic with period 2
2
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Periodic
Example 12: Periodic signals
39
5/2,....2,1,1,2...,25
)(5sin)(5cos)()(
5sin5cos)( 5
TkkT
TtTtjTtxtx
ttjjetx tj
This is periodic if there exists T > 0 such that:
This is true for sinusoidal signals:
Signal is periodic and fundamental period is
T = 5/2
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Example 13: Periodic signals
40
Sum of Periodic Signals
x(t) = x1(t) + x2(t)
x(t+T) = x1(t+n1T1) + x2(t+n2T2)
n1T1=n2T2 = To =Fundamental period
Example 14:
cos(t/3)+sin(t/4)
T1=(2)/(/3)=6; T2 =(2)/(/4)=8;
T1/T2=6/8 = ¾
n1T1=n2T2 = 6*4 = 3*8 = 24 = ToEE312 Signals and Systems Dr. Mohamed Zribi 41
Example 15: Sum of periodic Signals
x1(t) = cos(3.5t)
x2(t) = sin(2t)
x3(t) = 2cos(7t/6)
Is v(t) = x1 (t) + x2(t) + x3(t) periodic?
What is the fundamental period of v(t)?
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x1(t) = cos(3.5t) f1 = 3.5/2 T1 = 2 /3.5 x2(t) = sin(2t) f2 = 2/2 T2 = 2 /2 x3(t) = 2cos(t7/6) f3 = (7/6)/2 T3 = 2 /(7/6) T1/T2 = 4/7 Ratio or two integers T1/T3 = 1/3 Ratio or two integers Summation is periodic
n1T1 = n2T2 = n3T3 = 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) and 3(T1) = 1(T3) ; Hence:
21(T1) = 12(T2)= 7(T3); Thus, fundamental period: To = 21(T1) = 21(2 /3.5)=12(T2)=12
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Example 16: Sum of periodic Signals – may not always be periodic!
T1=2/= 2; T2 =2/ T1/T2= Note: T1/T2 = is an irrational number
x(t) is aperiodic
tttxtxtx 2sincos)()()( 21
2
2
2
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Example 17: Sum of periodic Signals – may not always be periodic!
T1=2/= 2; T2 = 1 There is no common factor between T1 and T2
x(t) is aperiodic
1 2( ) ( ) ( ) cos sin 2x t x t x t t t
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Product of Periodic Signals
x(t) = xa(t) * xb(t)
It can be checked that
x(t) = sin(t/3)+sin(t/4)
Thus, To=24.
EE312 Signals and Systems Dr. Mohamed Zribi
7If ( ) 2sin( ) cos( )24 24
x t t t
since 2sin( ) cos( ) sin( ) sin( )a b a b a b
46
Discrete Time Periodic Signals
A discrete time signal x[n] is periodic with period N if and only if
][][ Nnxnx for all n .
Definition:
N
][nx
n
Meaning: a periodic signal keeps repeating itself forever!
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Periodic Signals for DT Signals
For DT we must have
Here the smallest N can be 1
The smallest positive value of N is N0 which is called the fundamental period
0, ][][ NnnxNnx
Period must beinteger!
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A signal is periodic if repeats after T values:
x [n] = x [n+ N0] = x [n+2 N0] = x [n+3 N0] = …
N0 is the period of the signal
Periodic Signals for DT Signals
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[ ] 2cos 0.2 0.9x n n
Consider the Sinusoid:
It is periodic with period since 10N
][29.02.0cos2
9.0)10(2.0cos2]10[nxn
nnx
for all n.
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Example 18:
General Periodic Sinusoid
n
NkAnx 2cos][
Consider a Sinusoid of the form:
It is periodic with period N since
][22cos
)(2cos][
nxknNkA
NnNkANnx
for all n.
with k, N integers.
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1.03.0cos5][ nnxConsider the sinusoid:
It is periodic with period since 20N
][231.03.0cos5
1.0)20(3.0cos5]20[nxn
nnx
for all n.
We can write it as:
1.0
2032cos5][ nnx
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Example 19:
nN
kjAenx
2
][
Consider a Complex Exponential of the form:
for all n.
It is periodic with period N since
Periodic Complex Exponentials
][
][
22
)(2
nxeAe
AeNnx
jkn
Nkj
NnN
kj
1
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njejnx 1.0)21(][
Consider the Complex Exponential:
We can write it as
njejnx
2012
)21(][
and it is periodic with period N = 20.
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Example 20:
Example 21: Periodic Signals
727/2
,....2,1,1,2...,27
)(7sin)(7cos][][
)7sin()7cos(][ 7
kNkN
kkN
NnjNnNnxnx
njnenx nj
This is periodic if there exists N>0 such that :
This is true for sinusoidal signals:
Signal is periodic and fundamental period is N=2
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Periodicity Properties of DT Signals
(*) :conditiony Periodicit 000
itymust be un
NjωnjωN)(njω eee
!!otherwise! periodicnot number, rational a is 2
when periodic is exp DT So
integers. bemust and that (**) and (*) from conditions thehave We
0
Nm
Nm
*)*(* 2
ly equivalentOr
2 havemust weminteger somefor s,other wordIn (**) .2 of multipleinteger an is if holds This
0
0
0
Nm
mNN
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Periodicity Properties of DT Signals
mN02 then isfrequency lfundamenta The
outfactor common theTake
**)*(* 2 then is period lfundamenta The0
mN
!signals.)! sinusoidal DTfor validalso ist developmen same (The
*)*(*in as 2
express toneed weexp.complex a of freq. fund. thefind toTherefore 0
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Example 22: Periodic Signals
2 nIf x[n] cos( )12
4 nIf x[n] cos( ) 12
00 0
2 12 n x[n] cos( ) cos( n) no factors in common,12 12 2 1212so by using (****) , N 1 12 periodic with fund period 12.1
00 0
4 2 14 n x[n] cos( ) cos( n) ,12 12 2 12 612then using (****) , N 1 6 is periodic with fundamental period 6.2
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Comparison of Periodicity of CT and DT Signals
Consider x(t) and x[n]
x(t) is periodic with T=12,
x[n] is periodic with N=12.
2πt 2πn (t) cos ( ) [n] cos ( ) 12 12
x x
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if and
x(t) is periodic with 31/4.
In DT, we can’t have fractional periods. Thus for x[n] we have
then N=31.
If and
x(t) is periodic with 12, but x[n] is not periodic, because there is no way to express it as in (***)
31t8cos)(
tx 318cos][ nnx
314
20
)61cos()( ttx )
61cos(][ nnx
121
20
Comparison of Periodicity of CT and DT Signals
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6. Causal vs. Anti-causal vs. Non-causal
• Causal signals are signals that are zero for all negative time,
• Anticausal are signals that are zero for all positive time.
• Noncausal signals are signals that have nonzero values in both
positive and negative time
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In other words: A signal that does not start before t=0 is a causal signal.
x(t)=0, t < 0
A signal that starts before t=0 is a noncausal signal.
A signal that is zero for t > 0 is called an anticausal signal.
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Example 23: Causal vs. Anti-causal vs. Non-causal
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causal signal
Anti-causal signal
Non-causal signal
0
63
Example 24:
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7. Even and Odd Signals
If even signal (symmetric wrt y-axis)
If odd signal (symmetric wrt origin)
][][or )()( nxnxtxtx
][][or )()( nxnxtxtx
odd
t
x(t)
t
evenx(t)
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Even and Odd Signals
)()( txtx
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An even signal is identical to its time reversed signal, i.e. it can be reflected in the origin and is equal to the original:
Example 25:x(t) = cos(t)
x(t) = c
An odd signal is identical to its negated, time reversed signal, i.e. it is
equal to the negative reflected signal
Example 26:
x(t) = sin(t)
x(t) = t
Even and Odd Signals
)()( txtx
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Example 27: Even and Odd Signals
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even
odd
68
xe(t) = xe(-t) and xo(t) = - xo(-t)
evenodd
Example 28: Even and Odd Signals
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Even and Odd CT FunctionsEven Functions Odd Functions
g t g t g t g t
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g gg
2e
n nn
g g
g2o
n nn
g gn n g gn n
Even and Odd DT Functions
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Combination of Even and Odd Functions
Function type Sum Difference Product Quotient
Both even Even Even Even Even
Both odd Odd Odd Even Even
Even and odd Neither Neither Odd Odd
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Even vs. Odd Signals
• Prove that product of two even signals is an even signal.
• Prove that product of two odd signals is an even signal.
• What is the product of an even signal and an odd signal?
)()()()()()(
)()()(
21
21
21
txtxtxtxtxtx
txtxtx
Eventxtxtxtx
txtxtxtxtxtx
)()()()(
)()()()()()(
21
21
21
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Two Even Functions
Example 29: Products of Even and Odd Functions
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An Even Function and an Odd Function
Example 30: Products of Even and Odd Functions
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An Even Function and an Odd Function
Example 31: Products of Even and Odd Functions
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Two Odd Functions
Example 32: Products of Even and Odd Functions
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Two Even Functions
Example 33: Products of Even and Odd Functions
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An Even Function and an Odd Function
Example 34: Products of Even and Odd Functions
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Two Odd Functions
Example 35: Products of Even and Odd Functions
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Function type and the types of
derivatives and integrals
Function type Derivative Integral
Even Odd Odd + constant
Odd Even Even
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Integrals of Even and Odd Functions
0
g 2 ga a
a
t dt t dt
g 0a
a
t dt
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Accumulation of Even and Odd Functions
1
g g 0 2 gN N
n N nn n
g 0N
n Nn
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A signal can be even, odd or neither.
Any signal x(t) can be written as a combination of an even and odd signals.
1( ) ( ) ( )2
( ) ( ) ( )1( ) ( ) ( )2
e
e o
o
x t x t x t
x t x t x t
x t x t x t
EE312 Signals and Systems Dr. Mohamed Zribi
Decomposition of signals to Even and Odd Signals
84
Even and Odd Parts of Functions
1The of a signal is 2ex t x t x t even part
1The of a signal is 2ox t x t x t odd part
A signal whose even part is zero is odd and a signal
whose odd part is zero is even.
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))()((21)( txtxtxo ))()((
21)( txtxtxe
Example 36:
Even part
Odd part
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Given:
Example 37: Even-Odd Signals
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Symmetric across the vertical axis Anti-symmetric across the vertical axis
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Given x(t) find xe(t) and xo(t) x(t)
5
4___
5
2___
5
2___
-5-5
Example 38: Even-Odd Signals
EE312 Signals and Systems Dr. Mohamed Zribi
( )ex t( )ox t
89
Given x(t) find xe(t) and xo(t)4___
5
2___
5
2___
4e-0.5t
2___2e-0.5t
-2___
2___2e-0.5t2e+0.5t-2e+0.5t
Example 39: Even-Odd Signals
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( )ex t
( )ox t
( )x t
90
Example 40: Even-Odd Signals
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The even part of f(t) is as follows:
91 EE312 Signals and Systems Dr. Mohamed Zribi
The odd part of f(t) is as follows:
92
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The addition of the even part and the odd part gives f(t)
93
• x(t) is a finite length signal if it is nonzero over a
finite interval a < t < b
• x(t) is infinite length signal if it is nonzero over all
real numbers.
• Remark: Periodic signals are infinite length.
8. Finite vs. Infinite Length Signals
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In other words,
• f (t) is a finite-length signal if it is nonzero over a finite
interval t1 < f (t) < t2
• Infinite-length signal, f (t), is defined as nonzero over all real numbers:
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9. Energy of Signals
2x xE t dt
The energy of a signal x is:
All physical activity is mediated by a transfer of energy.
No real physical system can respond to an excitation unless it has energy.Signal energy of a signal is defined as the area under the square of the magnitude of the signal.
The units of signal energy depends on the unit of the signal.
If the signal unit is volt (V), the energy of that signal is expressed in V2.s.
2xand, xE n
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9a. Energy of Signals (over finite time intervals)
Total energy of a CT signal x(t) over [t1, t2] is:
where |.| denote the magnitude of the (complex) number/signal.
Total energy of a DT signal x[n] over [n1, n2]:
2
1
2)(t
tdttxE
2
1
2][n
nnnxE
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9b. Energy of Signals (over infinite time intervals)
For many signals, we’re interested in examining the energy over
an infinite time interval (-∞, ∞):
If the sums or integrals do not converge, the signal energy is
infinite.
dttxdttxET
TT22 )()(lim
n
N
NnN nxnxE 22 ][][lim
EE312 Signals and Systems Dr. Mohamed Zribi
For DT signal
For CT signal
98
Example 41: Energy of Signals
2
2 2 2
4 4
00
( ) ( )
| ( ) | ( ( ))
1/ 4
1/ 4
t
t
t t
x t e u t
E x t dt e u t dt
e dt e
EE312 Signals and Systems Dr. Mohamed Zribi 99
1 12 2 3
00
( ) [ ( ) ( 1)]
| ( ) | 1/ 3 1/ 3
x t t u t u t
E x t dt t dt t
EE312 Signals and Systems Dr. Mohamed Zribi
Example 42: Energy of Signals
100
Example 43: Determine the energy of this signal?
EE312 Signals and Systems Dr. Mohamed Zribi 101 EE312 Signals and Systems Dr. Mohamed Zribi
Example 44: Energy of a Signal
102
EE312 Signals and Systems Dr. Mohamed Zribi
Example 45: Energy of a Signal
103
Example 46: Energy of a Signal
2
2
0
[ ] (1/ 2) [ ]
| [ ] |
((1/ 2) [ ])
(1/ 4)
1/ (1 1/ 4)4 / 3
n
n
nn
nn
x n u n
E x n
u n
EE312 Signals and Systems Dr. Mohamed Zribi 104
By dividing the energy by (t2-t1) and (n2-n1+1), respectively, gives
the average power over finite time intervals,
2
112
2
112
2)1(
1
2)(
1
][
)(
n
nnnn
t
ttt
nxP
dttxP
EE312 Signals and Systems Dr. Mohamed Zribi
For DT signal
For CT signal
105
10. Power of SignalsPower of Signals
(over infinite time intervals)
The corresponding power is:
TEdttx
TP
T
T
TT 2
lim)(21lim 2
12lim][
121lim 2
NEnx
NP
N
N
NnN
EE312 Signals and Systems Dr. Mohamed Zribi
For CT signal
For DT signal
106
Example 47: Energy and Power of Signals
2
2 2 2
4 4
00
212
1 12 4
( ) ( )
| ( ) | ( ( ))
1/ 4
1/ 4
lim | ( ) |
lim *0
t
t
t t
T
T T T
T T
x t e u t
E x t dt e u t dt
e dt e
P x t dt
EE312 Signals and Systems Dr. Mohamed Zribi 107
Example 48: Determine the suitable measure for this signal?
The signal (b) does not approach 0 as |t| and it is a periodic wave, therefore use the power equation where g2 is replaced with t2.
.
EE312 Signals and Systems Dr. Mohamed Zribi 108
Example 49: Determine the power
•Periodic signal with• Suitable measure of size is power
0 2 / oT
- First term on the right hand side equals C2/2- Second term is zero –integral appearing in this term is area under a sinusoid.-Area is at most the area of half cycle – positive and negative portion cancels-A sinusoid of amplitude C has a power of C2/2 regardless of angular frequency
EE312 Signals and Systems Dr. Mohamed Zribi 109
g (t) = C1 cos (1t + 1) + C2 cos (2t + 2) 1 ≠ 2
This signal is the sum of two sinusoid signals. Therefore, use the power equation. Therefore, Pg = (C1
2 / 2) + (C22 / 2)
This Can be generalized
Example 50: Determine the power
EE312 Signals and Systems Dr. Mohamed Zribi 110
Example 51: What is the suitable measure for this signal?
g (t) = Dejt
The signal is complex and periodic. Therefore, use the power equation averaged over T0.
|ejt| = 1 so that |Dejt|2 = |D|2 and
EE312 Signals and Systems Dr. Mohamed Zribi 111
Example 52: Power of a Signal
212
212
212
( ) 5 10cos(100 / 3)
lim | ( ) |
lim (5 10cos(100 / 3))
1 1lim 25 100cos (100 / 3) 100[ cos(200 2 / 3)]2 2
25 50 75
T
T T TT
T T T
T
T T T
x t t
P x t dt
t dt
t t dt
EE312 Signals and Systems Dr. Mohamed Zribi 112
Example 53: Power of a Signal
21
)]2cos(21
21[lim
)(coslim
)cos()cos(lim
|)(|lim
)cos()(
21
221
21
221
T
T oTT
T
T oTT
T
T otj
otj
TT
T
TTT
otj
dtt
dtt
dttete
dttxP
tetx
EE312 Signals and Systems Dr. Mohamed Zribi 113
Example 54: Power of a Signal
0*lim
|][|lim
3/4)4/11/(1
)4/1(
])[)2/1((
|][|
][)2/1(][
34
121
212
1
0
2
2
NN
N
NnNN
nn
nn
n
n
nxP
nu
nxE
nunx
EE312 Signals and Systems Dr. Mohamed Zribi 114
Power in a Sine wave:
where P is period of the sine wave.
The period can be taken:
Usually we take the former as it often makes the mathematics easier.
2sin ty t AT
2/2
/2
/2/22 2 2
/2/2
1/ 2 / 2
41 cos2 2 2
2sin
4sin4
T
T
TT
TT
P dtT T
A t A AP dtT T T T
tAT
T ttT
1 2 1 2from / 2 to / 2 or from 0 to t T t T t t T
Example 55: Power of a Signal
EE312 Signals and Systems Dr. Mohamed Zribi 115
Example 56: Exponential and Sinusoidal Signals
Complex periodic exponential and sinusoidal signals are of
infinite total energy but finite average power
As the upper limit of the integrand is increased as
However, always Thus,
0
02
TT
T
tjperiod dteE
periodE
1periodP
T
T
tj
Tdte
TP 1
21lim
20
Finite average power!
1)(
1
0
periodperiod ETTT
P
0
1TT
T
dt 00 )( TTTT
EE312 Signals and Systems Dr. Mohamed Zribi 116
• A signal with finite signal energy (0< E <∞) is
called an energy signal.
• A signal with infinite signal energy and finite
average signal power (0< P< ∞) is called a
power signal.
•An energy signal has zero power.
•A power signal has infinite energy.
11. Energy Signals and Power Signals
EE312 Signals and Systems Dr. Mohamed Zribi 117
Energy Signals and Power Signals
• Energy signals have finite energy. All energy signals
decay to zero as |t| .
• Power signals have finite and non-zero power. All
periodic signals are power signals.
EE312 Signals and Systems Dr. Mohamed Zribi 118
• Periodic signals and random signals are usually
power signals.
• Signals that are both deterministic and aperiodic
are usually energy signals.
• Finite length and finite amplitude signals are
energy signals.
Energy Signals and Power Signals
EE312 Signals and Systems Dr. Mohamed Zribi 119
Examples for signals with finite energy (a) and finite power (b):
Remark: The terms energy and power are not used in their conventional sense as electrical energy or power, but only as a measure for the signal size.
Example 57: Energy signals and power signals
EE312 Signals and Systems Dr. Mohamed Zribi 120
Energy signal iff 0<E<, and so P=0
Example 58:
Power signal iff 0 < P < , and so E=
Example 59:
)()( 2 tuetx t
nnx )1(][
EE312 Signals and Systems Dr. Mohamed Zribi
0 and 25.0 PE
11212lim][
121lim 2
NNnx
NP
N
N
NnN
12lim)1(lim][lim22 NnxE N
N
Nnn
NN
NnN
121
Neither energy nor power, when both E and P are infinite
Example 60:
)()( tuetx t
EE312 Signals and Systems Dr. Mohamed Zribi
2 2 2
00
2 2
0
lim ( ) 0.5
1 1lim ( ) lim2 2
T t tT T
T Tt
T TT
E x t dt e dt e
P x t dt e dtT T
122
• Are all energy signals also power signals? • No. Any signal with finite energy will have zero power.
• Are all power signals also energy signals? • No. Any signal with non-zero power will have infinite
energy.
• Are all signals either energy signals or power signals? • No. Any infinite-duration, increasing-magnitude
function will not be either. (For example, the signal x(t) =tis neither.)
Energy Signals and Power Signals
EE312 Signals and Systems Dr. Mohamed Zribi 123 EE312 Signals and Systems Dr. Mohamed Zribi
III. Important CT Signals
124
Building-block Signals
1. We will represent signal as sums of building-block signals.
2. Important families of building-block signals are the unit
step, unit ramp, unit parabolic, unit impulse, and
complex exponentials functions.
EE312 Signals and Systems Dr. Mohamed Zribi 125
1. The Unit Step Function
1 , 0
u 1/ 2 , 00 , 0
tt t
t
Precise Graph Commonly-Used Graph
EE312 Signals and Systems Dr. Mohamed Zribi 126
A definition of the unit step function is as follows:
2. The Unit Ramp Function
, 0ramp u u
0 , 0
tt tt d t t
t
•The unit ramp function is the integral of the unit step function.•It is called the unit ramp function because for positive t, its slope is one amplitude unit per time.
EE312 Signals and Systems Dr. Mohamed Zribi 127
Ramp FunctionsA shifted ram function with slop B is defined as
Unit ramp function being at t=0 by making B=1 and t0=0 and multiplying by u(t), giving
)()( 0ttBtg
.0,
0,0)()(
ttt
ttutr
Time, t
f(t)
r(t)=tu(t)
EE312 Signals and Systems Dr. Mohamed Zribi 128
The unit parabolic function
t
)(!2
2
tut
02/00
2 tifttif
tp
EE312 Signals and Systems Dr. Mohamed Zribi 129
3. The Unit Parabolic Function 4. The Signum Function
1 , 0
sgn 0 , 0 2u 11 , 0
tt t t
t
Precise Graph Commonly-Used Graph
The signum function, is closely related to the unit-step function.
EE312 Signals and Systems Dr. Mohamed Zribi 130
5. The Rectangular Pulse Function
Rectangular pulse, 1/ , / 2
0 , / 2a
a t at
t a
EE312 Signals and Systems Dr. Mohamed Zribi 131
The Unit Rectangle Function
1 , 1/ 2
rect 1/ 2 , 1/ 2 u 1/ 2 u 1/ 2
0 , 1/ 2
t
t t t t
t
The signal “turned on” at time t = -1/2 and “turned back off” at time t = +1/2.
Precise graph Commonly-used graph
EE312 Signals and Systems Dr. Mohamed Zribi 132
6. The Unit Triangle Function
1 , 1
tri0 , 1
t tt
t
The unit triangle is related to the unit rectangle through an operation called convolution. It is called a unit triangle because its height and area are both one (but its base width is not).
EE312 Signals and Systems Dr. Mohamed Zribi 133
7. The Unit Sinc Function
sinsinc
tt
t
The unit sinc function is related to the unit rectangle function through the Fourier transform.
EE312 Signals and Systems Dr. Mohamed Zribi 134
The CT Impulse Function or the dirac delta function
Relationship
0
1/
(t)
Δ 0
l imt t
dt
tdut
0
dttu
8. The CT Impulse Function
EE312 Signals and Systems Dr. Mohamed Zribi 135
The Dirac delta function ( )t
( ) 0 , 0 .(0 )
( ) 1
t fo r t
t d t
Normalization
Representation:
t
• This “unit impulse” function (sometimes called the Dirac Delta Function) is defined by the conditions:
EE312 Signals and Systems Dr. Mohamed Zribi 136
The Delta Function
The value of delta function can also be defined in the sense
of generalized function:
(t): Test Function
we talk about the values of integrals involving (t).
)0()()(
dttt
EE312 Signals and Systems Dr. Mohamed Zribi 137
The Unit Step and Unit Impulse Function
As approaches zero, g approaches a unit
step andg approaches a unit impulse
a t
t
The unit step is the integral of the unit impulse and
the unit impulse is the generalized derivative of the unit step
Functions that approach unit step and unit impulse
EE312 Signals and Systems Dr. Mohamed Zribi 138
CT Unit Impulse FunctionThe continuous unit impulse signal is defined:
Note that it is discontinuous at t=0The arrow is used to denote area (1), rather than
actual value ()
The continuous unit step signal is defined:
000
)(tt
t
tdtu )()(
0100
)(tt
tu
The step function is discontinuous at time t=0
dttdut )()(
1)(
dtt
EE312 Signals and Systems Dr. Mohamed Zribi 139
Derivative of the Unit Step Function
0 t
u(t)
Derivative
0 t
(t)
EE312 Signals and Systems Dr. Mohamed Zribi 140
( )du tdt
0 0( )
1 0if t
u tif t
Remark: The value of u(t) at t = 0 is not well defined, we generally adopt one by convention.
( ) ( )t
u t d
CT Unit Impulse Function (t)
EE312 Signals and Systems Dr. Mohamed Zribi 141
Example 61: Negative steps and deltas
0 0( )
1 0if t
u tif t
( )d u
tdt
t
-u
-1
t
(-1)Representation:
t
(-1)
or
EE312 Signals and Systems Dr. Mohamed Zribi 142
Graphical Representation of the Impulse Function
The area under an impulse is called its strength or weight. It is represented graphically by a vertical arrow. Its strength is either written beside it or is represented by its length. An impulse with a strength of one is called a unit impulse.
EE312 Signals and Systems Dr. Mohamed Zribi 143
Example 62: integrate ( ) ( ) ( )f t t u t
( ) ( ) [ ( ) ( ) ] ( ) ( )t t t
g t f d u d u t u d
t
g
1
1
f
0 0( ) ( )
0
t tif tu d u d
t if t
( ) ( )dg t u tdt
Due tojump
Standard derivativeaway from t = 0.
( )t u t
( ) ( ) 1g t u t t
EE312 Signals and Systems Dr. Mohamed Zribi 144
Translations and flips of steps and deltas0
( )1
if tu t
if t
t
1
t( ) ( )t t
A pulse function:
( ) ( ) ( ) ( )1 [ , ]0
u t a u t b u t a u b tif t a botherwise
ta b
11( )
0if t
u tif t
EE312 Signals and Systems Dr. Mohamed Zribi 145
Example 63:( ) ( ) ( 2)
( ) (2 )f t u t u t
u t u t
t
f
0 2
t0 2
( ) 2df t tdt
dfdt
Using the first expression,
Using the second,
( ) (2 ) ( ) 2 ( 1)
( ) (2 ) ( ) 2
df t u t u t tdt
t u t u t t
Different answer?
EE312 Signals and Systems Dr. Mohamed Zribi 146
A basic property of the delta
• If f(t) is continuous at 0, ( ) ( ) 0 ( )t f t t f t
• If f(t) continuous at , ( ) ( ) ( )t f t t f t
Let f(t) be a standard function.
t
(1)
f
t
( )f
=
1 1
( ) (2 ) ( ) 2 ( ) (2) (2) 2
( ) 2
df t u t u t t t u u tdt
t t
Apply to the previous example:
Consistent with previous answerEE312 Signals and Systems Dr. Mohamed Zribi 147
Consequence of our basic property
( ) ( ) (0) ( ) (0) ( ) (0)f t t dt f t dt f t dt f
( ) ( ) ( )f t t dt f
( ) ( ) ( ) ( ) ( ) ( )
( )( ) ( ) ( )
0
b
af t t dt u t a u t b f t t dt
f if a bf u a u b
if a or b
Similarly,
t
(1)
f
a b
EE312 Signals and Systems Dr. Mohamed Zribi 148
( )( ) ( )
0
b
a
f if a bf t t dt
if a or b
t
(1)
f
a bWhat if falls exactly in one of the limits of integration?In that case, for the integral to be well defined we must specify whether that point is included or not, as follows:
( ) ( ) ( ), ( ) ( ) 0.b b
a af t t a dt f a f t t a dt
( ) ( ) ( ), ( ) ( ) 0,b b
a af t t b dt f b f t t b dt
EE312 Signals and Systems Dr. Mohamed Zribi 149
Properties of Unit Impulse Function
dtttt )()( 0
Write t as t + t0
dtttt )()( 0 )( 0t
)()()( 00 tdtttt
Proof:
P1: The Sampling Property
EE312 Signals and Systems Dr. Mohamed Zribi 150
The sampling property “extracts” the value of a function at a point.
Properties of the Impulse Function
0 01a t t t ta
P2: The Scaling Property
This property illustrates that the impulse is different from
ordinary mathematical functions.
EE312 Signals and Systems Dr. Mohamed Zribi 151
Properties of Unit Impulse Function
dttat )()( )0(
||1
a
dttt
a)()(
||1
dttt
a)()(
||1
)(||
1)( ta
at
Proof:
P3:
EE312 Signals and Systems Dr. Mohamed Zribi 152
Proof:
dttat )()(
Write t as t/a
Consider a>0
dt
att
a)(1
)0(||
1
a
dttat )()(
Consider a<0
dt
att
a)(1
)0(||
1
a
)0(||
1)()( a
dttat
P4:
EE312 Signals and Systems Dr. Mohamed Zribi 153
Properties of Unit Impulse Function
dttttf )()]()([
dtttft )]()()[(
)0()0( f
dtttf )()()0(
dtttf )()]()0([
)()0()()( tfttf
Proof:
P5:
)()()()(or 000 tttftttf
EE312 Signals and Systems Dr. Mohamed Zribi 154
Properties of Unit Impulse Function
Example 64:
)()0()()( tfttf
0)( tt
)(||
1)( ta
at
)()( tt
EE312 Signals and Systems Dr. Mohamed Zribi 155
0
)()0(' ,)()('
tdttd
dttdt
0
)()( )()0( ,)()(
tn
nn
n
nn
dttd
dttdt
)0(')(')()()('
dtttdttt
)0()1()()( )()( nnn dttt
P6:
P7:
EE312 Signals and Systems Dr. Mohamed Zribi 156
Properties of Unit Impulse Function
dttttf )(')]()([
dttttf )(')]()([
dtttft )](')()[(
dtttfttft )}()(')]'()(){[(
dtttftdtttft )]()'()[()]'()()[(
dtttftdtttft )]()'()[()]()()[('
dtttfttft )()](')()()('[
)(')()()(')]'()([ ttfttfttf
P8:
Proof:
EE312 Signals and Systems Dr. Mohamed Zribi 157
Properties of Unit Impulse Function
)()'()]'()([)(')( ttfttfttf
)]'()0([ tf )()0(' tf
)()0(')(')0()(')( tftfttf
P9:
Proof:
EE312 Signals and Systems Dr. Mohamed Zribi 158
Properties of Unit Impulse Function
Summary of important properties of unit impulseSifting properties
2
1
00 0
21t
t otherwisettttf
dttttf
Sampling properties
000 tttxtttx
Scaling properties
abt
abat 1
2010 ,12
1
ttttxdttttx nnt
t
n
Extension of Sifting properties
EE312 Signals and Systems Dr. Mohamed Zribi 159
Example 65:
dttt )10(2
5
0
2 )10( dttt
20
0
2 )10( dttt
20
0
'2 )10( dttt
a)
b)
c)
d)
= 100
= 0
= 100
= - 20)(.1 2tdtd
EE312 Signals and Systems Dr. Mohamed Zribi 160
Successive integration of the unit impulse1. Successive integration of the unit impulse yields a
family of functions.
2. Later we will talk about the successive derivatives of (t)
EE312 Signals and Systems Dr. Mohamed Zribi 161
Remark:
Unit step Unit rampintegration
Unit ramp Unit parabolicintegration
Unit impulse Unit stepintegration
differentiation
differentiation
differentiation
EE312 Signals and Systems Dr. Mohamed Zribi 162
9. The Unit Periodic Impulse
The unit periodic impulse/impulse train is defined by
, an integerTn
t t nT n
The periodic impulse is a sum of infinitely many uniformly-spaced impulses.
EE312 Signals and Systems Dr. Mohamed Zribi 163
10. Exponential Signals
These signals occur frequently and serve as building blocks to construct many other signals
CT Complex Exponential:where a and C are in general complex.
Depending on the values of these parameters, the complex exponential can exhibit several different characteristics
atCetx )(
x(t) x(t)
C Ctt
a < 0a > 0 Real Exponential (C and a are real)
growing Exponential
Decaying Exponential
EE312 Signals and Systems Dr. Mohamed Zribi
atCetx )(
164
Real Exponential SignalsExponential signals are characterized by exponential
functions
Where e is the Naperian constant 2.718… and C and are real constants.
( ) tf t Ce
f(t)
Time, t
( ) <0tf t Ce
EE312 Signals and Systems Dr. Mohamed Zribi 165 EE312 Signals and Systems Dr. Mohamed Zribi
Example 66: Exponential Signals
166
11. Sinusoidal Signals
x(t) = A cos(ωt + Φ)
A is the maximum amplitude of the sinusoidal signal
ω is the radian frequency
Φ is the phase shift
EE312 Signals and Systems Dr. Mohamed Zribi 167
Sinusoidal SignalsA sinusoidal function
frequency in hertz or cycles per second, the phase shift in radians, the radian frequency is rad/s and the period
is (sec)Exponential functions, as in
Where B is the amplitude, is angular frequency in radians/second, and is the phase shift in radians.
0
2cos)cos()2cos()(T
tAtAftAtf
( ) ( )( ) cos( )2 2
j t j tB Bg t e e B t
f.2 fT /10
f.2
EE312 Signals and Systems Dr. Mohamed Zribi 168
Exponentially Modulated Sinusoidal Functions1. If s sinusoid is multiplied by a real exponential, we have an
exponentially modulated sinusoidthat also can arise as a sum of complex exponentials, as in
)2cos()( ftAetf t
( ) ( )1( ) cos( )
2 2t j t t j tB Bf t e e e e B t
Example 67:)1cos(3)( 2.0 tetf t
Turned on at t = +1by multiplying shiftedunit step u(t-1)
EE312 Signals and Systems Dr. Mohamed Zribi 169
Example 68:
Complex exponentials and unit steps can be combined to produce
causal and anti-causal decaying exponentials.
EE312 Signals and Systems Dr. Mohamed Zribi 170
12. Complex Exponentials
The complex exponential signals have the form:
1. for all t
2. where C and s, are complex numbers.
3. If s is complex then it can be written as s = + j,
where and are the real and imaginary parts of s.
( ) stx t Ce
EE312 Signals and Systems Dr. Mohamed Zribi 171
Complex ExponentialsCase 1: real sIf s = is real and C is real then
,And we get the family of real exponential functions.
Case 2: imaginary sIf s = j is imaginary and C is real then
and we get the family of sinusoidal functions.
( ) tx t Ce
( ) (cos sin ),j tx t Ce C t j t
EE312 Signals and Systems Dr. Mohamed Zribi
( ) stx t Ce
172
Case 3: s complexIf s = + j is complex and C is real then
and we get the family of damped
sinusoidal functions.
( )( ) (cos sin ),j t tx t Ce Ce t j t
EE312 Signals and Systems Dr. Mohamed Zribi
Complex Exponentials
173
For is plotted for
different values of s superimposed on the complex s-
plane.
( ) , { ( )} coss t tx t Ce x t Ce t
EE312 Signals and Systems Dr. Mohamed Zribi
Complex Exponentials
174
1. For is plotted for different values of s superimposed on the complex s-plane.
( ) , Im{ ( )} sins t tx t Ce x t Ce t
EE312 Signals and Systems Dr. Mohamed Zribi
Complex Exponentials
175
General Complex Exponential Signals
Here, C and s are general complex numbers
Say,
(Real and imaginary parts) Growing and damping sinusoids for r>0 and r<0
and s ( )j stC C e j x t Ce
Re{ ( )}x t
, 0t
( ) cos( )trx t Ce t
, 0t
envelope
( )Then ( ) st t j tx t Ce C e e cos( ) sin( )t tC e t j C e t
EE312 Signals and Systems Dr. Mohamed Zribi 176
( ) stx t Ce
Re{ ( )}x t
General Complex Exponential Signals
EE312 Signals and Systems Dr. Mohamed Zribi 177
( ) cos( )trx t Ce t
Example 69: Examples of General Complex
Exponential Signals2 3
2.5 0.5 0.5
2.5
( )
( ) ( )
( ) 2 cos(0.5 )
( ) 2 cos(0.5 )
j t j t
j t j t j t
j t
x t e e
x t e e e
x t e t
x t t
EE312 Signals and Systems Dr. Mohamed Zribi 178
Example 70:
Superposition of sinusoidal signals ejt to
produce x(t)x(t) = sin(t) + 0.2cos(2t) + 0.1sin(5t)
x(t)
tt
sin(t)
sin(2t)
sin(5t)
EE312 Signals and Systems Dr. Mohamed Zribi 179 EE312 Signals and Systems Dr. Mohamed Zribi
IV. Important DT Signals
180
1. The DT Impulse Function
Unit-sample sequence [n]
1 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8
n
[n]
[n] is sometimes called a discrete-time impulse; or an impulse
0001
][nn
n
EE312 Signals and Systems Dr. Mohamed Zribi 181
The DT Impulse Function
)(n )3( n1
n
… …
1
n
… …
0m = n m = n-3
Plots of Unite Sample Sequences
0 1 2 3
EE312 Signals and Systems Dr. Mohamed Zribi 182
The DT Impulse Function
1 , 00 , 0
nn
n
The discrete-time unit impulse (also known as the “Kronecker delta function”) is a function in the ordinary sense (in contrast with the continuous-time unit impulse). It has a sampling property,
0 0x xn
A n n n A n
but no scaling property. That is,
for any non-zero, finite integer .n an a
EE312 Signals and Systems Dr. Mohamed Zribi 183
2. The Unit Step SequenceThe unit step sequence is the discrete-time version of the unit step
in CT situations.
Definition of unit step sequence:
The unit step sequence u(n) is related to unit sample sequence by
.0,0
0,1)(
nn
nu
1
n
…
Plot of Unit Step Sequence
0 1 2 3
.)()(
n
mmnu U(n)
step sequence
EE312 Signals and Systems Dr. Mohamed Zribi 184
00
0
,A shifted step is: [ ]
0,B n n
Bu n nn n
The DT Unit Impulse and Step SignalsThe discrete unit impulse signal is defined:
Critical in convolution as a basis for analyzing other DT signals
The discrete unit step signal is defined:
Note that the unit impulse is the first difference of the step signal
Similarly, the unit step is the running sum of the unit impulse.
0100
][][nn
nnx
0100
][][nn
nunx
]1[][][ nunun
EE312 Signals and Systems Dr. Mohamed Zribi 185
DT Unit Impulse and Unit Step Functions
0
k]-[n][k
nu
][][][][ 000 nnnxnnnx
][]0[][][ nxnnx
(DT step is therunning sum of DT unit sample)
More generally for a unit impulse [n-n0] at n0 :
[n-k]- - -- - -
nk
0
Interval of summation
n>0
[n-k]- - -- - -
n k0
Interval of summation
n<0
Sampling property
EE312 Signals and Systems Dr. Mohamed Zribi 186
Since a sequence of Discrete-Time Signals can be represented in term of
Shifted Unit impulse as defined below :
∞
x[n] = Σx[k]δ[n-k]k=-∞
Then, the unit step sequence can be defined in term of Shifted Unit Impulse
as shown below :
∞
u[n] =Σδ[n-k]k= 0
EE312 Signals and Systems Dr. Mohamed Zribi
Remark:
187
3. The DT Periodic Impulse Function
Nm
n n mN
EE312 Signals and Systems Dr. Mohamed Zribi 188
4. The DT Unit Ramp Function
, 0ramp u 1
0 , 0
n
m
n nn m
n
EE312 Signals and Systems Dr. Mohamed Zribi 189
r[n]=nu[n]
The Shifted Ramp FunctionRamp SequenceA shifted ramp sequence with slop of B is defined by:The unit ramp sequence and shifted ramp sequences
Example 71: g[n]= 2(n-10)u[n].
)()( 0nnBng
MATLAB Code:n=-10:1:20;f=2*(n-10);stem(n,f);
EE312 Signals and Systems Dr. Mohamed Zribi 190
5. The DT Rectangle Function
1 ,
rect , 0 , an integer0 ,w
wN w w
w
n Nn N N
n N
EE312 Signals and Systems Dr. Mohamed Zribi 191
Exponential function
n=0 n
x [n]
x[n] = A n 0 < < 1
5. The DT Exponential Function
EE312 Signals and Systems Dr. Mohamed Zribi 192
Sinusoidal function
n=0 n
x [n]
x[n] = A cos(n + )
1
6. The DT Sinusoidal Function
EE312 Signals and Systems Dr. Mohamed Zribi 193
The Sinusoidal DT SequenceA sinusoidal sequence may be described as:
where A is positive real number (amplitude), N is the period, and alpha is the phase.
Example 72:A = 5, N = 16 and
MATLAB Code:n=-20:1:20;f=5*[cos(n*pi/8+pi/4)]; stem(n,f);
NnAnf 2cos)(
.4/
EE312 Signals and Systems Dr. Mohamed Zribi 194
Summary of Basic DT Sequences
Unit impulse Unit step
Exponential
Periodic
Sinusoidal
Random
EE312 Signals and Systems Dr. Mohamed Zribi 195
7. The DT Real Exponential Function
)0for n alternatioSign (1,01,10,1
of instead use tocustomary and convenient more isIt numbers real are and C where
for
eαCnx
nr
n
1 10 01 1
EE312 Signals and Systems Dr. Mohamed Zribi 196
DT Real Exponential Signals
(a) 1 0
(b)0 1 C > 0
(c) 1 0
(d) 1
C
real are and C :signals lexponentia Real][ nCnx
trivial0are or 1 when cases 197
C is constant and аlpha is a real number.
Example 73: Real Exponential DT Signal
EE312 Signals and Systems Dr. Mohamed Zribi
nCnx ][
198
EE312 Signals and Systems Dr. Mohamed Zribi
0 2 4 6 8 10 120
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y(k)=0.7k
y(11)=0.0198
Example 74: Real Exponential DT Signal
199
Real exponential sequence is defined as:Example for C = 10 and alpha = 0.9, as n goes to infinity the sequence
approaches zero and as n goes to minus infinity the sequence approaches plus infinity.
nCnf )()(
)()()( nuaAnf nComposite sequence:
MATLAB Code:n=-10:1:10;f =10*(.9).^n;stem(n,f);axis([-10 10 0 30]);
Example 75: Real Exponential DT Signal
EE312 Signals and Systems Dr. Mohamed Zribi 200
8. Complex Exponential and Sinusoidal DT Signals
When C and are real numbers real exponential DT signals
When =ej0 complex exponential DT signal
n nx n C x n C
00 0Note that: cos sinj ne n j n
EE312 Signals and Systems Dr. Mohamed Zribi 201
DT Complex Exponential Signals
Unlike CT complex exponential signals ej0n is not distinct for distinct values of 0.
DT complex exponential signals are periodic with period N such that
njnj ee 200 0, 0±2, 0±4, … are identical. We need only consider a frequency interval of 2, e.g. 0≤ 0 ≤ 2.
number rational 2
21
0
00
00
Nm
mNeee
Nj
njNnj
If N and m have no factors in common then N is the fundamental period of x(t).
EE312 Signals and Systems Dr. Mohamed Zribi 202
DT Complex Sinusoidal Signals
]}[Im{]}[Re{ )sin()cos(][
then ,CC
as formpolar in are and C If
00
0
nxjnxnCjnCCnx
ee
nnn
jj
)1 decaying and ,1 (growing ssinusoidal are expcomplex general DT of partsimaginary and Real
1
1
EE312 Signals and Systems Dr. Mohamed Zribi
complex are and C :signals lexponentiaComplex ][ nCnx
]}[Im{or ]}[Re{
nxnx
]}[Im{or ]}[Re{
nxnx
203
DT Complex Exponential Signals
0
0 0
;
cos( ) sin( ) Re{ [ ]} Im{ [ ]}
jwj
n nn
C C e e
C C w n j C w nx n j x n
][nx
1
1
]}[{mor ]}[Re{ nxInx
]}[{mor ]}[Re{ nxInx
EE312 Signals and Systems Dr. Mohamed Zribi 204
Sinusoidal sequence
n
x [n]
)cos(]}[Re{ 0 nnCnx
EE312 Signals and Systems Dr. Mohamed Zribi
1
Example 76: Complex Exponential
205
0 2 4 6 8 10 12 14 16 18-1
-0.5
0
0.5
1
sin(k*pi/6)
0 2 4 6 8 10 12 14 16 18-1
-0.5
0
0.5
1
cos(k*pi/6)
1
Example 77: Complex Exponential
]}[Re{ nx
]}[{m nxI
EE312 Signals and Systems Dr. Mohamed Zribi 206
x[n] = Cеjωn; ω frequency of complex exponential sinusoid, C is a constant
Example 78: Complex Exponential
EE312 Signals and Systems Dr. Mohamed Zribi
])[Im(
])[Re(
nx
nx1
207
0 2 4 6 8 1 0 1 2 1 4 1 6 1 8- 1
- 0 . 8
- 0 . 6
- 0 . 4
- 0 . 2
0
0 . 2
0 . 4
0 . 6
0 . 8
1
u ( k ) = c o s ( k * p i / 6 ) * 0 . 9 k
Example 79: Complex Exponential
])[Re( nx
1
EE312 Signals and Systems Dr. Mohamed Zribi 208
Exponentialy Modulated DT Sinusoidal SequenceBy multiplying an exponential sequence by sinusoidal sequence, we
obtain an exponentially modulated sequence described by:
Example 80:A = 10, N = 16, a = 0.9AndMATLAB Code:n=-20:1:20;f=10*[0.9 .^n];g=[cos(2*n*pi/16+pi/4)];h=f .*g;stem(n,h);axis([-20 20 -30 70]);
NnaAng n 2cos)()(
.4/
EE312 Signals and Systems Dr. Mohamed Zribi 209
Example 81:
0 2 4 6 8 10 12 14 16 18 20-1
-0.5
0
0.5
1
1.5
2
)3
kπcos(0.8)3
kπsin(0.82y(k) kk
EE312 Signals and Systems Dr. Mohamed Zribi 210
Example 82:
The dash line are the CT function. The CT function are obviously different but the DT function are not.
EE312 Signals and Systems Dr. Mohamed Zribi 211
Comparison of the Signals
0 0 and jw t jw ne e
EE312 Signals and Systems Dr. Mohamed Zribi 212
EE312 Signals and Systems Dr. Mohamed Zribi
V. Operation on Signals
213
Basic operations on signals
Operation on Signals
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g t Ag t Example 83: Amplitude Scaling,
EE312 Signals and Systems Dr. Mohamed Zribi 215
g t Ag t Example 84: Amplitude Scaling,
EE312 Signals and Systems Dr. Mohamed Zribi 216
Example 85: Amplitude Scaling
EE312 Signals and Systems Dr. Mohamed Zribi 217
Operation on Signals
Sometimes we need to change the independent variable axis for
theoretical analysis or for just practical purposes (both in CT
and DT signals). Some of these operations are:
Time shift
[ Delay x(t-2); Advance x(t+2) ]
Time reversal
Time scaling
)()( ottxtx
)()( txtx
)2/()( txtx
EE312 Signals and Systems Dr. Mohamed Zribi 218
Example 86: Time Shifting
t0 < 0 x(t-t0) is an advanced version of x(t)
x(t)
x(t-t0)
t
tt0
Time shift
EE312 Signals and Systems Dr. Mohamed Zribi 219
Example 87: Time Shifting
EE312 Signals and Systems Dr. Mohamed Zribi 220
Replace t by t- to.
If to>0, shift to the right. If to <0, shift to the left. e.g. to = -2 :
Time shift
EE312 Signals and Systems Dr. Mohamed Zribi
Example 88: Time Shifting
221
• The original signal x(t) is shifted by an amount t0 > 0 .
• Given x(t) = u(t+2) - u(t-2), find x(t-t0) and x(t+t0).
EE312 Signals and Systems Dr. Mohamed Zribi
Example 89: Time Shifting
222
Shifting the function to the right or left by t0
Example 90: Time Shifting
EE312 Signals and Systems Dr. Mohamed Zribi 223
If n0 > 0 x[n-n0] is the delayed version of x[n] (Each point in x[n] occurs later in x[n-n0])
x[n]
x[n-n0]
. . . . . .
. . . . . . . . . . .
n
nn0
Time shift
EE312 Signals and Systems Dr. Mohamed Zribi
Example 91: Time Shifting
224
n
nx
n
3nx
t
tx
4 t
4tx
8
Time shift
EE312 Signals and Systems Dr. Mohamed Zribi
Example 92: Time Shifting
225
Determine x(t) + x(2-t) , where x(t) = u(t+1) - u(t-2)
find x(2-t): Advance, then reverse in time.
Add the two functions: x(t) + x(2-t)
u(t+1)- u(t-2)
t=0
EE312 Signals and Systems Dr. Mohamed Zribi
Example 93: Time Shifting
226
0 0 , an integern n n n Time shifting
Example 94: Time Shifting
EE312 Signals and Systems Dr. Mohamed Zribi 227
Time Reversal
Reflection about t=0
x(t)
x(-t)
t
t
Time reversal
EE312 Signals and Systems Dr. Mohamed Zribi 228
Example 95: Time reversal
EE312 Signals and Systems Dr. Mohamed Zribi 229
Example 96: Time reversal
EE312 Signals and Systems Dr. Mohamed Zribi 230
Replace t by – to.
Referred the reflected signal as folding (view it from behind the paper. e.g to = 6:
Time reversal
EE312 Signals and Systems Dr. Mohamed Zribi
Example 97: Time reversal
231
n
nx
n
nx
t
tx
t
tx
Time reversal
EE312 Signals and Systems Dr. Mohamed Zribi
Example 98: Time reversal
232
Time Scaling
Example 99: Given x(t), find y(t) = x(2t). This speeds up x(t) (the graph is shrinking)
What happens to the period T?The period of x(t) is 2 and the period of y(t) is 1.
EE312 Signals and Systems Dr. Mohamed Zribi 233
x(t)
t
compressed!
stretched!
x(2t)
t
x(t/2)
t
Time scaling
EE312 Signals and Systems Dr. Mohamed Zribi
Time Scaling
234
Multiply t by a constant , say, β .
If β > 0, the signal is compressed. If β <0, the signal is expanded. e.g β = 2:
Time scaling
Time Scaling
EE312 Signals and Systems Dr. Mohamed Zribi 235
Time Scaling
Given y(t), find w(t) = y(3t) v(t) = y(t/3).
EE312 Signals and Systems Dr. Mohamed Zribi 236
Expands the function horizontally by a factor of |a|
Example 100: Time Scaling
/t t a
EE312 Signals and Systems Dr. Mohamed Zribi 237
/t t a
If a < 0, the function is also time inverted. The time inversionmeans flipping the curve 1800 with the g axis as the rotation axis of the flip.
Example 101: Time Scaling
EE312 Signals and Systems Dr. Mohamed Zribi 238
t
x(t)
T20
10
t
(t) = x(t/2)
0
t
(t) = x(2t)
Original signal
0
T1
Expanded signal(a = 0.5)
Compressed signal(a = 2)
2T22T1
2T1
2T2
EE312 Signals and Systems Dr. Mohamed Zribi
Example 102: Time Scaling
239
Example 103: Time Scaling
Example 104: Time Scaling
EE312 Signals and Systems Dr. Mohamed Zribi 240
n Kn
K an integer > 1
Example 105: Time Scaling
EE312 Signals and Systems Dr. Mohamed Zribi 241
/ , 1n n K K Time expansion
For all such that / is an integer, g / is defined.
For all such that / is not an integer, g / is not defined.
n n K n K
n n K n K
Remark : Time Expansion for DT Signals
EE312 Signals and Systems Dr. Mohamed Zribi 242
Combined Operations on Signals
We can use various combinations of the three operations just
covered: time shifting, time scaling, and time reversal. The
operations can often be applied in different orders, but care must
be taken.
EE312 Signals and Systems Dr. Mohamed Zribi 243
Combined Operations on Signals
To form x(at - b) from x(t) we could use two approaches:
1) Time-shift then time-scaleA. Time-shift x(t) by b to obtain x(t - b). i.e., replace every t by t
- b.B. Time-scale x(t - b) by a (i.e., replace t by at) to form x(at - b)
2) Time-scale then time-shiftA. Time-scale x(t) by a to obtain x(at).B. Time-shift x(at) by b/a (i.e., replace t with t – b/a) to yield
x(a[t – b/a]) = x(at – b)
EE312 Signals and Systems Dr. Mohamed Zribi 244
0g g t tt Aa
0
amplitudescaling, / 0g g g gt t tA t t a t ttt A t A A
a a
A multiple transformation can be done in steps
0
amplitudescaling, / 0
0 0g g g g gt t tA t t a t ttt A t A t t A t Aa a
The order of the changes is important. For example, if weexchange the order of the time-scaling and time-shifting operations, we get:
Amplitude scaling, time scaling and time shifting can be appliedsimultaneously.
Combined Operations on Signals
EE312 Signals and Systems Dr. Mohamed Zribi 245
g t Agt t0
a
Example 106: Operation on Signals
EE312 Signals and Systems Dr. Mohamed Zribi 246
Find y(t) = x(2t + 3).
Example 107: Operation on Signals
EE312 Signals and Systems Dr. Mohamed Zribi 247
Example 108: Operation on Signals
x(t+1)
t10-1
1
x(1.5t+1)
t2/30-2/3
1
Find x(3t/2+1)
t
x(t)
1
1 20
EE312 Signals and Systems Dr. Mohamed Zribi 248
Example 109: Operation on Signals
Given the signal x(t):
Let us find x(t+1):
Let us find x(-t+1):
t
x(t)
1
1 20
(time reversal of x(t+1)) t
1
10-1
(It is a time shift to the left)
x(t+1)
t
x(-t+1)
10-1
1
EE312 Signals and Systems Dr. Mohamed Zribi 249
Example 110: Operation on Signals
0 1 2 3 43 2 1
0 1 2 3 4
tx
t
tx
2/2 tx
t
03 2 1 1 2 3 4 t
3 2 1
1
1
EE312 Signals and Systems Dr. Mohamed Zribi 250
Example 111: Operation on Signals
x(2t
)
EE312 Signals and Systems Dr. Mohamed Zribi 251
Example 112: Operation on Signals
Given x2(t), find y(t) = 1 - x2(t).
Remember: This is x(t) =1
EE312 Signals and Systems Dr. Mohamed Zribi
y(t) = 1 - x2(t)
252
Example 113: Operation on Signals
Multiplication of two signals: x2(t)u(t)
Step unit function
EE312 Signals and Systems Dr. Mohamed Zribi 253 EE312 Signals and Systems Dr. Mohamed Zribi
Example 114: Operation on Signals
254
Properties of the CT Unit Step Function
kttuttuttu )]([)]([)( 02
00
0),/()( 00 aattutatu
)(1)( tutu
EE312 Signals and Systems Dr. Mohamed Zribi 255
CT Unit Ramp Function
)()()()( 000
0
0
ttuttddtutft
t
t
Unit ramp function can be achieved by:
to to+1
1
Non-zero only for t>t0
0)()()( dttdttut
Remark:
EE312 Signals and Systems Dr. Mohamed Zribi 256
Note: u(-t+3)=1-u(t-3)
Example 115:
Example 116:
EE312 Signals and Systems Dr. Mohamed Zribi 257
Signals can be combined to make a rich population of signals
Unit steps and ramps can he combined to produce pulse signals.
EE312 Signals and Systems Dr. Mohamed Zribi 258
u(t) - u(t-1) r(t) - 2r(t-1)+r(t-2)
1
1
t t1 2
1
Example 117:
EE312 Signals and Systems Dr. Mohamed Zribi 259
Example 118:Describe analytically the signal shown in
Solution: Signal is (A/2)t at , turn on this signal at t = 0 and turn it off again at t = 2. This gives,
)].2()([2
)( tututAtf
20 t
A
0 2
t
f(t)
EE312 Signals and Systems Dr. Mohamed Zribi 260
Example 119:
)2(5)1(]1[)()(3)( tututttututf
EE312 Signals and Systems Dr. Mohamed Zribi 261
Example 120:
Plot
• t<-2 f(t)=0• -2<t<-1 f(t)=3[t+2]• -1<t<1 f(t)=-3t• 1<t<3 f(t)=-3• 3<t f(t)=0
)3(3)1(]1[3)1(]1[6)2(]2[3)( tututtuttuttf
EE312 Signals and Systems Dr. Mohamed Zribi 262
Example 121:
rect(t/T)
Can be expressed as u(T/2-t)-u(-T/2-t) Draw u(t+T/2) first; then reverse it!
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/t T/t
1
-T/t T/t
1
-T/t T/t
1
-T/t T/t
EE312 Signals and Systems Dr. Mohamed Zribi 263
Example 122:
Write the expression of the plotted function
Answer:f(t)=0 for t<-2f(t)=3(t+2) for -2<t<-1f(t)=-3t for -1<t<1f(t)=-3 for 1<t<3f(t)=0 for 3<t
EE312 Signals and Systems Dr. Mohamed Zribi
)3(3)1(]1[3 )1(]1[6)2(]2[3
)]3()1([3 )]1()1([3)]1()2()[2(3)(
tututtuttut
tututututtututtf
264
Representation of DT Signals using Sequences
Discrete-time system theory is concerned with processing
signals that are represented by sequences.
1 2
3 4 5 6 7
8 9 10-1-2-3-4-5-6-7-8
n
x [n]
EE312 Signals and Systems Dr. Mohamed Zribi 265
It is possible to re-generate an arbitrary signal by sampling it with shifted unit impulse:
This is called the sifting property:
[ ] [ ] [ ]k
x n x k n k
Sifting property of DT impulseweights
shifted impulse
x[n] = - - - + x[-4] δ[n+4] + x[-3] δ[n+3] + x[-2] δ[n+2] + + x[-1] δ[n+1] + x[0] δ[n] + x[1] δ[n-1] + - - -
EE312 Signals and Systems Dr. Mohamed Zribi
Representation of DT Signals using Sequences
266
The Unit Impulse can be shifted or delayed. The shifted Unit Impulse is denoted as :-δ[n-k] => The unit impulse is shifted to right by kδ[n+k] => The unit impulse is shifted to left by k
Example 123:The sequence, p[n] is expressed as :p[n] = a-3 δ[n + 3] + a1 δ[n – 1] + a2 δ[n – 2] + a7 δ[n – 7] Graphical representation of Shifted unit Impulse of p[n]:
EE312 Signals and Systems Dr. Mohamed Zribi
Representation of DT Signals using Sequences
267
1
2
3 4 5 6
7
8 9 10-1-2-3-4-5-6-7-8
n
x [n]
a1
a2 a7
a-3
k
knkxnx ][][][
]7[]3[]1[]3[][ 7213 nananananx
EE312 Signals and Systems Dr. Mohamed Zribi
Example 124:
268
Think of a DT signal as a sequence of individual impulses
Consider x[n]
4-3 13
0...2-1
-2... n-4
EE312 Signals and Systems Dr. Mohamed Zribi
Example 125:
269
x[n] is actually a sequence of time-shifted and scaled impulses
-2
x[-2]δ[n+2]-1x[-1]δ[n+1]
0
x[0]δ[n]
1
x[1]δ[n-1]
0
2
x[2]δ[n-2]
10
EE312 Signals and Systems Dr. Mohamed Zribi 270
Example 126:
Describe analytically the following sequence.
1
n
A Pulse Sequence
0 1 2 3
f[n]
-1-2
This pulse sequence can be describe by
f[n] = u[n]– u[n-3]
The first step sequence turn on the pulse at n = 0, and second step turns it off at n = 3.
EE312 Signals and Systems Dr. Mohamed Zribi 271
Remember These Geometric Series
EE312 Signals and Systems Dr. Mohamed Zribi 272