Introduction
Signals and Systems
Aims
• Introducing the mathematical descriptions and representations of
signals and systems.
• Developing mathematical tools for analyzing signals and systems.
• Analysis will be both in time and frequency domains.
• Analysis will be for both continuous-time and discrete-time signals and
systems
• The analytical developments build a foundation for
mathematical understanding of other topics in engineering
sciences, such as Communications, Signal Processing
and Control.
Module Information
• Textbook:
o Oppenheim, Willsky, “Signals & Systems,” second edition.
• Module page on KEATS is frequently updated and provides all
information about:
o
o
o
o
Taught material and tutorials
Assessment methods
Office hours (online by prior appointment)
Course reader (Textbook)
• Background material:
o College level mathematics
• Tutorials:
o Teach skills and expose you to examples (could be mixed with
lectures)
•
•
•
•
Regular attendance in weekly classes held on MS Teams
Active engagement in class discussions
Review lecture material before the classes
Try the tutorial problems before the classes
What we expect from you
•
•
•
•
•
Come to all lectures, tutorials (even the early and the last ones!).
Take notes during lectures and revise at home.
Make use of the course materials (including a textbook).
Read around the subject/try things for yourself.
Plan your time carefully: the more you leave things until later, the
more likely you are to struggle.
• Review the material at home on the same day of
lecture and amend your notes so that they are
complete for your later revision. If you did not
understand a point, ask me as soon as possible.
Signals and Systems (SaS)
• SaS is about how to use mathematical tools/techniques
to analyse and synthesis systems that process signals.
Signals are the carriers of information.
Mathematically signals are functions of some
independent variables (often time).
Systems process input signals to generate output signals.
Signals (Examples)
time
Speech Signal:
The periodicity implies a vowel letter.
Vowels in English: a,e,i,o,u
time
Electrocardiogram (ECG) Signal:
electrical activity of the heart
Recorded by the chest electrodes
Signals
In this lecture, we are interested in two types of signals, as functions of time.
Discrete-Time (DT)
Continuous-Time (CT)
• Most signals are CT
• Some signals are DT
Like current or voltage in an electronic circuit
or an audio signal in its original form.
Denoted as
where
Like an audio signal when stored in a CD.
Denoted as
integer-valued variable
real-valued variable
𝑥[𝑛]
𝑥(𝑡)
0
3
4
Note: 𝑥(1.3) is defined.
𝑡
-1 0
1 2
Note: 𝑥 1.3 is not defined.
𝑛
Signals and Systems
Digital Communication System Example
Message
Source
Transmitter
x i (t )
Channel
Message
Sink
y (t )
Receiver
• The transmitted signal has to be designed so that it gets through the channel.
• Channel is characterized by a mathematical model either in time domain or in
the frequency domain, so as the input signal to the channel.
• We need to find out how the channel respond to a given input, so as to design
the entire system!
Signals and Systems
Radar Ranging Exapmle
• By measuring the time delay
and given the signal
propagation speed :
=
The range R can be calculated.
Radar pulse (signal) contains
the plane range information.
Radar Transceiver System
R
The reflected signal s 𝑡 is initiated or
transmitted by plane and received
(back) by the radar with some delay and
added distortion. The relation between
input s 𝑡 and output 𝑟 𝑡 :
𝑟 𝑡 = 𝑠 𝑡 − 𝜏 + 𝑛(𝑡)
Signals and Systems
DT-System: Image Processing Example
System
(Distorting
Channel)
Original Image
Recovering
System
Distorted
Image
• Is the distorting system reversible?
Can we recover the original image signal from the
distorted one?
• We need to have a mathematical model to the distorting channel
and analytical tools to find out the answer.
• If the distorting system is reversible, what is the
Recovered Image
mathematical model of the recovering system, so that
?
Complex Signals
• We are are interested in complex-valued signals, where the values of
functions
or
are complex numbers, in particular:
CT signals of the form
DT signals of the form:
Where and are complex numbers.
and
are also complex numbers.
Complex Numbers (a reminder)
• The set of complex numbers is denoted by
and is defined as
• Forms of representation of a complex number :
Cartesian:
: real part of ;
: imaginary part of ]
Polar:
: length or modulus of ;
: argument of
The most convenient representation depends on the analysis.
An important and useful formula:
Euler’s Formula:
Complex Numbers (a reminder)
• Using the Euler’s formula:
•
Cartesian:
Polar:
Im (Imaginary axis)
𝑧
𝑦
•
𝑟= 𝑧
𝜃
Complex conjugate of :
∗
A key property:
∗
=
𝑥
Polar and cartesian representations
on the Complex Plane
Re (Real axis)
Complex Signals
• Examples:
For
Euler:
For
Euler:
=
Signal Energy and Power
• Power and energy are used as measures to characterise signals.
Example: Instantaneous power
of a resistor is given by:
where
is the current and
is the voltage across the resistor.
The total energy over a time interval
is:
The average energy over a time interval
is:
Generic Signal Energy and Power
(over finite Time Interval)
• We consider complex-valued signals.
• Total energy of a CT signal
over time interval
is:
The time averaged Power:
• Total energy of a DT signal
The time averaged Power:
over discrete time interval
is:
Generic Signal Energy and Power
(over Infinite Time Interval)
• Energy:
CT:
DT:
Generic Signal Energy and Power
(over Infinite Time Interval)
• Power:
CT:
DT:
Generic Signal Energy and Power
(Signal Classes in terms of Power and Energy)
• Signals with finite total energy (
Example: For
,
• Signals with finite average power (
Example: For
But:
) results in a zero average power:
,
and
) results in an infinite total energy.
(Infinite);
(Finite)
Last Lecture Overview
• Lecture structure and arrangements
• Aims and objectives
• Leaning outcomes
• How to be successful in this module
• Introductions to signals and systems with illustrative examples
• Complex signals
• Complex numbers (a reminder/review)
• Signal energy and power
Outline
• Basic Signal Operations
• Linear Combination of Basic Signal Operations
• Decimation and Expansion
• Periodic Signals
• Periodicity and scaling
• Signal Decomposition in Even and OddSignals
• Right-sided and Left-sided Signals
Basic Signal Operations
1. Time Shift
Time shift : For any
and
: set of integer numbers; : set of real numbers)
CT:
DT:
)
𝑥(𝑡)
-6 -4 -2 0 2 4 6
𝑦 𝑡 = 𝑥(𝑡 − 2)
𝑡
-6 -4 -2 0 2 4 6 8
=2
𝑡
Basic Signal Operations
2. Time Reversal
Time reversal : Multiplying the time variable by
CT:
DT:
An interpretation: flipping over the vertical axis
𝑥(𝑡)
-6 -4 -2 0 2 4 6
𝑦 𝑡 = 𝑥(−𝑡)
𝑡
-6 -4 -2 0 2 4 6
𝑡
Basic Signal Operations
3. Time Scaling
Time scaling: Multiplying the time variable by a constant
,
1
:
:
Decimated (speed up)
Expanded (slowed down)
𝑦 𝑡 = 𝑥(2𝑡)
𝑥(𝑡)
-6 -4 -2 0 2 4 6
:
𝑡
-6 -4 -2 0 2 4 6
𝑡
Linear Combination of 3 Basic Signal Operations
• Linear operation on the time variable as:
Recommended order of operation: Shift, Scale, Time Reverse (flip)
Illustrative Example
𝑥(𝑡)
-6 -4 -2 0 2 4 6
𝑥(𝑡 − 4)
𝑡
-6 -4 -2 0 2 4 6 8 10
𝑡
𝑥(2𝑡 − 4)
-6 -4 -2 0 2 4 6 8
𝑡
Basic Signal Operations
(Some insights)
• For
If
If
If
If
:
• Example:
as the signal of an Audio Tape Recording:
: is the same tape recording played backwards.
: is the same tape recording played at twice the original speed.
:is the same tape recording played at half of the original speed.
Decimation and Expansion in DT Signals
• Decimation: The decimated DT signal
is defined as:
• Expansion: The expanded DT signal
is defined as:
=
M and : integers
M: decimation factor.
Example:
Example:
𝑥[𝑛]
𝑥[𝑛]
𝑛
𝑛
𝑥[𝑛]
𝑥[𝑛]
𝑦 𝑛
𝑦 𝑛
𝑦 𝑛
𝑦 𝑛
𝑛
𝑛
Periodic Signals
• Definition (CT): A CT signal is periodic if a constant
such that
can be found
• Definition (DT): A DT signal is periodic if an integer constant
found such that
can be
• Definition (CT & DT): Signals do not satisfy the periodic conditions are
called aperiodic (non-periodic) signals.
Periodic Signals
• Definition (CT): A CT signal is periodic if
there exists a constant
such
that:
⋯
𝑥(𝑡)
1
⋯
-𝑇
-𝑇⁄2 -𝜏
𝜏 𝑇 ⁄2
0
2𝑇
𝑇
• Definition (DT): A DT signal is periodic if
an integer constant
can be found such that:
𝑥(𝑡)
• Definition (CT & DT): Signals do not
satisfy the periodic conditions are called
aperiodic (non-periodic) signals.
1
0 𝜏
aperiodic
-𝜏
𝑡
𝑡
Periodic Signals
Fundamental Period & Fundamental Frequency :
CT: We say that is the fundamental period of a periodic signal
if, it is the smallest value of
, satisfying
.
Then,
is called the fundamental (radian) frequency
.
is the number of fundamental periods per second and is called
fundamental frequency
.
DT: We say that
is the fundamental period of a periodic signal
if, it is the smallest integer value of
, satisfying
Then,
is called the fundamental (radian) frequency
.
Last Lecture Overview
• Signal Energy and Power
•
•
•
•
Defined energy and power of CT and DT signals over finite and infinite time intervals
For complex signals: 𝑥(𝑡) = 𝑥 𝑡 𝑥(𝑡)∗ and 𝑥[𝑛] = 𝑥[𝑛]𝑥[𝑛]∗
Signals with finite total energy (𝐸 < ∞) results in a zero average power.
Signals with finite average power (𝑃 < ∞) results in an infinite total energy.
• Basic Signal Operations
• Time Shift
• Time Reversal
• Time Scaling
• Linear Combination of Basic Signal Operations
• Recommended order of operation: Shift, Scale, Time Reverse (flip)
• Decimation and Expansion in DT Signals
• Scaling up/down in DT signals (Notice the integer variable and decimation/expansion variables in
DT)
• Periodic Signals
• Fundamental period
• CT: 𝑇 : Smallest 𝑇 > 0, satisfying 𝑥 𝑡 = 𝑥 𝑡 + 𝑇 , ∀𝑡 ∈ ℝ.
• DT: 𝑁 : Similar, but integer valued.
• Fundamental Frequency
• 𝜔 =
for CT and 𝜔 =
for DT signals.
Outline
• Periodic Signals
• Examples
•
•
•
•
•
•
•
•
Periodicity and scaling
Signal Decomposition in Even and Odd signals
Right-sided and Left-sided Signals
Unit Impulse Functions in CT and DT (Mathematical Definition)
Intuitive Illustration of Unit Impulse Function and its delayed version
Unit Step Function in CT and DT
Relationship between unit impulse function and unit step function
Sampling property of impulse function
Examples (1)
• Is
periodic?
Solution: Is there an integer N such that:
?
Since, there exist positive integer such that
is periodic.
But, we need the smallest integer
for
is the
fundamental period and, hence, the fundamental frequency is
.
,
Examples (2)
• Is
,
periodic?
Solution: Is there a constant
, such that:
But:
Then, for
Since, there exist a constant
is periodic.
?
, for
,
such that
But, we need the smallest
for
period and, hence, the fundamental frequency is
,
is the fundamental
Examples (3)
• Is
periodic?
Solution:
Since, there exist constant
periodic.
such that
But, we need the smallest
for
is the
fundamental period and, hence, the fundamental frequency is
,
is
Example(4)
• Show that
is aperiodic, but
is periodic with fundamental
period
Example 1):
Example 2):
Solution to Example 2
• Is there an integer
such that:
other words:
? In
, for some integer ?
Multiply both sides by
Now, what is (or is there?) the smallest integer that satisfies this relation
for all integer values of , i.e.,
is divisible by
Claim
; because:
is divisible by 16 for all
integer values of , and it is the smallest (check)!
Hence, the DT signal is periodic and the fundamental period is
Periodicity and Scaling
• If
• If
is periodic with fundamental period
is periodic with fundamental period of
.
is periodic with fundamental period
, is periodic and the fundamental period is the smallest
positive integer such that
is divisible by .
Example: For a periodic
with =6,
is periodic
with fundamental period
, because is the smallest positive
integer such that
is divisible by =6.
Even and Odd Signals
• CT:
• DT:
is even if
is odd if
] is even if
] is odd if
=
=
=
𝑥 𝑡 = 𝑡 − 40
CT_even
𝑥 𝑡 = 0.1𝑡
CT_odd
=
• Other than all-zero signal, no other
signal is both even and odd.
𝑥 𝑡 =𝑒 .
CT_neither even nor odd
Signal Decomposition
• Any CT signal
can be decomposed in even part
odd part
as:
+
,
where:
Proof: simply substitute for
and
and
.
• Any DT signal
can be decomposed in even part
odd part
as:
+
,
where:
Proof: similar.
and
and in
and in
Signal Decomposition
• Example:
𝑥 𝑡
𝑥 𝑡
1⁄2
1
1
𝑡
-1
𝑡
1
𝑡
𝑥 𝑡
𝑥 −𝑡
1⁄2
1
-1
1
𝑡
-1
-1⁄2
Right-sided and Left-sided signals
𝑥 𝑡
• Right-sided signal is zero for
𝑡
𝑇
• Left-sided signal is zero for
For any
positive or negative.
𝑥 𝑡
𝑇
𝑡
Unit Impulse Function
• Unit impulse
, a.k.a the Dirac-delta function, is defined
(mathematically) as:
•
, where
.
1
0
Representation of unit impulse
The number next to the impulse is its area.
• The unit impulse is not defined by its value, but is defined by how it acts
inside an integral when multiplied by a smooth function
as:
Choosing
we get
.
Narrow Pulse Approximation to Unit Impulse
• To get an intuitive picture, consider a set of rectangular pulses
having width of and height of so that all having an area of 1:
𝑝 (𝑡)
1
𝜖
• Then
=
𝜖
→
𝑡
, each
Intuitive Picture of Unit Impulse Definition
• As the rectangular pulse gets narrower and taller,
as a result, we get in the limit:
1
𝜖
𝑝 (𝑡)
𝑓(𝑡)
=
=
𝑡
𝜖
𝑓(𝑡) 𝑝 (𝑡)
𝑓(0)
𝜖
Furthermore:
𝜖
𝑡
Delayed Unit Impulse Function
• Similarly, for delayed unit impulse function
=
1
𝜖
:
𝑝 𝑡−𝑡
𝑓(𝑡)
𝛿 𝑡−𝑡
1
0
𝑡
𝑡
𝑡
𝜖
0
𝑡
=
→
→
𝑓(𝑡) 𝑝 𝑡 − 𝑡
𝑓(𝑡 )
𝜖
=
𝑡
0
𝜖
𝑡
Unit Step Function
• Unit step function is defined as:
• Unit step function is the integration of the unit impulse function:
1
1
0
Unit Impulse
Unit Step
Successive Integration of the Unit Impulse
Function
𝑛=6
1
1
0
Unit Impulse
Unit Step
1st Integration
Unit Step
𝑥 𝑡 = 𝑡𝑢(𝑡)
2nd Integration
Unit Ramp
1
𝑥 𝑡 = 𝑡 𝑢(𝑡)
2!
3rd Integration
Unit Parabola
𝑥 𝑡 =
1
𝑡
𝑢(𝑡)
𝑛−1 !
n-th Integration
In general
Discrete-Time Unit Impulse and Step Functions
𝛿[𝑛]
• DT unit impulse function (signal)
is defined as
• DT unit step function (signal) is
defined as
𝑢[𝑛]
Relationships Between Impulse and Step
Functions
•
•
=
• Also:
Properties of
and
Sampling property
•
•
Proof: Because it satisfies the
definition of
:
Proof:
=
• Special case (
• Similarly:
•
.
=0):
Last Lecture Overview
•
•
•
•
•
Periodic signals, some examples
Periodicity and scaling
Even and Odd signals and signal decomposition
Right-sided and left-sided signals
Unit Impulse and Unit Step Functions
• Definitions and properties
• Relations between the two
• Some important relations:
Sampling property:
Corollary:
)
Outline
• Representation property:
• Properties of unit impulse (continued)
• Sinusoidal signals
• Complex Exponentials
• Periodic Complex Exponentials
• CT vs DT Periodic Complex Exponentials
• Energy and Power
• Harmonically related Periodic Complex Exponentials
• Systems
• System Properties
Illustration of Sampling Property
𝑥(𝑡)
CT
𝑥[𝑛]
DT
3
2
𝑥(𝑡 )
𝑡
0
x
-4
𝑡
𝛿 𝑡−𝑡
x
=
𝑛
𝛿[𝑛 − 𝑛 ]
𝑛 =3
0
𝑡
𝑡
4
1
1
0
01
𝑥(𝑡)𝛿 𝑡 − 𝑡
=
𝑛
3
𝑥[𝑛]𝛿[𝑛 − 𝑛 ]
2
𝑛 =3
𝑥(𝑡 )
0
𝑡
𝑡
0
3
𝑛
Properties of
and
Corollary
•
)
Proof: Using the sampling property:
•
Proof: Using the sampling property:
=
)
].
).
• Special case (
• Special case (
):
)
• In general:
• In general:
𝑥 𝑡 𝛿 𝑡 − 𝑡 𝑑𝑡 =
𝑥(𝑡 ),
0,
if 𝑡 ∈ [𝑎, 𝑏]
if 𝑡 ∉ [𝑎, 𝑏]
):
Sifting (Representation) property of
+
• Proof: Noting that the only non-zero term in this sum occurs when
=
, i.e.,
,
we can write:
=
Example (unit step function):
.
• An illustrative Example:
understanding sifting property
using sampling property
+
𝑥 −3
𝑥0
𝑥3
+
.
.
.
𝑛 = −2
.
.
.
+
𝑛 =3
+
𝑛 =4
Sifting (Representation) property of
𝛿 𝑡 − 𝜏 = 𝛿 −𝜏 + 𝑡
• Proof: Note:
=
=
• Example (unit step function):
1
0
𝑡
𝜏
Properties of
Sifting property (more result)
• Using
:
:
because
0 for
Using change of variables of
Also, using the fundamental theorem of calculus:
Properties of
and
Representaion property (Importance)
• Why do we use such rather complex representation for
in terms of impulse functions?
and
Because, these representation are central for deriving the
Convolution Integral and Convolution Sum, that enable us to
determine the response of a linear time-invariant system to any input
signal from its response to impulse functions (to be studied later in
this class).
sinusoidal signals
•
or
,
cos (𝜃)
where is in seconds, is in
radians/second and is in radians.
It is common to write:
,
where the unit of is cycles/second
or Hertz (Hz).
The sinusoidal signal
is
periodic with fundamental period
(Why?)
s the fundamental frequency in radian/Second.
s the fundamental frequency in Hertz.
𝑇 =
2𝜋
𝜔
𝐴=1
𝜃 = 0.8
sinusoidal signals
(meaning of
•
slows down the rate of oscillation (increases the fundamental
period)
•
•
Exactly the opposite happens.
is constant, i.e., zero rate of oscillation, and the fundamental
period is not defined (i.e., could be any value!).
Last Lecture Overview
• Unit Impulse and Unit Step Functions
• Some important relations:
Sampling property:
)
Representation property:
( )
Sinusoidal signals
Outline
• Complex Exponentials
• Periodic Complex Exponentials
• CT vs DT Periodic Complex Exponentials
• Energy and Power
• Harmonically related Periodic Complex Exponentials
• Systems (First: a review of signals)
Complex Exponentials
• Definition (CT):
, for all
s
and
Complex numbers
• The general form of
are
:
Euler’s Formula
• Definition (DT):
for all
and
Complex numbers.
• The general form of
•
are
:
Euler’s Formula
CT Complex Exponentials with Real
1
𝑥 𝑡 = 𝑒
2
• Real s (s
𝐶 = 1; 𝜎 = −
:
1
2
1
𝑥 𝑡 = 𝑒
2
𝐶 = 1; 𝜎 =
1
2
A family of real exponential functions.
• Imaginary s (s
ℛ𝑒{𝑥 𝑡 }
)
ℑ𝑚{𝑥 𝑡 }
𝐶=1
𝜔 = 2𝜋
𝜎=0
=
:
A family of sinusoidal functions.
• Complex s (s
ℛ𝑒{𝑥 𝑡 }
(
)
=
:
A family of damped sinusoidal functions.
ℑ𝑚{𝑥 𝑡 }
𝐶=1
𝜔 = 2𝜋
1
𝜎=−
2
DT Complex Exponentials:
•
:
A family of Sinusoidal real and imaginary parts, not necessarily periodic.
•
is a growing exponential in DT
A family of exponentially growing sinusoidal real and imaginary parts.
•
is a decaying exponential in DT
A family of exponentially decaying sinusoidal real and imaginary parts.
Periodic Complex Exponentials
• Periodicity conditions for:
Is there an integer , such that:
?
•
Claim: periodic with fundamental period
Proof: Is
Yes, because:
Or:
)?
The fundamental frequency is:
.
)
Yes, if
Euler’s Formula
(
)
)
Periodic in time with the smallest period of
An immediate result:
(
?
,
Then, the condition for periodicity is
.
,
and with
is in its reduced form:
the fundamental period:
Periodicity in Frequency Domain for DT
Complex Exponentials
• DT complex exponentials is periodic in frequency with integer
multiples of
:
Proof:
For periodicity in frequency, we should show:
,
because:
This means:
,
The range of variations of can be limited to any real interval of length .
The notion of low and high frequency DT domain is different from CT domain.
• Note: Periodicity in frequency in DT cannot be extended to CT,
because:
CT versus DT Periodic Complex Exponentials (1)
𝟎 : Signals are all distinct for distinct values of
• CT)
.
𝟎 : Signals are not distinct, as the signal with frequency
• DT)
is
identical to the signals with frequencies
,
, ….., i.e.,:
,
Hence, in DT only a frequency interval of length
is considered and
usually intervals of
or
are used.
𝟎 : the larger the magnitude of
• CT)
, the higher is the rate of
oscillation in the signal.
𝟎 : rate of oscillation does not continually increase as the
• DT)
magnitude of
increases.
CT versus DT Periodic Complex Exponentials (2)
• DT)
𝟎
: variations in the rate of oscillation
Rate of oscillations
increases
Rate of oscillations
decreases
Constant signal:
𝒆𝒋𝝎𝟎𝒏 = 𝒆𝒋𝟎𝒏 =1
0
𝜋
Fastest Oscillations:
𝒆𝒋𝝎𝟎𝒏 = 𝒆𝒋𝝅𝒏 =(𝒆𝒋𝝅 ) = (−1)
Change of sign at each
point of time
2𝜋
Constant signal:
𝒆𝒋𝝎𝟎𝒏 = 𝒆𝒋𝟐𝝅𝒏 =1
𝜔
Energy and Power
• Periodic signals and in particular complex periodic exponential signal
are signals with infinite total energy and finite average power.
For
:
Then, total energy over
is infinite, i.e.,
But, the finite average power over one period:
.
=1.
Since, each period of signal are exactly the same, then averaging over multiple
periods yields 1, i.e.,:
Harmonically Related Complex Exponentials (1)
• Definition (CT): set of periodic complex exponentials, all of which are
periodic with a common period of .
For
to be periodic with period
,
Let us define:
Then, to satisfy
, i.e., for
,
, we must have
We say: a set of harmonically related complex exponentials is a set of
periodic exponentials with fundamental frequencies that are all integer
multiples of a single positive frequency
, and formally is shown as:
,
Harmonically Related Complex Exponentials (2)
Note that each
, i.e., the -th harmonic, in this set is periodic
with fundamental frequency
and fundamental period of
.
Note also that the -th harmonic
period , as well:
goes through exactly
interval of .
is still periodic with the
of its fundamental periods during any time
Illustration of Harmonics
𝜑 𝑡
𝜑 𝑡
𝜑 𝑡
𝜑 𝑡
Harmonically Related Complex Exponentials (3)
• Definition (DT): set of periodic complex exponentials, all of which are
periodic with a common period of
For
to be periodic with period
,
Hence, signals that are at frequencies with integer multiples of
, i.e.,
, form a set of harmonically related periodic complex exponentials:
,
Harmonically Related Complex Exponentials (4)
• Note: in DT, since
is also periodic in frequency domain, i.e.,
=
,
all of the harmonically related exponentials are not distinct.
Specifically:
Why are complex exponentials are so
important?
• The majority of signals can be represented as sum of basic complex
exponentials. Periodic complex exponentials are building blocks for
many other signals.
• Basic complex exponentials are eigenfunctions of a popular class of
systems called, linear time-invariant (LTI) systems.
• Computing the output signal of LTI systems is simple, if the input
signals can be represented as sum of of basic complex exponentials.
All to be seen later!
Systems
First: A Review of Last Material on Signals
Signals- A Review ( Basic Operations)
𝑥(𝑡)
• Time shift :
• CT:
• DT:
𝑥(𝑡 − 2)
)
𝑡
-6 -4 -2 0 2 4 6
• Time Reversal
• CT:
• DT:
-6 -4 -2 0 2 4 6 8
𝑥(−𝑡)
𝑥(𝑡)
𝑡
-6 -4 -2 0 2 4 6
• Time Scaling
•
,
•
1 : Decimated (speed up)
-6 -4 -2
•
: Expanded (slowed down)
𝑡
-6 -4 -2 0 2 4 6
𝑥(2𝑡)
𝑥(𝑡)
0 2 4 6
𝑡
𝑡
-6 -4 -2 0 2 4 6
𝑡
Signals- A Review ( Linear Combination of Basic Operations)
• Linear operation on the time variable as:
Recommended order of operation: Shift, Scale, Time Reverse (flip)
𝑥(𝑡)
-6 -4 -2 0 2 4 6
For
If
If
If
If
𝑥(𝑡 − 4)
𝑡
-6 -4 -2 0 2 4 6 8 10
𝑡
𝑥(2𝑡 − 4)
:
-6 -4 -2 0 2 4 6 8
𝑡
Signals- A Review (Decimation and Expansion in DT)
• Decimation:
M and : integers
M: decimation factor.
𝑥[𝑛]
𝑥[𝑛]
𝑛
𝑦 𝑛
𝑦 𝑛
𝑛
• Expansion:
𝑥[𝑛]
𝑛
=
𝑥[𝑛]
𝑦 𝑛
𝑦 𝑛
𝑛
Signals- A Review (Periodicity)
• Definition (CT): A CT signal is periodic if a constant
found such that
can be
• Definition (DT): A DT signal is periodic if an integer constant
can be found such that
• Some key relations to help to show periodicity or otherwise:
, for
, for
Signals- A Review (Periodicity – A nontrivial Example)
• Is
periodic? If so find
• Solution: Is there an integer
words:
(
(
such that:
)
)
? In other
, for some integer ?
Multiply both sides by
Now, what is (or is there?) the smallest integer
values of , i.e.,
is divisible by
that satisfies this relation for all integer
Claim
; because:
is divisible by 16 for all integer values of
, and it is the smallest (check)!
Hence, the DT signal is periodic and the fundamental period is
Example(4)
• Show that
is aperiodic, but
is periodic with fundamental
period
Example 1):
Example 2):
Signals- A Review (Odd and Even/Signal Decomposition)
• CT:
is even if
is odd if
=
=
• DT:
] is even if
] is odd if
=
=
𝑥 𝑡
• CT:
+
,
+
1
𝑡
-1
-1
1
𝑡
1
𝑡
𝑥 𝑡
𝑥 −𝑡
1
,
;
1⁄2
1
;
• DT:
𝑥 𝑡
1⁄2
𝑡
-1
-1⁄2
Signals- A Review (Unit Impulse and Unit Step Functions)
• CT:
, where
.
𝛿 𝑡
1
0
• CT:
• DT:
• DT:
1
𝑡
Signals- A Review (Properties of impulses)
• Sampling (CT):
A Result:
• Sampling (DT):
A result:
• Representation (CT):
• Representation (DT):
)
Signals- A Review (CT Complex Exponentials)
• The general form of
:
1
𝑥 𝑡 = 𝑒
2
𝐶 = 1; 𝜎 = −
ℛ𝑒{𝑥 𝑡
1
2
1
𝑥 𝑡 = 𝑒
2 1
𝐶 = 1; 𝜎 =
2
ℑ𝑚{𝑥 𝑡
𝐶=1
𝜔 = 2𝜋
𝜎=0
ℛ𝑒{𝑥 𝑡
ℑ𝑚{𝑥 𝑡
𝐶=1
𝜔 = 2𝜋
1
𝜎=−
2
Signals- A Review (DT Complex Exponentials)
• The general form of
𝒋𝜽
𝒋𝝎
•
𝒏
•
:
𝒏
𝒋𝜽
𝒏 𝒋𝝎𝒏
𝒏 𝒋(𝝎𝒏 𝜽)
𝒏
:
A family of Sinusoidal real and imaginary parts, not necessarily periodic.
•
is a growing exponential in DT
A family of exponentially growing sinusoidal real and imaginary parts.
•
is a decaying exponential in DT
A family of exponentially decaying sinusoidal real and imaginary parts.
Signals- A Review (Periodicity in Complex Exponentials: CT vs DT)
• CT ( 𝟎 ): Signals are all distinct for distinct values of .
• DT( 𝟎 ): Signals are not distinct, as the signal with frequency
is
identical to the signals with frequencies
,
, ….., i.e.,:
,
Hence, in DT only a frequency interval of length
is considered and
usually intervals of
or
are used.
• CT ( 𝟎 ): the larger the magnitude of , the higher is the rate of
oscillation in the signal.
• DT( 𝟎 ): rate of oscillation does not continually increase as the
magnitude of
increases.
Signals- A Review (Periodicity in DT Complex Exponentials)
• DT(
𝟎
): variations in the rate of oscillation
Rate of oscillations
increases
Rate of oscillations
decreases
Constant signal:
𝒆𝒋𝝎𝟎𝒏 = 𝒆𝒋𝟎𝒏 =1
0
𝜋
Fastest Oscillations:
𝒆𝒋𝝎𝟎𝒏 = 𝒆𝒋𝝅𝒏 =(𝒆𝒋𝝅 ) = (−1)
Change of sign at each
point of time
2𝜋
Constant signal:
𝒆𝒋𝝎𝟎𝒏 = 𝒆𝒋𝟐𝝅𝒏 =1
𝜔
Signals- A Review (Harmonically Related Complex Exponentials)
• CT: Set of harmonically related complex exponentials is a set of periodic
periodic exponentials with fundamental frequencies that are all integer
multiples of a single positive frequency
, and formally is shown as:
,
Note that each
, i.e., the -th harmonic, in this set is periodic with
fundamental frequency
and fundamental period of
.
Note also that the -th harmonic
as well:
.
goes through exactly
is still periodic with the period
,
of its fundamental periods during any time interval of
Signals- A Review (Harmonically Related Complex Exponentials)
• DT: Signals that are at frequencies with integer multiples of
, i.e.,
, form a set of harmonically related periodic complex exponentials:
,
• Note: in DT, since
is also periodic in frequency domain, i.e.,
=
all of the harmonically related exponentials are not distinct. Specifically:
Illustration of Harmonics
𝜑 𝑡
𝜑 𝑡
𝜑 𝑡
𝜑 𝑡
Signals and Systems
5CCS2SAS
Lecturer: M. R. Nakhai
Email: reza.nakhai@kcl.ac.uk
Outline
• Systems and Systems Properties
• Causality
• Linearity
• Time Invariance
• Linear Time Invariant (LTI) Systems
• Memoryless
• Invertibility
• Stability
• Convolution
Systems
• A system is a quantitative description of a physical process that
transforms Input Signals to Output Signals.
𝑥 𝑡
Input Signal
𝑦 𝑡
Continuous-Time (CT) Output Signal
System
𝑥[𝑛]
Input Signal
𝑦[𝑛]
Output Signal
Discrete-Time (DT)
System
System Representations
(Examples)
• CT System: Electric Circuit
• System: Image Distorting
Distorting
System
Input
Signal
Output
Signal
Representation: differential equation
𝑥[𝑛]
Input Signal
𝑦[𝑛]
Output Signal
Representation: difference equation
• Note: difference equation representation is not
helpful on its own for designing a distorting
system.
More new representations and tools are needed for system design and manipulations (Later)!
System Properties
1. Causal and Anti-causal Systems
(Definitions apply to both CT & DT systems)
• A system is causal if the output at time (or ) depends only on the
input at time
(or
), i.e., input in the past and/or present;
otherwise, the system is anti-causal.
𝑦 𝑡
𝑥 𝑡
System
• Examples:
causal/anti-causal/why?
causal/anti-causal
Why?
causal/anti-causal
Why?
:causal/anti-causal/why?
: causal/anti-causal/why?
Illustration example in CT
System Properties
1. Causal and Anti-causal Systems
(Definitions apply to both CT & DT systems)
• A system is causal if the output at time (or ) depends only on the
input at time
(or
), i.e., input in the past and/or present;
otherwise, the system is anti-causal.
• Examples:
𝑦 𝑡
𝑥 𝑡
System
Causal, because
output depends only on past input samples)
Causal
Anti-causal
: Anti-causal,
because
is a future input sample.
: Causal, because
is the input and
is a constant.
Example: illustration in CT
Remarks on Causality
• A system is causal if its output at any time depends only on the values
of its input up to that time.
In other words, a causal system is not an anticipator of the future values.
• In a causal system, effect (i.e., the output) occurs after the cause (i.e.,
the input).
• Example: an anti-causal economic system means its output as
investment decision today depends on tomorrow’s stock market prices!
• The causality of certain systems can be immediately determined from
the output to an impulse function as the input! How? (details later).
System Properties
2. Linear Systems (LS)
(Definitions apply to both CT & DT systems)
• Definition: A system is linear if it is
additive and scalable, i.e.,
for all input signals and all
• Illustration of Linearity
𝑥 𝑡
Linear
System
𝑦 𝑡
:
𝟏
𝟏
:
𝟐
𝟐
.
𝑥 𝑡
• Examples:
I.
Is the system
linear?
II.
Is the system
linear?
𝑎𝑥 𝑡 + 𝑏𝑥 (𝑡)
Linear 𝑦 𝑡
System
Linear
System
𝑎𝑦 𝑡 + 𝑏𝑦 (𝑡)
: 𝒂𝒙𝟏 𝒕 + 𝒃𝒙𝟐 (𝒕) ⟶ 𝒂𝒚𝟏 𝒕 + 𝒃𝒚𝟐 (𝒕)
Example-1
• Is the system
• Solution: Let
and
𝟏
𝟏
𝟐
𝟐
linear?
be the inputs. The resulting outputs are:
:
:
Find the response to the input
,
,
which can be written as:
Means:
𝟏
𝟐
Hence, the system is Linear.
𝟏
𝟐
which is:
Example-2
• Is the system
• Solution: Let
and
𝟏
𝟏
:
𝟐
𝟐
:
linear?
be the inputs. The resulting outputs are:
The response to the input:
is:
Clearly:
Means: 𝟏
𝟐
𝟏
Hence: the system is not Linear.
,
𝟐
,
which can be written as:
Systems (Causality; Linearity: a quick review)
• Causality: A system is causal if the output at time (or ) depends
only on the input at time
(or
), i.e., input in the past
and/or present; otherwise, the system is anti-causal.
• Linearity: A system is linear if it is additive and scalable, i.e.,
for all input signals and all
.
Key Property of Linear Systems:
Superposition
• In general:
CT LS:
If :
, then:
DT LS:
If :
, then:
For all
.
System Properties
3. Time-Invariance (TI)
(Definitions apply to both CT & DT systems)
• Definition: A system is time invariant if any time-shift in any input
signal results in the same time-shift in the output signal.
Mathematically:
A CT system is TI:
If
Then
for any real
A DT system is TI:
If
Then
for any
If:
𝑥 𝑡
𝑥[𝑛]
,
TI
System
𝑦 𝑡
𝑦[𝑛]
Then:
𝑥 𝑡−𝑡
,
𝑥[𝑛 − 𝑛 ]
TI
System
𝑦 𝑡− 𝑡
𝑦[𝑛 − 𝑛 ]
Illustration of TI system
Example-1
• Is the system described as
• Solution:
:
Let
be the input. Then:
From system description, we can write:
Clearly:
Means:
Hence, the system is time-invariant.
time-invariant?
Example-2
• Is the system described as
• Solution:
:
Check if
produces
time-invariant?
?
Finding a counter-example that violates above TI condition is enough:
Let
, for all (why?)
𝑥 𝑛 𝛿 𝑛−𝑛 =
For
:
𝑥 𝑛 𝛿 𝑛−𝑛
(Why?)
(Why?)
Means:
, for all inputs; hence, the system is not TI.
A Fact in TI Systems
• If the input signal to a TI system is periodic, then the output signal is
also periodic with the same period as the input signal.
Proof (is for CT and for DT is similar by following similar steps):
T
,
Periodicity implies:
)
, where is the period of
By TI condition:
)
But
)
that produces
Hence,
is periodic with the same period
.
Example (multiplier)
• Consider a multiplier system, multiplying the input signal
another signal g and producing an output signal
, as:
g(𝑡)
𝑥(𝑡)
×
y(𝑡)=𝑥(𝑡)g(𝑡)
a) Is this system linear? Why?
b) Is this system time invariant? Why?
by
Solution-(a)
• Let
and
be the inputs. The resulting outputs are:
𝟏
𝟏
:
𝟐
𝟐
:
The response to the input:
is:
,
,
which can be written as:
Means:
𝟏
𝟐
Hence, the system is Linear.
𝟏
𝟐
Solution-(b)
•
Let
Then:
:
be the input.
)
From system description, we can write:
)
Clearly:
Means:
Hence, the system is not Time-Invariant
and it is time-arraying!
)
Example (adding a constant)
• Consider the system:
,
where
is a constant.
Is this system linear? Why?
Solution (adding a constant)
• Let
and
be the inputs. The resulting outputs are:
𝟏
𝟏
:
𝟐
𝟐
:
The response to the input:
is:
Clearly:
Means:
𝟏
𝟐
𝟏
Hence, the system is not linear.
,
𝟐
Linear Time-Invariant (LTI) Systems
• A powerful model for analyzing the behavior of many practical systems
• A key fact: Given the response of an LTI system to some inputs, we can find
its outputs to many other signals.
• Example: In an LTI system:
Given:
Input
LTI System
Output
:
Find the system output to
, i.e.,
=?:
?
Solution
Given:
• Describe
in terms of
using scaling, addition
and time- shift:
• Apply LTI properties to
find the output:
=
System Properties
4. Memoryless
• Definition: A system is memoryless if the output at any continuous
time t (or any discrete time n) depends only on the input at the same
time t (or n).
• Examples:
Is the system
Is the system
Is the system
memoryless? Why?
memoryless? Why?
memoryless? Why?
System Properties
4. Memoryless
• Definition: A system is memoryless if the output at any continuous time (or
any discrete time n) depends only on the input at the same time (or n).
• Examples:
Is the system
memoryless? Why?
The system is memoryless, because the output
at any time depends only on the
input
at the same time there are no terms like
or
etc. in the
CT system description.
Is the system
memoryless? Why?
The system is not memoryless, because the output at any time depends also on the
input at time
i.e., because of the term
in the DT system description.
Is the system
memoryless? Why?
Yes; because the output at any time
depends only on the input at the same time .
System Properties
5. Invertible
• Definition: A system (CT or DT) is invertible if there is a one-to-one mapping
from any set of distinct input signals to a set of distinct output signals.
• How to prove the invertibility of a system?:
Find an inverse formula (description) from the output to the input, i.e.,
describing the input
as a function of the output
.
• How to prove the non-invertibility of a system?:
Give a counter example, violating the invertibility definition (above).
• Examples:
Is the CT system
invertible? Why?
Is the DT system
invertible? Why?
Is the system
invertible? Why?
System Properties
5. Invertible (Examples)
Is the CT system
invertible? Why?
Yes; because by rearranging the terms, we can write the input
in terms of) the output
, as:
Is the DT system
invertible? Why?
Yes; because by rearranging the terms, we can write the input
the output
, as:
Is the system
as a function (or
as in terms of
invertible? Why?
Using a counter example we can show that the system is not invertible: Consider
two distinct inputs
and
. Since the outputs to
are not distinct, i.e.,
the system is not invertible.
System Properties
5. Stable
• We say that a signal
(or
in DT) is bounded if there exists a finite
constant
, such that
for all .
• Definition: A system (CT or DT) is stable if the output signal
to any
bounded input signal
, i.e.,
for all , is always bounded, i.e.,
, for all .
• Example 1: Is the system
stable? Why?
Consider a bounded input
for all .
such that
for all and find out if
Triangle Inequality
since for any bounded input the output is always bounded, the system is stable.
,
System Properties
5. Stable (Example)
• Example 2: Is the system
stable? Why?
Consider a unit step function as the input, i.e.,
]. Clearly,
for all , and, hence, is bounded. Then:
=
Hence, as
.
Since the output is not bounded for the bounded input, the system is
not stable.
Fundamental Property of LTI Systems
• DT Systems: For an arbitrary input signal
by (Convolution/Superposition Sum):
, the output signal
is given
,
where
is the unit impulse (sample)response of the LTI system (i.e.,
the response of the system to a unit impulse (sample) input).
• CT Systems: For an arbitrary input signal
by (Convolution/Superposition Integral):
, the output signal
is given
,
where
is the unit impulse response of the LTI system (i.e.,
response of the system to a unit impulse input).
is
is the
Signals and Systems
5CCS2SAS
Lecturer: M. R. Nakhai
Email: reza.nakhai@kcl.ac.uk
Last lecture Review
• Systems Properties
• Causality
• Linearity
• Time Invariance
• Linear Time Invariant (LTI) Systems
• Memoryless
• Invertibility
• Stability
• So far we have seen basic definitions and how to apply
them to several examples directly.
• We will see how we can find out these properties simply
from impulse response of the system!
• Convolution Sum
Outline
• Convolution Sum
• Derivation of Convolution Sum
• How to compute convolution sum?
• DT convolution properties
• Convolution Integral
• Derivation of convolution integral
• How to compute convolution integral?
• CT convolution properties
Derivation of Convolution Sum
1. A Reminder: Sifting Property of
+
Hence:
: Basic signal
Coefficient (value of
at time
• An illustrative example
for sifting property:
.
.
.
Coefficient
Basic Signal
x[ 1]
[ n 1] = 𝑥 −1 ,
0,
𝑛 = −1
𝑛 ≠ −1
Impulse (Time-shifted to
value of signal at
)
Derivation of Convolution Sum
Definition of Impulse response:
DT LTI
Time-Invariance Property of LTI Systems :
DT LTI
Scaling Property of LTI Systems :
DT LTI
The Additivity/Superposition
Property of LTI Systems:
(Sifting/Representation Property )
DT LTI
Convolution Sum and Notations
• We denote convolution sum in a DT LTI
system as:
DT LTI System
• In general, the convolution sum between two DT functions
is defined and denoted as
and
How to Compute the Convolution Sum:
?
1. Rewrite/replot
and
in terms of , i.e., replace
form
and
, because the summation is on
2. For each and every value of ,
, do:
with
to
1. Obtain
, using the techniques we have seen in earlier lectures
2. Calculate
for all values of
3. Calculate the sum on , i.e.,
3. Go to step 2 and repeat steps 2.1, 2.2, 2.3 and step 3 for another
value of , until the output signal
is calculated for all values of ,
.
How to Compute the Convolution Sum?
• Example: Given the input
system, find the output
and the impulse response
in a DT LTI
.
ℎ[𝑛]
𝑥[𝑛]
𝑛
𝑛
For
Add for
Multiply
Flip (reverse in time)
ℎ[𝑘]
𝑥[𝑘]
𝑘
Replot and versus
because the summation is over
𝑘
How to Compute the Convolution Sum?
ℎ[𝑛]
𝑥[𝑛]
𝑛
𝑛
ℎ[𝑘]
𝑥[𝑘]
𝑘
𝑘
Add for
Multiply
Shift + Flip
• Illustration of calculation of
at
in more details as an example:
1 1
0 0 0 2 1 0 0 0
DT Convolution Properties
• Commutative:
• Associative:
DT Convolution Properties
• Distributive:
DT Convolution Properties
• Delay Accumulation:
If :
Then:
,
for any
Proof:
and
.
Let a change of variable:
then:
,
.
Derivation of Convolution Integral for CT systems
Staircase Approximation Model (1)
Recall our tall narrow pulse intuitive picture of CT impulse function
, where
=
:
𝑝 (𝑡)
1
𝜖
𝜖
𝑡
Derivation of Convolution Integral for CT systems
Staircase Approximation Model (2)
1
𝜖
1
𝜖
(Definition)
CT LTI
(TI)
CT LTI
𝜖
𝑘𝜀
𝑘𝜀
𝑥 𝑘𝜀
𝑥 𝑡
(Scaling)
𝑘𝜀
𝑥 𝑘𝜀
CT LTI
(Superposition)
CT LTI
𝑘𝜀
Derivation of Convolution Integral for CT systems
Staircase Model in Limit (when
)
• Staircase approximate input model
• Resulting approximate output
:
• In limit, as
:
:
,
and finally
Sifting/Representation
Property of
:
Convolution Integral:
,
,
leading to:
,
Convolution Integral and Notations
• We denote the convolution integral
in a CT LTI system as:
CT LTI System
• In general, the convolution integral between two CT functions
and
is defined and denoted as:
Review of last lecture
• Convolution sum and computation
• Properties of DT convolution
•
•
•
•
Commutative
Associative
Distributive
Delay Accumulation
• Convolution integral a
• Derivation through staircase model in limit
Outline
• How to compute convolution integral?
• CT convolution properties
• LTI system properties and impulse response
• Fourier Series
• Eigenfunctions and Eigenvalues of LTI systems
• Transfer Function
• Frequency Responce
How to Compute the Convolution integral:
?
1. Rewrite/replot
and
in terms of , i.e., replace with to
form
) and
), because the integration is on
2. For each and every value of ,
, do:
1. Obtain
, using the techniques we have seen in earlier lectures
2. Multiply to obtain
over all values of
3. Integrate on to calculate the integral
3. Go to step 2 and repeat steps 2.1, 2.2, 2.3 and 3 for another value of
, until the output signal
is calculated for all values of ,
.
How to Compute the Convolution integral?
• Example: Find the output of an LTI system with impulse response
to the input signal
Solution: We identify cases in terms of , where the integrand can be
expressed similarly in shape for all values of
Case 1:
, where the integrand
, hence:
,
Case 2:
, where
Therefore, the output for
hence:
:
Graphic illustration
of the solution,
with
Case 2:
Case 1:
−𝟏
𝟏
How to Compute the Convolution integral?
• Another example (to be tried by you):
Find the output of an LTI system with impulse response
to the input signal
Hint:
find the output with impulse response
then advance the
output in time by . (using the commutative property of convolution
and the time-invariance property of LTI systems, convince yourself why
this can be done!)
CT Convolution Properties
• Commutative:
• Associative:
CT Convolution Properties
• Distributive:
CT Convolution Properties
• Delay Accumulation:
If :
Then:
for any
,
and
.
LTI System Properties in terms of Impulse Response
1. Memoryless
• An LTI system is memoryless if and only if, for some scaling factor
DT:
CT:
0
0
DT
CT
LTI System Properties in terms of Impulse Response
2. Causal
• An LTI system is causal if and only if:
DT:
, for all
CT:
, for all
,
,
for all
for all
DT
CT
LTI System Properties in terms of Impulse Response
3. Stable
• An LTI system is stable (i.e., bounded inputs result in bounded
outputs) if and only if
DT:
CT:
0
DT
0
CT
LTI System Properties in terms of Impulse Response
4. Invertible
• An LTI system with impulse response
[or
] is invertible if and
only if there exists another LTI system with impulse response
[or
, such that:
DT:
CT:
Inverting of an LTI system is also referred to as Deconvolving.
Illustration of Invertibility in LTI systems:
Convolving
(Filtering)
Deconvolving
(Un-Filtering)
Associativity of
Convolution
Invertibility Condition
LTI System Properties in terms of Impulse Response
4. Invertible (Example)
• Given an LTI system with impulse response
as:
1
Convolving (Filtering)
Show that
can be recovered from
Deconvolving
(Un-Filtering)
Show:
using
, as:
?
......
LTI System Properties in terms of Impulse Response
4. Invertible (Example Solution)
.
But, using the sampling property of
.
Alternative solution using transform to come Later!
Stability, Causality, Invertibility
More Questions
• Is the LTI system with impulse response
causal? Why?
stable,
• Is the LTI system with impulse response
stable, causal? Why?
• Back to the earlier Invertible (Example):
Is
the inverse system of the LTI system
Why? (i.e., Is the expression
?
true?)
Memoryless Proof (DT LTI systems)
If
, then for any input
:
.
If the system is memoryless:
The output
depends only on the current input, i.e., does not depend on
In
, where
.
, or equivalently,
is a scaling factor. Hence:
, which implies:
Causality Proof (DT LTI systems)
• Input/Output (I/O) relation in an LTI system:
For causality,
cannot depend on
Then:
, for
, (or
Let
, then:
, for
Conversely: If
, for
Let
, then:
Hence,
only depends on
for
.
).
, then:
.
.
for
.
Stability Proof (DT LTI systems)
• Let
where
, then for any bounded input, i.e.,
is a constant upper bound:
is bounded.
Conversely, we can show that with
input, e.g., for
, for a bounded
such that:
is unbounded!
Fourier Series
Eigenfunctions and Eigenvalues of LTI Systems
• Objective 1: We would like to identify a set of signals {
that:
, such
• AS each -th signal of this set, denoted as
, passes through any LTI
system, the produced output is the same signal scaled by a scale factor,
denoted by
LTI
• Then:
• Definition: We say that
is the -th eigenfunction of the LTI
system and the scaling factor is the -th eigenvalue of the LTI
system.
Eigenfunctions and Eigenvalues of LTI Systems
• Objective 2: We want to represent any signal
of eigenfunctions, as:
as a linear combination
,
where ’s are scalers.
Then, using the superposition property of LTI systems, the output
any LTI system to the input
can be determined:
LTI
Hence, the solution to finding the response of LTI systems is to how to
determine eigenvalues .
of
Eigenfunctions of LTI Systems – CT Case
• Insightful examples for some specific LTI systems:
Any function is an eigenfunction for the LTI system with impulse
response
:
Any periodic function with period is an eigenfunction for the delay
introducing LTI system with impulse response
:
Eigenfunctions of LTI Systems - Complex
Exponentials
• Complex exponential function
, where
is an
eigenfunction of any continuous-time LTI system, and
,
where
is the impulse response of the system, is the corresponding
eigenvalue.
is known as the transfer function of the CT LTI system.
is defined by impulse response,
, of the system, but, is independent
of time variable and is a function in , only.
Hence,
can be regarded as a scaler in time-domain.
Eigenfunctions of LTI Systems - Proof
• The proof is simple and straightforward:
:
Commutative Property of convolution
Eigenvalue,
known as
Transfer Function
Eigenfunction
Signals and Systems
5CCS2SAS
Lecturer: M. R. Nakhai
Email: reza.nakhai@kcl.ac.uk
Office: S2.09
Office Hours: Thursdays 2:30pm-4:30pm
Review of last lecture
• Proved convolution integral and formulated its computation steps
(step-by-step) and demonstrated by solving a sample example in CT
domain.
• Described CT convolution properties (Commutative, Associative,
Distributive, Delay Accumulation)
• Restated the LTI system properties (Memoryless, Causality, Stability,
Invertibility) in terms of impulse response of the system in the form of
theorems with proofs.
• Introduced the important concept of Eigenfunctions and Eigenvalues
of both DT and CT systems.
Complex exponentials are the eigen functions of the LTI systems
Outline
• Eigenfunctions and the importance of eigenfunctions (with examples)
• Frequency response of LTI systems
• Fourier Series representation of periodic signals
• Examples
Importance of eigenfunction – An Example
• Let
with impulse response
be the input to an LTI system
. Find the expression for the output
• Using the eigenfunction effect:
)
)
)
• Using the superposition property of LTI systems:
)
)
)
Eigenfunctions of LTI Systems - A Special Case
• Subclass of periodic complex exponentials:
, i.e., when
.
: Frequency Response
is known as the frequency response of the CT LTI system.
is periodic with period
, where:
is radian frequency and
is the frequency in cycles per second (Hz).
Eigenfunctions of LTI Systems – DT Case
• DT complex exponential function , where
, is an eigenfunction
of any discrete-time LTI system, and
, where
is the discrete-time impulse response of the system, is the corresponding
eigenvalue.
is known as the transfer function of the DT LTI system.
is defined by impulse response,
, of the system, but, is
independent of time variable and is a function in , only.
Hence,
can be regarded as a scaler in time-domain.
Eigenfunctions of LTI Systems - Proof
• Following the same steps of proof as in CT:
: Commutative Property of convolution
Eigenvalue,
known as
Transfer Function
Eigenfunction
Eigenfunctions of LTI Systems - A Special Case
• Subclass of periodic complex exponentials (
, where
):
is an integer and
: Frequency Response
is known as the frequency response of the DT LTI system.
is periodic with period in discrete-time domain
is also periodic with period of
in frequency
Importance of eigenfunction – An Example
• Let
with frequency response
output
. be the input to an LTI system
, plotted below. Find the expression for the
• Using the eigenfunction effect:
1
− 𝜋-
-
-
𝜋
Importance of eigenfunction – An Example
(continued)
• Using the superposition property of LTI systems:
2
.
Importance of eigenfunction – Summary
CT:
DT:
CT and DT Fourier Series for Periodic Signals
• We now focus on a restricted set of complex exponential
functions(eigenfunctions):
CT:
, when
:pure imaginary
i.e., signals of the form:
DT:
, when
with
: pure phase (
i.e., signals of the form:
with
)
CT Fourier Series Representation of Periodic Signals
• Fourier series expansion of periodic signal
:
: Fundamental period (i.e., the smallest)
: Fundamental (radian) frequency
Representation of
exponentials:
,
as a linear combination of restricted complex
: Fourier Series coefficients
The complex coefficient
fundamental frequency
measures the portion of
.
that is at the -th harmonic of the
) : indicates DC (constant) component of
.
: first harmonic index;
: second harmonic index; etc…….
Computation of Fourier Series Coefficients
• Objective: Given
, calculate Fourier Series coefficients:
: Multiply both sides by
:
]
Integrate both sides over one period:
Fourier series Pair in CT Domain:
• With
, where
is the fundamental period of signal
Synthesis Equation:
Analysis Equation (integration is taken over any period interval):
:
Example 1
• Periodic square signal with fundamental period
and
:
For
For
:
:Average or DC component of
Example 2
• Consider a periodic signal as
t):
Using Euler’s formula:
Identify fundamental period and frequency:
(achieved with
(achieved with
Hence (using least common multiple):
and hence:
Therefore:
: (for
: (for
No DC component
and
: (for
: (for
The rest of coefficients are zeros.
Example 3 (An exercise for you)
• Repeat example 2, for
First: find the fundamental period and frequency.
Second: use Euler’s formula to write
in terms of complex
exponentials with harmonics of the fundamental frequency.
Finally, find the Fourier series coefficients using the synthesis
equation.
Signals and Systems
5CCS2SAS
Lecturer: M. R. Nakhai
Email: reza.nakhai@kcl.ac.uk
Office: S2.09
Office Hours: Thursdays 2:30pm-4:30pm
Review of Last Lecture
• Importance of eigenfunctions
• Frequency response of LTI systems
• Fourier series of CT periodic signals
Importance of eigenfunction – Summary
CT:
DT:
Fourier series Pair in CT Domain:
• With
, where
is the fundamental period of signal
Synthesis Equation:
Analysis Equation (integration is taken over any period interval):
:
Example 4: Impulse Train or Sampling Function
• Find the Fourier Series coefficients for the Periodic train of impulses:
1
⋯
-2𝑇
-𝑇
0
⋯
𝑇
2𝑇
The fundamental period of
is
(why?).
Outline
• The use of Fourier series
• Fourier series representation of discrete-time periodic signals
• Finding Fourier series coefficients (Example)
• Properties of Fourier series
• Fourier transform
• Fourier transform of non-periodic continuous-time signals
(An introduction)
The Use of Fourier Series
Harmonically Related Complex Exponentials
(Reminder of a previous lecture topic)
•
is periodic with period and fundamental frequency
• Consider the set of all following signals with period :
,
All of these signals that are at frequencies with integer multiples of , i.e.,
, form a set of harmonically related periodic complex exponentials.
In DT, since
is also periodic in frequency domain, i.e.,
=
all of the harmonically related exponentials are not distinct. Specifically:
Harmonically Related Complex Exponentials
(Reminder of a previous lecture topic)
• And in more general term:
, where is any integer number
For instance:
,
, and so on……
• Let us consider the representation of a periodic DT signal
linear combination of signals
, as:
in terms of a
Since
signals are distinct only over a range of N successive values of
the summation need only include terms over this range, indicated as:
DT Fourier Series Representation of Periodic DT Signals
• Let
By
be periodic with fundamental period , represent
, we mean that , for example, could take values:
, or
, etc…..
Now, the question is: what are the
coefficients?
The key fact to find the
coefficients:
as:
DT Fourier Series Representation of Periodic DT Signals
The first line is obvious.
The second line can be easily seen using the following general formula:
Back to the question:
Multiplying by
, summing over
and rearranging, we have:
DT Fourier Series Representation of Periodic DT Signals
Hence (changing variable to
for consistency in representations):
Fourier series Pair in DT Domain:
• With
, where
is the fundamental period of signal
:
Synthesis Equation:
Analysis Equation:
it makes no difference which sample to be the first one in summation.
DT Fourier series Coefficients
• The
coefficients are also referred to as the spectral
coefficients of the periodic signal
.
• The
coefficients decomposes a periodic signal
with
period into sum of harmonically-related complex
exponentials.
•
: Distinct for DT. There are only distinct samples or
pieces of information, in time-domain, i.e.,
, or
in frequency-domain.
• Hence, only any consecutive values of
coefficients are used
in the synthesis equation.
Example: Finding DT Fourier series coefficients
1
Factorise exponential with half of
the exponent to build Sine function
2𝜋
𝜔 =
𝑁
(
(
)
.
For
Using:
∑
:
.
)
𝑎 =
, 𝑎≠1
Properties of Fourier Series
See Table 3.1 (P208) & Table 3.2 (P223)
• Linearity:
(CT) If:
Then:
(DT) If:
Then:
Properties of Fourier Series
• Time Shift:
(CT) If:
Then:
(DT) If:
Then:
Note:
CT:
, where
is the fundamental period of signal
DT:
, where
is the fundamental period of signal
Properties of Fourier Series
• Time Reversal:
(CT) If:
Then:
(DT) If:
Then:
• Proof (CT): Let g
Let:
g
Properties of Fourier Series
• Conjugation:
(CT) If:
Then:
(DT) If:
Then:
• Proof (CT): Let g
Let:
g
Properties of Fourier Series
• Multiplication:
(CT) If:
Then:
(DT) If:
Then:
Properties of Fourier Series
• Differentiation and Integration:
If:
Then:
If:
Then:
For a proof simply apply differentiation and integration, respectively,
to both sides of synthesis equation.
Properties of Fourier Series
• Parseval Relation:
(CT) If:
Then:
(DT) If:
Then:
• Outline of proof:
Fourier Transform
Continuous-Time (CT) Fourier Transform (F.T.)
• Fourier series analysis requires two conditions to be held:
• The signal must be periodic:
There exists a
such that
.
• The magnitude square of the signal must be integrable:
• Now, the question is:
What about non-periodic signals?
Observations from Fourier Series (F.S.)
𝑇
𝑘=1
(𝜔 = 𝜔 )
(
)
(
)
: Envelop of the scaled F.S. coefficients
As
,
whilst the shape of the envelop remains
unchanged.
As
, F.S. coefficients
𝑇
approaches the envelop of
.
𝑘=2
(𝜔 = 2𝜔 )
(
)
Envelop of the scaled F.S. coefficients
: Discrete Frequency points
As T increases, discrete frequency points
become more densely populated in continuous
frequency points in .
Finally: as
,
.
Non-Periodic Signals
• Non-periodic signal
can be treated as a periodic signal with
• The corresponding F.S. coefficients approach to the envelop function
.
•
is called Fourier transform of the non-periodic signal
.
.
Derivation of Fourier Transform (1)
𝑥(𝑡)
Express periodic
in F.S.:
,
,
where,
-𝑇
𝑡
𝑇
𝑥 (𝑡)
-𝑇
-𝑇
𝑇
𝑇
Identical
𝑇
𝑇
here
−
2
Define:
Then:
2
𝑡
Derivation of Fourier Transform (2)
Periodicity in large period limit:
For
,
Substitute for
In limit as
in the synthesis equation of
;
:
Fourier Transform & Inverse Fourier Transform
• Also, we say
• Sone notations:
Fourier Transform Pair:
Signals and Systems
5CCS2SAS
Lecturer: M. R. Nakhai
Email: reza.nakhai@kcl.ac.uk
Office: S2.09
Office Hours: Thursdays 2:30pm-4:30pm
Review of Last Lecture
• Fourier series representation of discrete-time periodic signals
• Properties of Fourier series (CT & DT)
• Fourier transform of non-periodic continuous-time signals
• Derivation of CT Fourier transform
Review of Last Lecture
Fourier series Pair in CT Domain:
• With
, where
is the fundamental period of signal
:
Synthesis Equation:
Analysis Equation (integration is taken over any period interval):
Fourier series Pair in DT Domain:
• With
, where
is the fundamental period of signal
:
Synthesis Equation:
Analysis Equation:
it makes no difference which sample to be the first one in summation.
CT: Fourier Transform & Inverse Fourier Transform
• Also, we say
• Sone notations:
Fourier Transform Pair:
Outline
• Fourier transform and Fourier series
• Relation between Fourier transform and Fourier series coefficients
• Fourier transform (Continued):
• Examples
• Properties of Fourier transform
• Examples
Fourier Transform(F.T.) and Fourier Series (F.S.)
• Fourier transform applies to both periodic and non-periodic signals, whereas,
Fourier series applies only to periodic signals.
• Relation between F.T. and F.S.:
Let
be periodic with fundamental frequency
, where is
fundamental period. Then:
Now, apply F.T. to
Relation between F.T. and F.S (continued)
To justify the last equality:
Relation between F.T. and F.S (continued)
The Fourier transform
of a periodic signal
with
Fourier series coefficients
is:
A train of impulses occurring at the harmonically-related
frequencies
for which the area of the impulse at
the -th harmonic frequency
is
times the -th
Fourier series coefficient .
Relation between F.T. and F.S
(A Summary)
• Fourier transform applies to both periodic and non-periodic
signals, whereas, Fourier series applies only to periodic
signals.
• Relation between F.T. and F.S.:
Let
where
be periodic with fundamental frequency
is fundamental period. Then:
,
Fourier Transform Examples
Example 1: Impulse
Example 2: Shifted Impulse
Example 3: Find the Fourier transform of
the signal
.
1
(
)
-
Example 4
Fourier transform of a square pulse:
Properties of Fourier transform
Linearity:
If:
Then:
Time Shift:
If:
Proof:
then:
let
, then:
Properties of Fourier transform
• Interpretation of
If:
in time-shift property:
then:
Hence, a time-shift in time-domain contributes to a linear phase-shift
in frequency-domain.
The magnitude of Fourier transform remains unchanged.
Linearity + Time-shift (An example)
Example 5: Find the Fourier transform of
Express
.
1.5
1
in terms of square pulses:
0
2
1
3
4
3
4
3
4
1
Use the linearity and time-shift properties:
- 1.5
0
1.5
1
(
)
- 0.5 0 0.5
2
Signals and Systems
5CCS2SAS
Lecturer: M. R. Nakhai
Email: reza.nakhai@kcl.ac.uk
Office: S2.09
Office Hours: Thursdays 2:30pm-4:30pm
• Textbook:
Oppenheim, Willsky, “Signals & Systems,”
second edition.
• Midterm Exams (Coursework):
%30: Tuesday 3-March-2020,
15:00-17:00 in Bush House (S)2.04.
All material until the end of Fourier Series
will be examined.
• Final Exam:
%70, All taught material will be examined.
• Sample Exercises:
Problem sets to be solved at home by you.
Hand-written solutions will be provided later
after you have tried.
Correction in HW2, Q1d, solution:
is not memoryless, because output signal at time t
does not depend on the input at the same time t.
Review of Last Lecture
• Continuous-Time (CT) Fourier Transform (FT)
• Relation between FT and Fourier Series (FS) representation of
continuous-time periodic signals
• Properties of Fourier transform
• Linearity
• Time Shift (TS)
• Examples of some basic and useful Fourier transform pairs.
Review of Last Lecture
CT: Fourier Transform & Inverse Fourier Transform
• Also, we say
• Sone notations:
Fourier Transform Pair:
Review of Last Lecture
Fourier series Pair in CT Domain:
• With
, where
is the fundamental period of signal
:
Synthesis Equation:
Analysis Equation (integration is taken over any period interval):
Review of Last Lecture
Relation between F.T. and F.S
The Fourier transform
of a periodic signal
with
Fourier series coefficients
is:
A train of impulses occurring at the harmonically-related
frequencies
for which the area of the impulse at
the -th harmonic frequency
is
times the -th
Fourier series coefficient .
Outline
• Fourier transform properties (continued)
• Examples
• System analysis using Fourier transform (time permitted)
Properties of Fourier transform
• Conjugation:
If:
then:
An immediate result:
If
is a real function of time, i.e.,
Then:
:
,
Hermitian Function in
: Even symmetry for magnitude of
: Odd symmetry for the phase of
Example 6: Back to example 3 and the
Fourier transform of the signal
We showed that:
Clearly
and
verified:
is a real function of time
holds; also, it can be
: even symmetry
-
: odd symmetry
-
Properties of Fourier transform
• Conjugation:
If:
then:
An immediate result:
If
is a real function of time, i.e.,
Then:
:
:
1
𝑋(𝑗𝜔)𝑒
𝑑𝜔
2𝜋
1
𝑥∗ 𝑡 =
𝑋 ∗ 𝑗𝜔 𝑒
𝑑𝜔
2𝜋
Change of variable: Ω = −𝜔
1
𝑥∗ 𝑡 =
𝑋 ∗ −𝑗Ω 𝑒
𝑑Ω
2𝜋
𝑥 𝑡 =
,
Even symmetry for the real part of
: Odd symmetry for the imaginary part of
Properties of Fourier transform
• An important conceptual result:
For real function of time
, the negative frequency component
does not contain any additional information beyond the positive
frequency component.
Hence, the information about positive frequency, i.e.,
sufficient
, is
Properties of Fourier transform
• Conjugation:
If:
then:
Proof:
; Let
then:
; Hence:
FT:
,
Properties of Fourier transform
• Conjugation:
If:
then:
Another immediate result:
If:
is both a real function of time, i.e.,
, and an even
function of time, i.e.,
Then:
will be both a real and an even function of frequency, i.e.,
and
.
Let
∗
And main Conjugation property
Properties of Fourier transform
• Conjugation:
If:
then:
One more immediate result:
If:
is both a real function of time, i.e.,
, and an odd
function of time, i.e.,
Then:
will be purely imaginary function of frequency, i.e.,
.
Let
∗
And main Conjugation property
Properties of Fourier transform
• A conclusion
For a real function
If:
Then:
:
Ev{
Ev{
Od{
}
}
}
Od{
}
Properties of Fourier transform
• Example 8: Find the Fourier transform of
Note:
2Ev{
But, we know from example 3, for real function
Hence:
}
and:
Using the Linearity
(scaling) property
2Ev{
Properties of Fourier transform
• Time Scaling
If:
then:
Linear scaling in time by a factor of
results in a linear scaling in
frequency by a factor of “ , and vice-versa; including an amplitude
scaling of
.
• An immediate result:
For
Compressing in time results in stretching in
frequency and vice-versa:
Compressing in time-domain
Stretching in frequency-domain
Scaling property is an example of the inverse relationship between
Time and Frequency.
Properties of Fourier transform
• Time Scaling
If:
then:
• Another immediate result:
Special case:
Reversing a signal in time, also reverses signal’s Fourier transform in
frequency.
Example 7 (A Low-Pass Filter):
Inverse Fourier transform of a square
in frequency domain:
As
𝒄
increases,
becomes narrower and taller and approaches an impulse as
(Another example of inverse relationship between time and frequency)
𝒄
Definition of “sinc” Pulse
2
Properties of Fourier transform - Duality
• The main concept:
If
,
then, if another signal
has the same shape of
, but in time
domain, we can quickly deduce the Fourier transform of
, i.e.,
, will have the same shape of
, but in the frequency domain.
Properties of Fourier transform – Illustration of Duality
F.T.
Duality
F.T.
Properties of Fourier transform
• Differentiation and Integration
If:
,
then:
and:
DC or Average value
Properties of Fourier transform
• Example 8: Applying the integration property to find
We know that:
1
for
:
But:
Therefore:
,
• Example 9: Applying differentiation property to find
• We know that:
F.T.{
, therefore:
as
for
.
:
Signals and Systems
5CCS2SAS
Lecturer: M. R. Nakhai
Email: reza.nakhai@kcl.ac.uk
Office: S2.09
Office Hours: Thursdays 2:30pm-4:30pm
• Textbook:
Oppenheim, Willsky, “Signals & Systems,”
second edition.
• Midterm Exams (Coursework):
%30: Tuesday 3-March-2020,
15:00-17:00 in Great Hall.
All material until the end of Fourier Series
will be examined.
• Final Exam:
%70, All taught material will be examined.
• Sample Exercises:
Problem sets to be solved at home by you.
Hand-written solutions will be provided later
after you have tried.
Review of Last Lecture
• Fourier transform:
•
•
• Fourier transform properties
• Linearity:
• Time shift:
≮
• Conjugation:
• For real
∗
(≮
∗
:
: Even symmetry for magnitude of
: Odd symmetry for the phase of
: Even symmetry for the real part of
: Odd symmetry for the imaginary part of
)
Review of Last Lecture
• Conjugation (continued):
• For real and even
•
will be both a real and an even function of frequency
• For real and odd
•
:
will be purely imaginary function of frequency
• For real
• Ev{
• Od{
:
:
}
}
• Time scaling:
• Compressing in time-domain Stretching in frequency-domain
• Reversing a signal in time, also reverses signal’s Fourier transform in
frequency.
Review of Last Lecture
• Differentiation and Integration:
•
( )
•
• Duality:
• If
, then, if another signal
has the same shape of
,
but in time domain, we can quickly deduce the Fourier transform of
, i.e.,
, will have the same shape of
, but in the frequency domain.
Outline
• Fourier transform properties (continued)
• Examples
• System analysis using Fourier transform (time permitted)
Properties of Fourier transform
• Parseval’s Relation
If:
,
Then:
Proof:
∗
Properties of Fourier transform– Parseval’s relation
•
: Total energy in the signal
Total energy can be found, either by computing the energy per unit
time, i.e.,
, and integrating over all time, or:
Alternatively, by computing the energy per unit (radian) frequency,
i.e.,
, and integrating over all frequencies.
For this reason,
spectrum of the signal
is also referred to as energy-density
.
Fourier Transform Examples
Example 8: Shifted Impulse in Frequency Domain
)
Using Inverse Fourier transform to find
:
)
)
)
Example 9:
Fourier transform of Cosine:
Periodic Impulse Train (Sampling Function)
Fourier series coefficients:
2𝜋
𝑇
Fourier transform:
−
4𝜋
2𝜋
−
𝑇
𝑇
2𝜋
𝑇
4𝜋
𝑇
Period in time:
Period in frequency :
Properties of Fourier transform
• Convolution
If:
Then:
where,
,
is the impulse response of the LTI system and
is the Fourier transform of the impulse
response and the frequency response of the LTI system
Properties of Fourier transform
• Derivation of the convolution property:
]
Time-Shift property of F.T.
Example 3: Find the Fourier transform of
the signal
.
1
(
)
-
Properties of Fourier transform
Example 10: Find
Using
:
;
Hence:
Inverse Fourier transform
and the linearity property of F.T.
Signals and Systems
5CCS2SAS
Lecturer: M. R. Nakhai
Email: reza.nakhai@kcl.ac.uk
Office: S2.09
Office Hours: Thursdays 2:30pm-4:30pm
• Textbook:
Oppenheim, Willsky, “Signals & Systems,”
second edition.
• Midterm Exams (Coursework):
%30: Tuesday 3-March-2020,
15:00-17:00 in Great Hall.
All material until the end of Fourier Series
will be examined.
• Final Exam:
%70, All taught material will be examined.
• Sample Exercises:
Problem sets to be solved at home by you.
Hand-written solutions will be provided later
after you have tried.
Review of Last Lecture
• Duality:
•
•
• Differentiation:
• Integration:
• Parseval Relation:
Total energy in the
Time domain
• Convolution:
Total energy in the
Frequency domain
Spectral Energy Density
Outline
• Fourier transform properties (Continued)
• More examples
• System analysis using the Fourier transform
• Examples
• Derivation of the Discrete-Time Fourier transform
• Examples
• Properties of the Discrete-Time Fourier transform
Properties of Fourier transform
• Multiplication
If:
Then:
Proof:
;
(
)
let:
:
Properties of Fourier transform
Example 11:
Consider the signal
where
is some
bandlimited signal with Fourier transform
Determine the
]
Multiplication
Property
Illustration
assuming:
1
-𝜔
0
𝜔
𝜔
0.5
𝜔
−𝜔
−𝜔 + 𝜔
−𝜔 − 𝜔
0
𝜔 −𝜔
𝜔
𝜔 +𝜔
System Analysis using Fourier transform
• Consider an important class of continuous-time LTI systems whose input and
output relationship satisfy a linear differential equation with constant
coefficients.
We would like to find the frequency response of such LTI systems.
First order LTI system:
Taking F.T. from both sides, we can find the frequency response:
Frequency response of the LTI system:
Linearity and
differentiation Properties
Inverse Fourier transform
Impulse response of the LTI system:
System Analysis using Fourier transform
• Example 12 (Second order LTI system):
Find the frequency response, the impulse response and the system output
when the input is
.
Take I.F.T. on both sides:
Frequency response
Let
and apply partial fraction:
System Analysis using Fourier transform
• Example 12 (continued):
Impulse response
Inverse Fourier transform
and linearity property of F.T.
System Analysis using Fourier transform
• Example 12 (continued):
[
Let Let
][
]
and apply partial fraction:
;
;
.
Partial Fraction
Coefficients (Residues)
System Analysis using Fourier transform
• Example 12 (continued)
Use the Fourier transform Pairs:
Apply Inverse Fourier transform and the linear property of F.T.:
System Output
Partial Fraction (A math reminder)
1. Find :
2. Find :
3. Finding A:
1. Multiply by
2. First derivative:
3. Insert
:
Discrete-Time Fourier Transform (DTFT)
•
-
: Periodic with period
for
, when
-
-
Discrete-Time Fourier Transform - Derivation
• Discrete-time Fourier series expansion of
:
• Define:
Analysis Equation
Periodic in
with period
Discrete-Time Fourier Transform - Derivation
Then:
As
, then
Synthesis Equation
;
Integration over any
:
interval in
Discrete-Time Fourier Transform Pairs
• Discrete-Time Fourier Transform (DTFT) Equations:
DTFT (Analysis Equation):
Inverse DTFT (Synthesis Equation):
Notation:
Signals and Systems
5CCS2SAS
Lecturer: M. R. Nakhai
Email: reza.nakhai@kcl.ac.uk
Office: S2.09
Office Hours: Thursdays 2:30pm-4:30pm
Review of Previous Lecture
• Fourier transform:
•
•
• Fourier transform properties
• Linearity:
• Time shift:
≮
• Conjugation:
• For real
∗
(≮
∗
:
: Even symmetry for magnitude of
: Odd symmetry for the phase of
: Even symmetry for the real part of
: Odd symmetry for the imaginary part of
)
Review of Previous Lecture
• Conjugation (continued):
• For real and even
•
will be both a real and an even function of frequency
• For real and odd
•
:
will be purely imaginary function of frequency
• For real
• Ev{
• Od{
:
:
}
}
• Time scaling:
• Compressing in time-domain Stretching in frequency-domain
• Reversing a signal in time, also reverses signal’s Fourier transform in
frequency.
Review of Previous Lecture
• Duality:
•
•
• Differentiation:
• Integration:
• Parseval Relation:
Total energy in the
Time domain
• Convolution:
Total energy in the
Frequency domain
Spectral Energy Density
Review of Last Lecture
• Multiplication:
Convolution Formula in
frequency domain
• System Analysis using Fourier transform
Fourier transform
Partial fraction
Inverse Fourier transform
• Discrete-Time Fourier Transform Derivation:
DTFT (Analysis Equation)
IDTFT (Synthesis Equation)
Review of Last Lecture
• Multiplication:
Convolution Formula in
frequency domain
• System Analysis using Fourier transform
Fourier transform
Partial fraction
Inverse Fourier transform
• Discrete-Time Fourier Transform Derivation:
DTFT (Analysis Equation)
IDTFT (Synthesis Equation)
Outline
• Discrete-Time Fourier transform (Continued)
• More examples
• Properties of Discrete-Time Fourier transform
• Examples
Discrete-Time Fourier Transform Equations
• Why DTFT is periodic with period
• Note: This is not true for CTFT, i.e.,
in , i.e.,
?
, because:
Discrete-Time Fourier Transform – Examples
Example 1 (Unit sample
Example 2 (Time-shifted unit sample
Discrete-Time Fourier Transform – Examples
Example 3 (Right-sided decaying exponential:
Using geometric expansion formula:
and
.
Discrete-Time Fourier Transform – Examples
Example 4 (Rectangular Pulse):
Let:
Example 4 (Rectangular Pulse – Illustration)
1
DTFT Pair
-𝑁 0 𝑁
𝑛
2𝑁 + 1
Periodic in
with period
𝜔
Discrete-Time Fourier Transform – Examples
Example 5 (Complex exponential
Show:
):
Using IDTFT:
Integration over
one period
CTFT:
notice periodicity in
in DTFT
Discrete-Time Fourier Transform – Examples
Example 6 (Ideal Discrete-Time Low-Pass Filter)
Find the impulse response for the frequency response
Properties of Discrete-Time Fourier Transform
• Discrete-Time Fourier Transform (DTFT) Equations:
DTFT (Analysis Equation):
Inverse DTFT (Synthesis Equation):
Notation:
Properties of Discrete-Time Fourier Transform
• Periodicity:
• Linearity:
If:
Then:
• Time Shift:
If:
Then:
Proof:
Using a change of
variable:
Properties of Discrete-Time Fourier Transform
• Phase Shift (in time domain):
If:
Then:
Proof: Multiply both sides of DTFT synthesis equation by
Using a change of
variable:
:
Properties of Discrete-Time Fourier Transform
Example 7:
Consider
:
⋯
⋯
−2𝜋
−𝜋
𝜋
0
2𝜋
𝜔
Then:
(
)
:
with
,
⋯
⋯
−2𝜋
−𝜋
0
𝜋
2𝜋
𝜔
Properties of Discrete-Time Fourier Transform
• Time Reversal:
If:
Then:
• Conjugation:
If:
Then:
Proof:
Using a change of
variable:
Using a change of
variable:
Review of Last Lecture
• Discrete-Time Fourier Transform:
: DTFT (Analysis Equation)
: Inverse DTFT (Synthesis Equation)
• Examples
• Properties of Discrete-Time Fourier Transform
Periodicity:
Linearity:
Time Shift:
Phase Shift (in time domain):
Time Reversal:
Conjugation:
• Examples
(
)
(
∗
∗
)
Outline
• Properties of Discrete-Time Fourier transform (Continued)
Conjugation (continued)
Time Expansion
Differentiation
Parseval’s Relation
Convolution
Multiplication
• More examples
• Second Midterm Exam (Entire taught material until here)
• Sampling
Analog to Digital (A/D) Conversion
Properties of Discrete-Time Fourier Transform
• Conjugate Symmetry (Real Signals):
If:
Then:
Consequences:
and
≮
∗
∗
and
and
are even functions of
odd functions of
≮
Properties of Discrete-Time Fourier Transform
• Conjugate Symmetry (
Then:
: Real and Even;
):
is purely real and even in :
: Even
; Time Reversal Property
: A contradiction
purely real
for real
• Conjugate Symmetry (
: Real and Odd;
Then:
is purely imaginary and odd in :
: Odd
):
; Time Reversal Property
𝑋 𝑒
= −𝑋 𝑒
⟹ ℜ𝑒 𝑋 𝑒
= −ℜ𝑒 𝑋 𝑒
: A contradiction
for real 𝑥 𝑛 ⟹ ℜ𝑒 𝑋 𝑒
=0⟹𝑋 𝑒
purely imaginary
Properties of Discrete-Time Fourier Transform
• For real
:
If:
Then:
: Purely Real
: Purely Imaginary
Proof:
Time Reversal, linearity properties and
Properties of Discrete-Time Fourier Transform
• Time Expansion:
Reminder of time scaling property in CT:
But, in general, e.g.,
, for
is not defined in DT.
To rectify this problem in DT, let be a positive integer and define:
Then,
is obtained from
successive values of
.
by placing
zeros between
Properties of Discrete-Time Fourier Transform
• Example 8
Illustration of time expansion
with
:
is obtained by placing
3-1=2 zeros between successive
values of
.
- ( )
- - -
- - -
Properties of Discrete-Time Fourier Transform
DT Fourier transform of
But, according to definition
:
is zero, unless
then:
Change of variable
Hence:
If:
Then:
Signal expansion (spreading out) in DT time, when
, results in a
compression in frequency (i.e., signal’s DTFT is compressed by a factor of ).
Properties of Discrete-Time Fourier Transform
• Example 9
Illustration of compression
in frequency with
,
which is as a
consequence of
expansion in
time domain
with
:
⋯
⋯
−𝜋
−2𝜋
𝑧∗
−
𝜋
2
0
𝜋
2
𝜋
2𝜋
𝜔
2𝜋
𝜔
( )
⋯
⋯
−2𝜋
−𝜋
−
𝜋
2
0
−
𝜋
4
𝜋
𝜋 2
4
𝜋
Properties of Discrete-Time Fourier Transform
• Differentiation:
If:
Then:
Proof:
Multiplication by
in time domain
Differentiation in frequency domain scaled by
Properties of Discrete-Time Fourier Transform
• Parseval’s Relation:
If:
Then:
Total energy in the
Time domain
Total energy in the
Frequency domain
Spectral Energy Density
Properties of Discrete-Time Fourier Transform
• Convolution:
If:
Then:
• Multiplication:
If:
Then:
Properties of Discrete-Time Fourier Transform
• Example 10 (DT LTI system with impulse response
: Frequency Response of the DT LTI system
Find the output
when the the input is
)
:
(
Periodic with period of
(
)
:
in
)
Scalar: not a function of
Linearity Property
Recall: Complex exponentials are
Eigenfunctions of LTI systems:
: Eigenfunction
: Eigenvalue
Discrete-Time Fourier Transform – Examples
Example 6 (Ideal Discrete-Time Low-Pass Filter)
Find the impulse response for the frequency response
Properties of Discrete-Time Fourier Transform
• Example 11 (Cascading two LPFs):
Given the following impulse responses :
Find the impulse response of the composite
cascaded filter:
Properties of Discrete-Time Fourier Transform
• Example 11 (Solution):
Properties of Discrete-Time Fourier Transform
• Example 11 (Solution-continued with illustration):
1,
𝐻 𝑒
=
0,
1,
𝐻 𝑒
=
0,
1,
𝐻 𝑒
=
0,
ℎ𝑛 =
sin
𝜋
𝜋
− ≤𝜔≤
2
2
𝜋
< 𝜔 ≤𝜋
2
𝜋
𝜋
− ≤𝜔≤
4
4
𝜋
< 𝜔 ≤𝜋
4
𝜋
𝜋
− ≤𝜔≤
4
4
𝜋
< 𝜔 ≤𝜋
4
𝜋
𝑛
1
𝑛
4
= sinc
𝜋𝑛
4
4
⋯
⋯
−2𝜋
−𝜋
−
𝜋
2
𝜋
2
0
𝜋
𝜔
2𝜋
⋯
⋯
−2𝜋
−𝜋
−
𝜋 0
4
𝜋
4
𝜋
2𝜋
𝜔
𝐻 𝑒
⋯
⋯
−2𝜋
−𝜋
−
𝜋 0
4
𝜋
4
𝜋
2𝜋
𝜔
Discrete-Time Fourier Transform – Examples
Example 1 (Unit sample
Example 2 (Time-shifted unit sample
Properties of Discrete-Time Fourier Transform
• Example 12:
Find DTFT of
Signals and Systems
5CCS2SAS
Lecturer: M. R. Nakhai
Email: reza.nakhai@kcl.ac.uk
Office: BH(S)5.05
Office Hours: Fridays 2pm-4pm
• Textbook:
Oppenheim, Willsky, “Signals & Systems,”
second edition.
• Midterm Exams (Coursework):
%15 First (Monday 11-Feb.-2019/ 13:00-15:00/
in King’s Building-Great Hall)
%15 Second (Monday 18-March-2019/
15:00-17:00 in King’s Building-Great Hall)
Exam will cover the entire taught material from
the beginning to the end of Fourier Transform
(both CT & DT are included).
• Final Exam:
%70, All taught material will be examined.
Sampling
Review of Previous Lecture
• Discrete-Time Fourier Transform:
: DTFT (Analysis Equation)
: Inverse DTFT (Synthesis Equation)
• Examples
• Properties of Discrete-Time Fourier Transform
Periodicity:
Linearity:
Time Shift:
Phase Shift (in time domain):
Time Reversal:
Conjugation:
• Examples
(
)
(
∗
∗
)
Review of Last Lecture
• Properties of Discrete-Time Fourier Transform
Conjugation (continued)
Conjugate Symmetry (Real Signals):
o
and
are even functions of
o
and
odd functions of
Conjugate Symmetry (Real & Even Signals):
o
is purely real and even in
Conjugate Symmetry (Real & Odd Signals)
o
is purely imaginary and odd in
For real signals
o
: Purely Real
o
: Purely Imaginary
Review of Last Lecture
• Time Expansion:
is obtained from
by placing
zeros between
successive values of
.
Signal expansion (spreading out) in DT time, when
, results in a
compression in frequency (signal’s DTFT is compressed by a factor of ).
• Differentiation:
If:
Then:
• Parseval’s Relation:
Review of Last Lecture
• Convolution:
• Multiplication:
• Illustrative examples on properties
Second Midterm Exam (Entire taught material until here)
Outline
• Sampling
Analog to Digital (A/D) Conversion
Frequency domain analysis of A/D conversion
Sampling Theorem
Analog to Digital (A/D) Convertor
• Analog to Digital A/D
Converting System :
CT Signal
• A/D Convertor:
CT signal
: Integer
can be viewed as DT, if we define:
.
Illustration of A/D conversion
•
is a set of impulses
bounded by the envelop
•
is still a
continuous-time signal.
•
can be viewed as a
discrete-time signal if we
define
, i.e.,
the samples of
at
integer multiples of .
Frequency Domain Analysis of A/D Convertor
• How dose
• How dose
look like in the frequency domain?
look like in the frequency domain?
• How does
relates to the Fourier transform
representation of the original signal
, i.e.,
?
Periodic Impulse Train (Sampling Function)
Fourier series coefficients:
2𝜋
𝑇
Fourier transform:
−
4𝜋
2𝜋
−
𝑇
𝑇
2𝜋
𝑇
4𝜋
𝑇
Period in time:
Period in frequency:
Frequency Domain Analysis of A/D Convertor
• How dose
look like in the frequency domain?
The periodic impulse train
is referred to as the sampling
function.
is also a train of impulses with period in frequency .
The period is called the sampling period.
The fundamental frequency of
, i.e.,
, is referred to as
sampling frequency
, i.e.,
.
Frequency Domain Analysis of A/D Convertor
•
•
: train of periodic impulses
in time.
:
Period in time-domain
•
Fourier series coefficients
•
:
Anther train of periodic impulses with
period
in frequency.
Properties of Fourier transform
• Multiplication
If:
Then:
Proof:
;
(
)
let:
:
Frequency Domain Analysis of A/D Convertor
• How dose
look like in the frequency domain?
Signals and Systems
5CCS2SAS
Lecturer: M. R. Nakhai
Email: reza.nakhai@kcl.ac.uk
Office: BH(S)5.05
Office Hours: Fridays 2pm-4pm
• Textbook:
Oppenheim, Willsky, “Signals & Systems,”
second edition.
• Midterm Exams (Coursework):
%15 First (Monday 11-Feb.-2019/ 13:00-15:00/
in King’s Building-Great Hall)
%15 Second (Monday 18-March-2019/
15:00-17:00 in King’s Building-Great Hall)
Exam will cover the entire taught material from
the beginning to the end of Fourier Transform
(both CT & DT are included).
• Final Exam:
%70, All taught material will be examined.
Review of Last Lecture
• Sampling
Analog to Digital (A/D) Conversion
Frequency domain analysis of A/D conversion
Outline
• Sampling (Continued)
Frequency domain analysis of A/D conversion (Continued)
Sampling Theorem
Reconstruction
• Communication Systems
Modulation
Time-domain and frequency-domain analysis
Demodulation
Time-domain and frequency-domain analysis
Frequency Domain Analysis of A/D Convertor
• Illustration of the convolution:
The replicas overlap (aliasing)
at edges (aliasing effect) if:
2𝜋
𝑇
⋯
−
4𝜋
𝑇
−
2𝜋
𝑇
⋯
2𝜋
𝑇
1
𝑇
⋯
-
4𝜋
𝑇
⋯
2𝜋
𝑇
*
=
for
Convolution of
with
results in a periodic replicas of
with period .
The replicas do not overlap if
1
Frequency Domain Analysis of A/D Convertor
• As the sampling period
decreases (or sampling frequency
increases) such that
The replicas do not overlap
1
2𝜋
𝑇
⋯
−
2𝜋
𝑇
2𝜋
𝑇
1
𝑇
⋯
-
4𝜋
𝑇
⋯
2𝜋
𝑇
*
=
4𝜋
−
𝑇
⋯
Frequency Domain Analysis of A/D Convertor
• As the sampling period
increases (or sampling frequency
decreases) such that
1
-
The replicas overlap on edges
(aliasing effect)
2𝜋
𝑇
⋯
4𝜋
𝑇
−
2𝜋
𝑇
2𝜋
𝑇
4𝜋
𝑇
=
−
⋯
*
1
𝑇
⋯
⋯
-
2𝜋
𝑇
Sampling Theorem
• In these illustrations:
is the highest frequency that the signal
has in frequency domain.
is the sampling (radian) frequency (or sampling rate)
The overlap effect at edges of the replicas is called Aliasing.
If aliasing happens, the original CT signal
cannot be fully recovered
from its samples.
So, the aliasing must be avoided for the signal recovery purposes.
To avoid aliasing, the sampling period cannot be too large or equivalently
the sampling rate
cannot be too low!
• Sampling Theorem:
What is the minimum sampling rate so that there is No Aliasing?
Sampling Theorem
• (Definition) Band-limited Signal:
A signal
is band-limited if
where
is referred to as the bandwidth of the signal.
-
Band-limited signal
Band-unlimited signal
Sampling Theorem
• Sampling Theorem: Let
be a band-limited signal with
in
frequency domain:
: The frequency that must be exceeded by the sampling
frequency
is referred to as the Nyquist rate.
: Half of the Nyquist rate is referred to as the Nyquist frequency.
Sampling Theorem
• Example 1:
The maximum frequency of the signal
is
kHz.
Determine the minimum sampling rate, i.e., the
Nyquist rate, so that the sampling theorem is satisfied.
Since
, then:
Hence, the Nyquist rate:
-
[rad]
kHz.
Example 7 (A Low-Pass Filter):
Inverse Fourier transform of a square
in frequency domain:
As
𝒄
increases,
becomes narrower and taller and approaches an impulse as
(Another example of inverse relationship between time and frequency)
𝒄
Reconstruction of a signal from its samples
1
1
𝑇
⋯
,
𝜔
𝑇
1
-
⋯
Reconstruction of a signal from its samples
(Using Interpolation)
•
fits the continuous-time
between the sample points
and represents an interpolation formula.
For an ideal low-pas filter
:
Reconstruction by Interpolation: Illustration
𝑥 𝑡 =
𝜔𝑇
𝜋
𝑥 𝑛𝑇 sinc
𝜔 (𝑡 − 𝑛𝑇)
𝜋
o 𝑥 𝑡 is plotted for:
o 𝜔 =𝑊= :
o 𝑥 𝑡 =∑
𝑥 𝑛𝑇 sinc
−𝑛
: indicates the samples of 𝑥 𝑡 at sampling
intervals of 𝑇, i.e., 𝑥 𝑛𝑇 , 𝑛 = −5, … . . , 5
and the blue curves interpolates these
samples to perfectly reconstruct 𝑥 𝑡 ,
since the sampling theorem is satisfied.
Reconstruction of a signal from its samples
(Using Interpolation)
• Using the impulse response of an ideal Low-pass filter, interpolation
exactly reconstructs the original band-limited signal
if the sampling
frequency
satisfies the condition of the sampling theorem.
Reconstruction of a signal from its samples
• If the sampling theorem is satisfied, i.e., the aliasing does not occur
, the original CT signal
can be recovered from the
samples by an ideal low-pass filter.
• If the sampling theorem is not satisfied, i.e., the aliasing does occur
, then the original CT signal
cannot be recovered from
the samples by an ideal low-pass filter (illustration in the next slide).
Reconstruction of a signal from its samples
• The sampling theorem is not satisfied,
i.e.,
,
The reconstructed signal does not
recover the original signal
by ⋯
an ideal LPF.
1
1
𝑇
⋯
-
2𝜋
𝑇
𝑇
1
-
Communication Systems
Elements of Communication Systems
• Modulation: Embedding the information-bearing signal into second
signal.
Information-bearing signal is referred to as the modulating signal.
The second signal on which the modulating signal is embedded is referred to
as the carrier signal.
• Let
be the modulating signal and
The modulated signal
be the carrier signal. Then:
is expressed as:
• Aim: Transmission of an information-bearing signal over a
geographical distance from point A to point B.
Transmission of an information-bearing signal
(e.g., Voice Signal)
• Frequency range over which a voice signal bears information:
200 Hz to 4 KHz.
• Communication channel over which the voice signal is transmitted from
point A to point B.
Long distance transmission channels:
Microwave: useable frequency range: 300 MHz – 300 GHz.
Satellite: useable frequency range: from few hundred MHz - 40 GHz.
• Voice signals must be shifted to higher frequency ranges in order to pass
through the available transmission channels.
Amplitude Modulation (AM)
• Carrier signal is used to transport the information-bearing signal (e.g.,
voice)
through the transmission channel.
• Complex exponential signal is used as the carrier signal:
is referred to as Carrier Frequency (e.g., within the
range
300 MHz – 300 GHz in microwave transmission channel).
is the carrier phase.
• Assuming
, we can describe the modulated signal
as:
Implementation of the Modulator
Properties of Fourier transform
• Multiplication
If:
Then:
Proof:
;
(
)
let:
:
Frequency Domain Analysis of AM
• The key technique is the multiplication property of the Fourier transform.
Frequency Domain Analysis of AM – Illustration
• The spectrum of the modulated signal
, i.e.,
, is the spectrum of the
modulating signal
, i.e.,
, but
shifted in frequency by the amount of
the carrier frequency .
1
2𝜋
*
≈
=
1
≈
𝜔
Demodulation of the Modulating Signal
• Demodulation: Recovering the modulating signal
signal
• Frequency Domain Analysis:
from the modulated
Demodulation of the Modulating Signal – Illustration
• The spectrum of
is shifted back to
the original position of the spectrum of
on the frequency axis.
1
≈
2𝜋
≈
1
-
𝜔
Signals and Systems
5CCS2SAS
Lecturer: M. R. Nakhai
Email: reza.nakhai@kcl.ac.uk
Office: BH(S)5.05
Office Hours: Fridays 2pm-4pm
Review of Last Lecture
• Sampling Theorem
• Reconstruction of CT signal form its samples
• Communication Systems
• Modulation and Demodulation using complex exponential
Outline
• Communication Systems (Continued)
Modulation and Demodulation using sinusoidal signals
• The Z-Transform
Amplitude Modulation & Demodulation
(using complex exponetial carrier)
(
)
Modulation
(
)
Demodulation
Amplitude Modulation (AM)
• Sinusoidal Carrier - Consider the following carrier signal:
range
is referred to as Carrier Frequency (e.g., within the
300 MHz – 300 GHz in microwave transmission channel).
is the carrier phase.
• Assuming
, we can describe the modulated signal
as:
Frequency Domain Analysis of AM
]
]
• The spectrum of the modulated signal
is the scaled spectrum of the
modulating signal
by a factor of , but shifted to the left and the right in
the frequency domain by an amount of the carrier frequency .
Frequency Domain Analysis of AM – Illustration
1
: can be recovered
𝜋
𝜋
*
≈
≈
=
1
2
≈
≈
−𝜔
𝜔
• The spectrum of the modulated signal
, i.e.,
, is the spectrum of the modulating signal
, i.e.,
, but shifted in frequency to the left and the right by the amount of the carrier frequency
and
scaled in amplitude by a factor of .
Frequency Domain Analysis of AM – Illustration
1
: cannot be recovered!
-
𝜋
𝜋
1
2
−𝜔
*
=
• The spectrum of the modulated signal
, i.e.,
, is the spectrum of
the modulating signal
, i.e.,
, but shifted in frequency to the
left and the right by the amount of the
carrier frequency
and scaled in
amplitude by a factor of .
𝜔
Demodulation of the Modulating Signal
• When
by multiplying
,
can be recovered from the modulated signal
by the same sinusoidal carrier signal
Recovered
Demodulation of the Modulating Signal
• Frequency domain Analysis:
]
]
Demodulation of the Modulating Signal – Illustration
1
2
≈
≈
−𝜔
𝜔
*
𝜋
≈
≈
=
𝜋
1
2
≈
-
≈
−2𝜔 −2𝜔 + 𝜔
≈
≈
−2𝜔 − 𝜔
11
44
2𝜔 − 𝜔
2𝜔 2𝜔 + 𝜔
2
LPF:
-
Original CT Signal Spectrum :
-
=
1
Amplitude Modulation & Demodulation
(using sinusoidal carrier)
LPF
2
-
Modulation
Demodulation
Signals and Systems
5CCS2SAS
Lecturer: M. R. Nakhai
Email: reza.nakhai@kcl.ac.uk
Office: BH(S)5.05
Office Hours: Fridays 2pm-4pm
Review of Last Lecture
• Communication Systems (continued)
• Modulation and demodulation using Sinusoidal signals
• Frequency domain analysis and illustrations
• The Z-Transform
• An introduction
Outline
• The Z-Transform
•
•
•
•
•
•
•
•
Relation with Discrete Time Fourier Transform
Region of convergence (RoC)
Pole & Zero concept and plot on the complex z-plane.
Illustrative examples on RoC and Pole – Zero plots
Properties of the Z-Transform
Properties of the RoC
LTI system analysis using the Z – Transform
Characterization of the important LTI system properties using the Z – Transform
Z – Transform
Discrete-Time Fourier Transform Pairs
• Discrete-Time Fourier Transform (DTFT) Equations:
DTFT (Analysis Equation):
Inverse DTFT (Synthesis Equation):
Notation:
Complex Numbers (a reminder)
• The set of complex numbers is denoted by
and is defined as
• Forms of representation of a complex number :
Cartesian:
: real part of ;
: imaginary part of ]
Polar:
: length or modulus of ;
: argument of
The most convenient representation depends on the analysis.
An important and useful formula:
Euler’s Formula:
Complex Numbers (a reminder)
• Using the Euler’s formula:
•
Cartesian:
Polar:
Im (Imaginary axis)
𝑧
𝑦
•
𝑟= 𝑧
𝜃
Complex conjugate of :
∗
A key property:
∗
=
𝑥
Polar and cartesian representations
on the Complex Z – plane
Re (Real axis)
Eigenfunctions of LTI Systems – DT Case
• DT complex exponential function
any discrete-time LTI system and
, where
, is an eigenfunction of
where
is the discrete-time impulse response of the system and is the
corresponding eigenvalue.
is known as the transfer function of the DT LTI system.
For
, i.e., for =
1 and
, the above summation
corresponds to the DTFT of
.
More generally, with no restriction on
to be unity, the above summation
is referred to as the Z – Transform of
.
The Z – Transform
• Definition: The Z – Transform of a discrete-time signal
generalization of the DTFT of
and is defined as:
is a
where is a complex number, in its general form, on the complex Z –
plane :
• Notation: We denote the Z – Transform of
as:
DTFT as a special case of the Z – Transform
• DTFT of
can be described a special case
of the Z – Transform, when =1, i.e.,:
Im (Imaginary axis)
𝑧=𝑒
𝑟=1
𝜔
Re
(Real axis)
• Pictorially, DTFT of
can be viewed as
the Z – Transform of
, evaluated at the
points on the unit circle on the complex Z –
plane.
Unit Circle
Complex Z – plane
Relationship between DTFT and Z – Transform
• When
, the Z – Transform
Hence,
With =1:
The weighting
whether
1 or
is equivalent to:
can be interpreted as the DTFT of the DT signal
:
is growing or decaying with increasing , depending on
1.
Region of Convergence of the Z – Transform
• Definition: The Region of Convergence (RoC) of the Z – Transform is a
set of the points for which
converges to bounded values, i.e.,
the following relation is satisfied for
:
This convergence is not expected to happen for all values of =
Clearly, the convergence may happen for some values of =
not for the others.
.
, but
Hence, finding the Z – Transform of signal is followed by marking the
corresponding RoC on the Z – Plane.
Pole – Zero Plot and the Region of Convergence
• Example 1: Find the Z – Transform of the right-sided exponential,
specify poles and zeros of
and mark the RoC on the
the Z – Plane.
converges and is well-defined, if:
RoC
Pole – Zero Plot and the Region of Convergence
• Example 1 (continued):
Pole (“ x “) : the value of for which
RoC
Zero (” “ ) :the value of for which
Hence, for
,
there is a zero at
and a
pole at
For
, the RoC includes
the unit circle.
For
, the RoC does not
Include the unit circle.
x
Pole – Zero plot and RoC (i.e., the shaded area)
with
Pole – Zero Plot and the Region of Convergence
• Example 2: Find the Z – Transform of the left-sided exponential,
specify poles and zeros of
and mark the RoC
on the the Z – Plane.
Change of variable:
RoC
converges and is well-defined, if:
Pole – Zero Plot and the Region of Convergence
• Example 2 (continued):
Note: The Z – transform equation
is the same as Example 1 and the only
difference between the two is the
RoC.
x
RoC
Pole – Zero plot and RoC (i.e., the shaded area)
with
Some Other Common Z – transforms
• Example 3:
RoC: All
• Example 4:
RoC: if
if
Change of variable:
𝑛−𝑚 =𝑑
RoC:
All , except
All , except
(if
(if
, because
because
all , except
all , except
Another Common Z – transform (Table 10-2 for more)
• Example 5:
Then:
Properties of Z – transform (Table-10.1)
• Given the RoC of the original function(s), the RoC of the function obtained
after applying the following properties must be carefully determined.
• Linearity:
If:
Then:
• Time Shift:
If:
Then:
Properties of Z – transform
• Scaling in the Z – Domain:
If:
Then:
Proof:
Special case (
):
Interpretation of
: The locations of all poles and zeros
of
in the Z – plane rotate by an angle of .
Properties of Z – transform
• Time Reversal:
If:
Then:
Proof:
Change of variable:
𝑚 = −𝑛
• Conjugation:
If:
Then:
Properties of Discrete-Time Fourier Transform
• Time Expansion:
Reminder of time scaling property in CT:
But, in general, e.g.,
, for
is not defined in DT.
To rectify this problem in DT, let be a positive integer and define:
Then,
is obtained from
successive values of
.
by placing
zeros between
Properties of Discrete-Time Fourier Transform
• Example 8
Illustration of time expansion
with
:
is obtained by placing
3-1=2 zeros between successive
values of
.
- ( )
- - -
- - -
Properties of Z – Transform
• Time Expansion:
If:
and:
Then:
• Convolution:
If:
Then:
,
Properties of the Region of Convergence
• Property 1: The RoC consists of a ring in the -plane centered about the
origin. (The RoC is a ring or a disk centered at the origin of the -plane)
This is due to the fact that the -transform of
be interpreted as the DTFT of
]
at values
can
and for this DTFT to converge, i.e.,
i.e.,
] must be absolutely summable.
Hence, convergence depends only on the modulus
and not on
Hence, RoC consists of concentric rings.
The boundary of the RoC may extend inwards to the origin or outwards
to .
Properties of the Region of Convergence
• Property 2:
The RoC does not contain any poles.
This is due to the fact that the -transform
of
infinite, and hence, by definition does not converge.
• Property 3: If
except possibly
at a pole is
is of finite duration, then the RoC is the entire -plane,
and/or
.
Finite duration:
o
and
does not include
o
and
o
and
:
and
and
.
: RoC includes
: RoC includes
RoC
.
.
Properties of the Region of Convergence
• Property 4: The DTFT of
unit circle in the -plane
exists if and only if the RoC includes the
This is due to the fact that the DTFT of
, evaluated on the unit circle.
is the the -transform of
• Property 5: If
is a right-sided sequence, i.e.,
prior to
some value of , and if the circle with a radius of
is in the RoC,
then all finite values of for which
will be in the RoC.
In other words, for a right-sided sequence
, the RoC extends
outward from the outermost pole of
.
This is due to the fact that if
] is absolutely summable for
values of with
, then it will indeed be so for values of
for which
.
Properties of the Region of Convergence
• Property 6: If
is a left-sided sequence, i.e.,
after some
value of
, and if the circle with a radius of
is in the RoC,
then all values of for which 0
will be in the RoC.
In other words, for a left-sided sequence
, the RoC extends inward
from the innermost pole of
.
Similarly, if
] is absolutely summable for values of with
, then it will indeed be so for values of for which
If
, then the value
is not included in RoC.
If
, then the value
is included in RoC.
.
Properties of the Region of Convergence
• Property 7: If the -transform of
is rational, i.e.,
,
where
and
are polynomials in , and if
is right-sided,
then, the RoC is the region in the -plane outside of the outermost pole
of
.
• This property can be justified by applying the partial fraction to
and applying the result of the property 5 to the resulting partial
fraction terms to find the respective RoCs and considering their
intersection, which is the RoC of partial term with the outermost pole
of
.
For example if
, the outermost pole is and
hence, the RoC is the region
.
LTI System Analysis Using Z – Transform
• Convolution is the key property for analyzing the LTI systems:
: System Function (or Transfer Function)
: Impulse response of the LTI system
: -transform of the input DT signal
: -transform of the output signal
If the unit circle is in the RoC for
, then, evaluating on the unitcircle (i.e.,
),
reduces to the frequency response.
System Properties Using Z – Transform
• Several important properties of the LTI systems can be directly
characterized by the poles, zeros and region of convergence of the
system function,
.
• Causality: A discrete-time LTI system is causal if and only if the RoC of
its system function
is the exterior of a circle, including infinity.
Proof: An LTI system is causal if and only if:
o
is a right-sided signal.
o According to the Property 5, the RoC of
in the -plane.
o Since the powers of in
positive,
is included in RoC.
is the exterior of a circle
never becomes
System Properties Using Z – Transform
• Stability: A discrete-time LTI system is stable if and only if the RoC of its
system function
includes the unit circle, i.e., the values of with
Proof: An LTI system is stable, if and only if its impulse response is
absolutely summable:
hence, equivalently, if and only if the RoC of its transfer function
includes the unit circle.
System Properties Using Z – Transform
• A causal discrete-time LTI system with rational system function is stable
if and only if all of the poles of
are inside the unit circle.
• Proof:
o Causality implies that the RoC of
is the exterior of a
circle (including infinity).
o Property 2: The RoC does not contain any poles.
o Stability implies that the RoC of
includes the unit circle.
Hence, all of the poles of
lie inside the unit circle.
Properties of Discrete-Time Fourier Transform
DT Fourier transform of
But, according to definition
:
is zero, unless
then:
Change of variable
Hence:
If:
Then:
Signal expansion (spreading out) in DT time, when
, results in a
compression in frequency (i.e., signal’s DTFT is compressed by a factor of ).
Properties of Discrete-Time Fourier Transform
• Example 9
Illustration of compression
in frequency with
,
which is as a
consequence of
expansion in
time domain
with
:
⋯
⋯
−𝜋
−2𝜋
𝑧∗
−
𝜋
2
0
𝜋
2
𝜋
2𝜋
𝜔
2𝜋
𝜔
( )
⋯
⋯
−2𝜋
−𝜋
−
𝜋
2
0
−
𝜋
4
𝜋
𝜋 2
4
𝜋
1
1
-
≈
1
𝑇
⋯
-
≈
2𝜋
𝑇
⋯
2𝜋
𝑇
-
1
1
-
-
𝜔
1
1
-
-
𝜋
𝜋
2𝜋
𝑇
⋯
4𝜋
𝑇
−
2𝜋
𝑇
2𝜋
𝑇
1
2
1
𝑇
⋯
−𝜔
⋯
𝜔
-
4𝜋
𝑇
=
−
⋯
2𝜋
𝑇
*
1
2𝜋
𝑇
⋯
4𝜋
𝑇
−
2𝜋
𝑇
2𝜋
𝑇
1
𝑇
⋯
-
4𝜋
𝑇
=
−
⋯
⋯
2𝜋
𝑇
*
1
2𝜋
𝑇
⋯
−
2𝜋
𝑇
2𝜋
𝑇
1
𝑇
⋯
-
4𝜋
𝑇
⋯
2𝜋
𝑇
*
=
4𝜋
−
𝑇
⋯
2𝜋
𝑇
−
4𝜋
𝑇
−
2𝜋
𝑇
2𝜋
𝑇
-
2𝜋
𝑇
4𝜋
𝑇
⋯
⋯
−2𝜋
−𝜋
𝜋
0
(
2𝜋
𝜔
)
⋯
⋯
−2𝜋
−𝜋
0
𝜋
2𝜋
𝜔
- -
- - -
- - -
⋯
⋯
−𝜋
−2𝜋
−
𝜋
2
0
𝜋
2
𝜋
2𝜋
𝜔
2𝜋
𝜔
( )
⋯
⋯
−2𝜋
−𝜋
−
𝜋
2
0
−
𝜋
4
𝜋
𝜋 2
4
𝜋
−𝜋
⋯
⋯
−2𝜋
0
𝜋
2𝜋
𝜔
⋯
⋯
−2𝜋
−𝜋
𝜋
0
(
2𝜋
𝜔
)
⋯
⋯
−2𝜋
−𝜋
0
𝜋
2𝜋
𝜔
Discrete-Time Fourier Transform – Examples
Example 6 (DTFT of periodic DT signals using DT Fourier series coefficients):
Consider DTFT of a periodic signal with period :
Using:
Using the superposition property:
distinct terms
- -
- - -
- - -
1
-𝑁 0 𝑁
𝑛
2𝑁 + 1
𝜔
-
-
-
-
1
-
Key Property of Linear Systems
• Superposition:
A system is linear if it is additive and scalable:
If:
Then:
In General:
,
If:
Then:
,
Time Invariance (TI)
CT:
A system
and any time shift
If
Then
DT: A system
and any time shift
If
Then
is TI if for any input
:
is TI if for any input
:
Corollary
Fact: If the input to a TI System is periodic, then the output is periodic
with the same period.
Proof:
Consider a TI system
Consider a periodic input
Then, by TI property:
But, since
, then from
,
we must have
=
,
which means that output is periodic with the same
period .
Example (DT): An LTI System
Fact: If the response of an LTI system to some inputs are known, then,
the response to many inputs are Known
Known
Then
Known
Linearity and Causality
• A linear system is causal, if and only if it satisfies the condition of
initial rest:
Representing DT Signals with Sums of Unit Samples
Analytically (Sifting Property):
Coefficients
Shifted Unit Samples
Using basic blocks to build variations of signals
Sifting Property: An Illustrative Example
x[ 1]
value of signal at n=-1
e.g.,
=1 (here)
[ n 1] = 𝑥 −1 ,
0,
𝑛 = −1
𝑛 ≠ −1
Impulse (Time-shifted to n=-1)
=
….+
+
+
+⋯
System Modelling
Difference Equation Representation Example
• The blurring system can be modelled by a difference equation such as:
• Is this system invertible? How do you find out, given the system model?
• The deblurring system (if exists) may be modelled by a difference
equation, such as:
• To design the deblurring system the difference equation representation
may not be suitable and some alternative modelling and tools in time
domain or frequency domain may be needed!
Linear Combination of 3 Basic Signal Operations
• More Examples:
3
9 𝑡
𝑡
-3
3
𝑡
-3
3
𝑡
𝑡
1
3
𝑡
Successive Integration of the Unit Impulse
Function
• Successive integration of the unit impulse function yields a family of
function, i.e.,
…
, as follows:
1st Integration
Unit Step
2nd Integration
Unit Ramp
3rd Integration
Unit Parabola
n-th Integration
In general
• An Illustrative Example for
Approximating
with a tall narrow rectangle
pulse with width and
height , (
:
• Another illustrative example
for sifting property:
.
.
.
Importance of Unit Impulse
• Is a widely used idealization in science and engineering.
• Impulse of current delivers a unit amount of charge to a
circuit, instantaneously:
then for
.
• Impulse of force in time delivers an instantaneous
momentum to a mechanical system.
𝛿 𝑡−𝑡
𝑝 𝑡−𝑡
1
0
0
𝑡
𝑡
𝑡
𝑡
• An illustrative Example:
understanding sifting property
using sampling property
+
+
.
.
.
𝑛 = −2
.
.
.
+
𝑛 =3
+
𝑛 =4
sinusoidal signals
or
where is in seconds,
is in radians/second and
It is common to write:
or Hertz (Hz).
The sinusoidal signal
(Why?)
,
is in radians.
, where the unit of
is cycles/second
is periodic with fundamental period
s the fundamental frequency.
sinusoidal signals
(meaning of
•
slows down the rate of oscillation (increases the fundamental period)
•
Exactly the opposite happens.
•
is constant, i.e., zero rate of oscillation, and the fundamental
period is not defined (i.e., could be any value!).
1
⋯
-2𝑇
-𝑇
0
⋯
⋯
𝑇
2𝑇
X(𝑗𝜔)
0.5
𝑥 𝑡
𝜔
1
−𝜔
1
𝑡
-𝜔
𝑥 −𝑡
1
-1
1⁄2
𝑡
1
-1
-1⁄2
𝑡
𝜔
0
𝜔 −𝜔
𝜔
𝜔 +𝜔