the introduction of Signal and Systems...

1. Prepared By:
SYED HAMZA FAISAL

2. Signals & Systems
• Signals and systems introduction and mathematical background
• Signal classification and energy, basic operations with signals, useful signals
• Systems - examples and classification, LTI systems
• LTI systems and the impulse response - Convolution
• Linear systems: zero-input and zero-state response
• The trigonometric Fourier series as a convenient signal representation for LTI systems analysis
• The exponential Fourier series - Fundamental frequency and properties
• Beyond Fourier Series - The Fourier Transform
• Existence condition, inverse transform, useful Fourier transforms
• Properties of the Fourier transform
• LTI systems and the Fourier transform
• Laplace Transform, Properties of Laplace transform
• Inverse Laplace transform
• Z-transform , Region of convergence, The inverse Z-transform , More on the Z-transform ,Left
and right hand signals , Stable and unstable signals ,Causal and anti-causal signals

3. Signals
Signals are functions of independent variables that
carry information. For example:
Electrical signals
– Voltages and currents in a circuit
Acoustic signals
– Acoustic pressure (sound) over time
Mechanical signals
– Velocity of a car over time
Video signals
– Intensity level of a pixel (camera, video) over time

4. How is a Signal Represented?
• Mathematically, signals are represented as a function of one or
more independent variables.
• For instance a black & white video signal intensity is dependent on
x, y coordinates and time t f(x,y,t)
• On this course, we shall be exclusively concerned with signals
that are a function of a single variable: time
t
f(t)

5. What is System?
• Systems process input signals to produce output
signals
• A system is combination of elements that
manipulates one or more signals to accomplish a
function and produces some output.
system output
signal
input
signal

6. Examples of Systems
– A circuit involving a capacitor can be viewed as a
system that transforms the source voltage (signal)
to the voltage (signal) across the capacitor
– A communication system is generally composed of
three sub-systems, the transmitter, the channel and
the receiver. The channel typically attenuates and
adds noise to the transmitted signal which must be
processed by the receiver
– Biomedical system resulting in biomedical signal
processing
– Control systems

7. Some Application Areas

8. Communication Systems
o A communication system conveys information from its
source to a destination.
o Examples:
o Telephone
o TV
o Radio
o Cell phone
o Satellite

9. Communication Systems
o A communication system is composed of the following:
Fig. 1 Block Diagram of Communication System

10. Basic Components of a Communication
System
• Input Transducer
o Source: Analog or digital
o Example: Speech, music, written text, pictures
o Input Transducer: Converts the message produced
by a source to a form suitable for the
communication system.
o Example:
o Speech waves Microphone Voltage

11. Transmitter
o Prepare the Input signal for actual
transmission over the communication channel
e.g. Modulation
o Examples: TV station, radio station, web
server

12. Channel
o Physical medium that does the transmission
o Examples: Air, wires, coaxial cable, radio
wave, laser beam, fiber optic cable
o Every channel introduces some amount of
distortion, noise and interference

13. Receiver
o Job of receiver also includes to undo all the harmful
degradation introduced by channel e.g. noise
introduced by channel
o Demodulation: Extracts message from the
received signal
o Operations: Amplification, Demodulation, Filtering
o Examples: TV set, radio, web client

14. Output Transducer
o Converts electrical signal into the form desired
by the system
o Examples: Loudspeakers, PC

15. Continuous-Time & Discrete-Time

16. Modulation
o Modulation is an important step of communication
system. Modulation is defined as the process whereby
some characteristic (amplitude, frequency, phase of a
high frequency signal wave (carrier wave) is varied in
accordance with instantaneous value intensity of low
frequency signal wave (modulating wave.)
16
)cos( cctwA 

17. Modulation
o Modulation is a process that causes a
shift in the range of frequencies in a signal
o Two Type of communications
o In baseband communication baseband
signals are sent without any shift in the range
of frequencies
o Any communication that uses modulation of a
high-frequency carrier signal is called carrier
communication
17

18. Pulse Code Modulation(PCM)
• Pulse code modulation is used to convert an analog
data to digital signal(digitization).
• A PCM encoder has three processes
1.The analog signal is sampled
2.The sampled signal is quantized.
3.The quantized values are encoded as stream of bits.

19. Components of PCM encoder

20. Quantization and encoding of a
sampled signal

21. Components of a PCM decoder

22. Modes of Communications
• There are two basic modes of Communications
• Broadcast(Single Tx and Multiple Rxs)
– Radio and Tv
• Point-To-Point Communication
– Single Transmitter single Receiver
– Telephone systems
– Deep Space Communication(link b/w earth station and robot
navigating the surface of distant planet)
– Pathfinder Robot landed on Mars on July 4, 1997, a historic
day in the National Aeronautics and Space
Administration’s(NASA’s)

23. Classification of Signals
• Continuous & Discrete-Time Signals
• Even and Odd Signals
• Periodic and Non-periodic Signals
• Energy and Power Signals
• Deterministic Signals and Random Signals

24. Continuous & Discrete-Time Signals
• Continuous-Time Signals
• Most signals in the real world are
continuous time, as the scale is
infinitesimally fine.
• Eg voltage, velocity,
• Denote by x(t), where the time interval
may be bounded (finite) or infinite
• Discrete-Time Signals
• discrete time signals are defined only at
discrete instants of time.
• E.g. pixels, daily stock price (anything
that a digital computer processes)
• Denote by x[n], where n is an integer
value that varies discretely
• Sampled continuous signal
• x[n] =x(nT) is sample time
x(t)
t
x[n]
n

25. Even and Odd Signals
Even Functions Odd Functions
g t  g t  g t  g t 

26. Even and Odd Parts of Functions
 
   g g
The of a function is g
2
e
t t
t
 
even part
 
   g g
The of a function is g
2
o
t t
t
 
odd part
Ex 1.1 see book

27. Various Combinations of even and
odd functions
Function type Sum Difference Product Quotient
Both even Even Even Even Even
Both odd Odd Odd Even Even
Even and odd Neither Neither Odd Odd

28. Discrete Time Even and Odd Signals
 
   g g
g
2
e
n n
n
 
  
   g g
g
2
o
n n
n
 

   g gn n     g gn n  

29. Combination of even and odd
function for DT Signals
Function type Sum Difference Product Quotient
Both even Even Even Even Even
Both odd Odd Odd Even Even
Even and odd Even or Odd Even or odd Odd Odd

30. Periodic and Non-periodic Signals
• Given x(t) is a continuous-time signal
• x (t) is periodic if x(t) = x(t+Tₒ) for any T and any integer
n
• Example
– x(t) = A cos(wt)
– x(t+Tₒ) = A cos[w(t+Tₒ)] = A cos(wt+wTₒ)= A
cos(wt+2p) = A cos(wt)
– Note: Tₒ =1/fₒ ; w=2pfₒ

31. Periodic and Non-periodic Signals
Contd.
• For non-periodic signals
x(t) ≠ x(t+Tₒ)
• Example of non periodic signal is an
exponential signal
• See problem 1.3

32. Important Condition of Periodicity for
Discrete Time Signals
• A discrete time signal is periodic if
x(n) = x(n+N)
• For satisfying the above condition the
frequency of the discrete time signal
should be ratio of two integers
i.e. fₒ = k/N

33. Energy and Power Signals
Energy Signal
• A signal with finite energy and zero power is
called Energy Signal i.e.for energy signal
0<E<∞ and P =0
• Signal energy of a signal is defined as the area
under the square of the magnitude of the
signal.
• The units of signal energy depends on the unit
of the signal.
 
2
x xE t dt


 

34. Energy and Power Signals Contd.
Power Signal
• Some signals have infinite signal energy. In
that caseit is more convenient to deal with
average signal power.
• For power signals
0<P<∞ and E = ∞
• Average power of the signal is given by
 
/2
2
x
/2
1
lim x
T
T
T
P t dt
T

 

35. Energy and Power Signals Contd.
• For a periodic signal x(t) the average
signal power is
• T is any period of the signal.
• Periodic signals are generally power
signals.
 
2
x
1
x
T
P t dt
T
 

36. Signal Energy and Power for DT
Signal
•The signal energy of a for a discrete time signal x[n] is
 
2
x x
n
E n


 
•A discrtet time signal with finite energy and zero
power is called Energy Signal i.e.for energy signal
0<E<∞ and P =0

37. Signal Energy and Power for DT
Signal Contd.
The average signal power of a discrete time power signal
x[n] is
 
1
2
x
1
lim x
2
N
N
n N
P n
N



 
 
2
x
1
x
n N
P n
N 
 
For a periodic signal x[n] the average signal power is
The notation means the sum over any set of
consecutive 's exactly in length.
n N
n N

 
 
 
 
