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SIGNALS & SYSTEMSSIGNALS & SYSTEMSLECTURER:LECTURER:
MUZAMIR ISA MUZAMIR ISA 049798139 049798139
[email protected]@kukum.edu.my
PLV:PLV:MUHAMMAD HATTA HUSSEINMUHAMMAD HATTA HUSSEIN
[email protected]@kukum.edu.my
EVALUATIONEVALUATION
CourseworkCoursework : : 50 % 50 %
30 % Practical:30 % Practical:(i) 70 % from Lab Report(i) 70 % from Lab Report(ii) 30% from Lab Test(ii) 30% from Lab Test
20 % :20 % :(i) 15 % from Written Test 1 & Written Test (i) 15 % from Written Test 1 & Written Test
22(ii) 5 % from Tutorial(ii) 5 % from Tutorial
Final ExamFinal Exam : : 50 %50 %
REFERENCESREFERENCES
Simon Haykin, Barry Van Veen; Signal & Simon Haykin, Barry Van Veen; Signal & System, 2System, 2ndnd Edition, 2003, Wiley (main Edition, 2003, Wiley (main textbook)textbook)
MJ Robert; Signal & System, 2003, MJ Robert; Signal & System, 2003, McGraw HillMcGraw Hill
Charles L Philips et.al; Signal, System and Charles L Philips et.al; Signal, System and Transform, Pearson.Transform, Pearson.
Signals and SystemsSignals and Systems
Signals are variable that carry Signals are variable that carry informationinformation
Systems process input signals to Systems process input signals to produce output signalsproduce output signals
What Are “Signals”?What Are “Signals”?
A function of one or more variable, A function of one or more variable, which conveys information on the which conveys information on the nature of a physical phenomenon.nature of a physical phenomenon.
A function of time representing a A function of time representing a physical or mathematical quantities.physical or mathematical quantities.
e.g. : Velocity, acceleration of a car, e.g. : Velocity, acceleration of a car, voltage/current of a circuit.voltage/current of a circuit.
Even Signal Odd Signal
Deterministic Signal Random Signal
Classification of SignalsClassification of Signals
Continuous-Time and Discrete-Time Signals
Even and Odd Signals
Periodic and Nonperiodic Signals
Deterministic and Random Signals
Energy and Power Signals
Continuous Time (CT) and Discrete-Time (DT) Continuous Time (CT) and Discrete-Time (DT) SignalsSignals
Continuous-time signalsContinuous-time signalsExamples: Signals in cars and circuits
Signals described by differential equations, e.g., dy/dt = ay(t) + bf(t)
Signal itself could have jumps (discontinuities) in magnitude
t
y(t)
Discrete-time signalsExamples: money in a bank account, daily stock
prices
No derivative exists
Signals described by difference equations, e.g., y(k+1) = ay(k) + bf(k)
k
y(k)
Even and Odd SignalsEven and Odd Signals
Periodic and A-periodic SignalsPeriodic and A-periodic Signals
Right and Left-Sided Signals
Bounded and Unbounded Signals
OPERATION ON SIGNALSOPERATION ON SIGNALS
Operations performed on the independent variable Time scaling
y(t) = x(at) Reflection
y(t) = x(-t) Time shifting
y(t) = x(t – t0) where t0 is the time shift.
TIME SCALINGTIME SCALING
y(t) = x(at) ; y(t) = x(at) ;
Compress the signal x(t) by a.
This is equivalent to plotting the signal x(t) in a new time axis tn at the location given by t = atn or tn = t/a
REFLECTION OR FOLDINGREFLECTION OR FOLDING
y(t) = x(- t)y(t) = x(- t)
Just scaling operation with a = -1. It Just scaling operation with a = -1. It creates the folded signal x(- t) as a creates the folded signal x(- t) as a mirror image of x(t) about the vertical mirror image of x(t) about the vertical axis through the origin t = 0.axis through the origin t = 0.
TIME SHIFTINGTIME SHIFTING
y(t) = x(t – a)y(t) = x(t – a)
Displaces a signal x(t) in time without Displaces a signal x(t) in time without changing its shape. Simply shift the changing its shape. Simply shift the signal x(t) to the right by a. This is signal x(t) to the right by a. This is equivalent to plotting the signal x(t) in a equivalent to plotting the signal x(t) in a new time axis tn at the location given new time axis tn at the location given by t = tby t = tnn - a or t - a or tnn = t + a. = t + a.
EXAMPLEEXAMPLE
A CT signal is shown, sketch and label A CT signal is shown, sketch and label each of this signal;each of this signal;
a) x(t -1) a) x(t -1)
b) x(2t)b) x(2t)
c) x(-t)c) x(-t)-1 3
2
t
x(t)
-3 1
2
t
x(-t)
0 4
t
x(t-1)
2
-1/2 3/2
2
t
x(t)
A discrete-time signal, x[n-2]
A delay by 2
4
2
0 1 2 3 4 5 n
x(n-2)
A discrete-time signal, x[2n]
Down-sampling by a factor of 2.
4
2
0 1 2 3 n
x(2n)
A discrete-time signal, x[-n+2]A discrete-time signal, x[-n+2]
Time reversal and shifting
4
2
-1 0 1 2 n
x(-n+2)
A discrete-time signal, x[-n]
Time reversal
4
2
-3 -2 -1 0 1 n
x(-n)
Exercises
1.A continuous-time signal x(t) is shown below, Sketch and label each of the following signal
a.x(t – 2) b. x(2t) c.x(t/2) d. x(-t)
x(t)
t
4
0 4
Continue…Continue…
2.A discrete-time signal x[n] is shown below, Sketch and label each of the following signal
a. x[n – 2]b. x[2n] c. x[-n+2] d. x[-n]
x[n]
n
4
2
0 1 2 3
Basic Operation on SignalsBasic Operation on Signals
Operations performed on dependent Operations performed on dependent variablevariable Amplitude scalingAmplitude scaling AdditionAddition MultiplicationMultiplication DifferentiationDifferentiation IntegrationIntegration
Exponential SignalsExponential Signals
x(t) = Bex(t) = Beatat ; ; B is the amplitudeB is the amplitude
Decaying Exponential (a < 0)Decaying Exponential (a < 0)
Growing Exponential (a > 0)Growing Exponential (a > 0)
Sinusoidal SignalsSinusoidal Signals
x(t) = A cos(x(t) = A cos(t + t + ))
where where A = amplitudeA = amplitude
= frequency (rad/s)= frequency (rad/s)
= phase angle (rad)= phase angle (rad)
Unit Impulse Function
Narrow Pulse Approximation
Intuiting Impulse Definition
Uses of the Unit Impulse
Unit Step Function
Successive Integrations of the Unit Impulse Function
