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    Sns slide 1 2011 Sns slide 1 2011 Presentation Transcript

    • DEE2363 Signals And Systems by Mr. Koay Fong Thai [email_address] ; [email_address]
    • Announcement (1)
      • Schedule
      • Feedback to Lecturer (First two weeks of the semester)
        • The availability of Room, Students, Lecturer.
      • Plagiarism
      • 7 cases in August’08 Final/Resit Exams.
      • 1 expelled, 6 with Warning Letters and Re-take.
    • Announcement (2)
      • Badge
      • Warning Letter.
      • Absent
      • 3 times – Warning Letter and inform parents.
      • Less than 75% attendance  to be barred from final exam.
      • Use of Handphones and Laptops are STRICTLY PROHIBITED in Class and Lab.
    • Dos
      • Feel free to ask questions during the class (without disturbance), or
      • my consultation hours, every Wednesday and Friday, 9am – 11am.
      • Be sensitive to find my errors.
      • Work hard to get a good grade.
      • Be on time in the class.
      • Be on time to submit coursework (take note of the format).
    • Don’ts
      • Use of cell phones and computers in the class.
      • Play games!!
      • Sleep and/or talk in the class.
      • Copy tutorials/assignments (you will get zero if I catch you).
      • … etc, to be defined by me….
    • Assessment
      • Tutorial : 10%
      • Quiz : 10%
      • Assignment : 10%
      • Mid-Term Tests : 20%
      • 50% ****
      • Final Exam : 50% .
      • 100%
      Less than 25%  to be barred from final exam!!
    • Lecture and Tutorial Schedule
    • Lecture Schedule with Outcome
    • Broad Aims
      • To introduce the students to the idea of signal and system along with the analysis and characterization .
      • To introduce the students the transformation methods for both continuous-time and discrete-time signals and systems.
      • To provide a foundation to numerous other courses that deal with signal and system concepts directly or indirectly, for instance, communication, control, instrumentation, and so on; as well as to students of disciplines such as, mechanical, chemical and aerospace engineering.
    • Objectives
      • By the end of the course, you would have understood:
        • Basic signal analysis (mostly continuous-time)
        • Basic system analysis (also mostly continuous systems)
        • Time-domain analysis (including convolution)
        • Laplace Transform and transfer functions
        • Fourier Series and Fourier Transform
        • Sampling Theorem and Signal Reconstructions
        • Basic z-transform
    • Topics
      • Signals and Systems in the time domain
        • Impulse response, Convolution Integral/Sum, Differential/Difference Equations, Linear Constant-Coefficient Differential/Difference Equations.
      • Fourier Transform
        • Continuous/discrete Fourier Series and Transform, the properties and applications in signals and systems, Inverse Fourier Transform.
      • Laplace Transform
        • Properties, inverse transform; analysis of signals and systems.
      • z-Transform
        • Properties, inverse transform; analysis of signals and systems.
    • Reference Books
      • B.P. Lathi, “ Signal Processing and Linear Systems ”, 1998, Oxford University Press.
        • PSDC Library : TK5102.9 Lat 1998
      • M.J. Roberts, “ Signals and Systems: Analysis Using Transform Methods and MATLAB ”, 2004, McGraw Hill.
        • PSDC Library : TK5102.9 Rob 2004
      • Alan V. Oppenheim, Alan S. Willsky, S. Hamid Nawab, “ Signals and Systems ”, 2 nd Edition 1997, Pearson Education,.
        • PSDC Library : QA402 Opp 1997
      • Simon S. Haykin, B. Van Veen, “ Signals and Systems ”, 2 nd Edition 2005, John Wiley & Sons.
    • Introduction to Signals (CT and DT) By Koay Fong Thai [email_address]
    • Topics
      • Introduction
      • Size of a Signal
      • Classification of Signals
      • Some Useful Signal Operations
      • Some Useful Signal Models
      • Even and Odd Functions
    • Introduction
      • The concepts of signals and systems arise in a wide variety of areas:
        • communications,
        • circuit design,
        • biomedical engineering,
        • power systems,
        • speech processing,
        • etc.
    • What is a Signal?
        • SIGNAL
      • A set of information or data.
      • Function of one or more independent variables.
      • Contains information about the behavior or nature of some phenomenon.
    • Examples of Signals
      • Electroencephalogram (EEG) signal (or brainwave)
    • Examples of Signals (2)
      • Stock Market data as signal (time series)
    • What is a System?
        • SYSTEM
      • Signals may be processed further by systems, which may modify them or extract additional from them.
      • A system is an entity that processes a set of signals ( inputs ) to yield another set of signals ( outputs ).
    • What is a System? (2)
      • A system may be made up of physical components, as in electrical or mechanical systems (hardware realization).
      • A system may be an algorithm that computes an outputs from an inputs signal (software realization).
    • Examples of signals and systems
      • Voltages (x 1 ) and currents (x 2 ) as functions of time in an electrical circuit are examples of signals.
      • A circuit is itself an example of a system (T) , which responds to applied voltages and currents.
    • Signals Classification
      • Signals may be classified into:
        • 1. Continuous-time and discrete-time signals
        • 2. Analogue and digital signals
        • 3. Periodic and aperiodic signals
        • 4. Causal, noncausal and anticausal signals
        • 5. Even and Odd signals
        • 6. Energy and power signals
    • Signals Classification (2) – Continuous versus Discrete
      • Continuous-time A signal that is specified for every value of time t.
      • Discrete-time A signal that is specified only at discrete values of time t.
    • Signals Classification (3) – Analogue versus Digital (1)
      • Analogue, continuous
      • Analogue, discrete
    • Signals Classification (4) – Analogue versus Digital (2)
      • Digital, continuous
      • Digital, discrete
    • Signals Classification (5) – Periodic versus Aperiodic
      • A CT signal x(t) is said to be periodic if for some positive constant T o ,
      • The smallest value of T o that satisfies the periodicity condition is the fundamental period of x(t) .
    • Signals Classification (6) – Periodic versus Aperiodic (2)
      • Fundamental period = T o . Then, fundamental frequency is f o = 1/ T o in Hz or cycles per second.
      • or
      • Angular frequency, ω o = 2  / T o , radian per second.
      • Example, f(t) = C cos (2  f o t +  ) C : amplitude; f o : frequency;  : phase
        • Rewriting f(t) = C cos ( ω o t +  )
    • Signals Classification (7) – Periodic versus Aperiodic (3)
      • A DT signal x[n] is said to be periodic if for all positive integer N ,
      • The smallest value of N is the fundamental period of x[n] .
      • Fundamental angular frequency,  is defined by  = 2  / N.
    • Signals Classification (6) – Periodic versus Aperiodic (4)
      • For the signal,
      • Find the period and the fundamental frequency of the signal.
      • Solution (hint: sin  = cos (  -  /2) ):
    • Signals Classification (6) – Periodic versus Aperiodic (5)
    • Signals Classification (8) – Periodic versus Aperiodic (6)
      • Aperiodic (Nonperiodic) signals?
    • Signals Classification (9) – Causal vs. Noncausal vs. Anticausal
      • Causal ( 因果 ) signal:
        • A signal that does not start before t =0.
        • f(t) = 0; t <0
      • Noncausal signal:
        • A signal that starts before t =0, such as charge in capacitor before switch is turned on.
      • Anticausal signal:
        • A signal that is zero for all t  0.
    • Signals Classification (10) – Even versus Odd
    • Signals Classification (11) – Even versus Odd (2)
      • A signal x(t) or x[n] is referred to as an even signal if
        • CT:
        • DT:
      • A signal x(t) or x[n] is referred to as an odd signal if
        • CT:
        • DT:
    • Signal Classification (12) – Energy versus Power
      • Signal with finite energy (zero power)
      • Signal with finite power (infinite energy)
      • Signals that satisfy neither property are referred as neither energy nor power signals
    • Size of a Signal (1)
      • A number indicates the largeness or strength of the signal.
      • Such a measure must consider both amplitude and duration of the signal.
      • Measurement of the size of a human being, V with variable radius, r and height, h with assumption of cylindrical shape given by
    • Size of a Signal (2)
      • Assuming f(t) = sin t ,
        • f(t) could be a large signal,
        • yet its positive and negative areas cancel each other.
      • Then, indicates a signal of small size.
      • This can be solved by defining the signal size as the area under f 2 (t) ( f 2 (t) always > 0).
    • Size of a Signal, Energy (Joules)
      • Measured by signal energy E x :
      • Generalize for a complex valued signal to:
        • CT: DT:
      • Energy must be finite, which means
    • Size of a Signal, Power (Watts)
      • If amplitude of x(t) does not -> 0 when t -> ∞, need to measure power P x instead:
      • Again, generalize for a complex valued signal to:
        • CT:
        • DT:
    • Example
      • Determine the suitable measures of the signals in the figure below:
    • Example
    • Summary
      • By the end of the class, you would have understood:
        • Examples of signals
        • Signals classification
    • Signal Operations for CT Signals by Koay Fong Thai [email_address]
    • Signal Operations
      • Signal operations are operations on the time variable of the signal, involve simple modification of the independent variable .
        • Time Shifting
        • Time Scaling
        • Time Inversion (Reversal)
        • Combined operations
    • Signal Operations: Time Shifting
      • Shifting of a signal in time
      •  adding or subtracting the amount of the shift to the time variable in the function.
      • x(t)  x(t–t o )
        • t o > 0 ( t o is positive value), signal is shifted to the right (delay).
        • t o < 0 ( t o is negative value), signal is shifted to the left (advance).
      • x(t–2) ? x(t) is delayed by 2 seconds.
      • x(t+2) ? x(t) is advanced by 2 seconds.
    • Signal Operations: Time Shifting (2)
      • Subtracting a fixed amount from the time variable will shift the signal to the right that amount.
      • Adding to the time variable will shift the signal to the left.
    • Signal Operations: Time Shifting (3)
    • Signal Operations: Time Scaling
      • Compress es ( 压缩 ) and dilate s ( 膨胀 ) a signal by multiplying the time variable by some amount.
      • x(t)  x(  t)
        • If  >1, the signal becomes narrower  compression.
        • If  <1, the signal becomes wider  dilation.
      • Play audio recorded, f(t) in mp3 player at twice the normal recording speed?
        • f(2t) or f(t/2) ?
        • f(2t)
    • Signal Operations: Time Scaling (2)
    • Signal Operations: Time Scaling (3)
    • Signal Operations: Time Scaling (4)
    • Signal Operations: Time Inversion (Reversal)
      • Reversal of the time axis, or folding/flipping the signal (mirror image) over the y-axis.
    • Signal Operations: Time Inversion (Reversal) (2)
    • Signal Operations: Combined Operations
      • Signal f(at–b) can be realized in TWO ways:
        • Time-shift f(t) by b  f(t–b) , then, time-scaled f(t–b) by a  f(at–b)
        • Time-scale f(t) by a  f(at) , then, time-shift f(at) by b/a  f[a(t – b/a)]
        • If a <0, it’s time inversion operation.
    • Example
      • Given the signal as shown in figure below. Plot
      • i. x ( t + 1) ii. x (1- t )
      • iv.
    • Example (i)
      • Shift to the left by one unit along the t axis.
    • Example (ii)
      • Replace t with –t in x(t + 1)  x(-t + 1)  x(1-t).
      • It is obtained graphically by reflecting x(t + 1) about the y axis.
    • Example (iii)
      • The signal x(3/2 t)  compression of x(t) by a factor of 2/3.
    • Example (iv)
      • First, advance or shift to the left x(t) by 1 as shown in figure below.
      • Then, compress this shifted signal by a factor of 2/3.
    • Signal Operations
    • Some Useful Signal Models by Koay Fong Thai [email_address]
    • Signal Models: Unit Step Function
      • Continuous-Time unit step function, u(t) :
      • u(t) is used to start a signal, f(t) at t =0  f(t) has a value of ZERO for t <0
    • Signal Models: Unit Step Function (2)
      • f(t) = e -at x u(t)  a causal form of e -at .
      • x =
    • Signal Models: Unit Step Function (3)
      • Realize the rectangular pulse below:
    • Signal Models: Unit Impulse Function
      • Continuous-Time unit impulse function,  (t ) is defined by P.A.M. Diarc:
      • We can visualize an impulse as a tall and narrow rectangular pulse of unit area .
      • When   0, the height is very large, 1/  .
    • Signal Models: Unit Impulse Function (2)
      • A possible approximation to a unit impulse: An overall area that has been maintained at unity .
      • Multiplication of a function by an Impulse?
      • b  (t ) = 0; for all t  0 is an impulse function which the area is b .
      Graphically, it is represented by an arrow &quot;pointing to infinity&quot; at t=0 with its length equal to its area.
    • Signal Models: Unit Impulse Function (3)
      • May use functions other than a rectangular pulse. Here are three example functions:
      • Note that the area under the pulse function must be unity .
    • Signal Models: Unit Ramp Function
      • Unit ramp function is defined by:
        • r(t) = t  u(t)
      • Where can it be used?
    • Signal Models: Example
      • Describe the signal below:
    • Signal Models: Example (2)
      • x =t[u(t) – u(t-2)]
      • x = -2(t-3)[u(t-3) – u(t-2)]
      • t[u(t) – u(t-2)] - 2(t-3)[u(t-3) – u(t-2)]
    • Signal Models: Exponential Function e st
      • Most important function in SNS where s is complex in general, s =  + j 
      • Therefore, e st = e (  + j  )t = e  t e j  t = e  t (cos  t + j sin  t) (Euler’s formula: e j  t = cos  t + j sin  t )
      • If s  =  - j  ,
      • e s  t = e (  - j  )t = e  t e - j  t = e  t (cos  t - j sin  t)
      • From the above, e  t cos  t = ½( e st + e -st )
    • Signal Models: Exponential Function e st (2)
      • Variable s is complex frequency .
      • e st = e (  + j  )t = e  t e j  t = e  t (cos  t + j sin  t) e s  t = e (  - j  )t = e  t e - j  t = e  t (cos  t - j sin  t) e  t cos  t = ½( e st + e -st )
      • There are special cases of e st :
        • A constant k = ke0t (s=0   =0,  =0)
        • A monotonic exponential e  t (  =0, s =  )
        • A sinusoid cos  t (  =0, s =  j  )
        • An exponentially varying sinusoid e  t cos  t ( s =   j  )
    • Signal Models: Exponential Function e st (3)
    • Signal Models: Exponential Function e st (4)
      • In complex frequency plane:
    • Even and Odd Functions
      • A function f e (t) is said to be an even function of t if f e (t) = f e (-t)
      • A function f o (t) is said to be an odd function of t if f o (t) = - f o (-t)
    • Even and Odd Functions: Properties
      • Property:
      • Area:
        • Even signal:
        • Odd signal:
    • Even and Odd Components of a Signal (1)
      • Every signal f(t) can be expressed as a sum of even and odd components because
      • Example, f(t) = e -at u(t)
    • Even and Odd Components of a Signal (2)
      • Example, f(t) = e -at u(t)  casual?
    • Signal Models: Summary
      • Unit step function, u(t)
      • Unit impulse function,  (t)
      • Unit ramp function, r(t)
      • Exponential function, e st
      • Even and off function
      • All these functions are used for CONTINUOUS-TIME signals!!!!
    • Sampling By Koay Fong Thai [email_address]
    • Sampling Theorem
      • Sampling is the process of converting a continuous signal into a discrete signal .
    • Sampling Theorem (2)
      • The sampling frequency or sampling rate, ( f s )  the number of samples per second taken from a continuous signal to make a discrete signal.
      • It is measured in hertz (Hz) .
    • Sampling Theorem (3)
      • Sampling period or sampling time ( T )  The inverse of the sampling frequency which is the time between samples .
      • Given the sampling period T, the sampling frequency is given by
    • Sampling Theorem (4)
      • The discrete-time signal x[n] is obtained by “taking-samples” of the analog signal x c (T) every T second.
      • x[n] = x c (nT)
      • It is measured in hertz (Hz) .
      • The relationship between the variable t of analog signal and the variable n of discrete-time signal is
    • Sampling Theorem (5)
      • We refer to a system that implements the operation of the above equation as an ideal continuous-to-discrete-time (C/D) converter :
      • Block diagram representation of an ideal C/D Converter
    • Sampling Theorem (6)
      • Sampling is represented as an impulse train modulation followed by the conversion into a sequence .
      • Figure below illustrates a mathematical representation of sampling with a periodic impulse train followed by a conversion to a discrete-time sequence.
      (a) Overall system; (b) x s (t) for two sampling rates. The dashed envelope represents x c (t); (c) The output sequence for the two different sampling rates.
    • Sampling Theorem (7)
      • Figure below is the DT sequences of two CT signals at sampling frequency of 5 Hertz (samples/second).
    • Signal Operations for DT Signals By Koay Fong Thai [email_address]
    • Signal Operations
      • Signal operations are operations on the time variable of the signal, involve simple modification of the independent variable .
        • Time Shifting
        • Time Scaling
        • Time Inversion (Reversal)
        • Combined operations
    • Signal Operations: Time Shifting
      • Shifting of a signal in time
    • Signal Operations: Time Scaling
      • Compress es and expanse s a signal by multiplying the time variable by some integers.
      • x(k)  x(  k)
        • If  >1, the signal becomes narrower  compression  data losses  decimation ( 抽取 ) or downsampling ( 下降抽样 ).
        • If  <1, the signal becomes wider  expansion .
          • Insert missing samples using an interpolation formula  interpolation ( 插值 ) or upsampling ( 后续采样 )
    • Signal Operations: Time Scaling (2)
    • Signal Operations: Time Inversion (Reversal)
      • Reversal of the time axis, or folding/flipping the signal (mirror image) over the y-axis.
    • Some Useful Signal Models By Koay Fong Thai [email_address]
    • Signal Models: Unit Step Function
      • Discrete-Time unit step function/sequence, u(k) :
      • The unit step is the running sum of an impulse:
    • Signal Models: Unit Impulse Function
      • Discrete-Time unit impulse function/sequence,  (k) :
      • The unit impulse is the first-difference of a unit step:
    • Signal Models: DT Exponential Function  k
      • In CT SNS, CT exponential e st can be expressed in another form,  t where  t = e st .
      • However, in DT SNS, it is proven that  k is more convenient than e  k (k is the integer).
    • Signal Models: DT Exponential Function  k (2)
    • Signal Models : DT Exponential Function  k (3)
    • Questions and Answer
      • Any questions?