Signals and classification
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Signals and classification






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    Signals and classification Signals and classification Presentation Transcript

    • Topics     Introduction Classification of Signals Some Useful Signal Operations Some useful signal models 2
    • Introduction  The concepts of signals and systems arise in a wide variety of areas:  communications,  circuit design,  biomedical engineering,  power systems,  speech processing,  etc. 3
    • 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. 4
    • Examples of Signals  BRAIN WAVE 5
    • Examples of Signals  Stock Market data as signal (time series) 6
    • 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). 7
    • 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). 8
    • Examples of signals and systems   Voltage (x1) and current (x2) 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. 9
    • Some Useful Signal Models 10
    • 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 11
    • Signal Models: Unit Impulse Function A possible approximation to a unit impulse: An overall area that has been maintained at unity. Graphically, it is represented by an arrow "pointing to infinity" at t=0 with its length equal to its area.  Multiplication of a function by an Impulse?  bδ(t) = 0; for all t≠0 is an impulse function which the area is b. 12
    • 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. 13
    • Signal Models: Unit Ramp Function  Unit  ramp function is defined by: r(t) = t∗u(t)  Where can it be used? 14
    • Signal Models: Exponential Function est  Most important function in SNS where s is complex in general, s = σ+jϖ  Therefore, est = e(σ+jϖ)t = eσtejϖt = eσt(cosϖt + jsinϖt) (Euler’s formula: ejϖt = cosϖt + jsinϖt) s∗ = σ-jϖ,  es∗ t = e(σ-jϖ)t = eσte-jϖt = eσt(cosϖt - jsinϖt)  If  From the above, e cosϖt = ½(e +e ) σt st -st 15
    • Signal Models: Exponential Function est (2)    Variable s is complex frequency. est = e(σ+jϖ)t = eσtejϖt = eσt(cosϖt + jsinϖt) es∗ t = e(σ-jϖ)t = eσte-jϖt = eσt(cosϖt - jsinϖt) eσtcosϖt = ½(est +e-st ) There are special cases of est : 1. 2. 3. 4. A constant k = ke0t (s=0  σ=0,ϖ=0) A monotonic exponential eσt (ϖ=0, s=σ) A sinusoid cosϖt (σ=0, s=±jϖ) An exponentially varying sinusoid eσtcosϖt (s= σ ±jϖ) 16
    • Signals Classification  Signals      may be classified into: 1. Continuous-time and Discrete-time signals 2. Deterministic and Stochastic Signal 3. Periodic and Aperiodic signals 4. Even and Odd signals 5. Energy and Power signals 17
    • Continuous v/S Discrete Signals  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. 18
    • Deterministic v/s Stochastic Signal  Signals that can be written in any mathematical expression are called deterministic signal.  (sine,cosine..etc)  Signals that cann’t be written in mathematical expression are called stochastic signals.  (impulse,noise..etc) 19
    • Periodic v/s Aperiodic Signals  Signals that repeat itself at a proper interval of time are called periodic signals.  Continuous-time signals are said to be periodic.  Signals that will never repeat themselves,and get over in limited time are called aperiodic or non-periodic signals. 20
    • Even v/s Odd Signals 21
    • Even v/s Odd Signals 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: 22
    • Even and Odd Functions: Properties  Property:  Area:  Even signal:  Odd signal: 23
    • 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-atu(t) 24
    • Energy v/s Power Signals  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 25
    • Size of a Signal, Energy (Joules)  Measured by signal energy Ex:  Generalize  CT:  Energy for a complex valued signal to: DT: must be finite, which means 26
    • Size of a Signal, Power (Watts)  If amplitude of x(t) does not → 0 when t → ∞, need to measure power Px instead:  Again, generalize for a complex valued signal to:  CT:  DT: 27
    • OPERATIONS ON SIGNALS  It includes the transformation of independent variables.  It is performed in both continuous and discrete time signals.  Operations that are performed are- 28
    • 1.ADDITION &SUBSTRACTION    Let two signals x(t) and y(t) are given, Their addition will be, z(t) = x(t) + y(t) Their substraction will be, z(t) = x(t) – y(t) 29
    • 2.MULTIPLICATION OF SIGNAL BY A CONSTANT  If a constant ‘A’ is given with a signal x(t) z(t) = A.x(t)  If A>1,it is an amplified signal. If A<1,it is an attenuated signal.  30
    • 3.MULTIPLICATION OF TWO SIGNALS  If two signals x(t) and y(t) are given,than their multiplication will be z(t) = x(t).y(t) 31
    • 4.SHIFTING IN TIME  Let a signal x(t),than the signal x(t-T) represented a delayed version of x(t),which is delayed by T sec. 32
    • 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   to > 0 (to is positive value), signal is shifted to the right (delay). to < 0 (to 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. 33
    • 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. 34
    • Signal Operations: Time Shifting  Shifting of a signal in time 35
    • 5.COMPRESSION/EXPANSION OF SIGNALS    This is also known as ‘Time Scaling’ process. Let a signal x(t) is given,we will examine as x(at) where a =real number and how it is related to x(t) ? 36
    • Time Scaling 37
    • Signal Operations: Time Inversion  Reversal of the time axis, or folding/flipping the signal (mirror image) over the y-axis. 38
    • THANKS....................... FOR YOUR ATTENTION ! 39