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UNIT 2

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PERIODIC WAVEFORMS, SIGNAL SYNTHESIS AND

LAPLACE TRANSFORM

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SIGNALS

  • Variables that carry information
  • Described as a function of single or multiple independent variables
  • Convey info on nature of a physical phenomenon
  • Can be one/multi-dimensional
    • Ex: heart-rate, temperature, human speech, etc.

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TYPES OF SIGNALS

  • Continuous-time/Discrete-time
  • Periodic/Non-Periodic
  • Causal/Non-Causal
  • Deterministic/Random
  • Even/Odd

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CONTINUOUS/DISCRETE-TIME SIGNALS

  • CONTINUOUS-TIME
    • Analog signal
    • Defined for a continuum of values of time t
    • Denoted as
    • Ex: Speech signal
  • DISCRETE-TIME
    • Defined for discrete values of time n
    • Amplitude between two time instants is not defined
    • Denoted as
    • Ex: monthly sales of a corporation

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Source: https://www.signalandsystems.wikidot.com

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PERIODIC/NON-PERIODIC SIGNALS

  • PERIODIC
    • Repeats itself over and over
    • Time between the repeats: Period
    • Signal remains unchanged when time-shifted by 1 period
    • or
  • NON-PERIODIC
    • Does not repeat itself after a fixed period

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Source: https://www.signalandsystems.wikidot.com

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CAUSAL/NON-CAUSAL SIGNALS

  • CAUSAL
    • Does not start before time t=0

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  • NON-CAUSAL
    • Signal starts before time t=0

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Source: https://www.signalandsystems.wikidot.com

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DETERMINISTIC/RANDOM SIGNALS

  • DETERMINISTIC
    • Behaviour is predictable with respect to time
    • No uncertainty about its value at any time
    • Can be expressed mathematically
  • RANDOM
    • Behaviour is predictable with respect to time
    • Uncertainty exists about its value at any time
    • Cannot be expressed mathematically
    • Thermal noise

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EVEN/ODD SIGNALS

  • EVEN
    • Identical with its reflection about origin
    • Symmetrical about vertical axis
    • and
  • ODD
    • Negative of its value at the instant -t or -n
    • Anti-symmetrical about vertical axis
    • and

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Source: https://www.signalandsystems.wikidot.com

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BASIC SIGNALS: UNIT STEP

CONTINUOUS-TIME

DISCRETE-TIME

Image Source: intmath.com

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BASIC SIGNALS: UNIT IMPULSE

CONTINUOUS-TIME

DISCRETE-TIME

Image Source: researchgate.net

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BASIC SIGNALS: UNIT RAMP

CONTINUOUS-TIME

DISCRETE-TIME

Image Source: researchgate.net

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SYSTEM

  • SYSTEM:
    • System processes input signal to produce output
    • Transforms input signal to output signal

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    • Ex: CD player transforms binary data on CD into acoustic signal

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TYPES OF SYSTEMS

  • Causal/Non-Causal
  • Static/Dynamic
  • Time-variant/Time-invariant
  • Linear/Non-Linear
  • Stable/Unstable

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CAUSAL/NON-CAUSAL SYSTEMS

  • CAUSAL
    • Response of the system depends only on present and/or past values of the input
    • Response does not depend on future values
  • NON-CAUSAL
    • System response also depends on future values of the input

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STATIC/DYNAMIC SYSTEMS

  • STATIC
    • Response of the system depends only on the input at that particular time and not on past/future values
    • Memory-less system
  • DYNAMIC
    • System response also depends on input at that particular time and other times
    • Possess memory

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LINEAR/NON-LINEAR SYSTEMS

  • LINEAR
    • Principle of Superposition: Response of sum of weighted inputs is same as sum of weighted responses
  • NON-LINEAR
    • Principle of Superposition is not obeyed

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STABLE/UNSTABLE SYSTEMS

  • BIBO STABILITY
    • Every bounded input produces bounded output
    • System response to time-shifted input is shifted by the same amount of time

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  • UNSTABLE SYSTEM

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LAPLACE TRANSFORM (LT)

  • Conversion of time-domain function to corresponding frequency-domain function
  • Test signals such as ramp, parabolic are not Fourier Transformable. Hence, LT used widely

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  • Frequency-domain, computation is easier
  • Initial conditions are automatically considered in LT

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LAPLACE TRANSFORM OF �BASIC FUNCTIONS

Image Source: https://tutorial.math.lamar.edu

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LAPLACE TRANSFORM PROPERTIES

Image Source: https://tutorial.math.lamar.edu

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SIGNAL SYNTHESIS

BASIC BLOCK 1 : Unit Step signal

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Image Source: Network analysis and synthesis, Pankaj Swarnkar

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Continue

  • Synthesize the function:
  • F(t)= ktu(t)
  • F(t)= k(t-1)[u(t-1)-u(t-2)]

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SIGNAL SYNTHESIS

BASIC BLOCK 2 : Unit Ramp signal

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Image Source: Network analysis and synthesis, Pankaj Swarnkar

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Signal Synthesis

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SIGNAL SYNTHESIS

BASIC BLOCK 2 : Unit Ramp signal (cond..)

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Image Source: Network analysis and synthesis, Pankaj Swarnkar

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Signal Synthesis

  • Sin wt
  • Sin wt U(t)
  • Sin w(t-to)
  • Sin w(t-to) U(t)
  • Sin wt U(t-to)
  • Sin w(t-to)U(t-to)

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SIGNAL SYNTHESIS

BASIC BLOCK 3 : Sinusoidal signal

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Source: Network analysis and synthesis, Pankaj Swarnkar

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Continue

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Synthesize the given half sine waveform

V(t)

Vm

0 π

t

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Synthesize the following waveform

0 1 2 3 4 5 t

1

2

F(t)

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LAPLACE TRANSFORM OF �PERIODIC SIGNALS

Laplace transform of one period

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Example:

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one period

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Laplace transform of one period

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Laplace transform of periodic waveform

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SIGNAL SYNTHESIS

BASIC BLOCK 3 : Sinusoidal signal

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Source: Network analysis and synthesis, Pankaj Swarnkar