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ECE310 – Lecture 7

ECE310 – Lecture 7. Mathematical Description of Signals 01/26/01. Important Continuous Signals. Sinusoids Exponentials Continuous and differentiable. Discontinuous. Singularity Signals and Related. Unit step function – u(t) Unit ramp function – ramp(t) Unit rectangle function – rect(t)

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ECE310 – Lecture 7

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  1. ECE310 – Lecture 7 Mathematical Description of Signals 01/26/01

  2. Important Continuous Signals • Sinusoids • Exponentials • Continuous and differentiable

  3. Discontinuous

  4. Singularity Signals and Related • Unit step function – u(t) • Unit ramp function – ramp(t) • Unit rectangle function – rect(t) • The signum – sgn(t) • Unit triangle – • Unit sinc function – sinc(t) • Unit impulse function – d(t) • Unit comb function – comb(t)

  5. u(t) rect(t) sgn(t) ramp(t) Integral of the step function: u(t) The convolution of rect(t) with itself sinc(t) The continuous Fourier transform of rect(t) Relationships

  6. Impulse Function • Also called Dirac-delta function • Strictly speaking, it is not a function • The actual value of the impulse at t=0 is infinite. • The area under an impulse is called its strength or weight • A unit impulse function has a strength of unity 1 t=0

  7. How to Get Impulse

  8. Some Transformations

  9. Important Properties • Sifting or Sampling • Time scaling

  10. Some Examples

  11. Unit Comb Function • The strengths of the impulses are unity • The spacing between impulses is unity • The average value of the function is unity -2 -1 0 1 2

  12. Some Transformations

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