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Transformation of Functions in ECE310 Lecture: Continuous-time vs. Discrete-time Functions

Understand the concepts of Continuous-time and Discrete-time functions, with examples and exercises on Amplitude scaling, Time shifting, and Time scaling in function transformations.

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Transformation of Functions in ECE310 Lecture: Continuous-time vs. Discrete-time Functions

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  1. ECE310 – Lecture 3 Transformation of Functions 01/17/01

  2. CTF vs. DTF • Continuous-time function • g(t) • t can be any real number • Discrete-time function • Sampling rate is Ts • g(nTs)  g[n]

  3. CTF g(t) = 3+t2-2t3 g(2t) g(1-t) g(2x) = 5e-20x g(x) g(x2) DTF g[n] = 3+n2-2n3 g[2n] g[1-n] g[2n] = 5e-20n g[n] g[n2] The Argument of Function

  4. CTF Amplitude scaling g(t)  Ag(t) Time shifting g(t)  g(t-t0) Shifting g(t) to the right by t0 units Time scaling g(t)  g(t/a) Expands the function horizontally by a factor of a If a is negative, the function is also time-inverted DTF Amplitude scaling g[n]  Ag[n] Time shifting g[n]  g[n-n0] Time scaling g[n]  g[n/n0] Transformation of Function

  5. Example

  6. Amplitude scaling by A General Transformation • g(t)  Ag((t-t0)/a) • g(t)  Ag(t)  Ag(t/a)  Ag((t-t0)/a) • g(t)  Ag(t)  Ag(t-t0/a)  Ag(t/a-t0/a) Time scaling by a Time shifting by a Time shifting by t0/a Time scaling by a

  7. Exercises -1 1 2 -1 1 2 0 2 2 -1 0

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