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Signals and Systems Fall 2003 Lecture #10 7 October 2003

Signals and Systems Fall 2003 Lecture #10 7 October 2003. 1. Examples of the DT Fourier Transform 2. Properties of the DT Fourier Transform 3. The Convolution Property and its Implications and Uses. DT Fourier Transform Pair. – Analysis Equation – FT.

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Signals and Systems Fall 2003 Lecture #10 7 October 2003

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  1. Signals and Systems Fall 2003 Lecture #10 7 October 2003 • 1. Examples of the DT Fourier Transform • 2. Properties of the DT Fourier Transform • 3. The Convolution Property and its • Implications and Uses

  2. DT Fourier Transform Pair – Analysis Equation – FT – Synthesis Equation – Inverse FT

  3. Convergence Issues Synthesis Equation: None, since integrating over a finite interval Analysis Equation: Need conditions analogous to CTFT, e.g. — Finite energy — Absolutely summable

  4. Examples Parallel with the CT examples in Lecture #8 shifted unit sample Same amplitude (=1) as above, but with a linear phase-ωn0

  5. More Examples Exponentially decaying function Infinite sum formula

  6. Still More 4) DT Rectangular pulse (Drawn for N1 = 2)

  7. DTFTs of Sums of Complex Exponentials Recall CT result: What about DT: a) We expect an impulse (of area 2π) at ω = ωo b) But X(ejω) must be periodic with period 2π In fact Note: The integration in the synthesis equation is over 2π period, only need X(ejω) in one 2π period. Thus,

  8. DTFT of Periodic Signals DTFS synthesis eq. From the last page: • Linearity of DTFT

  9. Example #1: DT sine function

  10. Example #2: DT periodic impulse train — Also periodic impulse train – in the frequency domain!

  11. Properties of the DT Fourier Transform — Analysis equation — Synthesis equation — Different from CTFT Periodicity: Linearity:

  12. More Properties Time Shifting: Frequency Shifting: — Important implications in DT because of periodicity Example

  13. Still More Properties Time Reversal: Conjugate Symmetry: and are even functions and are odd functions and real and even real and even real and odd purely imaginary and odd

  14. Yet Still More Properties • 7) Time Expansion • Recall CT property: • But in DT: x[n/2] makes no sense • x[2n] misses odd values of x[n] Time scale in CT is infinitely fine • But we can “slow” a DT signal down by inserting zeros: • k —an integer ≥ 1 • x(k)[n] — insert (k - 1) zeros between successive values • Insert two zeros in this example • (k=3)

  15. Time Expansion (continued) if n is an integer multiple of k • — Stretched by a factor • of k in time domain otherwise -compressed by a factor of k in frequency domain

  16. Is There No End to These Properties? 8) Differentiation in Frequency multiply by j on both sides • Differentiation • in frequency • Multiplication • by n • 9) Parseval’s Relation Total energy in time domain Total energy in frequency domain

  17. The Convolution Property Frequency response DTFT of the unit sample response Example #1: H Periodic

  18. Example #2: Ideal Lowpass Filter

  19. Example #3:

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