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Section 3.8 PROPERTIES OF FOURIER REPRESENTATIONS

Section 3.8 PROPERTIES OF FOURIER REPRESENTATIONS. Non-periodic (k, w ). (3.19). (3.35). (T: period). (3.20). (3.36). Periodic (k, W ). (3.10). (3.31). (3.32). (N: period). (3.11). Continuous ( w, W). Discrete [k]. Freq. property. Linearity and symmetry. E.

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Section 3.8 PROPERTIES OF FOURIER REPRESENTATIONS

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  1. Section 3.8 PROPERTIES OF FOURIER REPRESENTATIONS Non-periodic (k,w) (3.19) (3.35) (T: period) (3.20) (3.36) Periodic (k,W) (3.10) (3.31) (3.32) (N: period) (3.11) Continuous (w, W) Discrete[k] Freq. property

  2. Linearity and symmetry E Example 3.30, p255: Find the frequency-domain representation of z(t). • Which type of freq.-domain representation? • FT, FS, DTFT, DTFS ?

  3. Periodic signals, continuous time. Thus, FS.

  4. Symmetry: We will develop using continuous, non-periodic signals. Results for other 2 cases may be obtained in a similar way. a) Assume  Further assume

  5.  Further assume

  6. b) Assume • Convolution: Applied to non-periodic signals.

  7. E Example 3.31: Conclusion: Convolution in time domain  Multiplication in freq. domain.

  8. E Example 3.32: From results of example 3.26, p264. Recall that (Example 3.25, p244)

  9. The same convolution properties hold for discrete-time, non-periodic signals. Convolution properties for periodic (DT or CT) and periodic with non-periodic signals will be discussed in Chapter 4.

  10. Differentiation and integration: (Section 3.11) • Applicable to continuous functions: time (t) or frequency (w or W) • FT (t, w) and DFTF (W) Differentiation in time: E Find FT of

  11. E Find x(t) if

  12. If x(t) is periodic, frequency-domain representation is Fourier Series (FS): Differentiation in frequency:

  13. Example 3.40, p275 E This differential equation (it has the same mathematical form as (*), and thus the functional form of G(jw) is the same as that of g(t) ) has a solution given as:

  14. Constant c can be determined as: Integration: • In time: applicable to FT and FS • In frequency: applicable to FT and DTFT

  15. For w=0, this relationship is indeterminate. In general, Determine the Fourier transform of u(t). E Problem 3.29, p279: Fund x(t), given E

  16. Review Table 3.6. Commonly used properties. Problem 3.22(b), p271. E

  17. Time and frequency shift Time shift: Note: Time shift  phase shift in frequency domain. Phase shift is a linear function of w. Magnitude spectrum does not change.

  18. Table 3.7, p280: Example 3.41: Find Z(jw) E

  19. Problem 3.23(a), p282: E

  20. Frequency shift:

  21. Note: • Frequency shift  time signal multiplied by a complex sinusoid. • Carrier modulation. Table 3.8, p284: Example 3.42, p284: Find Z(jw). E

  22. Example 3.43, p285: E

  23. Multiplication READ derivation on p291! Inverse FT of

  24. * * Periodic convolution: - p296, CT, periodic signals:

  25. * * - p297, DT, periodic signals: • Scaling

  26. Find Example 3.48, p300: E Example 3.49, p301: E

  27. Time scaling: • Time shift: • Differentiation:

  28. Parseval’s relationship:

  29. Table 3.10, p304: Example 3.50, p304: E

  30. Time-bandwidth product • Compression in time domain  expansion in frequency domain • Bandwidth: The extent of the signal’s significant contents. It is in • general a vague definition as “significant” is not mathematically defined. • In practice, definitions of bandwidth include • absolute bandwidth • x% bandwidth • first-null bandwidth. • If we define

  31. Duality

  32. Example 3.52, p308: E Problem 3.44, p309: E

  33. Table 3.11, p311:

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