1. PROPERTIES OF FOURIER REPRESENTATIONS
Section 3.8
PROPERTIES OF FOURIER REPRESENTATIONS
Time property
Periodic
(t,n)
Nonperiodic
C.T.
(t)
Fourier Series (FS)
Four Transform (FT)
D.T.
[n]
Discrete-Time Fourier Series (DTFS)
Discrete-Time Fourier Transform (DTFT)
Non-periodic
(k,w)
(3.19)
(3.35)
(T: period)
(3.20)
(3.36)
Periodic
(k,W)
(3.10)
(3.31)
(N: period)
(3.32)
(3.11)
Discrete[k]
Continuous (w, W)
Freq.
property

2. E Linearity and symmetry 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. Conclusion: Convolution in time domain  Multiplication in freq. domain.
Example 3.31:

8. E From results of example 3.26, p264. Example 3.32:
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. E 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. E
Example 3.40, p275
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

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

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: E
Example 3.41: Find Z(jw)

19. E
Problem 3.23(a), p282:

20. Frequency shift:

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

22. E
Example 3.43, p285:

23. Multiplication
READ derivation on p291!
Inverse FT of

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

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

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

27. Time scaling:
Time shift:
Differentiation:

28. Parseval’s relationship:

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

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

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

34. Table 3.11, p311: