Download presentation

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:

Similar presentations

© 2020 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google