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

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 ?

Periodic signals, continuous time. Thus, FS.

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

 Further assume

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

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

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

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.

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

E Find x(t) if

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

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:

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

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

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

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.

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

E Problem 3.23(a), p282:

Frequency shift:

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).

E Example 3.43, p285:

Multiplication READ derivation on p291! Inverse FT of

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

- p297, DT, periodic signals: * Scaling

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

Time scaling: Time shift: Differentiation:

Parseval’s relationship:

Table 3.10, p304: E Example 3.50, p304:

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

Duality

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

Table 3.11, p311: