4. The Continuous time Fourier Transform 4.1 Representation of Aperiodic signals: The Continuous time Fourier Transform 4.1.1 Development of the Fourier transform representation of the continuous time Fourier transform

(1) Example 4 The continuous time Fourier transform ( From Fourier series to Fourier transform )

(2) Fourier transform representation of Aperiodic signal 4 The continuous time Fourier transform (2) Fourier transform representation of Aperiodic signal For periodic signal : For aperiodic signal x(t) :

4 The continuous time Fourier transform

4 The continuous time Fourier transform When T , So

Relation between Fourier series and Fourier transform: 4 The continuous time Fourier transform Fourier transform: or Relation between Fourier series and Fourier transform:

4 The continuous time Fourier transform

4.1.2 Convergence of Fourier transform 4 The continuous time Fourier transform 4.1.2 Convergence of Fourier transform Dirichlet conditions: (1) x(t) is absolutely integrable. (2) x(t) have a finite number of maxima and minima within any finite interval. (3) x(t) have a finite number of discontinuity within any finite interval. Furthermore, each of these discontinuities must be finite.

4.1.3 Examples of Continuous time Fourier Transform 4 The continuous time Fourier transform 4.1.3 Examples of Continuous time Fourier Transform Example 4.1 4.2 4.3 4.4 4.5 Example (1) Example (2)

4.2 The Fourier Transform for Periodic Signal 4 The continuous time Fourier transform 4.2 The Fourier Transform for Periodic Signal Periodic signal: thus Example 4.6 4.7 4.8

4.3 Properties of the Continuous time Fourier Transform 4.3.1 Linearity If then

4.3.2 Time Shifting If then 4 The continuous time Fourier transform Example 4.9

4.3.3 Conjugation and Conjugate Symmetry 4 The continuous time Fourier transform 4.3.3 Conjugation and Conjugate Symmetry (1) If then (2) If then

4 The continuous time Fourier transform (3) If then

4.3.4 Differentiation and Integration 4 The continuous time Fourier transform 4.3.4 Differentiation and Integration (1) If then (2) If then Example 4.12

4.3.5 Time and Frequency Scaling 4 The continuous time Fourier transform 4.3.5 Time and Frequency Scaling If then Especially,

4.3.6 Duality If then 4 The continuous time Fourier transform Example 4.13

4 The continuous time Fourier transform

4.3.7 Parseval’s Relation If then 4 The continuous time Fourier transform 4.3.7 Parseval’s Relation If then Example 4.14

h(t) H(j) 4.4 The Convolution Property Consider a LTI system: x(t) 4 The continuous time Fourier transform 4.4 The Convolution Property Consider a LTI system: h(t) H(j) x(t) y(t)=x(t)*h(t) X(j ) Y(j)=X(j)H(j) 4.4.1 Examples Example 4.15 4.16 4.17 4.19 4.20

4 The continuous time Fourier transform

The multiplication(modulation) property: 4 The continuous time Fourier transform 4.5 The Multiplication Property The multiplication(modulation) property: s(t) p(t) r(t) Example 4.21 4.22 4.23

4 The continuous time Fourier transform

4.5.1 Frequency-Selective Filtering with Variable Center Frequency 4 The continuous time Fourier transform 4.5.1 Frequency-Selective Filtering with Variable Center Frequency A Bandpass Filter :

4 The continuous time Fourier transform

4.6 Tables of Fourier Properties and of Basic Fourier Transform Pairs 4 The continuous time Fourier transform 4.6 Tables of Fourier Properties and of Basic Fourier Transform Pairs Table 4.1 Table 4.2

4.7 System Characterized by Linear Constant- 4 The continuous time Fourier transform 4.7 System Characterized by Linear Constant- Coefficient Differential Equation Constant-coefficient differential equation: Fourier transform: Define: Example 4.24 4.25 4.26

4 The continuous time Fourier transform Problems: 4.3 4.4(a) 4.10 4.11 4.14 4.15 4.24 4.25 4.32(a)(b) 4.35 4.36