1. Chapter 2. Fourier Representation of Signals and Systems

2. Overview Fourier transform Signals and systems
Frequency content of a given signal
Signals and systems
Linear time-invariant system

3. Concept – Dirac Delta Function
Unit impulse function
Unit step function

4. Concept – Impulse Response
The response of the system to a unit impulse
A function of time
Impulse
response
h[t]
Input
x [t]
Output
y[t] 1 2 3 1 2 3

5. Concept – Linear Time Invariant System
A common model for many engineering systems
Linearity
Time invariance 1 2 3 1 2 3
0.7
0.49
0.35 1 2 3 1 2 3

6. Concept – Convolution Computes the output for an arbitrary input
LTI system
Impulse
response
h[t]
Input
x [t]
Output
y[t]

7. Concept – Euler's formula

8. Concept – Fourier Transform
A mathematical operation that decomposes a signal into its constituent frequencies
남성과 여성의 목소리 차이가 주파수에서 보임

9. 2.1 The Fourier Transform Definitions Notations
Fourier transform of the signal g(t) : analysis equation
Inverse Fourier transform : synthesis equation
Notations

10. 2.1 The Fourier Transform Dirichlet’s conditions
1. The function g(t) is single-valued, with a finite number of maxima and minima in any finite time interval.
2. The function g(t) has a finite number of discontinuities in any finite time interval.
3. The function g(t) is absolutely integrable
For physical realizability of a signal g(t), the energy of the signal defined by
must satisfy the condition
Such a signal is referred to as an energy signal.
All energy signals are Fourier transformable.

11. 2.1 The Fourier Transform Continuous Spectrum
A pulse signal g(t) of finite energy is expressed as a continuous sum of exponential functions with frequencies in the interval -∞ to ∞.
We may express the function g(t) in terms of the continuous sum infinitesimal components,
The signal in terms of its time-domain representation by specifying the function g(t) at each instant of time t.
The signal is uniquely defined by either representation.
The Fourier transform G(f) is a complex function of frequency f,

12. 2.1 The Fourier Transform The spectrum of a real-valued signal
: complex conjugate
: even function
: odd function

13. 2.2 Properties of the Fourier Transfrom
1. Linearity
(Superposition)
2. Dilation
3. Conjugation Rule
4. Duality
5. Time Shifting
6. Frequency Shifting
7. Area Under g(t)
8. Area Under G(f)

14. 2.2 Properties of the Fourier Transfrom
9. Differentiation in the Time Domain 10. Integration in the Time Domain 11. Modulation Theorem 12. Convolution Theorem 13. Correlation Theorem 14. Rayleigh’s Energy Theorem

15. 2.2 Properties of the Fourier Transfrom
Property 1 : Linearity (Superposition)
then for all constants c1 and c2,
Property 2 : Dilation
(proof) If a>0,
: reflection property

16. 2.2 Properties of the Fourier Transfrom
Property 3 : Conjugation Rule
Property 4 : Duality

17. 2.2 Properties of the Fourier Transfrom
Property 5 : Time Shifting
Property 6 : Frequency Shifting

18. 2.2 Properties of the Fourier Transfrom
Property 7 : Area Under g(t)
Property 8 : Area Under G(t)

19. 2.2 Properties of the Fourier Transfrom
Property 9 : Differentiation in the Time Domain
Property 10 : Integration in the Time Domain
Assuming G(0)=0,

20. 2.2 Properties of the Fourier Transfrom
Property 11 : Modulation Theorem
The multiplication of two signals in the time domain is transformed into the convolution of their individual Fourier transforms in the frequency domain.

21. 2.2 Properties of the Fourier Transfrom
Convolution
f(t)*g(t) = g(t)*f(t) : signal = system

22. 2.2 Properties of the Fourier Transfrom
Property 12 : Convolution Theorem
Property 13 : Correlation Theorem

23. 2.2 Properties of the Fourier Transfrom
Property 14 : Rayleigh’s Energy Theorem
Total energy of a Fourier-transformable signal equals the total area under the curve of squared amplitude spectrum of this signal.