Chapter 2. Fourier Representation of Signals and Systems

1. Chapter 2. Fourier Representation of Signals and Systems

2. Overview • Fourier transform • 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. 0 1 2 3 0 1 2 3 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]

5. 0 1 2 3 0 1 2 3 Concept – Linear Time Invariant System • A common model for many engineering systems • Linearity • Time invariance 0.7 0.49 0.35 0 1 2 3 0 1 2 3

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

7. Concept – Euler's formula

8. Concept – Fourier Transform • A mathematical operation that decomposes a signal into its constituent frequencies

9. 2.1 The Fourier Transform • Definitions • 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.