Nội dung

Nguyễn Công Phương SIGNAL PROCESSING Fourier Representation of Signals Contents I. Introduction II. Discrete – Time Signals and Systems III. The z – Transform IV. Fourier Representation of Signals V. Transform Analysis of LTI Systems VI. Sampling of Continuous – Time Signals VII.The Discrete Fourier Transform VIII.Structures for Discrete – Time Systems IX. Design of FIR Filters X. Design of IIR Filters XI. Random Signal Processing sites.google.com/site/ncpdhbkhn 2 Fourier Representation of Signals ⋮ sites.google.com/site/ncpdhbkhn 3 Fourier Representation of Signals 1. Sinusoidal Signals and their Properties a) Continuous – Time Sinusoids b) Discrete – Time Sinusoids c) Frequency Variables and Units 2. Fourier Representation of Continuous – Time Signals 3. Fourier Representation of Discrete – Time Signals 4. Summary of Fourier Series and Fourier Transforms 5. Properties of the Discrete – Time Fourier Transform sites.google.com/site/ncpdhbkhn 4 Continuous – Time Sinusoids (1) x(t ) = A cos( 2π F0t + θ ), − ∞ < t < ∞ Ω 0 = 2π F0 1 2π T0 = = F0 Ω 0 e ± jϕ = cos ϕ ± j sin ϕ A cos θ • • • • • T0 x( t ) A 0 t A: the amplitude θ: phase (radians, rad) F0: frequency (Hertz, Hz) Ω0: angular frequency (rad/s) T0: period (s) A jθ jΩ0t A cos(2π F0t + θ ) = ( e e + e − jθ e − jΩ0t ) 2 sites.google.com/site/ncpdhbkhn 5 Continuous – Time Sinusoids (2) x1 (t ) = cos 2π F1t 0 T1 t x2 (t ) = cos 2π F2t 0 T2 sites.google.com/site/ncpdhbkhn t 6 Continuous – Time Sinusoids (3) s1 (t ) = e jΩ0t = e j 2π F0t The fundamental/first harmonic s2 (t ) = e j 2 Ω0t = e j 2π 2 F0t The second harmonic ⋮ sk (t ) = e jkΩ0t = e j 2π kF0t ∫ t0 +T0 t0 sk (t ) s (t)dt = ∫ * m t0 +T0 t0 The kth harmonic e jk Ω0t − jmΩ 0t e sites.google.com/site/ncpdhbkhn T0 , k = m dt =  0, k ≠ m 7 Continuous – Time Sinusoids (4) x1 (t ) = A0 cos(2π F0t ) + A1 cos(2π 3F0t ) + A2 cos(2π 5F0t ) 0 t 0 t x2 (t ) = B0 cos(2π F0t ) + B1 cos(2π 2.83F0t ) + B2 cos(2π 7.14 F0t ) sites.google.com/site/ncpdhbkhn 8 Fourier Representation of Signals 1. Sinusoidal Signals and their Properties a) Continuous – Time Sinusoids b) Discrete – Time Sinusoids c) Frequency Variables and Units 2. Fourier Representation of Continuous – Time Signals 3. Fourier Representation of Discrete – Time Signals 4. Summary of Fourier Series and Fourier Transforms 5. Properties of the Discrete – Time Fourier Transform sites.google.com/site/ncpdhbkhn 9 Discrete – Time Sinusoids (1) T T T T 0 n 0 n 0 n 0 n sites.google.com/site/ncpdhbkhn 10