Chapter 4 The Continuous-Time Fourier Transform
1 The Continuous-Time Fourier Transform Chapter 4
Chapter 4 Fourier transform 94.1 Representation of Aperiodic Signals: The Continuous-Time fourier Transform T-T/2T0 T T/2 T t a, Tysinckko,r) Tar=2T sinc(oT)
2 Chapter 4 Fourier Transform §4.1 Representation of Aperiodic Signals : The Continuous-Time Fourier Transform 1 x(t) -T -T/2 –T1 0 T1 T/2 T t ( ) 0 1 1 sin 2 c k T T T ak = ( ) 0 1 1 2 sin k Tak T c T = =
Chapter 4 Fourier transform Figure 4.2 200 (b)T=87 (l)T=16T 7个→an=2/T↓□谱线变密
3 Chapter 4 Fourier Transform T 0 = 2 /T 谱线变密 ( ) 4 1 a T = T ( ) 8 1 b T = T ( ) 1 c T = 16T Figure 4.2
Chapter 4 Fourier transform Consider an aperiodic signals x y(t)=0 > T 0 ●● T T 0 T 4
4 Chapter 4 Fourier Transform –T1 0 T1 t x(t) ( ) 1 x t = 0 , t T -T –T1 0 T1 T t x(t) ~ Consider an aperiodic Signals
Chapter 4 Fourier transform Fourier transform pair factor x() J X(o)elon do Synthesis equation 2丌J-∞ X(o)=丁 x(te Ja t dt Analysis equation 1. A linear combination of complex exponentials 2.x(o)- Spectrum(频谱)ofx() x()<"→X(io) 5
5 Chapter 4 Fourier Transform ( ) ( ) j t X j x t e dt + − − = ( ) ( ) 1 2 j t x t X j e d + − = Synthesis equation Analysis equation Fourier Transform Pair 1. A linear combination of complex exponentials. factor 2. X(j) ——Spectrum（频谱） of x(t) x(t)⎯→X(j) F
Chapter 4 Fourier transform Consider a periodic signal (+T)=x(t) Defining(t) ∫R()ast≤t+T 0 others Xio k ka Eljko) The Fourier coefficients a, of x()are proportional to samples of the Fourier transform of one period ofx(o) 6
6 Chapter 4 Fourier Transform ( ) ( ) 0 1 1 0 X jk T X j T a k k = = = The Fourier coefficients of are proportional to samples of the Fourier transform of one period of k a x(t) ~ x(t) ~ Consider a periodic signal x(t T) x(t) ~ ~ + = Defining ( ) ( ) + = 0 others ~ x t t 0 t t 0 T x t
Chapter 4 Fourier transform 54.1.2 Convergence of Fourier Transforms 1.x(O) is square integrable x(t dt < oo 2. Dirichlet Conditions 7
7 Chapter 4 Fourier Transform §4.1.2 Convergence of Fourier Transforms 2. Dirichlet Conditions: ( ) + − x t dt 2 1. is square integrable x t( )
Chapter 4 Fourier transform 54.1.3 Fourier Transforms of Typical Signals Example 4.1 x()=e"u()a>0 F a+ya XO z/2∠X(o) 2a/2 x/4 I/ 丌/2 8
8 Chapter 4 Fourier Transform §4.1.3 Fourier Transforms of Typical Signals Example 4.1 ( ) = ( ) 0 − x t e u t a at ( ) a j e u t at F + − ⎯→ 1 2a / 2 1/ a −a a X(j) −a a X(j) / 2 − / 2 / 4 − / 4
Chapter 4 Fourier transform Emp42x()=e“a>0 a>0) 2a Example4.3 x(0)=8(t) 6()<>1 1<>2no(o)
9 Chapter 4 Fourier Transform Example 4.2 ( ) = 0 − x t e a a t ( ) 2 2 2 0 + ⎯→ − a a e a a t F Example 4.3 x(t) = (t) ( )⎯→1 F t 1⎯→2 () F
Chapter 4 Fourier transform (0)=m X(ja)e/o do Synthesis equation 2 兀 X(1o)=」x()e 10 t dt Analysis equation x(t)X(jio) 1.e"a(t) (a>0 a+a at 2a 2.e10) + 3.6()<>1 1<→2n6(o)
10 Chapter 4 Fourier Transform ( ) ( ) j t X j x t e dt + − − = ( ) ( ) 1 2 j t x t X j e d + − = Synthesis equation Analysis equation x(t)⎯→X(j) F ( ) ( ) 1 1. 0 at F e u t a a j − ⎯→ + ( ) 2 2 2 2. 0 a t F a e a a − ⎯→ + ( ) ( ) 3. 1 1 2 F F t ⎯→ ⎯→