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§3 Discrete-Time Fourier Transform

Normally, is a complex function. §3 Discrete-Time Fourier Transform. §3.1 CTFT( C ontinuous T ime F ourier T ranslation ). 3.1.1 Definition of CTFT. §3 Discrete-Time Fourier Transform. In rectangular form:. In polar form:. Dirichlet conditions: exists for.

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§3 Discrete-Time Fourier Transform

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  1. Normally, is a complex function. §3 Discrete-Time Fourier Transform §3.1 CTFT(Continuous Time Fourier Translation ) 3.1.1 Definition of CTFT

  2. §3 Discrete-Time Fourier Transform In rectangular form: In polar form:

  3. Dirichlet conditions: exists for §3 Discrete-Time Fourier Transform • Finite number of finite discontinities and a finite • number of maxima and minima in any finite interval. (b) Absolutely integrable.

  4. §3 Discrete-Time Fourier Transform Conjugation and Conjugate Symmetry (1) If then (2) If (real) then

  5. §3 Discrete-Time Fourier Transform (3) If (real) then and

  6. e-at u(t) 1 t |F(j)| 1/a  §3 Discrete-Time Fourier Transform 3.1.2 CTFT of Typical Signals • Examplea>0

  7. 1 -τ/2 τ/2 F(j) τ 2π/  4π/  §3 Discrete-Time Fourier Transform • Example 

  8. §3 Discrete-Time Fourier Transform Example From CTFT pair and conjugate symmetry, we can get

  9. §3 Discrete-Time Fourier Transform • ExampleBand-limitted signal

  10. δ(t) 0 t F(j) 1 t  0 §3 Discrete-Time Fourier Transform Example • Unit impulse δ(t) Unit impulse has uniform frequency density in whole frequency range, that means it has infinite wide band.

  11. §3 Discrete-Time Fourier Transform Example 1  2πδ() Constant 1 represents direct current signal, and its spectrum is non-zero only at  = 0, which is a δ()

  12. F[cos 0t] (π) (π) F[sin 0t] (jπ) -0 0 0 -0 (-jπ) §3 Discrete-Time Fourier Transform • Sin and cos function F[ej0t] F[cos0t]= F[(ej0t + e-j0t)/2]= π[δ( + 0)+ δ( - 0)] F[sin0t]= F[(ej0t - e-j0t)/2j]= jπ[δ( + 0) - δ( - 0)]

  13. δT(t) -20 - 0 0 0 20  -2T -T 0 T 2T t §3 Discrete-Time Fourier Transform Example • Unit impulse sequence ω0 = 2π/T

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