# Chapter6 sampling

King Mongkut's University of Technology Thonburi
27 de Apr de 2012
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### Chapter6 sampling

• 1. Signals & Systems Chapter 6 Sampling INC212 Signals and Systems : 2 / 2554
• 2. Overview  Sampling theorem  Signal reconstruction  Interpolation formula  Aliasing INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
• 3. The sampling theorem INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
• 4. The sampling theorem INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
• 5. The sampling theorem INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
• 6. The sampling theorem F (ω ) = 0 for ω > 2πB bandlimited Fs ≥ 2B Hz f (t ) = f (t )δ T (t ) = ∑ f (nT )δ (t − nT ) n 1 ∞ 2π F (ω ) = ∑ F (ω − nω s ); ω s = = 2πFs T n = −∞ T INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
• 7. The sampling theorem ∞ T 2 1 x(t ) = ∑ ck e jkω s t , − ∞ < t < ∞ ; ck = ∫ x(t )e − jkω s t dt k = −∞ T −T 2 ∞ Trigonometric form δ T (t ) = ∑ ck e jkω s t , − ∞ < t < ∞ k = −∞ 1 1 T 2 T 2 1 1 c0 = ck = ∫ δ T (t )e − jkω s t dt = T −T 2 ∫ δ T (t )(1)dt = T −T 2 T T 2 Ak = 2 ck = , k = 1,2,3, 1 ∞ jkω s t 2π T δ T (t ) = ∑ e , ω s = T k = −∞ T θk = 0 δ T (t ) = 1 [1 + 2( cos ωs t + cos 2ωst + cos ωst + ) ], ωs = 2π = 2πFs T T INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
• 8. The sampling theorem f (t ) = f (t )δ T (t ) 1 f (t ) = [ f (t ) + 2 f (t ) cos ω s t + 2 f (t ) cos 2ω s t + 2 f (t ) cos ω s t + ] T F 2 f (t ) cos ω s t ↔ F (ω − ω s ) + F (ω + ω s ) F 2 f (t ) cos 2ω s t ↔ F (ω − 2ω s ) + F (ω + 2ω s ) 1 ∞ F (ω ) = ∑ F (ω − nω s ) T n = −∞ INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
• 9. Effect of undersampling and oversampling f (t ) = sinc 2 (5πt )  ω  F (ω ) = 0.2 tri   20π  INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
• 10. Effect of undersampling and oversampling  ω  f (t ) = sinc (5πt ) 2 F (ω ) = 0.2 tri   20π  Fs = 10 Hz → T = 0.1 sec INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
• 11. Effect of undersampling and oversampling  ω  f (t ) = sinc (5πt ) 2 F (ω ) = 0.2 tri   20π  Fs = 5 Hz → T = 0.2 sec INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
• 12. Effect of undersampling and oversampling  ω  f (t ) = sinc (5πt ) 2 F (ω ) = 0.2 tri   20π  Fs = 20 Hz → T = 0.05 sec INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
• 13. Effect of undersampling and oversampling  ω  f (t ) = sinc (5πt ) 2 F (ω ) = 0.2 tri   20π  F (ω ) = 0 for ω > 10π ω s ≥ 20π ; Fs ≥ 10 Hz; T ≤ 0.1 INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
• 14. Effect of undersampling and oversampling F (ω ) = 0 for ω > 2πB ⇒ Bandlimited to B Hz Fs ≥ 2 B Hz The minimum sampling rate = 2B The Nyquist rate The sampling interval = 1/2B The Nyquist interval INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
• 15. Signal Reconstruction  Zero-order hold INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
• 16. Signal Reconstruction  The Interpolation Formula INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
• 17. Signal Reconstruction  The Interpolation Formula h(t ) = 2 BT sinc( 2πBt )  ω  H (ω ) = T rect  Assuming the Nyquist rate; 2 BT = 1  4πB  h(t ) = sinc( 2πBt ) f (t ) = h(t ) * f (t ) ∞ = h(t ) * ∑ f (nT )δ (t − nT ) n = −∞ ∞ = ∑ f (nT )h(t − nT ) n = −∞ ∞ = ∑ f (nT ) sinc(2πB(t − nT )) n = −∞ ∞ ∴ f (t ) = ∑ f (nT ) sinc(2πBt − nπ ) n = −∞ INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
• 18. Aliasing INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
• 19. Aliasing  Amplitude spectrum of time-limited signal not be bandlimited ωs = 2 B INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
• 20. Aliasing  Anti-aliasing x (t) x [n] Lowpass Sampling filter  The sampling frequency may be as large as 10 or 20 times B. INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
• 21. INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling