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