1. Chapter 3 Sampling

2. Introduction Digital signal processing system
Most signals in nature are in analog form
Needs an analog-to-digital conversion process
Sampling
Acquisition of continuous signal at discrete-time intervals
Conversion of continuous signal to discrete-time signal
A-D convertor
Digital system
Fig. 3-1.

3. Sampling theory Sampling with interval T
Sampling with a periodic impulse train converts to a discrete-time sequence
Discrete signal
Analog signal
Sampler
Fig. 3-2.

4. Same sampled valued at each sampling point
Fig. 3-3.

5. Mathematical representation of sampling theorem
Multiplying continuous signal with impulse train
Fig. 3-4.

6. Sampled signal in time domain
Impulse signal property gives:
Applying it to Eq.(3-1)
(3-1)
(3-2)
(3-3)

7. Using convolution properties of Fourier transform
By example 2-8
Convolution in frequency domain
Shifting the signal to each position of impulses
(3-4)
(3-5)
(3-6)

8. Sampling in frequency domain
Repeats the spectrum of sampled signal at period of
Scaled by
If , each repeated signal is preserved
If , each repeated signal is overlapped
Fig. 3-5.

9. Sampling sinusoidal signal of frequency,
with three sampling intervals
Fig. 3-6.

10. Sampling frequency required to reconstruct original signal
Nyquist frequency
(3-7)
(3-8)

11. Reconstruction of continuous signal from samples
Reconstruction of continuous signal from sampled signal
Using low-pass filter in time domain
By Eq.(3-3)
Interpolation with ideal low-pass filter
Impulse response
(3-9)
(3-10)

12. Reconstructed continuous signal
(3-11)
Fig. 3-7.

13. Reconstructed signal with
Using convolution of sinc function
(3-12)
(3-13)
Fig. 3-8.