1. Continuous-time Signal Sampling
ELEC 309
Prof. Siripong Potisuk

2. Why Digital Domain Uniform representation of analog signals
i.e., a sequence of 0’s and 1’s.
Operation of digital circuits not dependent on precise values of the digital signals
Require only bistable circuits and storage medium to process, store, and transmit signals
Possibility of perfect signal regeneration
Low cost of digital processor hardware

3. Benefits (continued)
Any desirable accuracy achievable by increasing the binary wordlength subject to cost limitation
The same digital computer technology used for general information processing can be used for DSP
Applicability to very low frequency signals (seismic applications)

4. Disadvantages Increased system complexity (A/D and D/A)
Limited range of frequencies available for processing because of the sampling requirement
Digital systems mostly constructed using active devices that consume electrical power
Advantages far outweigh disadvantages

5. Analog-to-Digital Conversion
Discretize the independent variable or time of an analog signal (Sampling)
Discretize the dependent variable or amplitude of an analog signal by rounding off to the nearest integer (Quantization)
Each quantization level represented using binary encoding scheme (Encoding)

6. Why Sampling
Necessary for digital processing of analog signals, e.g., x(t)
Taking snapshots of x(t) every Ts seconds
Each snapshot is called a sample
Ts is the so-called sampling interval, i.e., the time interval between each sample (second/sample)
Prefer regularly spaced samples, though not necessary

7. Analog-to-Digital Conversion (A/D)

8. Mathematical Description of Sampling
The sequence of samples is given by
is the sampling frequency or rate
The unit of sampling frequency is samples/second, but often expressed in terms of Hz to match the highest frequency in Hz of the analog signal

9. Sampling Interval Selection
By sampling, we throw out a lot of information in between samples
A set of samples can represent more than one distinct signal

10. Sampling Interval Selection
Given a set of sampling points, under what conditions can we reconstruct the original CT signals from which those samples came?
In other words, all values of x(t) between sampling points are lost because of sampling
Need to consider the frequency contents of the CT signal being sampled  Fourier Transform

11. Impulse Sampling

12. Frequency-Domain Analysis of Sampling
Using the Multiplication property of CTFT,
and
where
Thus,

13. Frequency-Domain Analysis of Sampling
Assuming x(t) is band-limited, i.e.,
No overlap between shifted
spectra!

14. Reconstruction of the CT signal from its samples
If there is no overlap between
shifted spectra, we can use a
lowpass filter to reconstruct the
original CT signal
Original Spectrum
Reconstructed Spectrum

15. Nyquist Sampling Theorem

16. Practical Sampling

17. Aliasing Effect

18. Distortion of filtered spectrum caused by Aliasing

20. Common Samples obtained from sampling two
sinusoids of different frequencies

21. Anti-aliasing Filter
Band-limiting operation done on the analog signal before sampling to prevent aliasing
Passing it through an analog lowpass filter, e.g. Butterworth, Chebyshev, Elliptical
Alternatively called Guard Filter
Remove all frequency components outside the range [-Fc, Fc] where Fc < fd

22. Signal Reconstruction
Three interpolation Methods
Band-limited interpolation
Zero-order Hold interpolation
First-order Hold Interpolation (linear)

24. Band-limited Interpolation in the Time Domain

25. Zero-order Hold Interpolation

26. First-order Hold Interpolation