Continuous-time Signal Sampling ELEC 309 Prof. Siripong Potisuk

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

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)

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

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)

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

Analog-to-Digital Conversion (A/D)

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

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

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

Impulse Sampling

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

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

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

Nyquist Sampling Theorem

Practical Sampling

Aliasing Effect

Distortion of filtered spectrum caused by Aliasing

Common Samples obtained from sampling two sinusoids of different frequencies

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

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

Band-limited Interpolation in the Time Domain

Zero-order Hold Interpolation

First-order Hold Interpolation