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1、Continuous-time Signal Sampling,ELEC 309Prof.Siripong Potisuk,Why Digital Domain,Uniform representation of analog signals i.e.,a sequence of 0s and 1s.Operation of digital circuits not dependent on precise values of the digital signalsRequire only bistable circuits and storage medium to process,stor
2、e,and transmit signalsPossibility of perfect signal regenerationLow cost of digital processor hardware,Benefits(continued),Any desirable accuracy achievable by increasing the binary wordlength subject to cost limitationThe same digital computer technology used for general information processing can
3、be used for DSPApplicability 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 requirementDigital systems mostly constructed using active devices that consume electri
4、cal powerAdvantages 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 represen
5、ted 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 secondsEach snapshot is called a sampleTs is the so-called sampling interval,i.e.,the time interval between each sample(second/sample)Prefer regularly
6、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 rateThe unit of sampling frequency is samples/second,but often expressed in terms of Hz to match the highest frequency in Hz of
7、the analog signal,Sampling Interval Selection,By sampling,we throw out a lot of information in between samplesA 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
8、which those samples came?In other words,all values of x(t)between sampling points are lost because of samplingNeed to consider the frequency contents of the CT signal being sampled Fourier Transform,Impulse Sampling,Frequency-Domain Analysis of Sampling,where,and,Using the Multiplication property of
9、 CTFT,Thus,Assuming x(t)is band-limited,i.e.,Frequency-Domain Analysis of Sampling,No overlap between shiftedspectra!,Reconstruction of the CT signal from its samples,If there is no overlap betweenshifted spectra,we can use alowpass filter to reconstruct theoriginal CT signal,Original Spectrum,Recon
10、structed 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 prev
11、ent aliasingPassing it through an analog lowpass filter,e.g.Butterworth,Chebyshev,EllipticalAlternatively called Guard FilterRemove all frequency components outside the range-Fc,Fc where Fc fd,Signal Reconstruction,Three interpolation MethodsBand-limited interpolationZero-order Hold interpolationFirst-order Hold Interpolation(linear),Band-limited Interpolation in the Time Domain,Zero-order Hold Interpolation,First-order Hold Interpolation,
