Continuous-time Signal Sampling Prof. Siripong Potisuk

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

The Sampling Process 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 In other words, all values of x(t) between sampling points are lost because of sampling An unfortunate choice of T S can result in ambiguity In other words, a set of samples can represent more than one distinct analog signal

Sampling Interval Selection Given a set of sampling points, under what conditions can we uniquely reconstruct the original CT signals from which those samples came? Need to consider the frequency contents of the CT signal being sampled  Fourier Transform

Impulse Sampling

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

Assuming x(t) is band-limited, Frequency-Domain Analysis of Sampling 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 SpectrumReconstructed Spectrum

Nyquist Sampling Theorem

Aliasing Effect

Distortion of filtered spectrum caused by Aliasing

Common samples obtained from sampling two sinusoids of different frequencies

Example 2.3 (text) Consider the following analog signal. It is desired to sample the signal at a rate of 8000 Hz. (a)Sketch the spectrum of the sampled signal up to 20 kHz. (b)Sketch the spectrum of the recovered analog signal that was ideally low-pass filtered at a cutoff frequency of 4 kHz.

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

Analog Lowpass Butterworth Filter Maximally flat passband filter magnitude response decreases monotonically starting from 1 at DC (F = 0) As n, the order of the filter, increases, the transition band decreases (steeper roll-off)

Magnitude Response of a 2 nd order Butterworth Lowpass Filter

Second-order Unit-gain Sallen-key Lowpass Filter

Sampled-signal Spectrum With a Practical Anti-aliasing Filter

Example 2.4 & 2.5 (text) Consider a DSP system with a sampling rate of 8000 Hz. Its anti- aliasing filter is a 2 nd order Butterworth lowpass filter with a cutoff frequency of 3.4 kHz. Determine (a)the percentage of aliasing level at the cutoff frequency, (b)the percentage of aliasing level at 1000 Hz. (c)Repeat part (a) if the sampling rate of the system is 16,000 Hz.

Example 2.6 (text) Consider a DSP system with a sampling rate of Hz. Its anti- aliasing filter is a Butterworth lowpass filter with a cutoff frequency of 8 kHz. Determine the order of the anti-aliasing filter required to limit the percentage of aliasing level at the cutoff frequency to less than 1%.