Continuous-time Signal Sampling
Prof. Siripong Potisuk
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
Why Sampling
The sequence of samples is given by
x[n] x(nTs ), n I
1
FS is the sampling frequency or rate
Ts
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,
FT
2
where and S 2 Fs
T
Thus,
Frequency-Domain Analysis of Sampling
Assuming x(t) is band-limited, i.e.,
X ( j) 0, M
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
Let x(t ) be a band - limited signal such that
X ( j ) 0, M .
Then, x(t ) is uniquely determined by its samples
x(nT ), n I if
s 2 M Nyquist rate
2
where S sampling frequencyand
T
M Nyquist frequency.
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
t
h0 (t ) [ ( ) ( T )]d
0
u (t ) u (t T )
First-order Hold Interpolation