Digital Signal Processing Chapter 7 Sampling Theory Prof.
Yasser Mostafa Kadah
Slide 3
Textbook Luis Chapparo, Signals and Systems Using Matlab,
Academic Press, 2011.
Slide 4
Sampling Motivation Many of the signals found in biomedical
applications are analog If we wish to process these signals with a
computer it is necessary to sample, quantize, and code them to
obtain digital signals. Sampling in time + quantization in
amplitude Sampling issues How to sample in time (sampling period)
How to quantize in amplitude (number of bits)
Slide 5
Uniform Sampling Sampling function Periodic sequence of
impulses of period T s Sampled signal:
Slide 6
Uniform Sampling Period = T s
Slide 7
Uniform Sampling Sampling in the frequency domain:
Sampling
Slide 8
Sampling Rate Calculation Requirement: No Aliasing
Slide 9
Sampling Rate Calculation Nyquist criterion
Slide 10
Sampling Rate Calculation: Example
Slide 11
Reconstruction of the Original Continuous-Time Signal from
Samples Assuming band-limited signal and sufficient sampling,
original signal can be recovered by low-pass filtering
Slide 12
Sampling of Infinite Bandwidth Signals: Anti-Aliasing Filter
Anti-aliasing filter: a low-pass filter applied to the input signal
to make sure that the signal to be sampled has a limited bandwidth
Applied in all practical analog-to-digital converters
Slide 13
Sampling of Modulated Signals Nyquist sampling rate condition
applies to low-pass or baseband signal x(t) with maximum frequency
max Consider a modulated signal x m (t)= x(t) exp(j c t) where x(t)
is the baseband signal and exp(j c t) is the carrier with carrier
frequency c >> max To avoid aliasing the shifting in
frequency should be such that there is no overlapping of the
shifted spectra: Independent of carrier frequency General case: s
> B (Bandwidth of signal)
Slide 14
Sampling of Modulated Signals: Illustration 0 B X( ) 0 B s >
B Xs()Xs() After Sampling
Slide 15
Sampling of Modulated Signals: Biomedical Examples Sampling in
Magnetic Resonance Imaging (MRI) In 1.5T MRI, the signal frequency
has a small bandwidth but carried on around 64 MHz Practical
sampling rates used in many commercial systems is only 1 MHz.
Sampling in pulsed-wave Doppler ultrasound (PWD) The ultrasound
center frequency used is around 5-10 MHz and the Doppler shift
around that center frequency is only a few kHz. Practical sampling
in PWD is limited by physics to around 5-10 kSa/s and it is usually
sufficient
Slide 16
Practical Aspects: Quantization and Coding Amplitude
discretization of the sampled signal x s (t) is accomplished by a
quantizer consisting of a number of fixed amplitude levels
Quantization error Example: Doppler signal sampling Need at least
12-14 bits per sample Useful signal superimposed on large clutter
can be lost if lower quantization resolution is used
Slide 17
Problem Assignments Problems: 7.1, 7.2, 7.5, 7.6, 7.7, 7.10,
7.11, Turn in problems marked in red as homework Partial Solutions
available from the student section of the textbook web site