1. Digital Signal Processing – Chapter 7
Sampling Theory
Prof. Yasser Mostafa Kadah

2. Textbook
Luis Chapparo, Signals and Systems Using Matlab, Academic Press, 2011.

3. 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)

4. Uniform Sampling Sampling function
Periodic sequence of impulses of period Ts
Sampled signal:

5. Uniform Sampling 𝓕 Fourier Transform of a shifted delta function:
Hence,
ℱ 𝑛 𝛿(𝑡−𝑛 𝑇 𝑠 ) = 𝑛 𝑒 −𝑗Ω𝑛 𝑇 𝑠 = m 𝛿(Ω−𝑚 2𝜋 𝑇 𝑠 ) 𝓕
Period = Ts
Period
Ω𝑠= 2/Ts

6. Uniform Sampling Sampling in the frequency domain:
ℱ{𝑥 𝑡 ∙ 𝑛 𝛿 𝑡−𝑛 𝑇 𝑠 =𝑋(Ω)∗ℱ 𝑛 𝛿 𝑡−𝑛 𝑇 𝑠
=𝑋(Ω)∗ m 𝛿(Ω−𝑚 2𝜋 𝑇 𝑠 ) = m 𝑋(Ω−𝑚 2𝜋 𝑇 𝑠 )
Sampling

7. Sampling Rate Calculation
Requirement: No Aliasing

8. Sampling Rate Calculation
Nyquist criterion

9. Sampling Rate Calculation: Example

10. 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

11. 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

12. 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 xm(t)= x(t) exp(jct) where x(t) is the baseband signal and exp(jct) 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)

13. Sampling of Modulated Signals: IllustrationB
X()
After Sampling B
s > B
Xs()

14. 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 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

15. Practical Aspects: Quantization and Coding
Amplitude discretization of the sampled signal xs(t) is accomplished by a quantizer consisting of a number of fixed amplitude levels
Quantization error
Example: Doppler signal sampling
Need at least bits per sample
Useful signal superimposed on large clutter can be lost if lower quantization resolution is used

16. 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