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

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

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)

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

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

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

Sampling Rate Calculation Requirement: No Aliasing

Sampling Rate Calculation Nyquist criterion

Sampling Rate Calculation: Example

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

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

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)

Sampling of Modulated Signals: Illustration B X() After Sampling B s > B Xs()

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

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 12-14 bits per sample Useful signal superimposed on large clutter can be lost if lower quantization resolution is used

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