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

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

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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)

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Uniform Sampling Sampling function Periodic sequence of impulses of period T s Sampled signal:

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Uniform Sampling Period = T s

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Uniform Sampling Sampling in the frequency domain: Sampling

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Sampling Rate Calculation Requirement: No Aliasing

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Sampling Rate Calculation Nyquist criterion

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Sampling Rate Calculation: Example

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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

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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

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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)

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Sampling of Modulated Signals: Illustration 0 B X( ) 0 B s > B Xs()Xs() After Sampling

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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

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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

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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