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## Digital Signal Processing – Chapter 7

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**Digital Signal Processing – Chapter 7**Sampling Theory Prof. Yasser MostafaKadah**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**• Fourier Transform of a shifted delta function: • Hence, Period = Ts Period = 2/Ts**Uniform Sampling**• Sampling in the frequency domain: = Sampling**Sampling Rate Calculation**• Requirement: No Aliasing**Sampling Rate Calculation**• Nyquist criterion**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(jct)where x(t)is the baseband signal and exp(jct)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**s> B Xs() After Sampling B X() 0 0 B**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 quantizerconsisting 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