Signals and Systems Chapter 0

Digital Signal Processing Chapter 7Sampling Theory

Prof. Yasser Mostafa Kadah

TextbookLuis Chapparo, Signals and Systems Using Matlab, Academic Press, 2011.

Sampling MotivationMany of the signals found in biomedical applications are analogIf 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 amplitudeSampling issuesHow to sample in time (sampling period)How to quantize in amplitude (number of bits)

Uniform SamplingSampling functionPeriodic sequence of impulses of period Ts

Sampled signal:

Uniform Sampling

Period = Ts

Uniform SamplingSampling in the frequency domain:

Sampling

Sampling Rate CalculationRequirement: No Aliasing

Sampling Rate CalculationNyquist criterion

Sampling Rate Calculation: Example

Reconstruction of the Original Continuous-Time Signal from SamplesAssuming band-limited signal and sufficient sampling, original signal can be recovered by low-pass filtering

Sampling of Infinite Bandwidth Signals: Anti-Aliasing FilterAnti-aliasing filter: a low-pass filter applied to the input signal to make sure that the signal to be sampled has a limited bandwidthApplied in all practical analog-to-digital converters

Sampling of Modulated SignalsNyquist sampling rate condition applies to low-pass or baseband signal x(t) with maximum frequency maxConsider 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 >> maxTo avoid aliasing the shifting in frequency should be such that there is no overlapping of the shifted spectra:

Independent of carrier frequencyGeneral case: s > B (Bandwidth of signal)

Sampling of Modulated Signals: Illustration

0BX()

0Bs > BXs()After Sampling

Sampling of Modulated Signals: Biomedical ExamplesSampling in Magnetic Resonance Imaging (MRI)In 1.5T MRI, the signal frequency has a small bandwidth but carried on around 64 MHzPractical 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 CodingAmplitude discretization of the sampled signal xs(t) is accomplished by a quantizer consisting of a number of fixed amplitude levelsQuantization error

Example: Doppler signal samplingNeed at least 12-14 bits per sampleUseful signal superimposed on large clutter can be lost if lower quantization resolution is used

Problem AssignmentsProblems: 7.1, 7.2, 7.5, 7.6, 7.7, 7.10, 7.11, Turn in problems marked in red as homeworkPartial Solutions available from the student section of the textbook web site