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Chapter 7 Sampling

Sampling is a critical process for converting continuous-time (CT) signals into discrete-time (DT) signals, allowing for digital processing through computers and DSPs. This chapter explores the fundamental concepts of sampling, the sampling theorem, and methods for signal reconstruction from samples. Key topics include impulse-train sampling, zero-order hold techniques, interpolation methods, and the consequences of undersampling (aliasing). Proper sampling ensures accurate signal representation and recovery, making it essential for various applications, including sound recording and digital signal processing.

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Chapter 7 Sampling

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  1. Chapter 7 Sampling 崔琳莉

  2. SAMPLING –– A crucial step in converting CT signals to DT, so that we can use versatile digital computers or DSPs to process them. Example: Digital recording of sounds

  3. The issue of sampling applies to spatially varying signals

  4. Contents • The concept of sampling and the sampling theorem • The process of reconstructing a continuous-time signal from its samples

  5. 7.1 Representation of a Continuous-time Signal by its Samples: The Sampling Theorem 7.1.1 Impulse-train Sampling (1) Sampling

  6. p(t) x(t) xp(t) The period T is the sampling period, and is the sampling frequency. (2) Impulse-Train Sampling Sampling function

  7. Time domain:

  8. Frequency domain:

  9. (3) (Nyquist) Sampling theorem Let x(t) be a band-limited signal with X(j)=0 for ||> M . Then x(t) is uniquely determinedby its samples x(nT),n=0,1,2,…, if s>2M, where s=2/T . 2M is called Nyquist Rate. ( Minimum distortionless sampling frequency )

  10. (4) Recovery System for sampling and reconstruction:

  11. Zero-order hold x(t) x0(t) 7.1.2 Sampling with a Zero-order Hold (1) Sampling system construction:

  12. (2) Signal Recovery

  13. For example, if the cutoff frequency of H(jw) is ws/2, the reconstruction filter following a zero-order is shown in the figure as:

  14. *7.2 Reconstruction of a signal from its samples using interpolation • Interpolation (内插)is the fitting of a continuous signal to a set of sample values. • Interpolation is commonly used to reconstructing a function, either approximately or exactly, from samples.

  15. The zero-order hold is a simple interpolation procedure. • Another useful form if interpolation is linear interpolation, whereby adjacent sample points are connected by a straight line.

  16. band-limited interpolation

  17. Graphic Illustration of Time-domain Interpolation Original CT signal After sampling The LPF smoothes out sharp edges and fill in the gaps. After passing the LPF

  18. Commonly Used Interpolation Methods • Bandlimited Interpolation • • Zero-Order Hold (E.g. movie projection) • • First-Order Hold —Linear interpolation, Commonly used in plotting.

  19. The zero-order hold of interpolation is a rough interpolation; • Higher order holds can be a smoother interpolation strategy.

  20. 7.3 The Effect of Undersampling: Aliasing Aliasing: When s<2 M, the spectrum of x(t) is no longer replicated in Xp(jw) and thus isno longer recoverable by lowpass filtering. Note: Using band-limited interpolation,at the sampling instants for any choice of ws

  21. x(t)=cosw0t

  22. x(t)=cosw0t

  23. x(t)=cos(w0t+F) s=60 s=30

  24. s=1.50 s=1.20

  25. Application Example: ——the stroboscopic effect (频闪效应)

  26. Therefore, the first step in sampling is anti-alias filtering (AAF) –– a LPF to assure thatωs ≥ 2ωM

  27. The effect of anti-alias filtering (AAF) The AAF low-pass filter will rid of the information in x(t) beyond |ω| >ωs / 2, but will avoid the much more serious aliasing problem.

  28. 7.4 Discrete-Time Processing of Continuous-Time Signals Homework: 7.1 7.2 7.3 7.6 7.9

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