330 likes | 465 Views
Chapter 7 Sampling. 崔琳莉. 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. The issue of sampling applies to spatially varying signals. Contents.
E N D
Chapter 7 Sampling 崔琳莉
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
Contents • The concept of sampling and the sampling theorem • The process of reconstructing a continuous-time signal from its samples
7.1 Representation of a Continuous-time Signal by its Samples: The Sampling Theorem 7.1.1 Impulse-train Sampling (1) Sampling
p(t) x(t) xp(t) The period T is the sampling period, and is the sampling frequency. (2) Impulse-Train Sampling Sampling function
(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>2M, where s=2/T . 2M is called Nyquist Rate. ( Minimum distortionless sampling frequency )
(4) Recovery System for sampling and reconstruction:
Zero-order hold x(t) x0(t) 7.1.2 Sampling with a Zero-order Hold (1) Sampling system construction:
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:
*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.
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.
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
Commonly Used Interpolation Methods • Bandlimited Interpolation • • Zero-Order Hold (E.g. movie projection) • • First-Order Hold —Linear interpolation, Commonly used in plotting.
The zero-order hold of interpolation is a rough interpolation; • Higher order holds can be a smoother interpolation strategy.
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
x(t)=cos(w0t+F) s=60 s=30
s=1.50 s=1.20
Application Example: ——the stroboscopic effect (频闪效应)
Therefore, the first step in sampling is anti-alias filtering (AAF) –– a LPF to assure thatωs ≥ 2ωM
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.
7.4 Discrete-Time Processing of Continuous-Time Signals Homework: 7.1 7.2 7.3 7.6 7.9