Spring 2012

Signals and Systems

Chapter SS-7Sampling

Shou shui Wei

Sep08 – Dec08

Figures and images used in these lecture notes are adopted from“Signals & Systems” by Alan V. Oppenheim and Alan S. Willsky, 1997

Shou shui Wei©2012Outline

Representation of of a Continuous-Time Signal

by Its Samples: The Sampling Theorem

Reconstruction of of a Signal from Its Samples

Using Interpolation

The Effect of Under-sampling: Aliasing

Discrete-Time Processing of Continuous-Time

Signals

Sampling of Discrete-Time Signals

SDU-BME

Shou shui Wei©2012The Sampling Theorem

Representation of CT Signals by its Samples

Shou shui Wei©2012The Sampling Theorem

Representation of CT Signals by its Samples

Shou shui Wei©2012The Sampling Theorem

Representation of CT Signals by its Samples

Shou shui Wei©2012The Sampling Theorem

Impulse-Train Sampling:

Shou shui Wei©2012The Sampling Theorem

Impulse-Train Sampling:

Ex 4.8, pp. 299-300

Eq 4.70, p. 322

Ex 4.21, p. 323

Shou shui Wei©2012The Sampling Theorem

Impulse-Train Sampling: Ex 4.21, 4.22, pp. 323-4

Shou shui Wei©2012The Sampling Theorem

The Sampling Theorem:

Shou shui Wei©2012The Sampling Theorem

Exact Recovery by an Ideal Lowpass Filter:

Shou shui Wei©2012The Sampling Theorem

Sampling with Zero-Order Hold:

Shou shui Wei©2012The Sampling Theorem

Sampling with Zero-Order Hold:Ex 4.4, p. 293

Eq 4.27, p. 301

Shou shui Wei©2012Outline

Representation of of a Continuous-Time Signal

by Its Samples: The Sampling Theorem

Reconstruction of of a Signal from Its Samples

Using Interpolation

The Effect of Under-sampling: Aliasing

Discrete-Time Processing of Continuous-Time

Signals

Sampling of Discrete-Time Signals

Shou shui Wei©2012Reconstruction of a Signal from its Samples Using Interpolation

Exact Interpolation:

Ex 2.11, p. 110

Shou shui Wei©2012Reconstruction of a Signal from its Samples Using Interpolation

Exact Interpolation:

Shou shui Wei©2012Reconstruction of a Signal from its Samples Using Interpolation

Ideal Interpolating Filter & The Zero-Order Hold:

Shou shui Wei©2012Reconstruction of a Signal from its Samples Using Interpolation

Sampling & Interpolation of Images:

Shou shui Wei©2012Reconstruction of a Signal from its Samples Using Interpolation

Higher-Order Holds:

Shou shui Wei©2012Reconstruction of a Signal from its Samples Using Interpolation

Higher-Order Holds:

= *

=

Ex 4.4, p. 293

X

Shou shui Wei©2012Reconstruction of a Signal from its Samples Using Interpolation

First-Order Hold on Image Processing:

Shou shui Wei©2012Outline

Representation of of a Continuous-Time Signal

by Its Samples: The Sampling Theorem

Reconstruction of of a Signal from Its Samples

Using Interpolation

The Effect of Under-sampling: Aliasing

Discrete-Time Processing of Continuous-Time

Signals

Sampling of Discrete-Time Signals

Shou shui Wei©2012Effect of Under-sampling: Aliasing

Overlapping in Frequency-Domain: Aliasing

Shou shui Wei©2012Effect of Under-sampling: Aliasing

Overlapping in Frequency-Domain: Aliasing

Shou shui Wei©2012Effect of Under-sampling: Aliasing

Overlapping in Frequency-Domain: Aliasing

Shou shui Wei©2012Effect of Under-sampling: Aliasing

Overlapping in Frequency-Domain: Aliasing

Shou shui Wei©2012Effect of Under-sampling: Aliasing

Overlapping in Frequency-Domain: Aliasing

Shou shui Wei©2012Effect of Under-sampling: Aliasing

Overlapping in Frequency-Domain: Aliasing

Shou shui Wei©2012Effect of Under-sampling: Aliasing

Overlapping in Frequency-Domain: Aliasing

Shou shui Wei©2012Effect of Under-sampling: Aliasing

Strobe Effect:

Shou shui Wei©2012Outline

Representation of of a Continuous-Time Signal

by Its Samples: The Sampling Theorem

Reconstruction of of a Signal from Its Samples

Using Interpolation

The Effect of Under-sampling: Aliasing

Discrete-Time Processing of Continuous-Time

Signals

Sampling of Discrete-Time Signals

Shou shui Wei©2012Discrete-Time Processing of Continuous-Time Signals

Discrete-Time Processing of CT Signals:

Shou shui Wei©2012Discrete-Time Processing of Continuous-Time Signals

C/D or A-to-D (ADC) and D/C or D-to-A (DAC):

Shou shui Wei©2012Discrete-Time Processing of Continuous-Time Signals

C/D Conversion:

Shou shui Wei©2012Discrete-Time Processing of Continuous-Time Signals

C/D Conversion:

Shou shui Wei©2012Discrete-Time Processing of Continuous-Time Signals

C/D Conversion:

Shou shui Wei©2012Discrete-Time Processing of Continuous-Time Signals

C/D Conversion:

Table 4.2, p. 329 Eq 7.3, 7.6, p. 517

Shou shui Wei©2012Discrete-Time Processing of Continuous-Time Signals

D/C Conversion:

Overall System:

Shou shui Wei©2012Discrete-Time Processing of Continuous-Time Signals

Frequency-Domain Illustration: 1

Shou shui Wei©2012Discrete-Time Processing of Continuous-Time Signals

Frequency-Domain Illustration:

Shou shui Wei©2012Discrete-Time Processing of Continuous-Time Signals

CT & DT Frequency Responses:

Shou shui Wei©2012Discrete-Time Processing of Continuous-Time Signals

Digital Differentiator:Ex 4.16, p. 317

Shou shui Wei©2012Discrete-Time Processing of Continuous-Time Signals

Half-Sample Delay:Ex 4.15, p. 317

Shou shui Wei©2012Outline

Representation of of a Continuous-Time Signal

by Its Samples: The Sampling Theorem

Reconstruction of of a Signal from Its Samples

Using Interpolation

The Effect of Undersampling: Aliasing

Discrete-Time Processing of Continuous-Time

Signals

Sampling of Discrete-Time Signals

Shou shui Wei©2012Sampling of Discrete-Time Signals

Impulse-Train Sampling:

Shou shui Wei©2012Sampling of Discrete-Time Signals

Impulse-Train Sampling:

Shou shui Wei©2012Sampling of Discrete-Time Signals

Exact Recovery Using Ideal Lowpass Filter:

Shou shui Wei©2012Sampling of Discrete-Time Signals

DT Decimation & Interpolation: Down-samplingEq 5.45, p. 378: Time expansion

Shou shui Wei©2012Sampling of Discrete-Time Signals

Higher Equivalent Sampling Rate: Up-sampling

Shou shui Wei©2012Sampling of Discrete-Time Signals

Down-sampling

+ Up-sampling:

Shou shui Wei©2012In Summary

Shou shui Wei©2012In Summary

Discrete-Time Processing of CT Signals