Chapter 3 Sampling of - Prince of Songkla...

Sampling of Continuous-Time Signals

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Chapter 3

Sampling of Continuous-Time

Signal

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Outline

3 .0 Introduction3 .1 Periodic Sampling3 .2 Frequency-Domain Representation of Sampling3 .3 Reconstruction of a Bandlimited Signal from its Samples3 .4 Discrete-Time Processing of Continuous-Time Signals

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3 .5 Continuous-Time Processing of Discrete-Time Signals3 .6 Changing the Sampling Rate Using Discrete-Time Processing3 .7 Practical Considerations

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3 .0 Introduction

In this chapter, we discuss the process of periodic sampling in some detail, including the issue of aliasing. Particularly, the continuous-time signal processing can be implemented through a process of sampling, discrete-time processing and subsequent reconstruction of a continuous-time signal.

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3 .1 Periodic Sampling

The method of obtaining a discrete-time representation of a continuous-time signal is through periodic sampling, wherein a sequence x[n] is obtained from a continuous-time signal xc(t) :

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x [n]= xc t ∣t=nT=x nT −∞n∞

T is the sampling period.fs= /1 T is sampling frequency.

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Mathematical representation of sampling

Fig. Overall system1

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Fig. Sampling rates and output sequences.2

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. Frequency-Domain Representation 3 2 of Sampling

The modulating signal s(t) is a periodic impulse train :

where (t) is the unit impulse function or Dirac delta function.

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Consequently (from fig. ) :1

x st =xc t s t

=xc t ∑n=−∞

t−nT

x st =∑n=−∞

xcnT t−nT

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Fourier transform of xs(t)

The fourier transform of a periodic impulse train :

where s = 2/T is the sampling frequency in radians/s.

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where * denotes the operation of convolution.

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Nyquist Sampling Theorem

Let xc(t) be a bandlimited signal with

Then xc(t) is uniquely determined by its samples x[n] = xc(nT), n = , 0± ,1 ± , ... , if2

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. Reconstruction of a Bandlimited 3 3 Signal from its Samples

An impulse train xs(t) :

The output of an ideal lowpass filter :

Time domain

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Fig. Block diagram of an ideal bandlimited 3 signal reconstruction system.

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Fig. Frequency response of an IRF and 4 Impulse response of an IRF.

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Frequency domain

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. Discrete-Time Processing of 3 4 Continuous-Time Signals

The process includes three steps : sampling process, discrete-time signal process, and reconstructingprocess.

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Discrete-time processing of continuous-time signals.

The system is continuous-time system because both input and output are continuous-time signals. But the middle block is a discrete-time system.

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x [n]= xct ∣t=nT=x nT −∞n∞

C/D converter

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D/C converter

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Linear Time-Invariant Discrete-Time Systems

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Impulse Invariance

Fig. Continuous-time LTI system and 5 Equivalent system for bandlimited.

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. Continuous-Time Processing of 3 5 Discrete-Time Signals

The C/D converter samples yc(t) without aliasing :

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Fig. Continuous-time processing of 6 discrete-time signals.

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. Changing the Sampling Rate 3 6 Using Discrete-Time Processing

A Discrete-time signal of a continuous signal xc(t) :

x[n] = xc(nT)

When the sampling rate is changed, the new discrete-time signal :

x' [n] = xc(nT')

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Sampling Rate Reduction by an integer Factor

Representation of downsampler

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Discrete Fourier transform of x[n] :

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Increasing the Sampling Rate by an integer Factor

Where T' = T/L, from the sequence of samples.

x[n] = xc(nT)

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General system for sampling rate increase by L

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xe [n]= ∑k=−∞

x [k ][n−kL]

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. Practical Considerations3 7

If the input is bandlimited and the sampling rate exceeds the Nyquist rate. The ideal linear time-invariant discrete-time system can be used to implement linear time-invariant continuous-time system.

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The realistic model for digital processing of continuous-time (analog) signals

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Prefiltering to Avoid Aliasing

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Analog-to-Digital (A/D) Conversion

An ideal C/D converter converts a continuous-time signal into a discrete-time signal with infinite precision. But A/D converter is a physical device that converts a voltage or current amplitude into a binary code representing a quantized amplitude value.

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An ideal sample and hold

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Typical input and output signals for the sample and hold

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Analysis of Quantization Errors

The quantization error is defined as

For -bit quantizer :3

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A simple but useful model of the quantizer is depicted below. In this model the quantization error samples are thought of as an additive noise signal.

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D/A Conversion

The reconstruction is represented as :

The ideal reconstruction filter is :

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The relationship between xr(t) and x[n] is :

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The output would be :

The reconstructed output signal would be :

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