0.8 Lecture 9:sampling of ct signals  (Page 2/4)

Digital audio — reproduction system

A single channel of a digital audio reproduction system is illustrated with a block diagram. The reproduction system input is the digital data from the recording medium or from the incoming transmitted data. The output of the reproductions system is designed to reproduce the originally recorded/transmitted audio signal.

Sample-and-hold circuit

In this lecture we are concerned only with the sampling of a CT signal to produce a sampled CT signal. Later we will discuss how to form a DT signal from the sampled CT signal. We will not describe how to form a digital signal which involves converting infinite precision numbers to finite precision numbers, a process called analog-to-digital conversion. A schematic diagram of a sample-and-hold circuit that produces samples of a CT signal is shown below.

Definition

Sampling a one-dimensional signal x(t) at t = nT where T is the sampling period yields the samples x(nT). Sampling a two dimensional signal f(x, y) at x = nδx and y = mδy yields the samples f(nδx,mδy).

Key issues

We shall consider the sampling of one-dimensional signals only. The issues are as follows.

• How should we model the sampling process?
• Under which conditions can you recover x(t) from x(nT)?
• How do you recover x(t) from x(nT) when these conditions are met?
• What happens when these conditions are not met?

II. MODEL OF SAMPLING — IMPULSE MODULATION

1/ Definition

Let x(t) be a continuous time function and let s(t) be a uniform impulse train of period T,

$s\left(t\right)=\sum _n\delta \left(t-\mathrm{nT}\right)$

The sampled time function is

$\hat{x}\left(t\right)=x\left(t\right)\times s\left(t\right)=x\left(t\right)\sum _n\delta \left(t-\mathrm{nT}\right)$ $=\sum _{n}x\left(t\right)\delta \left(t-\mathrm{nT}\right)=\sum _{n}x\left(\mathrm{nT}\right)\delta \left(t-\mathrm{nT}\right)$

Multiplication of time functions is called modulation. Therefore, multiplication by an impulse train is called impulse modulation.

Therefore, we have

$\hat{x}\left(t\right)=x\left(t\right)\times s\left(t\right)=\sum _{n}x\left(\mathrm{nT}\right)\delta \left(t-\mathrm{nT}\right)$

2/ The essence of sampling is captured by impulse modulation

Note that with impulse modulation, the sampled signal is represented as a sequence of impulses whose areas are the sample values, i.e.,

$\hat{x}\left(t\right)=x\left(t\right)\times s\left(t\right)=\sum _{n}x\left(\mathrm{nT}\right)\delta \left(t-\mathrm{nT}\right)$

The only property of the impulses that have any consequences are their areas and these are the sample values. Hence, impulse modulation is an effective model of sampling; the sample values, and only the sample values, are preserved by impulse modulation.

3/ Physical samplers can be modeled with an impulse modulator and a filter

A sampler that produces rectangular pulses can be represented.

$y\left(t\right)=\hat{x}\left(t\right)*p\left(t\right)=\left(\sum _{n}x\left(\mathrm{nT}\right)\delta \left(t-\mathrm{nT}\right)\right)*p\left(t\right)$ $=\sum _{n}x\left(\mathrm{nT}\right)\left(\delta \left(t-\mathrm{nT}\right)*p\left(t\right)\right)=\sum _{n}x\left(\mathrm{nT}\right)p\left(t-\mathrm{nT}\right).$

III. THE CTFT OF A SAMPLED SIGNAL

The condition for recovering x(t) from _x(t) is more readily seen in the frequency domain.

1/ Derivation

$x\left(t\right)\stackrel{F}{⟺}X\left(f\right)$

$s\left(t\right)=\sum _n\delta \left(t-\mathrm{nT}\right)\stackrel{F}{⟺}S\left(f\right)=\frac{1}{T}\sum _n\delta \left(f-\frac{n}{T}\right)$

$\hat{x}\left(t\right)=x\left(t\right)\times s\left(t\right)\stackrel{F}{⟺}\hat{X}\left(f\right)=X\left(f\right)*S\left(f\right)$

$\hat{x}\left(t\right)=\sum _{n}x\left(\mathrm{nT}\right)\delta \left(t-\mathrm{nT}\right)\stackrel{F}{⟺}\hat{X}\left(f\right)=\frac{1}{T}\sum _{n}X\left(f-\frac{n}{T}\right)$

2/ Sampling Theorem

The Fourier transform of the sampled time function equals that of the unsampled time function repeated periodically at the sampling frequency fs = 1/T and scaled by 1 /T .