7.4 Practical Aspects of Sampling 441
FIGURE 7.11
Sample-and-hold circuit.
r
C R
++
− −
Ts
x(t) xsh(t)
n Two significant changes due to considering the pulses of width1 > 0 in the sampling are:
nThe spectrum of the ideal sampled signal xs(t)is now weighted by the sinc function of the frequency
response H(j)of the zero-order hold filter. Thus, the spectrum of the sampled signal using the sample-
and-hold system will not be periodic and will decay asincreases.
nThe reconstruction of the original signal x(t)requires a more complex filter than the one used in the
ideal sampling. Indeed, the concatenation of the zero-order hold filter with the reconstruction filter
should be such that H(s)Hr(s)= 1 , or that Hr(s)= 1 /H(s).
n A circuit used for implementing the sample-and-hold system is shown in Figure 7.11. In this circuit the
switch closes every Tsseconds and remains closed for a short time 1. If the time constant rC<< 1, the
capacitor charges very fast to the value of the sample attained when the switch closes at some nTs, and by
setting the time constant RC>>Tswhen the switch opens 1 seconds later, the capacitor slowly discharges.
The cycle repeats providing a signal that approximates the output of the sample-and-hold system explained
before.
n The DAC also uses a holder to generate an analog signal from the discrete signal coming out of the decoder
into the DAC. There are different possible types of holders, providing an interpolation that will make the
final smoothing of the signal a lot easier. The so-calledzero-order holdbasically expands the sample
value in between samples, providing a rough approximation of the discrete signal, which is then smoothed
out by a low-pass filter to provide the analog signal.
7.4.2 Quantization and Coding
Amplitude discretization of the sampled signalxs(t)is accomplished by a quantizer consisting of a
number of fixed amplitude levels against which the sample amplitudes{x(nTs)}are compared. The
output of the quantizer is one of the fixed amplitude levels that best representsx(nTs)according to
some approximation scheme. The quantizer is a nonlinear system.
Independent of how many levels, or equivalently of how many bits are allocated to represent each
level of the quantizer, there is a possible error in the representation of each sample. This is called
thequantization error. To illustrate this, consider a 2-bit or 2^2 -level quantizer shown in Figure 7.12.
The input of the quantizer are the samplesx(nTs), which are compared with the values in the bins
[− 21 ,− 1 ], [− 1 , 0], [0, 1 ], and [ 1 , 2 1 ], and depending on which of these bins the sample falls in
it is replaced by the corresponding levels− 21 ,− 1 , 0, or 1. The value of the quantization step 1 for
the four-level quantizer is
1 =
2 max|x(t)|
22
(7.23)
