Finally, the discrete-time Fourier transform of x(n) is

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Comparing Eq. (3.3)with Eq. (3.2),it follows that

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CHAP. 31

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SAMPLING

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Thus, X ( e J W is a frequency-scaled version of X , ( j Q ) , with the scaling defined by )

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) This scaling, which makes x ( ~ J " periodic with a period of 2 n , is a consequence of the time-scaling that occurs when x,(t ) is converted to x ( n ) .

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EXAMPLE 3.2.1

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Suppose that x a ( t ) is strictly bandlimited so that X a ( j Q ) = 0 for ( R J> Ro as shown in the figure below.

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If x a ( t ) is sampled with a sampling frequency Q , 2 2Q0, the Fourier transform of ~ X , ( j Q ) as illustrated in the figure below.

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, ~ ( tformed is )

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by periodically replicating

t xs""'

However, if R, < 2R0, the shifted spectra X , , ( j R - jkQ,) overlap, and when these spectra are summed to form X , ( j Q ) , the result is as shown in the figure below.

t ""'"'

This overlapping of spectral components is called aliasing. When aliasing occurs, the frequency content of xa(t)is compted, and X , ( j Q ) cannot be recovered from X,v(jQ).

As illustrated in Example 3.2.1, if x , ( t ) is strictly bandlimited so that the highest frequency in x , ( t ) is Qo, and if the sampling frequency is greater than 2Q0,

no aliasing occurs, and x , ( t ) may be uniquely recovered from its samples xa(nTv) with a low-pass filter. The following is a statement of the famous Nyquist sampling theorem:

Sampling Theorem: If x , ( t ) is strictly bandlimited,

then x , ( t ) may be uniquely recovered from its samples x,(nT,) if

The frequency Q0 is called the Nyquist frequency, and the minimum sampling frequency, Q , = 2'20, is called the Nyquist rate.

SAMPLING

[CHAP. 3

Because the signals that are found in physical systems will never be strictly bandlimited, an analog antialiasing filter is typically used to filter the signal prior to sampling in order to minimize the amount of energy above the Nyquist frequency and to reduce the amount of aliasing that occurs in the AID converter.

3.2.2 Quantization and Encoding

A quantizer is a nonlinear and noninvertible system that transforms an input sequence x(n) that has a continuous range of amplitudes into a sequence for which each value of x ( n ) assumes one of a finite number of possible values. This operation is denoted by ,W) = Qlx(n)l

The quantizer has L

+ I decision levels X I , xl, . . . . x ~ +that divide the amplitude range for x(n) into L intervals [

For an input x(n) that falls within interval l k , quantizer assigns a value within this interval, &, tox(n). This the process is illustrated in Fig. 3-3.

rdecision level

Fig. 3-3. A quantizer with nine decision levels that divide the input amplitudes into eight

quantizer output

quantization intervals and eight possible quantizer outputs. i r . Quantizers may have quantization levels that are either uniformly or nonuniformly spaced. When the quantization intervals are uniformly spaced,

A is called the quantization step size or the resolution of the quantizer, and the quantizer is said to be a uniform or linear quantizer.' The number of levels in a quantizer is generally of the form

in order to make the most efficient use of a (B 1)-bit binary code word. A 3-bit uniform quantizer in which the quantizer output is rounded to the nearest quantization level is illustrated in Fig. 3-4. With L = 2'" quantization levels and a step size A, the range of the quantizer is

Therefore, if the quantizer input is bounded, l*v(n)l5

Xmax

the range of possible input values may be covered with a step size

With rounding, the quantization error e(n) = Qlx(n)l - x(n)

'ln some applications, such as speech coding, the quantizer levels are adaptive (1.e..they change with time).