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5 and I H (elW)l= 0 otherwise.
FOURIER ANALYSIS 2n 3 n 5n ( b )  < lo1 < . 6 6 [CHAP.2
( a ) Iwl <
lo1 c . 2n 3 Unique, h ( n ) = ;6(n). The DTFT is constant with an amplitude of
a for lo1 < n and it decreases linearly to zero at w = , 4 zt. y(n) = n = 4, 2.0,2,. otherwise
. ., Beginning with index n = 3, the sequence values are [ I , 3 .s5.
i,  i, 2. i. g, i, I]. Yes.
3
Sampling
INTRODUCTION
Most discretetime signals come from sampling a continuoustime signal, such as speech and audio signals, radar and sonar data, and seismic and biological signals. The process of converting these signals into digital form is called analogtodigital (AID) conversion. The reverse process of reconstructing an analog signal from its samples is known as digitaltoanalog ( D / A )conversion. This chapter examines the issues related to A/D and D/A conversion. Fundamental to this discussion is the sampling theorem, which gives precise conditions under which an analog signal may be uniquely represented in terms of its samples. 3.2 ANALOGTODIGITAL CONVERSION
An A/D converter transforms an analog signal into a digital sequence. The input to the A/D converter, x,(t), is a realvalued function of a continuous variable, t . Thus, for each value o f t , the function x,(t) may be any real number. The output of the A/D is a bit stream that corresponds to a discretetime sequence, x(n), with an amplitude that is quantized, for each value of n, to one of a finite number of possible values. The components of an A/D converter are shown in Fig. 3 1. The first is the sampler, which is sometimes referred to as a continuoustodiscrete ( C P ) converter, or ideal AlD converter. The sampler converts the continuoustime signal x , ( t ) into a discretetime sequence x ( n ) by extracting the values of .u,(r) at integer multiples of the sampling period, T,, Because the samples x,(nTs) have a continuous range of possible amplitudes, the second component of the A/D converter is the quantizer, which maps the continuous amplitude into a discrete set of amplitudes. For a uniform quantizer, the quantization process is defined by the number of bits and the quantization interval A. The last component is the encoder, which takes the digital signal i ( n ) and produces a sequence of binary codewords. *d[) *(I ) 3 ~ ) Quantizer
Encoder
c(n) Fig. 31. The components of an analogtodigital converter.
3.2.1 Periodic Sampling
Typically, discretetime signals are formed by periodically sampling a continuoustime signal
The sample spacing T, is the sampling period, and f, = I / T, is the sampling frequency in samples per second. A convenient way to view this sampling process is illustrated in Fig. 32(a). First, the continuoustime signal is multiplied by a periodic sequence of impulses, to form the sampled signal
SAMPLING
[CHAP. 3
Then, the sampled signal is converted into a discretetime signal by mapping the impulses that are spaced in time by Ts into a sequence x(n) where the sample values are indexed by the integer variable n: This process is illustrated in Fig. 32(b). 2Ts
 2  1 Fig. 32. Continuoustodiscrete conversion. (a)A model that consists of multiplying x , ( I ) by a sequence of impulses. followed by a system that converts impulses into samples. (b) An example that illustrates the conversion process. The effect of the C/D converter may be analyzed in the frequency domain as follows. Because the Fourier transform of 6(t  nTs) is eJnnTs, Fourier transform of the sampled signal x,(t) is the Another expression for X s ( j O )follows by noting that the Fourier transform of s,(t) is
where 9, = 2n/T, is the sampling frequency in radians per second. Therefore,

