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1 I and y(n) = -. y(n) = 1 -a I +a
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(a) y(n) = $x(n). (b) y(n) = h ~ ~ e j " " / ~ .
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h(n) = hl(n)
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sin(nwl) . + hz(n) where hl(n) = --- 1s an ideal low-pass filter with a cutoff frequency of wl, and
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- an ideal high-pass filter with a cutoff frequency of w. is
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sin(nw) nwz
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A high-pass filter with a cutoff frequency w, = ~ / 3 .
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1 H (ejW)l= 1 for Iwl z
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 discrete-time signals come from sampling a continuous-time 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 analog-to-digital (AID) conversion. The reverse process of reconstructing an analog signal from its samples is known as digital-to-analog ( 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 ANALOG-TO-DIGITAL CONVERSION
An A/D converter transforms an analog signal into a digital sequence. The input to the A/D converter, x,(t),
is a real-valued 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 discrete-time 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 continuousto-discrete ( C P ) converter, or ideal AlD converter. The sampler converts the continuous-time signal x , ( t ) into a discrete-time 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. 3-1. The components of an analog-to-digital converter.
3.2.1 Periodic Sampling
Typically, discrete-time signals are formed by periodically sampling a continuous-time 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. 3-2(a). First, the continuous-time 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 discrete-time 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. 3-2(b).
-2Ts
- 2 - 1
Fig. 3-2.
Continuous-todiscrete 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 e-JnnTs, 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,
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