CH03 Analog-to-Digital Signal Conversion

CHAPTER 3 Analog-to-Digital Signal Conversion The process...

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C H A P T E R 3 Analog-to-Digital Signal Conversion The process of analog-to-digital signal conversion consists of converting a continuous time and amplitude signal into discrete time and amplitude values. Sampling and quantization constitute the steps needed to achieve analog-to-digital signal conversion. To minimize any loss of information that may occur as a result of this conversion, one must understand the underlying principles behind sampling and quantization. 3.1 Sampling Sampling is the process of generating discrete time samples from an analog signal. First, it is helpful to mention the relationship between analog and digital frequencies. Let us consider an analog sinusoidal signal x t ð Þ ¼ A cos o t þ f ð Þ . Sampling this signal at t ¼ nT s , with the sampling time interval of T s , generates the discrete time signal x ½ n ¼ Acos ð o nT s þ f Þ ¼ A cos ð y n þ f Þ ; n ¼ 0 ; 1 ; 2 ; . . . ; (3.1) where y ¼ o T s ¼ 2 p f f s denotes digital frequency with units being radians (as compared to analog frequency o with units being radians/sec). The difference between analog and digital frequencies becomes more evident by observing that the same discrete time signal is obtained from different continuous time signals if the product o T s remains the same. (An example is shown in Figure 3-1 .) Likewise, different discrete time signals are obtained from the same analog or continuous time signal when the sampling frequency is changed. (An example is shown in Figure 3-2 .) In other words, both the frequency of an analog signal f and the sampling frequency f s define the frequency of the corresponding digital signal y . 57
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10 15 20 1 1.5 2 1 1.5 2 1 0.5 0 0.5 1 1 0.5 0 5 0 10 15 20 5 0 0.5 1 1 0.5 0 0.5 1 1 0.5 0 0.5 1 x ( t ) = cos(2 π t ) x ( t ) = cos(2 π t ) Ts = 0.05s Ts = 0.025s Figure 3-2: Sampling of the same analog signal leading to two different digital signals. 1 1.5 2 1 1.5 2 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 0 5 10 15 20 0 5 10 15 20 x ( t ) = cos(2 π t ) x ( t ) = cos(4 π t ) Ts = 0.05s Ts = 0.025s Figure 3-1: Sampling of two different analog signals leading to the same digital signal. 58 Chapter 3
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It helps to understand the constraints associated with the preceding sampling process by examining signals in the frequency domain. The Fourier transform pairs in the analog and digital domains are given by Fourier transform pair for analog signals X ð j o Þ ¼ ð 1 ±1 x ð t Þ e ± j o t dt x ð t Þ ¼ 1 2 p ð 1 ±1 X ð j o Þ e j o t d o 8 > > > > < > > > > : (3.2) Fourier transform pair for discrete signals X ð e j y Þ ¼ X 1 n ¼±1 x ½ n e ± jn y ; y ¼ o T s x ½ n ¼ 1 2 p ð p ± p X ð e j y Þ e jn y d y 8 > > > > > < > > > > > : (3.3) As illustrated in Figure 3-3 , when an analog signal with a maximum bandwidth of W (or a maximum frequency of f max ) is sampled at a rate of T s ¼ 1 f s , its (a) (b) x ( t ) t 1 t 1 t 2 t 2 t 3 t 3 t t t 4 t 4 Spectrum Analog Signal Discrete Signal Spectrum X ( f ) f W −W Y ( f ) f W −W y ( t ) T s f s =1/T s Figure 3-3: (a) Fourier transform of a continuous-time signal and (b) its dis- crete time version.
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