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PC Chapter3 - Chapter 3 Pulse Modulation 3.1 Introduction...

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rate sampling : 1 period sampling : where (3.1) ) ( ) ( ) ( signal sampled ideal the denote ) ( Let s s s s n s T f T nT t nT g t g t g = - = -∞ = δ δ δ Chapter 3 Pulse Modulation 3.1 Introduction
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-∞ = -∞ = -∞ = -∞ = -∞ = -∞ = -∞ = - = = = - + = - = - - = - - n s m m s s s s n s m s s m s s m s s n s W n f j W n g f G W T W f f G mf f G f f G f f G nf T j nT g f G mf f G f t g mf f G f T m f T f G nT t t (3.4) ) exp( ) 2 ( ) ( 2 1 and for 0 ) ( If (3.5) ) ( ) ( ) ( or (3.3) ) 2 exp( ) ( ) ( obtain to (3.1) on Transform ier apply Four may or we (3.2) ) ( ) ( ) ( ) ( 1 ) ( ) ( ) g( have we A6.3 Table From 0 π π δ δ δ δ δ δ 2
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) ( of n informatio all contains ) 2 ( or for ) 2 ( by determined uniquely is ) ( (3.7) , ) exp( ) 2 ( 2 1 ) ( as ) ( rewrite may we (3.6) into (3.4) ng Substituti (3.6) , ) ( 2 1 ) ( that (3.5) Equation from find we 2 . 2 for 0 ) ( . 1 With t g W n g n W n g t g W f W W nf j W n g W f G f G W f W f G W f G W f W f f G n s < < - < < - - = < < - = = = -∞ = π δ 3
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) ( of formula ion interpolat an is (3.9) (3.9) - , ) 2 ( sin ) 2 ( 2 ) 2 sin( ) 2 ( (3.8) ) 2 ( 2 exp 2 1 ) 2 ( ) 2 exp( ) exp( ) 2 ( 2 1 ) 2 exp( ) ( ) ( have may we , ) 2 ( from ) ( t reconstruc To t g t n Wt c W n g n Wt n Wt W n g df W n t f j W W n g df f t j W n f j W n g W df ft j f G t g W n g t g n n n W W W W n -∞ = -∞ = -∞ = - - -∞ = - < < - = - - = - = - = = π π π π π π π π 4
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rate. sampling higher have or bandwidth signal limit the may we aliasing, avoid .To occurs aliasing sampling) (under limited - band not is signal the When 2 1 interval Nyquist 2 rate Nyquist ) 2 ( from recovered completely be can signal The . 2 . ) 2 ( by described completely be can , to limited is which signal 1.a signals limited - band strictly for Theorem Sampling W W W n g W n g W f W = = < < - 5
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Figure 3.3 ( a ) Spectrum of a signal. ( b ) Spectrum of an undersampled version of the signal exhibiting the aliasing phenomenon. 6
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Figure 3.4 ( a ) Anti-alias filtered spectrum of an information-bearing signal. ( b ) Spectrum of instantaneously sampled version of the signal, assuming the use of a sampling rate greater than the Nyquist rate. ( c ) Magnitude response of reconstruction filter. 7
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(3.14) ) ( ) ( ) ( ) ( have we , property sifting the Using (3.13) ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( (3.12) ) ( ) ( ) ( is ) ( of version sampled ously instantane The (3.11) otherwise T t 0, t T t 0 , , 0 2 1 , 1 ) ( (3.10) ) ( ) ( ) ( as pulses top - flat of sequence the denote ) ( Let s n s s n s s n s n s s s n s nT t h nT m t h t m d t h nT nT m d t h nT nT m d t h m t h t m nT t nT m t m t m t h nT t h nT m t s t s - = - - = - - = - = - = = = < < = - = -∞ = - -∞ = - -∞ = - -∞ = -∞ = δ δ δ δ τ τ τ δ τ τ τ δ τ τ τ δ 3.3 Pulse-Amplitude Modulation 8
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(3.18) ) ( ) ( ) ( (3.17) ) ( ) ( M (3.2) ) ( ) ( (3.2) Recall (3.16) ) ( ) ( ) ( (3.15) ) ( ) ( ) ( is ) ( signal PAM The -∞ = -∞ = -∞ = - = - = - = = k s s k s s m s s δ f H k f f M f f S k f f M f f mf f G f t g f H f M f S t h t m t s t s δ δ δ 9
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Pulse Amplitude Modulation – Natural and Flat-Top Sampling The circuit of Figure 11-3 is used to illustrate pulse amplitude modulation (PAM). The FET is the switch used as a sampling gate.
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