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Chapter 4 sampling of continous-time signals 4.5 changing the sampling rate using discrete-time processing 4.5.1 sampling rate reduction by an integer factor (downsampling,decimation) 4.5.2 increasing the sampling rate by an integer factor (upsampling,interpolation) 4.5.3 changing the sampling rate by a noninteger factor 4.5.4 application of multirate signal processing 4.1 periodic sampling 4.2 discrete-time processing of continuous-time signals 4.3 continuous-time processing of discrete-time signal 4.4 digital processing of analog signals

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4.1 periodic sampling 1. ideal sample （理想采样） T sample period fs=1/T:sample rate samples/second Ω s=2 π /T:sample rate, radians/second [] ) ( ) ( nT x t x n x c nT t c = = =
Figure 4.1 ideal continuous-time-to-discrete-time (C/D) converter We refer to a system that implements the operation of the above as an ideal continuous- to-discrete-time (C/D) converter （理想连续时间 到离散时间的转换器）

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Figure 4.2(a) mathematic model for ideal C/D −∞ = = n nT t ) ( δ 时间轴 归一化 t Æ t/T=n It is convenient to represent the sampling process mathematically in the two stages.
Let us now consider the Fourier transform of . Since it is the product of and , it’s Fourier transform is the convolution of the Fourier transform and . So it can be expressed as: ) ( t x s () c Xj Ω c Sj Ω ) ( t x c ) ( t s 11 2 ( ) (( ) ) ) ) sc s c kk k k TT T π ∞∞ =−∞ Ω = Ω− Ω = Ω− ∑∑ 证明见课堂笔记 −∞ = = Ω = Ω = k c T s j T k j X T j X e X ) / ) 2 ( ( 1 | ) ( ) ( / ω For , its Fourier transform is: ] [ n x 证明自己复习

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N N s Ω < Ω Ω N N s Ω Ω Ω 折叠频率 No aliasing aliasing −∞ = Ω
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## This note was uploaded on 02/29/2012 for the course EE 325 taught by Professor Lilili during the Fall '11 term at BYU.

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ç¬¬04ç«  è¿žç»­æ—¶&eacut

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