lecture3-AD-升采样降采样

1、 ms V ms V Filter Antialiasing k A A/D k A Digital Processing ms V D/A ms V Filter ReconstructionSometimes steps missing- Filter + A/D (ex: economics);- D/A + filter (ex: digital output wanted). General schemeDigital ProcessingFilter Antialiasing A/D.- ),(nnTxnxc1/ is the sampling frequency (samples

2、 per second) sfT2 / is the sampling frequency (radians per second) sT( )( ) ( )( )() (modulation)sccnx tx t s tx ttnT( )() () (sifting property)scnx tx nTtnT筛选性质frequency domainSpatial/time domain,band limited2( )()()() 2 (where ) snkss ttnTS jkTT )(*)(21)()()()(jSjXjXtstxtxcscsksckscskjXTkTjXjX)(1

3、)(2*)(21)(脉冲串的傅立叶变换：因为：所以：1()( ()scskXjXjkT 这个方程给出了连续时间信号和离散时间信号之间傅立叶变换之间的关系通过这个式子，我们可以观察到：()sXj 是由 的傅立叶变换 的周期重复构成的。( )cx t 被整倍数的采样频率所移位，然后叠加起来就得到了该样本冲击串的周期傅立叶变换()cXji.e., ()cXjcC there is no aliasingX ( ) be reconstructed by X( )ifTcan 生活中进行限带的例子：人能发出的声音的频率？电话系统中把人的声音带限到多少？Original imagedownsampled

4、 imageG 1/4G 1/8Gaussian 1/2G 1/4 G 1/8Gaussian 1/21/4 (2x zoom)1/8 (4x zoom)1/2Sampling Without AliasingSampling With Aliasing理想低通滤波器samplingIdeal low-passfilteringOriginal SignalDiscrete-Time SignalIdeal low-pass FilteringReconstructed Signal如果满足采样定理，如果调制脉冲串被一个合适的滤波器进行滤波，那么滤波器输出信号的傅立叶变换就和原始信号的傅立叶变

5、换完全一样。给定一个采样序列 xn，我们可以构成脉冲串( ) ()snx tx ntnT( )( ) *( ) ()rsrrnx tx th tx n h tnT如果脉冲串被输入到一个理想的连续时间的低通滤波器：Frequency ResponseImpulse ResponseTtTtTtthrsinc/)/sin()()()(nTthnxtxrnrTtTtTtthrsinc/)/sin()(nrTnTtTnTtnxtx/ )(/ )(sin)( () and ()0 for /2 we have:( )( )ccsrcif x nx nTXjTthenx tx t 原始连续信号采样信号重构

6、信号.)()(nTnjrrejHnxjX).()()(TjrreXjHjX)()(nTthnxtxrnrnrTnTtTnTtnxtx/ )(/ )(sin)( ()( )sin( ()/)( )()/sin( ()/)( )()/( )() time shifting property()()jtrrnjtnnjtnnjnTrnrXjx t edttnTTx nedttnTTtnTTx nedttnTTx n eHjHjX jT ()cdxnx nMx nMT利用“采样”一个序列可以降低一个序列的采样率，即定义一个新的序列为：What happened during down-sampling

7、?xdn就是对xn按降采样因子M降采样后的序列，我们的目的就是要找到两个序列的DTFT之间的关系。 )(nTxnxckcjTkTjXTeX21)( )(nMTxnMxnxcdrcjdMTrMTjXMTeX21)()()(jdjeXeX10112 ()()MjcdikkMiXeXjMTMTMT101122()MjcdikkiXeXjMTMTTMT12()jcdrrXeXjMTMTMT)(kMir - k+ 0iM-1k=0: 0M-1: 0M-1k=1: 2M-1: 0M-1k=2: 23M-1: 0M-1rikMrirMirMi 101122()MjcdikikXeXjMTMTT(2)/122

8、jiMckikX eXjTMTT1(2)/01()MjjiMdiXeX eM1(2)/20 M=21()21()()222jjidiletXeX eXXNT 4/2 / , and 0, otherwiseix nL nkLkZx nnkn() ()jj neekj njLkjLkX eDTFT x nx knkLex knkL ex k eX e LnLnnhi/)/sin(kiLkLnLkLnkxnx/ )(/ )(sin,.3,2,010LLLnnhhii,.3,2, ,/LLLnLnxnxi-15-10-505101502n-15-10-505101500.51n-15-10-5051

9、01502n-15-10-505101502n)(nx)(np)(nxp)(nxdreturn4 Dreturnh h )( jeX0 2 2 )( jeP0 2 2 s s s 3s 3 D 2 0 2 2 h h D1s s s 3s 3 )( jpeX 0 2 2 hD hD D1)( jdeX04812162024283236404448024n04812162024283236404448024n04812162024283236404448024n)(nx)(nxp)(nxIreturn4 Ireturn 0 2 2 h h )( jeX)( jIeX 0 2 2 Ih I 2I

10、6)( jpeXIh 0 2 2 Ih Ih 72 )( jeX0 2 2 72 712 716 712 716 76 )( jdeX0 2 2 76 3 Dreturn72 )( jeX0 2 2 72 712 716 712 716 7 )( jIeX0 2 2 7 2 I )( jIdeX0 2 2 7 , 2 DI , 3 , 1 , 0 , 4 , 2 , 0 ),()(nnnxnxp)2()(nxnxd DeXDeXsDkkjjps 2 , )(1)(10)( )()(DjpjdeXeX n)(nx0n)(nxp0n)(nxd02 D43 )( jeX0 2 2 43 45 411 45 411 1 )( jpe