離散2015數(shù)字信號處理第4章

1、1Mathematical represenion of sling xn xc (nT )s(t) xc (t)xs (t)xn xc (nT )4.1 Periodic slingperiodic sling: xn xc (nT )T: sling periodfs=1/T : sling frequency, sles/ss 2 /T : sling frequency, radians/sideal continuous-to-discrete-time (C/D) converterxc (t)xn xc (nT )roductionA D-T signal can be obta

2、ined by sling a C-T signal.Whappenedime and frequencywhen a C-T signal xc(t) is sled and converted to a D-T signal xn?If xc(t) can be recovered from xn without information loss?Whatconditions?4.0roductionSpeech signal and its sequence of slesx(t) Slingxn Take a segmentMain contentsSling TheoremDiscr

3、ete-time prosing of continuous-time (C-T) signalsContinuous-time prosing of discrete-time (D-T) signalsChanging the sling rate of D-T signalsMultirate signal prosingAntialiasing filterCh4.Sling of Continuous-Time Signals2理想抽樣對信號頻譜的影響續(xù)時(shí)間信號經(jīng)過理想抽樣后，其頻譜將以抽樣頻率 s 2 T 為間隔重復(fù)，即產(chǎn)生周期延拓。對實(shí)信號而言，頻譜周期性重復(fù)現(xiàn)象可從脈沖 調(diào)制的

4、角度解釋。由于脈沖序列具有相等大小 的各階諧波分量，因而當(dāng)其被某實(shí)信號調(diào)幅后，該信號的頻譜就被調(diào)制到脈沖序列的各階諧波上，出現(xiàn)基帶的頻譜搬移。Xs ( j) 即是這些調(diào)制頻譜的總和。采樣xs(t) 信號頻譜S ( j) 2 k (4.5)Tsk Xs( j) consists of periodically repeated copies of Xc( j). The copies of Xc( j) are shifted by eger multiples of s and then superim ed to produce Xs( j). 2sTX ( j) 1 X j( k ) (4

5、.6)sT csk X ( j) 1 X ( j) S ( j)s2 c周期函數(shù)沖擊串的沖擊串頻譜級數(shù)2s(t) ) a e jksts kTnk a 1 T 2 s(t)e jk t dt 1 T 2 )e jk t dtkT T 2T T 2ssnT1內(nèi)，僅當(dāng)n=0時(shí)有意義s(t) ) 1 e jkst nT k 2 S ( j) T ( ks )k e jkst CF 2 ( k )s區(qū)間TnT4.2 Frequency-represenionx (t)x (t) csxs (t) xc (t)s(t) xc (t) T )n xc (nT ) )(4.4)FTnx (t) CF X (

6、 j)s(t) CF S ( j)ccs(t) X ( j) 1 X ( j) S ( j)s2 cSling frequency and Sequencex1(t) sin(2 f1t) x2 (t) sin(2 f2t)f2 2 f1T1 2T2T T1 4不同的輸入信號對應(yīng)同一個(gè)序列x2n x1n x1n x2n T T1 4T T1T T1Mathematical represenion of slingx (t)cx (t) x (t)s(t) x (t) T )sccn xc (nT ) )(4 4)ns(t) nT )xn T T n1xs (t) differentxs (t)

7、xnxn 2/T : sling frequency同一個(gè)輸入信號 s對應(yīng)不同的序列T 2T13圖解 xs(t) VS xn, Xs( j) VS X(ej)ime- in frequency-xs (t)FT1 X ( j)sTs NNs TDTFTxn1 X (e j )T2 NTNT2the time axis is normalizedthe frequency axis is normalized by by a factor of Ta factor of 1/T, or multip d by TX ( j) X (e j ) X (e jT )(4.18) s T1 X s (

8、 j) X c j( ks ) (4 6)T k jT1 X (e) T Xc j ks (4.19)k X (e j ) 1 X j 2 k (4.20)Tc TT k 序列的頻譜是采樣信號頻譜的簡單頻率縮放，其頻率軸按 T 來定義，亦即頻率軸對采樣率fs 作歸一化。思考：數(shù)字頻率 2 對應(yīng)的模擬頻率是多少？Express X(ej) by Xs( j) and Xc( j)s sX ( j) x (t)e jt dt x (nT ) )e jt dt x (nT )e jTn (4.15)ccnnX (e j ) xne jnn xn xc (nT ) (4 16) X (e j ) x

9、(nT )e jn (4.17)cnX ( j) X (e j ) X (e jT )(4.18)s Txs (t) xc (nT ) (4 4)nxs(t) VS xc(t), Xs( j) VS Xc( j)ime-, xs(t) is the impulse traodulated result with xc(t)In frequency-, Xs( j) consists of periodically replicas of Xc( j) , with the copied period being 2sTIf s 2N, Xc( j) can be recovered from

10、Xs( j) , i.e., xc(t) can be recovered from xs(t), byXr ( j) Xs ( j)Hr ( j) , N c (s N )If s 2N, Xc( j) can not be recoveredRecovering xc(t) from xs(t)When s 2N , xc(t) can be recovered from xs(t) with an ideal loss filter1 Xs ( j)Ts NNs(s N )H ( j) X r ( j) X s ( j)Hr ( j)Ty response: H ( j) T , ccc

11、0, X ( j)ccutoffy: N c (s N ) Xr ( j) Xc ( j)N N s(t) nT )nx (t)x (t)Hr ( j) x (t) csr圖解時(shí)域采樣對頻域的影響Xc ( j)X ( j)caliasing phenomenonNN NNS ( j)S ( j)22TTssss1 X ( j)1 X s ( j)sTTs NNss NN s(s N )(s N )s N N , or s 2Ns N N , or s 2NExle 1 xc (t) cos(4 t),T 1/ 6X ( j)cX ( j), X (e j )determineN 4s44Xs

12、( j)Xc ( j) ( 4 ) ( 4 )TTTTTT 2 / T 12 s16 12 8 4 T 4 812 16 /2 6csX (e j ) X ( jT )sTTTTTT8 /3 2 4 /3 2 /32 / 3 4 / 3 28 / 3Practical Slingcontrolled by impulsexs (t) xc (t) ideal sling1 X ( j)sTNNs(s N )sX ( j)c1 Xs ( j)TNNsNNs(s N )輸出的頻譜進(jìn)行了周期延拓（只在s的各次諧波處存在）各次諧波處的譜幅度受到了sinc函數(shù)調(diào)制44.3 Reconstruction

13、of a bandlimited signal from its slesGiven xn, we can form an impulse train xs(t)xs (t) xn )(4.22)nInput xs(t) to an ideal loss filter with Hr ( j)and hr (t) , the output of the filter will bexr (t) xnhr )(4.23)nxnxs(t)x (t )xnxr (t) H ( j) / h (t)Ideal reconstruction systemReconstruction of a bandl

14、imited signal from its slesSling theorem: xc(t) can be recovered from its sles xn, if and only if s 2NIf the sling theorem is met, and the modulated impulse train xs(t) is filtered by an appropriateloss filter, the output of the filter will be xc(t)s(t ) nT )n H ( j)xc (t)xs (t)rrxr (t) xc(t) How to

15、 recoverxc(t) from xn?xs (t) s (t ) xc(t)xs(t)TNyquist Sling TheoremLet xc(t) is a continuous-time bandlimited signal withXc ( j) 0 for NThen xc(t) can be uniquely recovered from its discrete-time sles xn = xc(nT) , if and only ifNyquist rateApplicationfmaxfs設(shè)計(jì)數(shù)字系統(tǒng)時(shí)須先確定系統(tǒng)的采樣頻率語音處理3400Hz6 8kHz語音處理20k

16、Hz40kHz4MHz8MHz生物醫(yī)學(xué)1kHz2kHz地質(zhì)500Hz1kHz 2 2sTNExle 2 xc (t) cos(4 t),T 1/ 3jdetermine Xs ( j), X (e )Xc ( j)N 4s 2 / T 644 /2 3Xs ( j)csTTTTTT106 4 22 4 610 X (e j ) X ( jT )sTTTTTT10 /32 4 /3 2/32 /3 4 / 3 210 / 3 5 2sTX s(j) consists of periodically replicas of X c(j)s 2NX ( j) X ( j)H ( j)rsN c s

17、Ns 2NXr ( j) Xs ( j)H ( j)1 X ( j)1 X s ( j)cscTTs NNs s NN s(s N )(s )ccSummary of Sling and ReconstructionC/D converter:x (t) s(t) x (t)x n csD/C converter: xnxs(t)xr (t) xc (t)H ( j)Time-descriptionxc (t) xr (t) xnhr )n xn sin( () / T ) ) / Tnxn The ideal loss filterxr (t )olates betn the sles of

18、 xn toconstruct xr(t)xr(t) = xc(t), for t = nTTime-descriptionsin( t / T )hr)？hr (t) t / T(4.24)hr (0) 1hr (nT ) 0, n 1, 2,xr (mT ) xm xc (mT )xr (t) xnhr ) (4.23)nhr (t ) H ( j) T , c , ( )r0, NcsNcIFTh (t) sin(ct)sin( t / T ) (4.24)r t t / Tc / T / T xr (t) xnhr)(4.23)n xn sin( () / T )(4.25) ) / T nT