Quantization and Analog to Digital Conversion

Transcription

1 Chapter 10 Quatizatio ad Aalog to Digital Coverio I all thi cla we have examied dicrete-time equece repreeted a a equece of complex (ifiite preciio) umber. Clearly we have bee dicuig implemetatio of may of thee method o a computer. However, to repreet thee equece o a computer, we ca oly do o with fiite preciio uig bit. herefore we would eed to repreet thee (ifiite preciio) complex umber approximately uig a fiite umber of bit. hi repreetatio i called quatizatio ad i fudametal to the implemetatio of ay igal proceig techique o a digital computer Quatizer he quatizer i a o-liear ytem whoe repoe i illutrated i Figure It traform the (ifiite preciio) iput ito a fiite et of precribed value. he quatizer illutrated i Figure 10.1 i called a uiform quatizer, but oe ca alo have o-uiformly paced quatizatio level. We repreet the quatizatio operatio by ˆx[] = Q(x[]), (10.1) where ˆx[] i the quatized ample. For a uiform quatizer, we ca divide the iterval [ A,A] ito 2 b ub-iterval to be able to repreet iput i the rage [ A,A] through b bit a follow. We defie k th ub-iterval a, [ I k = A + ka (k + 1)A 2b 1, A + 2 b 1 he we ca defie the quatizatio fuctio Q( ) a, ˆx[] = Q(x[]) = A + ), k = 0,,2 b 1 (10.2) (k + 1/2)A 2 b 1, (10.3) 143

2 144 Chapter 10. Figure 10.1: ypical quatizer for aalog-to-digital (A/D) coverio. Coverio of iput x ito 8 level repreetable by 3 bit. where x[] I k. Clearly the error icurred i uch a repreetatio if = A/2 b 1 i i.e., or e[] = ˆx[] x[] [ /2, /2), (10.4) /2 e[] < /2, e[] /2 = A 2b. (10.5) Example 10.1 If a igal x[] [ 1,1], i.e. A = 1, ad b = 8, i.e. we have 2 b = 256 ub-iterval, the [ I k = 1 + k 128, 1 + k + 1 ), k = 0,, Alo = = yieldig e[]

3 10.2. Digital-to-Aalog coverio ad iterpolatio Digital-to-Aalog coverio ad iterpolatio Give that we repreet equece uig a quatizer, the repreetatio of x[] i replaced by ˆx[] = Q(x[]). Now if we wat to recotruct the cotiuou-time igal, we kow from optimal iterpolatio that, x r (t) = ( ) t x[] ic, where ic(x) = i(πx) πx, ad i the amplig period. If we oly have acce to the quatized dicrete-time igal, a atural recotructio would be x DA (t) = ( ) t ˆx[] ic (10.6) = ( ) t x[] ic + ( ) t e[] ic. } {{ } } {{ } x r(t) e r(t) Clearly the recotructio error igal due to quatizatio i e r (t). Aother apect of importace to A/D ad D/A coverio i to limit the amplitude of the iput igal to the iterval to which the quatizer operate. hi idea i illutrated i Figure 10.2 where the ampled igal i clipped to lie i iterval [ A,A]. Figure 10.2: Aalog-to-digital coverio with iput clippig Applicatio of quatizatio to compreio Beide the repreetatio of dicrete-time igal i a digital computer for igal proceig operatio, quatizatio could alo be ued for repreetig (efficietly) phyical igal for torage ad tramiio. For example, we had ee the perceptual model for the auditory ytem where the huma

4 146 Chapter 10. ear wa le tolerat to diturbace i the lower frequecy rage tha at higher frequecy rage. hi could be ued to more efficietly repreet a phyical audio igal digitally while reducig the effect of quatizatio error i huma perceptio. Example 10.2 Suppoe we ample the audio igal at 44.1kHz ad each ample i quatized uig a 16-bit quatizer. he we eed for oe ecod of audio, which ha ample, = 705,600 = bit, which form a baelie of compario. Itead, oe ca utilize the perceptual model ad plit the audio igal ito ub-bad a illutrated i Figure I thi cae, we ca allocate differet umber of bit for each of the output of the ub-bad to get a more compact (compreed) repreetatio of the audio igal. hi idea i called ub-bad codig ad form the bai for digital audio uch a MP3. Figure 10.3: Sub-bad codig of audio igal Example 10.3 For the ub-bad ytem repreeted i Figure 10.3, let H 0 ( e jω ) be a ideal low-pa filter with badwidth π/2, i.e. H 0 ( e jω ) = 0, ω > π/2 ( ad H 1 e jω ) ( i a ideal high-pa filter with H 1 e jω ) = 0, ω < π/2. he phyically thee filter focu o the bad kHz ad kHz repectively. Now, if we allocate a 16-bit quatizer to v 0 [] ad a 4-bit quatizer to v 1 [], we ee the followig repreetatio i bit. We have for oe ecod of audio ample of x[]. After dowamplig we have ample each of v 0 [] ad v 1 []. herefore the repreetatio ˆv 0 [] ad ˆv 1 [] would eed, = = = bit,

5 10.4. Overampled aalog-to-digital coverio 147 which i a reductio over jut quatizig the origial audio igal. hi repreet a avig i torage with little perceptual error ice we utilize the auditory model for huma ear. he baic priciple illutrated i the above example form the bai for mot moder digital audio coder. Such a ub-bad coder i implemeted uig may more ub-bad alog with careful choice of filter ad quatizer i digital audio tadard uch a i MP Overampled aalog-to-digital coverio We had ee i oe of the motivatio for multirate igal proceig that amplig at a rate higher tha the miimum rate eeded, (i.e. amplig at a rate larger tha twice the igal badwidth) might have ueful practical applicatio. I particular, we aw that amplig at a higher rate tha eceary (alo called overamplig) make it poible to ue a le triget aalog ati-aliaig filter. Moreover, oe ca do dowamplig ad filterig i the dicrete-time domai which i eaier to implemet i practice. I thi ectio we explore aother ue of overamplig which i i combiatio with quatizatio. We will ee that overamplig ad ubequet dicrete-time filterig ad dowamplig permit a icreae i the tepize of the quatizer, or, equivaletly, a reductio i the umber of bit required i the aalog-to-digital coverio. o explore the relatiohip betwee overamplig ad quatizatio tepize, we coider the ytem i Figure Aalog to digital coverio amplig rate coverio cut-off: ω c = π N x c(t) Cotiuou to Dicrete time x[] Qatizer Low pa filter N amplig rate = Figure 10.4: Sub-bad codig of audio igal We aume that the igal badwidth of x c (t) i Ω N, ad hece the miimum amplig rate eeded accordig to amplig theorem i 2Ω N. We aume that we ample at N time the required rate, i.e. ( 2π = 2Ω N ) 1 N. (10.7)

6 148 Chapter 10. hi cotat N i called the overamplig ratio ad i aumed to be a iteger. he aalyi of uch a ytem i o-trivial, though the ituitio i clear. If the quatizatio error i radom, the ome of it effect will be removed through thi procedure. Aother way to udertad the advatage i through decribig the cotiuoutime igal through differece. Whe we did quatizatio, we imply repreeted the dicrete-time igal at a give time itat, i.e. ˆx[] = Q(x[]). herefore, the umber of bit eeded for the ame maximal quatizatio error depeded o the rage of value take by x[], i.e. it dyamic rage. If x[] [ A, A], the for a maximal error of /2, i.e. x[] ˆx[] /2 we eeded 2 b = 2A/ or ( ) 2A b = log 2 bit. herefore, if we are able to reduce the dyamic rage A, we ca reduce the umber of bit b, eeded for the ame maximal error i quatizatio. he baic idea of reducig the dyamic rage lead u to coider a differetial quatizatio, i.e. itead of quatizig x[] we quatize d[] = x[] x[ 1]. Now it i clear that if we ample at a higher rate, the dyamic rage of d[] i maller.