http://vi.wikipedia.org/wiki/L%E1%BA%A5y_m%E1%BA%ABu_%28x%E1%BB %AD_l%C3%BD_t%C3%ADn_hi%E1%BB%87u%29

Ly mu (x l tn hiu)

Ly mu tn hiu. Cc tn hiu lin tc c mu xanh lc cn cc mu ri rc c mu xanh lam. Trong x l tn hiu, ly mu l chuyn i mt tn hiu lin tc thnh mt tn hiu ri rc. Mt v d ph bin l vic chuyn i ca mt sng m thanh (mt tn hiu lin tc) thnh mt chui cc mu (mt tn hiu thi gian ri rc). Mt mu cha mt gi tr hoc tp hp cc gi tr ti mt im trn trc thi gian (v/hoc khng gian).

L thuytLy mu c th c thc hin cho cc tn hiu khc nhau trong h ta khng thi gian, hoc vi cc h ta bt k, v kt qu tng t thu c trong h ta 2 hoc nhiu chiu. i vi cc tn hiu khc nhau theo thi gian, v d l mt tn hiu lin tc c ly mu, v vic ly mu c thc hin bng cch o cc gi tr ca tn hiu lin tc thi im mi giy , T c gi l khong thi gian ly mu. Nh vy, tn hiu sau khi c ly mu c a ra bi: , vi n = 0, 1, 2, 3, ...

L tng y, tn hiu c xc nh chnh xc ti thi im ly mu nT. Trong ton hc, c th c biu din bng php nhn hm tn hiu hm delta Dirac): vi hm rng lc Dirac (h qu ca

Tn hiu sau khi ly mu l sa: - chnh l tp hp cc gi tr ca hm rng lc c nhn vi bin tng ng ca hm s(t)

. Ph tn ca sa l chui Fourier ca tn hiu sa:

. Tn s ly mu hay t l ly mu fs c nh ngha l s lng cc mu thu c trong mt giy, hoc fs=1/T. T l ly mu c o bng hertz hoc s mu/giy. Trong mt s trng hp c th ti to li hon ton v chnh xc tn hiu ban u (ti lp hon ho). nh l ly mu Nyquist-Shannon cung cp iu kin (nhng khng phi lc no cng cn thit), theo c th ti lp hon ho tn hiu. nh l ly mu m bo rng cc tn hiu c tn s gii hn c th c ti to hon ton t phin bn mu ca n, nu t l ly mu ln hn gp i tn s ti a (fs>2fmax). Ti lp trong trng hp ny c th t c bng cch s dng cng thc ni suy Whittaker-Shannon.

Thc t

V d v sng fa(t) (mu ) c thu bng mt mch trch mu v gi. Do khng th to ra mt hm delta Dirac l tng, cho nn tn hiu c phn no nhanh hn so vi thi im ly mu thc t, hay ni cch khc, khi tn hiu bin i nhanh tng i so vi chuyn i ca mch ADC. Trong trng hp ny ngi ta s dng mt 'mch trch mu v gi' (sampling and hold). Hm rng lc Dirac c thay th bng mt xung ch nht vi di xung t0. Vic ly mu c thc hin bi mt mch trch mu v gi gi gi tr ca mt mu khng i trong mt di ca xung hnh ch nht. Trong ton hc, iu tng ng vi mt tch chp vi mt hm rect:

Ph thu c:

. Ph tn ny cha ng mt yu t ca hm sinc. (xem di). iu ny c th sai lch tn hiu phc hi v phi c khc phc bi mt b lc ti thit (reconstruction filter). Tn s bng mt na t l ly mu l mt chn trn ca tn s cao nht tng ng vi cc tn hiu ly mu. Tn s ny (mt na t l ly mu) c gi l tn s Nyquist ca h thng ly mu. C th quan st thy cc tn s ln hn tn s Nyquist fN trong tn hiu ly mu, nhng nhng tn s ny khng r rng. Mt phn tn hiu c tn s f khng th c phn bit vi cc thnh phn khc c tn s NfN+f v NfNf vi N l s nguyn dng khc 0, s khng r rng ny c gi l hin tng chng ph hay rng ca. x l vn ny mt cch cng mn cng tt, hu ht cc tn hiu tng t (analog) c lc vi mt b lc chng rng ca (thng l mt b lc thng thp vi tn s ct ti tn s Nyquist) trc khi chuyn i ly mu.

Cc bin dng trong thc tTrong thc t, cc tn hiu lin tc c ly mu bng cch s dng mt mch chuyn i tn hiu tng t sang tn hiu s (ADC), y l 1 thit b khng l tng. iu ny dn n xut hin mt lch trong qu trnh ti dng li tn hiu so vi l thuyt, gi chung l b mo. Cc loi mo c th xy ra, bao gm:

Rng ca hay hin tng chng ph (aliasing): mt iu kin tin quyt ca nh l ly mu l tn hiu phi c c gii hn bng tn (bandlimited). Tuy nhin, trong thc t, khng c tn hiu no trong thi gian v hn m c bng tn hu hn c. Jitter: l hin tng xung ng b (xung iu khin khong cch thi gian ly mu) b mo dng hoc b sai lch mt cch khng ng u. Nhiu, bao gm c nhiu do nhit hoc nhiu trong mch analog... Li do vn tc tng th (Slew rate), gy ra bi gi tr ng ra ca thit b chuyn i ADC khng th thay i nhanh. Lng t ha nh mt h qu ca chnh xc hu hn cc t i din cho cc gi tr chuyn i. Sai s m ha hay sai s lng t (Quantization): l li xy ra trong qu trnh s ha cc gi tr ca tn hiu mu. S bit m ha cng ln th sai s lng t s cng nh.

ng dngLy mu m thanhm thanh k thut s s dng iu ch m xung (PCM) v cc tn hiu k thut s ti to li m thanh, bao gm chuyn i t tn hiu tng t sang tn hiu s (ADC), chuyn i t tn hiu s sang tn hiu tng t (DAC), lu tr, v truyn. Tnh hu ch chnh ca mt h thng k thut s l kh nng lu tr, truy xut v truyn tn hiu m khng c bt k tn tht no v mt cht lng. Tn s ly mu Tai ngi c th nghe c m thanh tn s trong khong 20Hz-20kHz, theo nh l Nyquist th tn s ly mu ti thiu phi l 40kHz, v vy ngi ta thng ly mu vi tn s 44.1 kHz (CD), 48kHz (pro audio) hay 96kHz. Hin nay, cng ngh ly mu c xu hng s dng tn s ly mu vt qu cc yu cu c bn, v d 96kHz hay thm ch l 192kHz.[1] iu ny tri ngc vi nhng kt qu trong phng th nghim chng minh rng tai ngi khng th nghe c nhng m thanh tn s siu m, tuy vy trong mt vi trng hp, th nhng siu m c kh nng tng tc v iu chnh mt phn ca ph tn m thanh (bin dng do iu bin tng h; intermodulation distortion hay IMD). S bin dng iu bin ny khng c trong m thanh

thc v v vy n l 1 c im c trng ca m thanh nhn to.[2] Mt im li th l nu tn s ly mu cao vt mc th s gim c nhng yu cu ca b lc thng thp trong ADC v DAC. su s (bit depth) m thanh thng c lng t ha vi su s 8, 16 hoc 20 bit, trn mt l thuyt th cc i ca 't s tn hiu trn nhiu lng t' (signal to quantization noise ratio; SQNR) s cho ra dng sng sin chun (pure sine wave) l khong 49.93dB, 98.09dB v 122.17dB. [3] m thanh 8-bit ni chung l khng cn c s dng na do nhiu lng t (cc i ca t s SQNR thp), mc d nhng bin m 8-bit theo thut ton lut A v lut ng phn gii thnh 8 bit nhng li lm tng h s mo phi tuyn (total harmonic distortion; THD). Cht lng m thanh ca CD c m ha vi 16 bit. Nhiu nhit (thermal noise) lm hn ch s lng cc bit thc s c th c s dng trong qu trnh lng t ha. Vi h thng analog (tng t) c t s tn hiu trn tp m (SNR hay S/N) vt qu 120dB, do s cn hn 20 bit lng t ha.

Hm sincTrong ton hc, hm sinc, k hiu l sinc(x) hoc i khi l Sa(x), l mt hm gii tch c hai nh ngha tng ng tng t nhau.[1] Trong x l tn hiu s, hm s sinc chun c nh ngha l

. N c xem l nh ngha chun v tt c cc gi tr ca hm ti cc im x nguyn l bng nhau. Trong ton hc, khi nim hm sinc khng chun cn c gi l si(x) c nh ngha l

. im khc nhau duy nht gia hai nh ngha l s nhn gi tr ca bin vi tha s . Trong c hai trng hp, gi tr ca hm s ti im k d 0 c hiu l c gi tr gii hn bng 1. .

Khi nim "sinc" l s vit tt ca tn y hm ny trong ting Latin, sinus cardinalisc Phillip M. Woodward a ra ln u tin vo nm 1953.[2][3][4]

Giao vi trc honh

Cc i v cc tiu ca hm si(x) khng chun ng vi cc gi tr x ti giao im ca hm si(x)v cos(x). th hm sinc mu , th hm cosin mu xanh. Hm sinc khng chun ct trc honh ti nhng im l bi s ca k, vi k l s nguyn khc 0, v t cc tr ti cc im giao vi hm s cosin, tc l nhng im x= v sin()/ = cos(), lc th o hm ca [sin(x)/x]' = 0.

vi vi

Bin iHm sinc chun c th bin i nh sau:

V cng c th bin i theo hm Gamma chn:

bng cng thc Euler p dng cho hm

Cng theo cng thc Euler:

Chui Taylor:

Bin i FourierBin i Fourier ca mt hm sinc (tn s thng) l mt hm rect(f):

Vi hm rect c nh ngha nh sau:

. Bin i Fourier ca hm rect trn cng thu c mt hm sinc:

. ng dng thc t cng thc ny to thnh b lc sinc nh cc b lc thng thp hay brick-wall. Trng hp c bit trong bin i Fourier ny:

l mt tch phn suy rng v khng phi l mt tch phn Lebesgue hi t, nh:

Nhng tnh cht ca hm sinc chun c ng dng trong vic ti lp cc mu ca nhng hm c gii hn bng thng:

L mt hm ni suy, tc l sinc(0) = 1, v sinc(k) = 0 vi k l s t nhin khc 0. Hm xk(t) = sinc(tk) (k l s t nhin) hnh thnh mt h c s trc chun (orthonormal basic) ca cc hm c bng thng gii hn trong khng gian hm Lp: L2(R) vi tn s gc cao nht H= (c ngha l tn s chu k cao nht H=1/2).

o hmo hm bc n ca hm:

, vi x 0. C dng:

T suy ra:

Cc trGi tr gn ng ca honh x ti cc tr th n vi n 1 c th tnh bng cng thc:

Vi gi tr n l tng ng vi im cc tiu, n chn tng ng vi im cc i. th hm s i xng qua trc tung v t cc i ln nht ti v tr 0 = (0,1). Cc tr ca hm si(x) = sin(x)/x Cc i Cc tiu 0 4,4934095 1 0,219284 7,7252518 2 0,12873

10,904122 3 0,091452 14,066194 4 0,070973 17,220755 5 0,057989 20,371303 6 0,049049 23,519452 7 0,042493 26,666054 8 0,042998 29,811599 9 0,033531 32,956389 10 0,030334 36,100622 11 0,0276935 39,244432 12 0,025476 (2n+) ((2n+))1 (2n) ((2n))1