SAMPLING Sampling i the acquiition of a continuou ignal at dicrete time interval and i a fundamental concept in real-time ignal proceing. he actual ampling operation can alo be defined by the figure belo and a can be een i a MODULAION proce. he carrier ignal δ (t) i a train of impule. x(t) x (t) δ () t = δ ( t n ) n= he output of the modulator i x (t) and i given by: x () t = x() t δ () t = x() t δ ( t n ) n= If e recall that convolution by a hifted impule i the ignal hifted by the ame amount then () t = x( n ) ( t n ) n= x δ hi relation decribe the ampling proce in the time-domain. he ampling frequency F =.0/ mut be elected large enough uch that the ampling proce ill not reult in any lo of pectral information (no-aliaing). We ill invetigate the ampling operation further by taking the Fourier ranform of x () t. he Fourier ranform of the impule train: F { δ () t } = F δ ( t n ) = δ ( n ) n= π n=
π here = = πf here f i the ampling frequency in Hz. Alo e recall that multiplication in time i convolution in frequency domain. X π ( ) = X ( ) F{ δ ( t) } = X ( ) δ ( n ) π π n= but e already kno that X ( ) δ ( ) = X ( ) X ( ) = X ( ) n n= () Firt let x(t) be band limited in frequency uch that X() = 0 for > M. hi e can ho a in the diagram belo: X() - M M Next aume that the highet frequency in X() i le than one-half the ampling frequency: M < From equation () e note that the effect of the impule ampling of x(t) i to replicate the frequency pectrum of X() about the frequencie n = ±, ±, ± 3,.a hon belo:
X () / - - 0 Fig. 3. For thi cae e can recover the ignal x(t) exactly from it ample x[n] uing an ideal lo-pa filter. We call the recovery of a ignal from it ample DAA RECONSRUCION. No aume the cae hen the highet frequency in x(t) i greater than /. he ne plot ill be a hon belo; / - - 0 Fig. 3. For the cae in Fig. 3. x(t) can be recovered exactly from x (t) by mean of a lopa filter ith gain and a cutoff frequency greater than M and le than ( M ). Belo i the block diagram hoing the ue of the lopa filter:
δ () t x(t) x (t) h(t) H(j) x r (t) If e aume that the pectrum of x(t) i a hon belo X() - M M and aume that the ampling frequency > M then the pectrum of the ampled ignal ill be a hon belo: X (j) / - - - M 0 M No if e elect the cutoff frequency of the lopa filter to be beteen M and ( M ) the characteritic curve for the filter ill be a hon belo:
H(j) - c c he pectrum of the recontructed ignal x r (t) ill then be a follo: X r (j) - M M Note that in Fig. 3. frequency component overlap in the ampled ignal and x(t) can not be recovered by lopa filtering. A requirement for ampling i the folloing: Sampling frequency mut be at leat tice a great a the highet frequency M in the ignal that i ampled.
No let u conider practical ampling. A phyical ignal cannot be bandlimited in frequency. Hoever a phyical ignal doe exit uch that the amplitude of the frequency pectrum above a certain frequency i o mall that it i negligible. We ee that an other requirement for ampling i that: Frequency pectrum of the ignal be inignificant above the frequency here = πf and f i the ampling frequency. f Frequency i called the NYQUIS frequency. hi i alo knon a Shannon Sampling heorem. Sampling With a Zero Order Hold. Sampling theorem a explained in term of impule-train ampling. Hoever in practice, narro large-amplitude pule hich approximate impule are relatively difficult to generate and tranmit. It i often more convenient to generate the ampled ignal in a form knon a a zeroorder hold. hi ytem ample x(t) at a given time intant and hold that value until the next intant.
x(t) Zero Order Hold x o (t) he recontruction of x(t) from the output of a zero-order hold can be achieved by lopa filtering. Hoever the required filter no longer ha contant gain in the paband. Let u try to develop thi filter characteritic curve. We note that x o (t) can be generated by impule train ampling folloed by an LI ytem ith a rectangular impule repone a hon belo: p () t x(t) x p (t) h 0 (t) x o (t) 0 x(t) x p (t) x o (t) o recontruct x(t) from x o (t) e conider proceing x o (t) ith an LI ytem ith impule repone x r (t).
p () t H(j) x(t) x p (t) h 0 (t) 0 x o (t) h r (t) H r (j) r(t) We need to pecify H r (j) o that r(t) = x(t). We note that thi ill only be true if the cacade combination of h 0 (t) and h r (t) i the ideal lopa filter H(j). Or imilarly H(j) = H 0 (j).h r (j) But e kno that the pectrum of h 0 (t) i: H 0 j = in / ( j) e hence it i required that H r ( j) j e H ( j) = () in( / ) for example ith cutoff frequency of / the ideal magnitude and phae for the recontruction filter folloing a zero-order hold i a belo:
H r (j) π/ H r (j) -π/ We note that the actual realization of equation () i not poible ithout an approximation. In fact in many ituation the output of the zero-order hold i conidered an adequate approximation to the original ignal by itelf. Alternatively in ome application e may ih to perform a moother interpolation beteen ample value. o important form of data recontruction exit:. Interpolation (uitable for data tranmiion ytem). extrapolation (uitable for feedback ytem) Extrapolation i interpolation extended to point outide the convex hull of a dataet. Interpolation value at a point outide the convex hull of an input dataet i referred to a an extrapolated value.
Interpolation I the fitting of a continuou ignal to a et of ample value. One imple interpolation procedure i the zero order hold previouly dicued. An other ueful form of interpolation i linear interpolation here adjacent point are connected by a traight line a een belo: We have een before that if the ampling intant are ufficiently cloe, then the ignal i recontructed exactly. i.e. through the ue of a lopa filter exact interpolation can be carried out beteen the ample point. Conider in the time-domain the effect of the lopa filter x ( t) = x r or x ( t) = r p n= ( t) h( t) x( n ) h( t n ) For the ideal lopa filter H(j), h(t) i a follo c in( ct) h( t) = π t c
Interpolation Uing Matlab YI = INERP( X,Y,XI) hi matlab command ill find YI uing value of the underlying function Y. It carrie out interpolation for the point in vector XI. Vector X pecifie the point at hich the data Y i given Interpolation i the ame operation a table lookup. Decribed in table look up term INERP look up the element of XI in X and baed upon their location return value YI interpolated ithin the element of Y. Uage: YI = INERP(X,Y,XI, method ); nearet : nearet neighbor interpolation linear : linear interpolation pline : cubic pline interpolation cubic : cubic interpolation Example: x = 0:0; y= in(x); xi = 0:0.5:0; yi = interp(x,y,xi); plot(x,y, o,xi,yi, * ) x = >> y y = 0 3 4 5 6 7 8 9 0 Column through 0 0 0.845 0.9093 0.4-0.7568-0.9589-0.794 0.6570 0.9894 0.4 Column -0.5440
>> xi xi = Column through 0 0 0.500 0.5000 0.7500.0000.500.5000.7500.0000.500 Column through 0.5000.7500 3.0000 3.500 3.5000 3.7500 4.0000 4.500 4.5000 4.7500 Column through 30 5.0000 5.500 5.5000 5.7500 6.0000 6.500 6.5000 6.7500 7.0000 7.500 Column 3 through 40 7.5000 7.7500 8.0000 8.500 8.5000 8.7500 9.0000 9.500 9.5000 9.7500 Column 4 0.0000 >> yi yi = Column through 0 0 0.04 0.407 0.63 0.845 0.8584 0.8754 0.893 0.9093 0.773 Column through 0 0.55 0.333 0.4-0.0834-0.3078-0.533-0.7568-0.8073-0.8579-0.9084 Column through 30-0.9589-0.7890-0.69-0.4493-0.794-0.0453 0.888 0.49 0.6570 0.740 Column 3 through 40 0.83 0.9063 0.9894 0.8450 0.7007 0.5564 0.4 0.73-0.0660-0.3050 Column 4-0.5440 >>