11.3 Sa-mpling 269 'I'herefore for the integral, (11.30) In other words: the bi-Lsis functions of the Fourier series this into ( I 1.29) viclds Me orthogonal Putting k By cortiparing (11.28) ancl (11.31), we find the coxincction between the Fourier coefficients of y(t), -c1 (t) and r z j t ) Ck = AkBkT (11.32) Ideal S a r r i ~ l i n ~ Now that we are comfortable with the spectra of periodic signals and the delta inipulse train we c'a~iuse' it to tl ribc the sariiphg of continuorrs signals WP will first dertl with ided sarripling, where we ta,ke the precise values of the voritiiiuoiw fiirictioii at saniplc points This idealisation allows us t o ideiitify the fuiidanient,al principles particdarly siniply I ,atcr we will see that the con( for realisation can also be described by sirriplr cxteiisions ot the concept of ideal sampling To rnodel idcal sampling we start with a contiiiuous signal it( C) arid nuiltiply itz with an impulse train &LLL(-$), as shown in Figure 11.6 The result ~ ( tis)again il, series of impulses The weight*of thc individual impulses arc' the values of t h y signal at points t = k T , Ic E Z In Figure 11.6, the wcights are syiribolisrd hy the lerigt,hs of the arrowb For t3hespectrum X(3w) oi: the hampled signal -I(() we IW the multiplication theorem (9.75) t u obtain I t ( t ) = (t) - U T x(JW) * -dy(.JW) 2n - (;:j (11.33) Comvolulion of tl-tc spectr urn (p) of thc contiiiiious signal with tlie inpnlse train in the frequency-dorimin L L L ( ~ T / ~gives T ) a periodic coritiiirration of (p) at rnultiplw of 'Ln/T,as shown in h'ig 11.7 Tlie freyiienry w', = is called the sarnplzng ,jrcquency T h c spectrim of thc samplccl sigridl iii the f~rc~iiency-doniain is clearly periodic with period w, a i d there is a diiality her( with the Fourici- transform of periodic signals, ill bv tlic following svhenme %