11 Sampling and Periodic Signals 264 The asLonishingly simple result of this can also be expresscd as follows: ih.e Founer transform of a delta m p u b c : tram tn the teme d o m m i s a delta zmpulse trim the fr~qzicncW-doma2rr: (11.9) In order to keep thc derivation as simple as possible, we defiiied an impulse train with an interval of (11.1) To describe sampling procctlures with an arbitrary sampling interval T we use the delta impulse train (11.10) We also dcacribe these with the sha-symbol from (11.1),where the scaling property (8.19) is important (1 1.11) With thc siinilarity theorem we can derive from (11.9) the general relation (11.12) urier ~ a n ~ f o r ~eriodic e d Signals 7'hc connection (11.12) between clclta impulse tiains in the time-domain arid frequency-domain is v t ~ yclegarii and can greatly shorten otherwise cornpiicated calculation^ Its use, howevri rcquires some practice, so bcforc we t.niplc)y i t to cled with sampling continuous fiinctions we will try it out on sonie classical problems in particular, repr nting periodic sigiials in the time- arid frequencydomain Periodic sigimls have a lirir spectrum where the di5t itnce bt>tween tlit lines is given by the periods in the timc~-tloinain The weighting or individual lines can bc clctcrrnined using Fourirr scries We will 11ow use the delta impulse train to derive these First we cwiisider a periodic time signal s ( t ) with peiiod T It can be represented as convolutioii of a function T o ( t ) with a drlta, impulse train (see Figlire I 31 (11.13) T h e separation T of thc inipidsos is sct so that t,here ;we no gaps between repeti- tions of t h function 11'0(1)