Wiley signals and systems e book TLFe BO 301

286 11 Sampling and Pcriodic Signals Sanipling iii t,he frequency-domain is again idealized in (11.57) as mult,i~~licati(j~i wit,h a delta impulse train This corresponds to convoliitioii (11.58), that causes periodic continuation of $(t) in the time-domain If the signal G(t) is limited in time and its duration is less than T , sampling in the frequency-domain does riot lead to overlapping in the time-domain 'rhen y ( t ) ca.n be recovered from y ( t ) by multiplication with a time window of leng%liT In the frequency-domain, t,his correspoiids bo a11 interpolation (convolution) with the spectrum si (uT/2), that transforms the line spectrum into the smooth spectrum belonging to S i t ) If overlapping occurs in the time-domain, an error-free reconstruction is impossible In this context we use the term tzm~-domnza&using Saarrpling in the frequency-domain is always carried out if values of a spectrum are t o be stored in t,he memory of a cornpiiter, where continuous functions cannot exist, oiily a series of numbers The necessary restvict,iorito time-limited functions is, for example, achieved by splitting the signel into sections with finite duration The relationship bctween a series of finite length and their spectra is represented by the discrete E'ourier tmrisfo~m,(DFT) xercises xereise 11.1 Write the sketched signal with a sum of delta impulses as well a? with the _UL symbol -5 10 t Exercise 11.2 Determine the Foiirier transform X ( j w ) of ~ ( k )= UL(at) and sketch ~ ( tand ) X (jw)for a = a == and U = Note: use tmnsform pair (11.12) i, Exercise 11.3 Coiisider a delta impulse train shifted by to: