EEL5: Discree-Time Signals and Sysems Lecure #6: Coninuous-Time Signals Lecure #6: Coninuous-Time Signals. Inroducion In his lecure, we discussed he ollowing opics:. Mahemaical represenaion and ransormaions o coninuous-ime signals.. Some imporan coninuous-ime uncions. s we have seen so ar, signals are usually one-dimensional uncions o ime. We can broadly classiy hese signals ino wo caegories:. Coninuous-ime signals, denoed as x (), where usually denoes ime as he independen variable, and. Discree-ime signals, denoed as xn [, where n is an ineger and denoes he ime index. Signals in he real world are usually coninuous-ime. Discree-ime signals, on he oher hand, are convenien o represen and operae on in digial compuer sysems; oenimes, discree-ime signals are sampled versions o coninuous-ime signals. Because o he compuer revoluion o he pas several decades, he sudy and undersanding o discree-ime signals and sysems has gone up dramaically wih he rise o ever-more powerul compuers and DSP (digial signal processing) sysems. We have already seen (and, in some cases, heard) many examples o signals, boh coninuous-ime and discreeime. In oday s lecure, we begin our sudy o signals rom a more mahemaical poin o view.. Coninuous-ime signals. Signal ransormaions One o he mos imporan basic skills we require in our sudy o signals is he abiliy o undersand basic ransormaions wih respec o he independen variable. In his secion, we will examine ransormaions on coninuous-ime signals; laer, we will do he same or discree-ime signals. Le x () denoe a coninuous uncion o ime. In his class i will requenly be imporan o know how he uncion x changes when we change is argumen. The able below gives he qualiaive eec o some simple changes in argumen. x( ) uncion Relecion xa ( ), a > Compression x ( a), a > Sreching. We have seen one excepion namely, images which are wo-dimensional signals, where he independen variables are he horizonal and verical locaion o each pixel ( xy, ) and he dependen variable is each pixel s color (inensiy or gray-scale images). eec x ( a), a > Shi o he righ along he horizonal axis x ( + a), a > Shi o he le along he horizonal axis a x(), a > Magniicaion a x(), a < Reducion x () + a, a > Shi up along he verical axis x () a, a > Shi down along he verical axis - -
EEL5: Discree-Time Signals and Sysems Lecure #6: Coninuous-Time Signals Figures and illusrae some o hese on wo simple coninuous uncions. I is very imporan ha you undersand each o hese illusraions, and are able o perorm hem yoursel wihou he aid o a compuer or calculaor. Compound ransormaions ha perorm boh scaling and le/righ shiing are a lile rickier han each one by isel. Consider x () in Figure and he compound ransormaion x( ). To undersand wha his uncion looks like, we irs change i o: x( ) x -- () In his orm, we see ha we irs scale he uncion (in his case, compress i), and hen shi he scaled uncion by ime unis (no ime unis) o he righ; his ransormaion is illusraed in he boom righ corner o Figure. Figure (boom righ corner) illusraes anoher compound-ransormaion example: x( + 5) x[ ( 5) () gain, we see ha by acoring he scaling inormaion (in his case a relecion), he uncion is irs releced abou he y -axis, and is hen shied 5 ime unis o he righ (no o he le). B. Some useul coninuous-ime signals In his secion, we inroduce some very useul coninuous-ime signals. The irs o hese is he impulse or Dirac dela uncion δ(). I is deined as ollows: δ(), () δ () d () Noe ha he value o δ( ) is deined only implicily by equaion (). We can view he Dirac dela uncion as he limiing case o a square pulse p (), as illusraed in Figure below. p () ------ Figure ------ Noe ha no maer wha he value o, he inegral o p () will always be equal o one: p () d (5) For larger values o, p () becomes more narrow and aller, bu equaion (5) sill holds. Thereore, we can view δ() as he ollowing limi: lim p () δ() (6) One o he imporan properies o he δ() uncion is is siing or sampling propery. For any coninuousime uncion x (), - -
EEL5: Discree-Time Signals and Sysems Lecure #6: Coninuous-Time Signals x () x( ) - - - - x( ) x -- - - - - x ( ) x ( + ) - - - - x () x ( ) x( ( ) ) - - Figure - - - -
EEL5: Discree-Time Signals and Sysems Lecure #6: Coninuous-Time Signals. x () x( )... -. - -5 5 -. - -5 5. x( ). x --.. -. - -5 5 -. - -5 5 x ( 5) x ( + 5).... -. - -5 5 -. - -5 5. ( )x () x( + 5) x( ( 5) )... -. -. - -5 5 - -5 5 Figure - -
EEL5: Discree-Time Signals and Sysems Lecure #6: Coninuous-Time Signals x ()δ ( ) d x ( ) since he ime-shied dela uncion δ( ) is zero everywhere excep or. Le s look a a couple o examples: δ( ) d ( x () some consan) (8) (7) x ()δ ( ) d x( ) x ( )δ() d x( ) (9) () x ( )δ ( + ) d x( 5) One place where he dela uncion comes up is in he coninuous-ime Fourier ransorm represenaion o a sinusoid. For he sinusoid, x () cos( π o ), () he magniude specrum represenaion X () is given by, () X () -- [ δ( + ) + δ( ) as illusraed in Figure below. () x () cos( π o ) X () T --δ ( + ) --δ ( ) Figure noher coninuous-ime uncion o signiicance in our mahemaical represenaion o signals is he uni sep uncion u (), u () < () ploed in Figure 5. The uni sep uncion allows us o mahemaically represen signals ha are piece wise coninuous wih disconinuiies a a inie number o poins. Consider or example, he hree uncions ploed in Figure 6. Each o hese would be impossible o describe uncionally wihou he aid o he uni sep uncion; wih u (), however, each uncion has a sraighorward represenaion: x () [ u ( + ) u ( ) x () ( + ) [ u ( + ) u () + ( + ) [ u () u ( ) x () sin( π) [ u () u ( ) (5) (6) (7) - 5 -
EEL5: Discree-Time Signals and Sysems Lecure #6: Coninuous-Time Signals u () Figure 5: The uni-sep uncion 8 6 x () x ().5.5 -.5 - - -.5.5 -.5 - - - - x () - - - Figure 6 Finally, we look a sinusoidal uncions and heir mahemaical represenaion. In our previous lecures, we have already seen and heard many dieren sinusoids and hese uncions are perhaps he mos imporan in our undersanding o signals and sysem analysis. While we could use boh cosines or sines in our mahemaical represenaion, we will primarily use he cosine uncion by convenion rom here on ou. Consider x () deined by, x () cos( π o + α) and ploed in Figure 7 below. For equaion (8), we deine he ollowing quaniies: (8) o ampliude, α phase (rad), (9) cyclic requency (/sec or Herz (Hz) unis), () ω o π o radian requency (rad/sec unis), () T o o period (unis o seconds). () - 6 -
EEL5: Discree-Time Signals and Sysems Lecure #6: Coninuous-Time Signals cosα x () cos( π o + α) T Figure 7 sinusoidal wave is periodic; ha is, x () x ( T o ),. () In ac, he sinusoid is perhaps he mos imporan periodic waveorm ha we will sudy, since almos all oher periodic waveorms can be consruced as he ininie sum o sinusoids; or any periodic signal x (), we can wrie, x () a + a k cos( π o k) + b k sin( π o k) () k Equaion () is known as he Fourier series represenaion o he periodic signal x (). We will look a how o deermine he coeiciens a k and b k a lile laer. - 7 -