Transcription of A Very Basic Introduction To Time/Frequency …
1 AVeryBasicIntroductionToTime/FrequencyDo mainsParticleMarch 10,2004 AbstractA very briefintroductionto waves, terminology, time /frequencydomains,witha bit of mentionof IntroductionInthecontextof communications,a signalis basicallysomeinformationsomehow encodedas a wave. Everythingtravelsas a wave, so let'slookatsomeof thesimpleideasof presents a thevolume,or air-pressure,or strength,or energy,etc.,dependingonwhatthiswave example,if thisis anoceanwave,thentheverticalaxismay be thewave height. If thisis a soundwave in theair,thenit is it is thesoundwave in a wire,thenit' (heredepictedin termsof ) oftenrepresents thetimeordistance(orspace) thatthewave generallyhave a period, which canbe viewed as thetime/distanceof whenthewave repeatsitself| Figure1 theperiod is 2 .A closelyrelatedproperty of theperiod is thefrequency. Frequencyisthespeedat which thewave cyclesitsperiods,andis always expressedinproper speedunits,as in `cyclesper second'(commonlyknownas Hz),and`kilo-cyclesper second'(commonlyknownas kHz) `kilo'standsfor 1000,so if somethingis 8kHz,thenit's 1 =2 3 =22 5 =23 time /DistancePeriodAmplitudeFigure1: 1 : 4 HzSinWave; sin(2 4t)Figure2 illustratesaSinwave at , andis expressedinseconds.
2 Youcancount thenumber of timesthewave cycles,andyou'llseethatit's4, , youcaneasily ndthewave example,if thewave cyclesitself4 timesper second,thentheperiod is14seconds,orsimply1=F requency. Similarly, knowingtheperiod, youcan thewavelength, which is basicallythedistanceoccupiedby thewave period. Given thespeedof propagation(lightin a vacuum,orsoundin theair,etc.),we ,light/electricity travel around3 108metersper 'simaginethatFigure2 represents electricity in a wire,thenthewavelengthis just3 108 4 or just7:5 107meters,or simplyvelocity f ndvelocity if youknow 1 : 12 HzSinWave;13 sin(2 12t)Continuingonwiththediscussion,Figure 3 showsa wave of 12Hz, theperiod gotsmalleras FrequencyDomainFigure4 illustratestheideaof takingtwo waves thiscase,we've addedwaves fromFigures2 ways of viewingany type of a wave; in thetimedomain,or in Figure4 1 : 4Hz+ : FrequencyDomainof 4Hz+ ApplicationsIt turnsoutthatviewingwaves in frequencydomainis usuallya lotmoreusefulthanviewingthemin DiscreteFourierTransformHn=N 1Xk=0hke2 ikn=N6 InverseDiscreteFourierTransformhk=1NN 1Xn=0 Hne 2 ikn=N7 2D DiscreteCosineTransformJPEG,etc:DCT(i; j) =1p2NC(i)C(j)N 1Xx=0N 1Xy=0P ixel(x.)
3 Y)COS"(2x+ 1)i 2N#COS"(2y+ 1)j 2N#WhereC(x) =1p2ifxis 0, else1 ifx > 2D InverseDiscreteCosineTransformP ixel(x; y) =1p2NN 1Xi=0N 1Xj=0C(i)C(j)DCT(i; j)COS"(2x+ 1)i 2N#COS"(2y+ 1)j 2N#WhereC(x) =1p2ifxis 0, else1 ifx > ConclusionIn class:How
