Signals and the frequency domain ENGR 40M lecture notes July 31, 2017 Chuan-Zheng Lee, Stanford University

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1 Signals and he requency domain ENGR 40M lecure noes July 3, 07 Chuan-Zheng Lee, Sanord Universiy signal is a uncion, in he mahemaical sense, normally a uncion o ime. We oen reer o uncions as signals o convey ha he uncion represens some sor o phenomenon or example, an audio signal, he elecromagneic signal broadcas in FM radio, a currency exchange rae, or he volage somewhere in a circui. One o he grea advances o he 9h cenury was couresy o a French mahemaician called Joseph Fourier, who showed in his work abou hea low ha represening a signal as a sum o sinusoids opens he door o more powerul analyical ools. This idea gave rise o wha is now known as he requency domain, where we hink o signals as a uncion o requency, as opposed o a uncion o ime. Our goal oday is o deine, in some sense, wha ha means. Preliminaries Periodic signals. leas o begin, we ll mainly be concerned wih signals ha are periodic. Inormally, a periodic signal is one ha repeas, over and over, orever. To be more precise: signal x() is said o be periodic i here exiss some number T such ha, or all, x() = x(+t ). The number T is known as he period o he signal. The smalles T saisying x() = x( + T ) is known as he undamenal period. Sinusoids. Precisely wha do we mean by a sinusoid? The erm sinusoid means a sine wave, bu we don jus mean he sandard sin(). To enable our analysis, we wan o be able o work sine waves o dieren heighs, widhs and phases. So, o us, a single sinusoid means a uncion o he orm x() = sin(π + φ), or some (is ampliude), (is requency) and φ (is phase). x() / φ π The Fourier series Joseph Fourier s bold idea was o express periodic signals as a sum o sinusoids. Theorem. I x() is a well-behaved periodic signal wih period T, hen i can be wrien in he orm x() = 0 + n sin(πn + φ n ). () n= where = /T, and or some 0,,,... (which we call magniudes) and φ, φ,... (which we call phases). We call his sum he Fourier series o x().

2 Tha well-behaved cavea is worh expanding on, briely. In engineering, every pracical signal we ever deal wih is well-behaved and has a Fourier series. I you alk o a mahemaician, hey ll say ha we re sweeping a lo o deails under he carpe. They re o course righ, bu in real-world applicaions his ends no o boher us. I s worh relecing on why Fourier s claim was so signiican. I s no oo hard o believe ha some periodic signals can be represened by a sum o sinusoids. Bu included in he well-behaved caegory are signals wih sudden jumps (disconinuiies) in hem. Can a disconinuous signal really be expressed as a sum o smooh sinusoids? Prey much, yes. The cach is ha you migh need an ininie number o sinusoids. Time-domain and requency-domain represenaions noher ac relaing o Fourier series is ha he magniudes 0,,,... and phases φ, φ,... in () uniquely deermine x(). Tha is, i we can ind he 0,,,... and φ, φ,... corresponding o a periodic signal x(), hen, in eec, we have anoher way o describing x(). When we represen a periodic signal using he magniudes and phases in is Fourier series, we call ha he requency-domain represenaion o he signal. We oen plo he magniudes in he Fourier series using a sem graph, labeling he requency axis by requency. In his sense, his represenaion is a uncion o requency. To emphasize he equivalence beween he wo, we call plain old x() he ime-domain represenaion, since i s a uncion o ime. For example, here are boh represenaions o a square wave: Time-domain represenaion x() 0.8 Frequency-domain represenaion Here s some more erminology: ˆ In elecrical engineering, we call he erm 0 he DC componen, DC ose or simply ose. DC sands or direc curren, in conras o he sinusoids, which alernae, hough we use his erm even i he signal isn a curren. In a sense, he DC componen is like he zero requency componen, since cos(π 0 ) =. We oen hink o ose in his way, and plo he DC ose a = 0 in he requency-domain represenaion. The DC componen is oen easy o eyeball i s equal o he average value o he signal over a period. For example, in he signal above, he DC ose is. ˆ The sinusoidal erms are oen called harmonics, a erm borrowed rom music. The harmonics will have requencies,, 3, 4 and so on. ˆ We also call each harmonic, n sin(πn + φ n ), he requency componen o x() a requency n. For example, i = 0 Hz, we call he harmonic or which n = 3 he 30 Hz componen, relecing ha in his case sin(πn + φ n ) is a sinusoid o requency 30 Hz. I s also common o plo he phases on anoher graph, bu we won in his course.

3 Geing o know he requency domain Wha happens o our humble square wave i we don pick up on all o he ininiely many requencies? Say, i everyhing above 9 Hz disappeared? Time-domain represenaion x() Frequency-domain represenaion We sill have an approximaion o he square wave: i s go he general shape righ, i s jus no working so well a he corners. In ac, as you migh have expeced, ha disconinuiy gives he Fourier series a sum o coninuous uncions a bi o a hard ime. Noneheless, he more requencies we include, he closer o he rue square wave we ge. Here s an aphorism ha encapsulaes his observaion: I akes high requencies o make jump disconinuiies. More generally, he ollowing saemens provide some inuiion or how o hink in he requency domain. ˆ High requencies come in where he signal changes rapidly. he exreme, when a signal changes suddenly, ininiely high requencies come ino play. ˆ Low requencies come in where he signal changes slowly. he exreme, when i doesn change a all, we have he DC componen (zero requency). Now, you migh be hinking: This is all very well or periodic signals, bu in real lie, no signal is ruly periodic, no leas because no signal lass orever. Then wha do we do? Fear no a lo o he same insincs rom periodic signals also apply o signals ha are roughly repeiive in he shor erm, including audio signals and your hearbea. Circuis wih capaciors and inducors The ollowing is a perhaps surprising ac abou circuis involving capaciors and inducors. ny circui wih only sources, resisors, capaciors and inducors acs on requencies individually. Tha is, he circui akes each requency componen, muliplies each componen by a gain speciic o ha requency, and hen oupus he sum o he new requency componens. The requency domain hus gives us a richer way o undersand how hese circuis work. We ll sudy his in he coming lecures. In ac, here is a close cousin o he Fourier series, known as he Fourier ransorm, ha deals wih non-periodic signals much more rigorously. 3

4 Compuing Fourier coeiciens (no examinable) You migh be wondering: I s all very nice ha hese magniudes and phases exis, bu can we compue hem? We can, bu i urns ou ha he orm in () is raher awkward o do hese calculaions wih. noher way o wrie a Fourier series is x() = a 0 + [a n cos(πn) + b n sin(πn)]. () n= Showing ha () and () are equivalen is le as an exercise. The coeiciens can hen be ound using he ollowing inegrals, where T = /: a n = T T 0 x() cos(πn) d, b n = T T 0 x() sin(πn ) d. Then a 0, a, a,... and b, b,... can be convered o magniudes and phases o i he orm o (). We will no ask you o apply hese ormulae in his course. 4

5 Exercises Exercise. Which o hese could be he ime-domain represenaion o he signal whose requency-domain represenaion is below? Frequency-domain represenaion Candidae ime-domain represenaion x().5 Candidae ime-domain represenaion x().5 Candidae ime-domain represenaion 3 x() 3 4 Candidae ime-domain represenaion 4 x().5 Candidae ime-domain represenaion 5 x().5 Candidae ime-domain represenaion 6 x().5 5

6 Exercise. Which do you hink is he correc requency-domain represenaion o he signal whose imedomain represenaion is below? Time-domain represenaion x() 4 6 Candidae requency-domain represenaion Candidae requency-domain represenaion Candidae requency-domain represenaion Candidae requency-domain represenaion