15 CONTINUOUS TIME FOURIER SERIES CHAPTER 3.3-3.8
16 CTFS TRANSFORM PAIR ο‘ Suppose π¦(π’) can be expressed as a linear combination of harmonic complex exponentials β π π π πππ 0 π’ synthesis equation ο‘ π¦ π’ = Ο π=ββ ο‘ Then the FS coefficients {π π } can be found as ο‘ π π = 1 π π¦(π’) π βπππ 0 π’ ππ’ analysis equation π β« ο‘ π 0 - fundamental frequency ο‘ π = 2π/π 0 - fundamental period ο‘ π π known as FS coefficients or spectral coefficients
17 CTFS PROOF ο‘ While we can prove this, it is not well suited for slides. ο‘ See additional handout for details ο‘ Key observation from proof: Complex exponentials are orthogonal
18 VECTOR SPACE OF PERIODIC SIGNALS All signals Periodic signals, π 0
19 VECTOR SPACE OF PERIODIC SIGNALS Periodic signals, π 0 ο‘ Each of the harmonic exponentials are orthogonal to π πππ 0 π’ each other and span the space π¦(π’) π π of periodic signals π π2π 0 π’ ο‘ The projection of π¦(π’) onto a particular harmonic ( π π ) gives π 2 the contribution of that π ππ 0 π’ complex exponential to π 1 building π¦ π’ π 0 π π0π’ = 1 π β1 ο‘ π π is how much of each harmonic is required to construct the π π(βπ 0 )π’ periodic signal π¦(π’)
20 HARMONICS ο‘ π = Β±1 β fundamental component (first harmonic) ο‘ Frequency π 0 , period π = 2π/π 0 ο‘ π = Β±2 β second harmonic ο‘ Frequency π 2 = 2π 0 , period π 2 = π/2 (half period) ο‘ β¦ ο‘ π = Β±π β Nth harmonic ο‘ Frequency π π = ππ 0 , period π π = π/π (1/N period) 1 π π¦ π’ ππ’ , DC, constant component, average ο‘ π = 0 β π 0 = π β« over a single period
21 HOW TO FIND FS REPRESENTATION ο‘ Will use important examples to demonstrate common techniques ο‘ Sinusoidal signals β Eulerβs relationship ο‘ Direct FS integral evaluation ο‘ FS properties table and transform pairs
22 SINUSOIDAL SIGNAL 1 1 1 4 π π2π 0 π’ + π βπ2π 0 π’ + 2π π π3π 0 π’ β π βπ3π 0 π’ ο‘ π¦ π’ = 1 + ο‘ π¦ π’ = 1 + 2 cos 2ππ’ + sin 3ππ’ Read off coeff. directly ο‘ ο‘ First find the period ο‘ π 0 = 1 Constant 1 has arbitrary period ο‘ ο‘ π 1 = π β1 = 0 cos 2ππ’ has period π ο‘ 1 = 1 ο‘ π 2 = π β2 = 1/4 π 3 = 1/2π , π β3 = β1/2π sin 3ππ’ has period π 2 = 2/3 ο‘ ο‘ π π = 0 , else ο‘ ο‘ π = 2, π 0 = 2π/π = π ο‘ Rewrite π¦ π’ using Eulerβs and read off π π coefficients by inspection
23 PERIODIC RECTANGLE WAVE 1 π’ < π 1 ο‘ π¦ π’ = α π 0 π 1 < π’ < 2
24 SINC FUNCTION ο‘ Important signal/function in DSP and communication sin ππ¦ normalized ο‘ sinc π¦ = ππ¦ sin π¦ unnormalized ο‘ sinc π¦ = π¦ ο‘ Modulated sine function ο‘ Amplitude follows 1/x ο‘ Must use LβHopitalβs rule to get x=0 time
25 RECTANGLE WAVE COEFFICIENTS ο‘ Consider different βduty cycleβ for the rectangle wave 1 50% (square wave) ο‘ π = 4π 1 25% ο‘ π = 8π 1 12.5% ο‘ π = 16π ο‘ Note all plots are still a sinc shape ο‘ Difference is how the sync is sampled ο‘ Longer in time (larger T) smaller spacing in frequency ο more samples between zero crossings
26 SQUARE WAVE ο‘ Special case of rectangle wave 1/2 π = 0 with π = 4π ο‘ π π = α 1 sin(ππ/2) πππ‘π ο‘ One sample between zero-crossing ππ
27 PERIODIC IMPULSE TRAIN β ο‘ π¦ π’ = Ο π=ββ π(π’ β ππ) ο‘ Using FS integral Notice only one impulse in the interval ο‘
28 PROPERTIES OF CTFS ο‘ Since these are very similar between CT and DT, will save until after DT ο‘ Note: As for LT and Z Transform, properties are used to avoid direct evaluation of FS integral ο‘ Be sure to bookmark properties in Table 3.1 on page 206
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