DISCRETE TIME FOURIER SERIES CHAPTER 3.6 30 DTFS VS CTFS - - PowerPoint PPT Presentation

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DISCRETE TIME FOURIER SERIES CHAPTER 3.6 30 DTFS VS CTFS - - PowerPoint PPT Presentation

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29 DISCRETE TIME FOURIER SERIES CHAPTER 3.6 30 DTFS VS CTFS DIFFERENCES While quite similar to the CT case, DTFS is a finite series, , k < K Does not have convergence issues Good News: motivation and intuition from

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SLIDE 1

DISCRETE TIME FOURIER SERIES

CHAPTER 3.6 29

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SLIDE 2

DTFS VS CTFS DIFFERENCES

ο‚‘ While quite similar to the CT case,

ο‚‘ DTFS is a finite series, 𝑏𝑙 , k < K ο‚‘ Does not have convergence issues

ο‚‘ Good News: motivation and intuition from CT

applies for DT case

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SLIDE 3

DTFS TRANSFORM PAIR

ο‚‘ Consider the discrete time periodic signal 𝑦 π‘œ = 𝑦 π‘œ + 𝑂 ο‚‘ 𝑦 π‘œ = σ𝑙=<𝑂> π‘π‘™π‘“π‘˜π‘™πœ•0π‘œ

synthesis equation

ο‚‘ 𝑏𝑙 =

1 𝑂 Οƒπ‘œ=<𝑂> 𝑦 π‘œ π‘“βˆ’π‘˜π‘™πœ•0π‘œ

analysis equation

ο‚‘ 𝑂 – fundamental period (smallest value such that periodicity

constraint holds)

ο‚‘ πœ•0 = 2𝜌/𝑂 – fundamental frequency ο‚‘ Οƒπ‘œ=<𝑂>

indicates summation over a period (𝑂 samples)

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SLIDE 4

DTFS REMARKS

ο‚‘ DTFS representation is a finite sum, so there is

always pointwise convergence

ο‚‘ FS coefficients are periodic with period N

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DTFS PROOF

ο‚‘ Proof for the DTFS pair is similar to the CT case ο‚‘ Relies on orthogonality of harmonically related DT

period complex exponentials

ο‚‘ Will not show in class

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SLIDE 6

HOW TO FIND DTFS REPRESENTATION

ο‚‘ Like CTFS, will use important examples to

demonstrate common techniques

ο‚‘ Sinusoidal signals – Euler’s relationship ο‚‘ Direct FS summation evaluation – periodic

rectangular wave and impulse train

ο‚‘ FS properties table and transform pairs

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SLIDE 7

ο‚‘ 𝑦[π‘œ] = 1 +

1 2 cos 2𝜌 𝑂

π‘œ + sin

4𝜌 𝑂

π‘œ

ο‚‘ First find the period ο‚‘ Rewrite 𝑦[π‘œ] using Euler’s and

read off 𝑏𝑙 coefficients by inspection

ο‚‘ Shortcut here ο‚‘ 𝑏0 = 1, 𝑏±1 =

1 4 , 𝑏2 = π‘βˆ’2 βˆ—

=

1 2π‘˜

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SINUSOIDAL SIGNAL

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SLIDE 8

ο‚‘ 𝑦(𝑒) = cos πœ•0𝑒 ο‚‘ 𝑏𝑙 = α‰Š1/2

𝑙 = Β±1 π‘“π‘šπ‘‘π‘“

ο‚‘ 𝑦 π‘œ = cos πœ•0π‘œ ο‚‘ 𝑏𝑙 = α‰Š1/2

𝑙 = Β±1 π‘“π‘šπ‘‘π‘“

ο‚‘ Over a single period οƒ  must

specify period with period N

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SINUSOIDAL COMPARISON

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SLIDE 9

ο‚‘ Type equation here.

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PERIODIC RECTANGLE WAVE

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SLIDE 10

ο‚‘ Consider different β€œduty cycle” for

the rectangle wave

ο‚‘ 50% (square wave) ο‚‘ 25% ο‚‘ 12.5%

ο‚‘ Note all plots are still a sinc

shaped, but periodic

ο‚‘ Difference is how the sync is sampled ο‚‘ Longer in time (larger N) smaller

spacing in frequency οƒ  more samples between zero crossings

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RECTANGLE WAVE COEFFICIENTS

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SLIDE 11

ο‚‘ 𝑦[π‘œ] = σ𝑙=βˆ’βˆž

∞

πœ€[π‘œ βˆ’ 𝑙𝑂]

ο‚‘ Using FS integral

ο‚‘

Notice only one impulse in the interval 39

PERIODIC IMPULSE TRAIN