Fourier Series for Periodic Functions. Lecture #8 5CT3,4,6,7. BME 333 Biomedical Signals and Systems - J.Schesser

Transcription

1 Fourier Series for Periodic Functions Lecture #8 5C3,4,6,7

2 Fourier Series for Periodic Functions Up to now we have solved the problem of approximating a function f(t) by f a (t) within an interval. However, if f(t) is periodic with period, i.e., f(t)=f(t+), then the approximation is true for all t. And if we represent a periodic function in terms of an infinite Fourier series, such that the frequencies are all integral multiples of the frequency /, where = corresponds to the fundamental frequency of the function and the remainder are its harmonics. a f t e dt j t () jt j t j t j t a a a a j t j a( ) cos( ), where a and a f() t a [ e * e ] e a Re[ e ] f t a C C e C 3

3 Another Form for the Fourier Series t f() t C C cos( ) t t A A cos( ) B sin( ) where A C cos and B -C sin and A C 4

4 Fourier Series heorem Any periodic function f (t) with period which is integrable ( f (t)dt ) can be represented by an infinite Fourier Series If [f (t)] is also integrable, then the series converges to the value of f (t) at every point where f(t) is continuous and to the average value at any discontinuity. 5

5 Symmetries Properties of Fourier Series If f (t) is even, f (t)=f (-t), then the Fourier Series contains only cosine terms If f (t) is odd, f (t)=-f (-t), then the Fourier Series contains only sine terms If f (t) has half-wave symmetry, f (t) = -f (t+/), then the Fourier Series will only have odd harmonics If f (t) has half-wave symmetry and is even, even quarter-wave, then the Fourier Series will only have odd harmonics and cosine terms If f (t) has half-wave symmetry and is odd, odd quarterwave,then the Fourier Series will only have odd harmonics and sine terms 6

6 Properties of FS Continued Superposition holds, if f(t) and g(t) have coefficients f and g, respectively, then Af(t)+Bg(t)=>Af +Bg j t ime Shifting: f() t ae Differentiation and Integration delayed original jt jt jt j t f( tt ) a e e a e e ( a ) ( a ) e j t j ( a) derivative ( a) ( a) integral ( a) j original original 7

7 FS Coefficients Calculation Example E jt jt jt a [ ] 4 e dt Ee dt e dt j t E 4 e 4 j 4 j j j j [ e ] [ ] E Ee e e j j sin( ) j E 4 4 e, 4 4 E E a dt 4 4 f ( t), t f ( t) f ( t) E, t f ( t), t f ( t 4) Since a C, then sin E E 4 t f() t cos( )

8 Example Continued.5 term terms

9 Frequency Spectrum of the Pulse Function In the preceding example, the coefficients for each of the cosine terms was proportional to sin( 4) ( 4) We call the function Sa( x) sin x x the Sampling Function. If we plot these coefficients along the.8.6 frequency axis we have.4. the frequency spectrum of f(t)

10 Frequency Spectrum Note that the periodic signal in the time domain exhibits a discrete spectrum (i.e., in the frequency domain) Frequency (/4)

11 Another Example V his signal is odd, H/W symmetrical, and mean=; this means no cosine terms, odd harmonics and mean=. jt jt [ ] jt jt e e j j a Ve dt Ve dt V [ V V [ cos ] for odd, j j V 9 for odd. for even. f ( t) odd V odd V sin( t ) cos( t )

12 Fourier Series of an Impulse rain or Sampling Function xt () ( tn) jt a x() t e dt n () te jt dt for all xt () e jt 3

13 4 Fourier Series for Discrete Periodic Functions ] [ ] [ n j n n j n j n j n e x a e a e a e a a n x Notice the similarities to the previous example. Since x[n] is discrete, this becomes a summation

14 Homewor Problem () Compute the Fourier Series for the periodic functions a) f(t) = for <t<π, f(t)= for π<t<π b) f(t) = t for <t<3 Problem () Compute the Fourier series of the following Periodic Functions: f(t)= t, nπ < t < (n+)π for n =, (n+)π < t < (n+)π for n f(t)= e -t/π, nπ < t < (n+)π for n Use Matlab to plot f(t) usinga for maximum number of components, N=5,,, and. Show your code. Problem (3) Problems: 4./3 Find the Fourier series of the following waveforms (choose t c = o /4): o -( o + t c ) - ( o -t c ) - o / -t c t c o / o o -t c o + t c 5

15 Homewor Problem (4) Deduce the Fourier series for the functions shown (hint: deduce the second one using superposition): -π+a -a π+a π-a -π -π/3 -π/3 4π/3 5π/3 -π a π-a π π π/3 π/3 π π 5C.7. 6