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Example 9. Expand .f(x) : Ax2 + Bx + C (0 < x < 2tr) in Fourierseries. Figure 22 shows the periodic extension ofl(x) for a certain choice ofthe constants A, B, and C. Using the expansions of the functions x2 and x(0 < x < 2rc), given in Examples 6 and7, we find that
Ax2 + Bx + c :o*' * B* * c+ 4r I cosjn'r3 '"" - "'k, nz
for0< x<2x.
6a
- ArA - 2,8) t gnuI,.Lr.n
n= I
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Example 6. Expand f(x):1-(0' * 1^) in Fourier- se.r.ies' This
example bears a ,up",n"ili'isemblance to Example 4' but the diference is
immediately apparent ii *. -"oirtr"ct
the periodic extension of /(x) (see
Fig. l9). The criterior "i il l0 is applicable to this extended function'
At the points of aiscontiriuitf iit" f9.urf series converges to the arithmetic
mean of the right-hand "JiJfi-t'*O limits' i'e'' to the value t' The function
/(x) is neither even nor odd'
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Ftcune 19
Since
I r2too:;lo :h,x dx :* trt]::l
- *t: sinzxdr = 0 (n - 1,2,...),lNN JO
o, ::f" *cosn* d,I r=2n: a Ix sin zxlfin' 'r=o
I ?2nb,:;Jo xsinnxdx
= - *t, *' *il" + *f" "o, ,* d* = - 1,
we have
tr : r - z (sinx * Y * %1I +...),for0< x<2tt.
Example 7. Expand l@) - 12 (0 < x < 2n) in Fowier series. Thisexample resembles Example l, but the graph of the periodic extension of/(.r)immediately shows the difference (see Fig. 20). The criterion or sec. io is
(r3.s)
Frounr 20
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applicable, and at the points of discontinuity the series converges to thearithmetic mean of the right-hand and left-hand limits, i.e., to the valte 2tc2.The function/(x) is neither even nor odd.
Since
lfzr ^. lf13'lx=2r 8*2ao : iJo xzctx : i [-il,=o : T,
I f}n ) r2ta- : I I'-x2cosnxdx : -' I xsinnxdx
tc Jo lcn Jo
2 x=2n 1 rzn 4: -lxcoszxl - -:-^ | cosnxdx: a,NnL . 'r=0 1ln JO n.
112b,: ! [' xzsinnxdx
: - I [x2 cos n*l'='n + Z {'" x cos nx dx*n' 'r=o fin Jo
- -4n -Ll'"sinnxdx: -4n,n 7cn2 Jo nwe haYe
- 4r2 .t cos2x r sinlxxz: -T + 4(cosx - Trsln x + -,/f - 2- +...
_r cosj/rx _ 7E sin nx + . . .lnz n I (13.6)4r.2 3 /cos nx zc sin zx\:J_+o,1,\__F__ " )4nz Scoszx Ssinzx::*4 )
--47i >SErn'Ern
for0< x<2rc,
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% ### EX2DfourierHPfilter.m ### 2015.09.21 CB (updated 2017.01.20) % purpose of this code is perform a high-pass filtering of an image, akin% to what is shown in Fig.12.6 of Hobbie & Roth (4th Ed.) % Notes% o Caution: axes for FFT are not (presently) properly labeled% o originally called HRfig12x6.m clear% ==========================================================fileA= './Images/HRfig12x6'; % [no need for extension]mScale= 0; % boolean re linear (0) or log (1) axes for the magnitudeL= 15; % filter length for hi-pass% ==========================================================% ---imageA = imread(fileA,'jpg'); % load in an image% ---% if color, convert to B&Wif (size(imageA,3)>1), imageA= rgb2gray(imageA); end% ---fftA = fft2((imageA)); % compute FFT% ---% create a high-pass filter "mask" and apply (kludgy; likely better bookkeeping possible)fftA(1:L,1:L)= 0;fftA(1:L,end-L:end)= 0;fftA(end-L:end,1:L)= 0;fftA(end-L:end,end-L:end)= 0;% ---imageF= ifft2(fftA, 'symmetric'); % inverse FFT% place zero-frequency position in center?if (1==1), FA= fftshift(fftA); else FA= fftA; end% ---% plot originalfigure(1); clf;subplot(221); imagesc(imageA); title('Image A'); colorbar;if mScale==0 subplot(223); imagesc(abs(FA),[0 100000]); colormap gray; title('Mag.'); colorbar; % linear axeselse subplot(223); imagesc(db(FA)); colormap gray; title('Mag.'); colorbar; % log axesendsubplot(224); imagesc(angle(FA),[-pi pi]); colormap gray; title('Phase'); colorbar;xlabel('Freq. scale incorrect')subplot(222); imagesc(imageF); title('Hi-pass filtered version'); colorbar;
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