EE 321 Fourier Series Examples Fall 2016


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1 EE 31 Fourier Series Examples Fall A periodic sigal is give as xx(tt) = 3 cos(ππ8tt) + 7 si(ππ16tt). Fid the period, fudametal frequecy, ad Fourier series. Sketch the magitude spectrum. TT = ωω 0 = aa 0 = aa = bb =
2 . Fid the Fourier series for the sigal, x(t), show below. Plot the trucated Fourier series represetatio for = 0,1,3, ad 5. Period, T = sec. Fudametal frequecy = ωω 0 = ππ rad/s (ff TT 0 = 1 = 50 Hz). TT x ( t) = a 0 + a cos( ω 0t) + b si( ω0t) = 1 TT/ Trucated Fourier Series: x aa 0 = 1 TT xx(tt)dddd = 1 TT/ bb = 0 π a = si ( a = 0 for eve) π b = 0 (sice x(t) is a eve fuctio) 0 + a cos( ω 0t) + b si( 0t) = 1 ( t) = a ω
3 ow, suppose that the phase of the Fourier series terms are icorrect. Specifically, suppose that for all 1 < i the trucated Fourier series, there phases are off by exactly 180. This is show below, compared to the correct trucated Fourier series, for = 3,5, ad 15.
4 ote that the phase errors cause the recostructed waveforms to look quite differet from the origial square wave.
5 3. Covetioal AM geerated usig a switchig modulator. Problem: Use Fourier Series aalysis to explai how the output sigal i the circuit below, vv 0 (tt), ca be filtered to produce the evelope sigal of the form AA cc (cos(ππff cc tt) + μμμμ(tt)), where 0 μμ 1, ad the message sigal, mm(tt), is assumed scaled so that max tt mm(tt) < 1. Assume that the maximum value of m(t) is much less that A. The the sigal across the resistor is wellmodeled as vv 0 (tt) = AA cos(ππff cc tt) + mm(tt) gg(tt) where gg(tt) = 1, iiii cos(ππff cctt) > 0; 0, iiii cos(ππff cc tt) < 0. Hece, gg(tt) is a square wave, as i the previous problem, with period TT = 1, ωω ff 0 = ππ, cc TT bb = 0, ad gg(tt) = 1 + ssssss cos(ωω 0 tt). =1 As a specific example, suppose that the message sigal is a cosie, say mm(tt) = cos (ππff mm tt), where ff mm =,000 Hz, the carrier frequecy is ff mm = 50,000 Hz, ad AA = 100. I AM radio the carrier frequecies are at least 100 times larger tha the highest frequecy i the message (e.g., a carrier frequecy of at least 540 khz ad a maximum message frequecy of 5 khz).
6 4. Fid the Fourier series of the periodic sigal, x (t), show below. Solutio: Write x( t) = xe ( t) + xo ( t) ad fid the Fourier series of the eve part ad odd part of x (t), the combie. xx(tt) + xx( tt) xx(tt) xx( tt) xx ee (tt) =, xx oo (tt) = The Fourier series coefficiets are the computed as aa 0 = 0 (for both xx ee (tt) ad xx oo (tt)). For the eve part, bb = 0 ad aa = 8 si. For the odd part, aa = 0 ad bb = 1 cos. The complete Fourier series for x(t) is the aa 0 = 0, aa = 8 bb = 1 cos. si, ad
7 1. Use Matlab to plot the magitude ad phase spectra.. Determie how may terms are required i a trucated Fourier series to have 10% ormalized MSE. To have 0.5% ormalized MSE. 1 b Magitude: A = a + b, Phase: θ = ta, a fuctio fourier_series1(1,) % Computes the 1term plot of the magitude ad phase spectra % ad the term ormalized MSE a0=0; =[1:1:max(1,)]; a=8*si(0.5*pi*)./(pi*); b=*(1cos(0.5*pi*))./(pi*); figure(1) subplot(,1,1) stem([0 (1:1)],[a0 sqrt(a(1:1).^ + b(1:1).^)]) xlabel('idex, ') ylabel('a_') title('magitude Spectrum') subplot(,1,) stem([0 (1:1)],[0 ata(b(1:1),a(1:1))*180/pi]) xlabel('idex, ') ylabel('\theta_, Degrees') title('phase Spectrum') ormmse=(4.5(a0^+0.5*(a(1:)*a(1:)' + b(1:)*b(1:)')))/4.5; Percet_Error = ormmse*100 1 P [ a0 + a + b ] = 1 MSE orm = P 10% MMMMMM = 5 terms 0.5% MMMMMM = 85 terms
8 3. Determie the fractio of the sigal power i i) the first harmoic; i the first 3 harmoics. Percet Power i th harmoic = 0.5 (aa + bb ) PP 100% Usig P = 4.5, Percet Power i 1 st harmoic = 0.5 (aa 1 + bb 1 ) 100 PP ππ + ππ = 100 = 76.6% 4.5 Percet Power i first 3 harmoics = 0.5 (aa 1 + bb 1 + aa + bb + aa 3 + bb 3 ) 100% PP = 89.6% 4. Use Matlab to costruct the trucated Fourier series, x 0 + a cos( 0t) + b si( 0t) = 1 ( t) = a ω ω. 5. Also, use Matlab to costruct the trucated Fourier series from the magitude/phase form of the Fourier series coefficiets, Magitude: A = a + b, 1 b Phase: θ = ta, a x 0 + A cos( 0t ) = 1 ( t) = a ω θ. fuctio y=fs_example(,t,time,origial) % Fourier series example a0=0; fs=48000; % Samplig rate, i samples per secod. =[1:]; a=(8/pi)*si(*pi/)./; b=(/pi)*(1cos(*pi/))./; t=[t:1/fs:t+time]; le_=floor(1+*t*fs); y=a0*oes(1,legth(t)); for k=1: y=y+a(k)*cos(k**pi*t/t)+b(k)*si(k**pi*t/t); ed
9 y0=y; figure(1) plot(t(1:le_),y(1:le_)) % Costruct usig magitude/phase form of Fourier series A=sqrt(a.^ + b.^); theta=ata(b,a); y=a0*oes(1,legth(t)); for k=1: y=y+a(k)*cos(k**pi*t/ttheta(k)); ed
10 6. Fially, radomize the phase, θ, ad costruct the trucated Fourier series, x ~ ( t) = a ω θ, 0 + A cos( 0t ) = 1 where ~ θ is θ plus a uiform radom phase variable over [ π, π ]. fuctio y=fs_example(,t,time,origial) % Fourier series example a0=0; fs=48000; % Samplig rate, i samples per secod. =[1:]; a=(8/pi)*si(*pi/)./; b=(/pi)*(1cos(*pi/))./; t=[t:1/fs:t+time]; le_=floor(1+*t*fs); y=a0*oes(1,legth(t)); for k=1: y=y+a(k)*cos(k**pi*t/t)+b(k)*si(k**pi*t/t); ed y0=y; figure(1) plot(t(1:le_),y(1:le_)) % Costruct usig magitude/phase form of Fourier series A=sqrt(a.^ + b.^); theta=ata(b,a); y=a0*oes(1,legth(t)); for k=1: y=y+a(k)*cos(k**pi*t/ttheta(k)); ed figure() plot(t(1:le_),y(1:le_)) %ow radomize the phase theta1=theta+*pi*(rad(1,legth(theta))0.5); figure(3) y=a0*oes(1,legth(t)); for k=1: y=y+a(k)*cos(k**pi*t/ttheta1(k)); ed plot(t(1:le_),y(1:le_)) if(origial==0) y=y0; ed >> y=fs_example(19,0.004,,0); % Output the trucated Fourier series waveform >> y1=fs_example(19,0.004,,1); % Output the Fourier series with radom phase % Play out the soud (the samplig rate is 48,000 samples per secod) >> soud(y/50,48000) >> soud(y1/50,48000) ote: Each time the program is ru, a differet radom set of phases is geerated, ad a differet recostructed waveform geerated.
11 Each time the program is ru, a differet recostructed radom waveform geerated.
12 Speech Codig Example: Segmet of speech file (top) ad segmet of improved multibad excitatio (IMBE) coded speech (bottom). The origial speech file is 1bit PCM with a samplig rate of 8,000 samples/sec. Tollquality μlaw PCM uses 8 bits/sample, ad a trasmissio rate of 64 kb/s. The IMBE speech is ecoded at rate 4,800 bits/s.
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