Massachusetts Institute of Technology Dept. of Electrical Engineering and Computer Science Fall Semester, Introduction to EECS 2.

Transcription

1 Massachusetts Istitute of Techology Dept. of Electrical Egieerig ad Computer Sciece Fall Semester, Itroductio to EECS Prelab Exercises Pre-Lab#3 Modulatio, demodulatio, ad filterig are itegral to the frequecy divisio multiplexig FDM) commuicatio system you will be buildig i Lab 3. The purpose of these prelab exercises is to familiarize you with the implemetatio of these sigal processig operatios usig simple siusoidal sigals. B. Uderstadig Modulatio The purpose of this exercise is to uderstad the mechaics of modulatio ad demodulatio as well as to study the effects of these operatios o sigals i the time ad frequecy domais. To achieve this purpose we will study the modulatio ad demodulatio of a Hz siusoid by 0 Hz. Cosider the sigal x t) = cosπ * t). This is a siusoid with amplitude of ad frequecy of Hz. Create a Matlab file called modulatio.m ad eter the followig commads to create ad visualize xt) i the time domai. Your code should geerate Figure.

2 Fs = 000; t = 0:/Fs:5; % Sample Frequecy i Hz % Time vector form 0-5 secods i icremets of % 0.00 secods eg , 0.00, ) x = cos*pi**t); % Cosie with amplitude ad frequecy Hz figure) % Create Figure plott,x) % Plot Sigal xt) i Figure title'time-domai'); % Add Title Time-Domai to Figure xlabel'time sec)'); % Add Time sec) as x-axis label of Figure ylabel'amplitude'); % Add Amplitude as y-axis label of Figure axis[0 - ]) % Set x-axis rage [0-] sec ad y-axis rage [- ] Figure Now lets visualize the magitude of the spectrum of xt). Sice xt) is a real sigal, the magitude of its spectrum is a eve fuctio symmetric about 0 Hz). We expect to see a peak at Hz ad aother at - Hz i the magitude of the spectrum of xt). Add the followig commads to your Matlab file, ad reru it to

3 visualize the magitude of the spectrum of xt). Your code should ow geerate Figure. X = fftx); N = legthx); f = [-N/:N/-]*Fs/N); % perform FFT o sigal xt) % Make frequecy vector f % that spas frequecies from -500 Hz to 500 Hz figure) % Create Figure plotf,absfftshiftx:n)))) % Plot positive ad egative spectrum of xt) axis[ ]); % Set x-axis rage [-5 5] Hz ad y-axis [0 3000] title'frequecy Domai'); % Add Frequecy Domai as title of Figure xlabel'frequecy Hz)'); % Add Frequecy Hz) as x-axis label of Figure ylabel'magitude'); % Add Magitue as y-axis label of Figure Figure Now lets modulate xt) upwards by 0 Hz. To do this we eed to geerate the modulatig sigal y t) = cosπ *0t), ad the create the modulated sigal mt) = xt)yt). Recall from the modulatio lecture otes that

4 cos{π f + f ) t} + cos{π f f ) t} cos π ft)*cosπf t) = m t) = cosπ t) *cosπ *0 * t) = cosπ * t) + cosπ *9t). Add the followig commads to your Matlab file, ad reru it to visualize the i the time-domai represetatio of the modulated sigal mt). Your code should geerate a plot similar to that i Figure 3. y = cos*pi*0*t); % Create modulatig sigal yt); a cosie % with amplitude ad frequecy 0 Hz m = x.*y; % Multiply sigal xt) by modulatig sigal % yt) to create the modulated sigal mt). figure3) % Create Figure 3 plott,m) % Plot Sigal mt) i Figure 3 title'time-domai'); % Add Title Time-Domai to Figure 3 xlabel'time sec)'); % Add Time sec) as x-axis label of Figure 3 ylabel'amplitude'); % Add Amplitude as y-axis label of Figure 3 axis[0 - ]) % Set x-axis rage [0-] sec ad y-axis rage [- ] Figure 3

5 Now lets visualize the magitude of the spectrum of mt). Sice mt) is a real sigal, the magitude of its spectrum is a eve fuctio symmetric about 0 Hz). We expect to see a peak at +/- 9 Hz ad +/- Hz i the magitude of the spectrum of mt). Add the followig commads to your Matlab file, ad reru it to visualize the magitude of the spectrum of xt). Your code should ow geerate a plot similar to Figure 4. M = fftm); f = [-N/:N/-]*Fs/N); % perform FFT o sigal mt) % Make frequecy vector f % that spas frequecies from -500 Hz to 500 Hz figure4) % Create Figure 4 plotf,absfftshiftm:n))));% Plot positive ad egative spectrum of mt) axis[ ]); % Set x-axis rage [-0 0] Hz ad y-axis [0 500] title'frequecy Domai'); % Add Frequecy Domai as title of Figure 4 xlabel'frequecy Hz)'); % Add Frequecy Hz) as x-axis label of Figure 4 ylabel'magitude'); % Add Magitude as y-axis label of Figure 4

6 f +f = Hz Cosie f -f = 9 Hz Cosie Figure 4 At this poit you have completed modulatig the sigal xt) by 0 Hz. You have implemeted the block diagram show i Figure 4. Modulatio xt)= cosπt) X mt) = yt)xt) yt)=cosπ0t) Figure 5

7 C. Uderstadig Demodulatio Now let s demodulate the sigal mt) dowward by 0 Hz i order to recover the sigal xt). To do this we multiply mt)yt). Recall from the modulatio lecture that the ct) should have the form below. c t) = cosπ f f ) t) + cosπf t) + cosπ f + f ) t) 4 4 c t) = cosπ *9t) + cosπ * t) + cosπ * t) 4 4 Add the followig commads to your Matlab file, ad reru it to visualize the i the time-domai represetatio of the demodulated sigal ct) You should see a plot like that i Figure 6. c = m.*y; % Multipy modulated sigal mt) by yt) to % create demodulated sigal ct) figure5) % Create Figure 5 plott,c) % Plot Sigal ct) i Figure 5 title'time-domai'); % Add Title Time-Domai to Figure 5 xlabel'time sec)'); % Add Time sec) as x-axis label of Figure 5 ylabel'amplitude'); % Add Amplitude as y-axis label of Figure 5 axis[0 - ]) % Set x-axis rage [0-] sec ad y-axis rage [- ]

8 Figure 6 Now visualize the magitude of the spectrum of ct). Sice ct) is a real sigal, the magitude of its spectrum is a eve fuctio symmetric about 0 Hz). We expect to see a peak at +/- Hz, +/- 9 Hz, ad +/- Hz. Add the followig commads to your Matlab file, ad reru it to visualize the magitude of the spectrum of ct). Your code should ow geerate a plot similar to Figure 7. C = fftc); f = [-N/:N/-]*Fs/N); % perform FFT o sigal mt) % Make frequecy vector f % that spas frequecies from -500 Hz to 500 Hz figure6) % Create Figure 6 plotf,absfftshiftc:n))));% Plot positive ad egative spectrum of ct) axis[ ]); % Set x-axis rage [-30 30] Hz ad y-axis [0 500] title'frequecy Domai'); % Add Frequecy Domai as title of Figure 6 xlabel'frequecy Hz)'); % Add Frequecy Hz) as x-axis label of Figure 6 ylabel'magitude'); % Add Magitude as y-axis label of Figure 6

9 Figure 7 Note that we did ot recover xt); we recovered xt) as well as some higher frequecy compoets. To remove the higher frequecy compoets 9Hz ad Hz compoets) we eed to apply a low frequecy filter to ct). Filterig is the topic of the ext pre-lab exercises. A. Low Pass ad Bad Pass Filter Desig Recall from the filterig lecture otes that, i geeral, the output of a differece equatio y) ca deped o preset ad past values of the iput x) as well as past values of the output. This depedece is captured i the equatio below = = + = = M j N i i i N o M i x b j y a y N x b x b x b x b M y a y a y a y 0 ) ) ) ) ) ) ) ) ) ) ) f = Hz f -f = 9 Hz Cosie f +f = Hz Cosie

10 I this pre-lab, we will be implemetig filters usig differece equatios whose output y) depeds oly o preset ad past values of the iput x). That meas the coefficiets a j = 0. This reduces our differece equatio to that show below y ) = b x ) + b x ) + b x ) + + b y ) = 0 N i= 0 b x i) i N x N) Also recall from the filterig lecture otes that the values ad umber of the b i coefficiets determies the type ad performace of the filter implemeted by the differece equatio. Through the followig exercises you we will become familiar with Matlab commads for buildig filters fir commad); visualizig the frequecy respose of filters freqz commad); ad applyig filters to sigals filter commad). Let s desig a low pass filter with order 8 ad a cutoff frequecy f c =5 Hz to extract the 0Hz compoet of x ) = siπ *0* * Ts) + siπ *50* * Ts) + siπ *90* * Ts) Create the file filterig.m, ad add the followig commads to create the sigal x) ad its time-domai plot. You should see a plot such as that i Figure 8. Fs = 000; % Sample Frequecy i Hz t = 0:/Fs:5; % Time vector form 0-5 secods i icremets of % 0.00 secods eg , 0.00, ) x = si*pi*0*t) + si*pi*50*t)+si*pi*90*t); figure) % Create Figure plott,x) % Plot Sigal x) i Figure title'time-domai'); % Add Title Time-Domai to Figure xlabel'time sec)'); % Add Time sec) as x-axis label of Figure ylabel'amplitude'); % Add Amplitude as y-axis label of Figure axis[ ]) % X-axis rage [0-0.5] sec ad y-axis rage [-4 4]

11 Figure 8 Now let s visualize the magitude of the spectrum of x). Sice x) is a real sigal, the magitude of its spectrum is a eve fuctio symmetric about 0 Hz). We expect to see a peak at +/-0 Hz, +/- 50 Hz, +/- 90 Hz. Add the followig commads to your Matlab file, ad reru it to visualize the magitude of the spectrum of x). Your code should ow geerate Figure 9. X = fftx); N = legthx); f = [-N/:N/-]*Fs/N); % perform FFT o sigal x) % Make frequecy vector f % that spas frequecies from -500 Hz to 500 Hz figure) % Create Figure plotf,absfftshiftx:n)))) % Plot positive ad egative spectrum of x) axis[ ]); % X-axis rage [-00 00] Hz ad y-axis [0 3000] title'frequecy Domai'); % Add Frequecy Domai as title of Figure xlabel'frequecy Hz)'); % Add Frequecy Hz) as x-axis label of Figure ylabel'magitude'); % Add Magitue as y-axis label of Figure

12 Figure 9 Now let s create the low pass filter that will extract the 0Hz compoet i x). The low pass filter will have a order of 8 ad a cutoff frequecy f c =5 Hz. Before gettig to the code lets lear about the commads we will be usig to create filters ad visualize their frequecy respose. To create our filter we will use the fir commad. To produce a low pass filter usig the fir commad, use the followig format B_Coffeciets = fir Filter_Order, Cuttoff_Frequecy, low ) The variable Filter_Order specify the order of the filter i our case 8). The variable Cuttoff_Frequecy specifies the cutoff frequecy of the filter i our case 5 Hz). The strig argumet low, specifies to the fir commad to create a low pass filter. After executig this commad, the variable B_Coffeciets will be a vector whose elemets are the b i coefficiets of a differece equatio that implemets a low pass filter.

13 To obtai the frequecy respose of a filter we will use the freqz commad. I geeral, this commad computes the frequecy respose of filter specified by a i ad b i coeffeciets; i our case we oly have b i. Use the followig format to call the freqz H = freqzb_coffeciets, [], whole ); F=-56:55)*Fs/5; The variable B_Coffeciets will be a vector of b i coeffeciets produced by the fir commad. The argumet [] specifies to the freqz commad that there are o a i coeffeciets. The strig argumet whole specifies to the freqz commad to compute the frequecy respose for positive ad egative frequecies defied i the variable F. Now add the followig commads to your Matlab file to create the low pass filter ad view its frequecy respose. You should see a plot like that Figure 0 we added the Figure 0A to give you practice with viewig spectra o log scales). Filter_Order = 8; % Set filter order to be 8 Cuttoff_Frequecy = 5*/Fs); % Set Cuttoff Frequecy to 5 Hz %Followig are commads for creatig filter ad filter frequecy respose B_Coeffeciets= firfilter_order, Cuttoff_Frequecy,'low'); H = freqzb_coeffeciets, [], 'whole'); F=-56:55)*Fs/5; figure3) % Create Figure 3 plotf,absfftshifth))) % Plot positive ad egative spectrum of filter axis[ ]); % X-axis rage [-50 50] Hz ad y-axis [0 ] title'frequecy Respose'); % Add Frequecy Respose as title of Figure 3 xlabel'frequecy Hz)'); % Add Frequecy Hz) as x-axis label of Figure 3 ylabel'magitude'); % Add Magitude as y-axis label of Figure 3

14 Figure 0 Figure 0A Now let s apply this low pass filter to the sigal x). To filter sigals we will use the Matlab commads filter. Use the followig format to call the commad filter output_sigal = filterb_coffeciets, [], iput_sigal);

15 The variable B_Coffeciets is a vector of b i coeffeciets produced by the fir commad) of the differece equatio represetig the filter. The argumet [] specifies to the filter commad that there are o a i coeffeciets i the differece equatio. The variable iput_sigal refers to the iput sigal to be filtered. After executig this commad, the variable output_sigal holds the filtered sigal. Now add the followig code to your Matlab file i order to filter the sigal x). You should see a plot like that i Figure. ylpf = filterb_coeffeciets,[],x); %Apply the filter to the sigal x) figure4) % Create Figure 4 plot0:legthylpf)-)/fs, ylpf) % Plot Sigal y) i Figure 4 title'time-domai'); % Add Title Time-Domai to Figure 4 xlabel'time sec)'); % Add Time sec) as x-axis label of Figure 4 ylabel'amplitude'); % Add Amplitude as y-axis label of Figure 4 axis[. - ]) % X-axis rage [0. ] sec ad y-axis rage [- ] Figure Now let s desig a bad pass filter to extract the 50Hz frequecy compoet from x ) = siπ *0* * Ts) + siπ *50* * Ts) + siπ *90* * Ts). The bad

16 pass filter will have a order 8 ad cutoff frequecies f c,low = 35 Hz ad f c,high = 65 Hz. To desig a low pass filter usig the fir commad use the followig format B_Coffeciets = fir Filter_Order, [Cuttoff_Frequecy_Low Cuttoff_Frequecy_High]) The variable Filter_Order specify the order of the filter i our case 8). The variables Cuttoff_Frequecy_Low ad Cuttoff_Frequecy_High specify f c,low ad f c,high respectively i our case f c,low = 35 Hz ad f c,high = 65 Hz). After executig this commad, the variable B_Coffeciets will be a vector whose elemets are the b i coefficiets of a differece equatio that implemets a badpass filter. There is o chage i how we call the commad freqz to evaluate the frequecy respose of the badpass filter. Now add the followig commads your Matlab file to create the badpass filter ad view its frequecy respose. You should see a plot such as that i Figure we added the Figure A to give you practice with viewig spectra o log scales). Filter_Order = 8; % Set filter order to be 8 Cuttoff_Frequecy_Low = 35*/Fs); % Set f c,low to 35 Hz Cuttoff_Frequecy_High = 65*/Fs); % Set f c,high = 65 Hz %Followig are commads for creatig filter ad filter frequecy respose B_Coeffeciets= firfilter_order, [Cuttoff_Frequecy_Low Cuttoff_Frequecy_High]); H = freqzb_coeffeciets, [], 'whole'); F=-56:55)*Fs/5; figure5) % Create Figure 5 plotf,absfftshifth))) % Plot positive ad egative spectrum of filter axis[ ]); % X-axis rage [-00 00] Hz ad y-axis [0 ] title'frequecy Respose'); % Add Frequecy Respose as title of Figure 5 xlabel'frequecy Hz)'); % Add Frequecy Hz) as x-axis label of Figure 5 ylabel'magitude'); % Add Magitude as y-axis label of Figure 5

17 Figure Figure A Now let s apply this badpass filter to the sigal x). There is o chage to the way i which we ivoke the commad filter. Now add the followig code to your Matlab file i order to filter the sigal x). You should see a plot like that i Figure 3.

18 yhbpf = filterb_coeffeciets,[],x); %Apply the filter to the sigal x) figure6) % Create Figure 6 plot0:legthyhbpf)-)/fs, yhbpf) % Plot Sigal y) i Figure 6 title'time-domai'); % Add Title Time-Domai to Figure 6 xlabel'time sec)'); % Add Time sec) as x-axis label of Figure 6 ylabel'amplitude'); % Add Amplitude as y-axis label of Figure 6 axis[..3 - ]) % X-axis rage [0..3] sec ad y-axis rage [- ] Figure 3 B. Filterig ad Modulatig Sigals With Nozero Badwidth So far we have bee dealig with the modulatio ad filterig of siusoids; these sigals have a sigle frequecy compoet ad zero badwidth. I this exercise you will modulate ad filter a voice sigal; the voice sigal is sampled at Fs = 50kHz, has a badwidth of 4 khz, ad is cetered at 0 khz. Complete Tasks -6, ad for each task save ad had-i the plots you produce.

19 Task : Dowload the voice sigal voice_sample.mat from the course website The voice sigal is the word cat sampled at a samplig frequecy Fs = 50kHz ad modulated up to a frequecy of 0kHz. Plot the spectrum of the voice sigal spectrum. Task : Modulate the voice sigal so that it is cetered at 40kHz ad plot the spectrum of the modulated sigal. Task 3: Create a badpass filter of order 56 ad f c,low = 35kHz ad f c,high = 45kHz. Plot the frequecy respose of the badpass filter. Task 4: Filter the modulated sigal resultig from Task usig the badpass filter, ad plot the spectrum of the filtered sigal. Task 5: Demodulate the filtered sigal from Task 4 back to DC 0Hz). Plot the spectrum of the demodulated sigal. Task 6: Create a low pass filter of order 56 ad f c = 4.5 khz. Filter the demodulated sigal from Task 5 usig the low pass filter. Plot the spectrum of the low pass filtered sigal.