EE 47 BIOMEDICAL SIGNALS AND SYSTEMS Active Learning Exercises Part 2 29. For the system whose block diagram presentation given please determine: The differential equation 2 y(t) The characteristic polynomial Eigenvalues and - eigenfuctions x(t) - The transfer - 2.5 function 1.5 The impulse response Stability of the system Steady-state output for x(t)=u(t) Steady-state output for x(t)=cos(2t) Step response 3. Plot the impulse and step responses of the system in question-29 using functions you determined above step and impulse functions of the MATLAB 31. A linear time-invariant system is required to realize the impulse response shown in the figure. Amplitude 1 Impulse Response 8 6 4 2-2 -4-6.5.1.15.25.3.35.4 Time (sec) Figure for question 31 a) Design the system in block diagram, differential equation and transfer function form b) Determine the system parameters from the figure c) Modify the system so that its step response reaches its steady-state value in the shortest time d) Draw the step responses for the unmodified and modified system together 1
showing carefully all critical amplitude and time values for the plots 32. The antibiotic ampicillin (6 mg) needs to be added intravenously repeatedly in order to treat a certain infection, the drug is ineffective if it drops below a bloodstream concentration of.1 mg/ml and has some toxic side effects if its concentration exceeds.55 mg/ml. You have been asked to determine what happens to the concentration of the drug in the body as a function of time and to determine the best time interval between injections. The drug is expensive. You are given the following information: The body s fluid volume is 12 liters, A certain fraction is eliminated from the body by the kidneys, A certain fraction is eliminated from the body by metabolism, 4 hours after initial injection, the concentration of ampicillin in the blood is.5 mg/ml, 4 hours after the initial injection, the concentration of ampicillin in the urine is.2 mg/ml. Based on the information given: a) Develop a model to solve the above problem. You may develop a MATLAB script or an EXCEL spreadsheet to execute the model, graphing the drug concentration as a function of time, b) What are the fractions of antibiotic removed by the kidneys and by metabolism? c) What is the time interval that you would recommend? d) What are the upper and lower bounds to your answer? e) Which parameters affected your model the most? 33. Synthesis of a square-wave: Run MATLAB demo on MATLAB/ Graphics/ generating square wave from sine waves. 34. Calculate the Continuous-Time Fourier Series for x(t) = 2 + 2cos(t) + 2sin(t) with = 2 x(t) = cos(2t)sin(3t) and graph its amplitude and phase spectra Download and run FS_ALE_28_3_25.m 35. Obtain the complex (exponential) Fourier series representation of the following functions and draw their line spectra A square wave (FS_square_wave.m) A rectangular wave (FS_rectangular_wave.m) A triangular wave (FS_triangular_wave.m) 36. CTFS of an impulse train and a rectangular pulse sequence using properties of F series. Determine CTFS expansion of following functions: x1 ) n an impulse train t t nt ( (q1_fs_dirac.m) a shifted impulse train T x2( t) t nt (q2_fs_dirac.m) n 4 T T x3( t) t nt t nt (q3_fs_dirac.m) n 4 n 4 t x4 ( t) x3( ) d (q4_fs_dirac.m) R 37. For the RC low-pass filter shown determine: The impulse response V i C 2 V
The transfer function The frequency response function 38. For the RLC type band-pass filter shown Obtain and draw the impulse response Determine the frequency response function 39. Fourier transform of x(t) = (t+1) - (t-1) is a) 2jsin() b) 1 c) d) 2cos() e) None of the above, they are. FS 4. If x( t) X[ k] then Y[k] for y(t)=x(t-2)+x(t+2) is The Fourier transform of signal shown is a) X[ k]sin(2 F ) 2 j b) X[ k]cos(2k) kf c) 2X[ k]cos(2k F ) j2kf d) 2X[ k] e 41. The Fourier series coefficient X[3] for x(t) = 2 + cos(2t) + 3sin(5t) is a) b).5 c) 1.5j d) 2 42. Assume that the signal in question 3 is a voltage applied to 1- resistor. The total signal power is a) 6 W b) 9 W c) 4 W d) 3 W 43. In Fourier analysis a) Fourier series is defined for periodic signals only b) Continuous-time signals may have infinite number of independent harmonics c) Discrete-time periodic signals will have perfect conversion d) All of the above e) None of the above 5s 44. Input x(t) = 2cos(2t) is applied to H ( s). The output is s 5 a) 1cos(2t) b) 9.7cos(2t + 14) c) 2sin(2t) d) 5cos(2t 26.6) 45. A continuous-time square wave at 1 khz is to be synthesized from its 3
harmonics. For ideal reconstruction we must add a) First 3 harmonics b) First 1 harmonics c) First 2 harmonics d) All harmonics up to infinity 46. Which one of the following filter has a periodic frequency spectrum? a) A discrete-time low-pass filter b) An R-C type low-pass filter c) An RLC type filter d) A continuous-time band-pass filter 47. Design a system (in transfer or frequency response function form) to realize the asymptotic Bode diagram shown in the figure. Calculate the expected phase shifts between the output and input at 5 rad/sec and 5 rad/sec 48. (From 8.1 in Bruce) The electromyogram (EMG) signal recorded by large surface 4 H(j)dB electrodes overlying a skeletal muscle can be approximated (to the first degree) using a filter of the type H(z) = b/(1 + a1z-1 + a2z-2). Studies in the literature indicate that b = 2., a1 = -1.35 and a2 = 1.7 are realistic for a normalized sampling frequency. Determine the transfer function Determine the frequency response function and plot its magnitude versus frequency 49. Studying DFT of a signal: Type sigdemo1 in the command window Change the signal type and observe the spectrum Change the frequency of the signal and observe the spectrum Apply window types available on various signal types and observe their effects on the spectrum 5. Feature extraction by FFT: Type demo in the command window and click on the left pane on MATLAB Mathematics Run Using FFT in MATLAB Run FFT for Spectral Analysis 51. Exercises with FFT: Run FFT_tryout.m, study the program and try to interpret every step Experiment on changing the sampling frequency and duration of samples for FFT Change the signal type (i.e. instead of square wave use rectangular wave, sawtooth and sine wave) 2.1 1 (rps) 1 1 1k 4
clear %Fourier Series analysis for ALE on 29/3/21 Tstart=;Tstop=3;dT=(Tstop-Tstart)/11; t=tstart:dt:tstop; TF=1;fF=1/TF;wF=2*pi*fF;%Fundamental perion and frequencies x1=2+2*cos(wf*t)+2*sin(wf*t);%first time function x2=cos(2*pi*t).*sin(3*pi*t);%second time function t=1;%starting point for integration T1i=int16(TF/dT);T2i=2*T1i;%Duration of integration nmax=1;%maximum number of harmonics x1ci=x1(t:t+t1i);x1=dt*ff*sum(x1ci);%dc value of the signal X1m=abs(X1);X1a=angle(X1)*18/pi; x2ci=x2(t:t+t2i);x2=dt*sum(x2ci)/t2i;%dc value of the signal X2m=abs(X2);X2a=;%angle(X2)*18/pi; %Calculation of harmonics for k=1:nmax; x1c=x1.*cos(k*wf*t);x1ci=x1c(t:t+t1i);x1c(k)=2*dt*ff*sum(x1ci); x1s=x1.*sin(k*wf*t);x1si=x1s(t:t+t1i);x1s(k)=2*dt*ff*sum(x1si); X1(k)=(X1c(k)-j*X1s(k))/2;X1pm(k)=abs(X1(k));X1pa(k)=angle(X1(k))*18/pi; x2c=x2.*cos(k*.5*wf*t);x2ci=x2c(t:t+t2i);x2c(k)=4*dt*ff*sum(x2ci); x2s=x2.*sin(k*.5*wf*t);x2si=x2s(t:t+t2i);x2s(k)=4*dt*ff*sum(x2si); X2(k)=(X2c(k)-j*X2s(k))/2;X2pm(k)=abs(X2(k));X2pa(k)=angle(X2(k))*18/pi; %negative frequency components X1nm(nmax+1-k)=X1pm(k);X1na(nmax+1-k)=-X1pa(k); X2nm(nmax+1-k)=X2pm(k);X2na(nmax+1-k)=-X2pa(k); end X1m=[X1nm,X1,X1pm];X2m=[X2nm,X2,X2pm];X1a=[X1na,X1a,X1pa];X2a=[X2na,X2a,X2pa]; n=-nmax:nmax;%range of Harmonics subplot(3,2,1),stem(n,x1m,'filled'),grid,ylabel('amplitude'),title('line spectrum for question-1') subplot(3,2,2),stem(n,x2m,'filled'),grid,ylabel('amplitude'),title('line spectrum for question-2') subplot(3,2,3),stem(n,x1a,'filled'),grid,ylabel('phase angle (degrees)'),xlabel('harmonic number (n)') subplot(3,2,4),stem(n,x2a,'filled'),grid,ylabel('phase angle (degrees)'),xlabel('harmonic number (n)') subplot(3,2,5),plot(t,x1),grid,ylabel('amplitude'),xlabel('time (sec)'), title(['waveform for question-1, frequency =' num2str(ff),'hz']) subplot(3,2,6),plot(t,x2),grid,ylabel('amplitude'),xlabel('time (sec)'), title('waveform for question-2') 5