Homework+_2_+EENG2620+%281%29

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Jan 9, 2024

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UNT FALL 2021 R. Aouf EENG 2620 Homework #2 Due 2/25/2022 at 11:59 PM CT Name: Note: Please type. No handwork. No scanning, No pictures Type you answers in the “red” zone only, save the file then submit through Canvas under Homework 2 assignment Note: some MATLAB answers may be shown under a different form from your work. Problem 1. (Fourier Transform and Inverse Fourier Transform) Textbook 4.21. (a), (b), and Textbook 4.22. (b), (d). Solve 4.21 (a) three different ways: i. by hand - Follow the steps below by using this equation: - For simplicity, pick α=1, ω=1 x(t)= -Express the given expression above in exponential form: -Take this integral by hand on paper (Note: the integral limits, 0 to ∞, set by u(t)) - Type your answer here from your handwork (no need to show your work): X ( jw ) = ¿ ii. by MATLAB - RUN MATLAB code below to verify your answer. %Homework 2, Problem 1 4.21(a) clc; clear all ; close all ; syms t W ; %symbolic variables t:time, W:frequency u(t)=heaviside(t); %unit step %4.21 PART a x= (exp(-t)*cos(t)) *u(t); % expression for x(t) 4.21 part a X=fourier(x); %Fourier transform disp( '4.21 Part a, X(jw)=' ) pretty(X) Copy/paste result from the Command window: iii. by Table - As an alternative, what entry in the Fourier Pairs table below would you use to solve 4.21 (a)? ENTRY #____?

UNT FALL 2021 R. Aouf Fourier Transform Pairs Table

UNT FALL 2021 R. Aouf Solve 4.21 (b) three different ways:

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UNT FALL 2021 R. Aouf i. Follow the steps below to solve 4.21 (b) using this equation: - Write the given signal as x(t)= -Let -Take this integral on paper (Note: the integral limits, 0 to ∞, set by |t| ) - Type your answer here from your handwork (no need to show your work): X 1 ( jw ) = ¿ -Let -So - Type your answer here: X 2 ( jw ) = ¿ Type your answer here using linearity for X(jw)=X1(jw)+X2(jw): X ( jw ) = ¿ ii. by MATLAB RUN MATLAB code below to verify your answer. %Homework 2, Problem 1, 4.21 Part b clc; clear all ; close all ; syms t W ; %symbolic variables t:time, W:frequency %4.21 PART b x= exp(-3*abs(t))*sin(2*t) ; %given expression for x(t) 4.21 part b X=collect(fourier(x)); %Fourier transform disp( '4.21 Part b, X(jw)=' ) pretty(X) Copy/paste result from the Command window: iii. by Table - As an alternative, what entry in the Fourier Pairs table above would you use to solve 4.21 (b)?

UNT FALL 2021 R. Aouf ENTRY #____? Solve 4.22 (b) three different ways: i. by hand - Follow the steps below to using this equation: -Express the given expression in exponential form: -No need to take the integral, instead apply the Fourier transform of the shifted impulse. - Type your answer here from your handwork (no need to show your work): x ( t ) = ¿ ii. by MATLAB - RUN MATLAB code below to verify your answer. %Homework 2, Problem 1 4.22 b clc; clear all ; close all ; syms t W ; %symbolic variables t:time, W:frequency %4.22 PART b X= cos(4*W+pi/3) ; %<-enter expression for X(jw) 4.22 part b x=vpa(ifourier(X,t),2); %inverse Fourier Transform disp( '4.22 Part b, x(t)=' ) pretty(x) copy/paste result from the Command window: iii. by Table - As an alternative, what entry in the Fourier Pairs table above would you use to solve 4.22 (b)? ENTRY #____?

UNT FALL 2021 R. Aouf Solve 4.22 (d) three different ways: i. by hand - Using the impulse shifting property below for ω 0 = ± 1and ω 0 = ± 2π. -Express the given expression in exponential form: -Apply the Fourier transform of the shifted impulse to the given function below. __?no need to show work here -Simplify the answer using sine and cosine (no need to show work here): - Type your answer here from your handwork with sine and cosine form: x ( t ) = ¿ ii. by MATLAB - RUN MATLAB code below to verify your answer. %Homework 2, Problem 1 4.22(d) clc; clear all ; close all ; syms t W ; %symbolic variables t:time, W:frequency %4.22 PART d X=2*(dirac(W-1)-dirac(W+1))+3*(dirac(W-2*pi)+dirac(W+2*pi)) ; %<- enter expression for X(jw) 4.22 part d x=ifourier(X); %inverse Fourier Transform x=rewrite(x, 'sincos' ); disp( '4.22 Part d, x(t)=' ) pretty(x) copy/paste result from the Command window: iii. by Table - As an alternative, what 2 entries in the Fourier Pairs table above would you use to solve 4.22 (d)? ENTRY #____? and ENTRY#____?

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UNT FALL 2021 R. Aouf Problem 2. (Properties of Fourier Transforms) Textbook 4.23. (a), (c), and Textbook 4.25. (b), (c). Solve 4.23 (a) two different ways: i. by hand Follow the steps below to solve 4.23 (a) using this equation: -Find Xo(jw) for 0≤t≤1 -The given signal x(t)=x 0 (t)+x 0 (-t); So, using inversion, find X 0 (-jw)= (no need to show work here): -The Fourier transform of x(t) is: X(jw)= X 0 (jw)+X 0 (-jw)= (no need to show work here): -Simplify the answer using sine and cosine (no need to show work here): - Type your answer here from your handwork with sine and cosine form: X ( jw ) = ¿ ii. by MATLAB %Homework 2, Problem 2 4.23 Part a clc; clear all ; close all ; syms t W ; %symbolic variables t:time, W:frequency xo(t)=exp(-t)*rectangularPulse(t-1/2); %Complete x0(t) for 0≤t≤1 only Xo(W)=fourier(xo); %get Fourier transform Xo(W) of xo(t) %4.23 PART a %x(t)=xo(t)+xo(-t); %Using linearity X= Xo(W)+ Xo(-W); %Complete X= Xo(w)+ Xo(-w) X=rewrite(X, 'sincos' ); disp( '4.23 Part a, X=' ) pretty(X) copy/paste result from the Command window:

UNT FALL 2021 R. Aouf Solve 4.23 (c) two different ways: i. by hand Follow the steps below to solve 4.23 (c) using this equation: -Find Xo(jw) for 0≤t≤1 -The given signal x(t)=x 0 (t)+x 0 (t+1); So, using time shift, find e jw X 0 (jw)= (no need to show work here): -The Fourier transform of x(t) is: X(jw)= X 0 (jw)+ e jw X 0 (jw)= (no need to show work here): -Simplify the answer using sine and cosine (no need to show work here): - Type your answer here from your handwork with sine and cosine form: X ( jw ) = ¿ ii. by MATLAB %Homework 2, Problem 2, 4.23(c) clc; clear all ; close all ; syms t W ; %symbolic variables t:time, W:frequency xo(t)=exp(-t)*rectangularPulse(t-1/2); %Complete xo(t) for 0≤t≤1 Xo(W)=fourier(xo); %get Fourier transform Xo(W) of xo(t) %4.23 PART c %x(t)=xo(t)+xo(t+1); %Using linearity and time shifting X= Xo(W)+ exp(j*W)*Xo(W); %Complete X= Xo(w)+ ejw*Xo(w) disp( '4.23 Part c, X=' ) pretty(X) copy/paste result from the Command window:

UNT FALL 2021 R. Aouf Solve 4.25 (b) two different ways: i. by hand - Follow the steps below to using this equation: -For ω=0, type answer below: ----------? ii. by MATLAB %Homework 2, Problem 2, 4.25(b) clc; clear all ; close all ; syms t W ; %symbolic variables t:time, W:frequency x(t)=1+piecewise(t<-1, __ ,-1<=t<=0, __ , 0<t<=1, __ ,1<t<=2, __ ,2<t<=3, 1,t>3, __ ); % <-complete blanks in red first before running code. %PART b X= int(x,-1,3); disp( '4.25 Part b, X=)' ) Show completed MATLAB code above + copy/paste result from the Command window: Solve 4.25 (c) two different ways: -Substitute x(0) then type answer below: Answer:________________? ii. by MATLAB (no need to use MATLAB for this one really, but just for practice) %Homework 2, Problem 2 clc; clear all ; close all ; syms t W ; %symbolic variables t:time, W:frequency x(t)=1+piecewise(t<-1, 0,-1<=t<=0, 1, 0<t<=1, -t+1,1<t<=2, t- 1,2<t<=3, 1,t>3,0); %PART c disp( '4.25 Part c, Int(X(jω))' ) 2*pi*x(0)

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UNT FALL 2021 R. Aouf copy/paste result from the Command window: Problem 3. (Outputs of LTI systems for Complex Exponential) Textbook 4.31. Solve 4.31 (a) two different ways: i. by hand System h 1 (t) -Use the Fourier Transform Table above to convert cos(t) X(jω) = (no need to show work here) -Use the Fourier Transform Table above to convert the unit step u(t) H 1 (jω) = (no need to show work here) - Determine the output Y(jω) = X(jω)H 1 (jω) Y(jω) = (no need to show work here) -Multiplying in the frequency domain is tricky (Recall multiplying two polar numbers). So, you should end up with: Y(jω)= jπ[(δ(ω+1)/-|ω|+δ(ω-1)/-|ω|] since δ(ω)δ(ω ±1)=0. pick -|ω|=-(-ω)=ω for ω<0 and -|ω|=-(ω)=-ω for ω>0, then over a period ω=, yielding: Y(jω)= jπ[(δ(ω+1)-δ(ω-1)] Finally, use the Table to find the inverse Fourier transform y(t) y(t) = _______________? (show answer here) Repeat for a different system h 2 (t) -Use the Fourier Transform Table above to convert the unit step u(t) (no need to show work here) - Determine the output Y(jω) = X(jω)H 2 (jω). This can be simplified to: Y(jω)= jπ[(δ(ω+1)-δ(ω-1)] -U se the Table to find the inverse Fourier transform y(t) y(t) = _______________? (show answer here) Repeat for a different system h 3 (t) -Use the Fourier Transform Table above to convert the unit step u(t) (no need to show work here) - Determine the output Y(jω) = X(jω)H 3 (jω). This can be simplified to:

UNT FALL 2021 R. Aouf Y(jω)= jπ[(δ(ω+1)-δ(ω-1)] -U se the Table to find the inverse Fourier transform y(t) y(t) = _______________? (show answer here) ii. by MATLAB %Homework 2, Problem 3 Part (a) %Homework 2, Problem 3 clc; clear all ; close all ; syms t W ; %symbolic variables t:time, W:frequency x(t)=cos(t); u(t)=heaviside(t); X(W)=fourier(x) h1(t)=u(t); H1(W)=fourier(h1) Y(W)=X(W)*H1(W) y(t)=ifourier(Y,t); disp( '4.31 Part a,i' ) y=rewrite(y, 'sincos' ) d(t)=dirac(t); h2(t)=-2*d(t)+5*exp(-2*t)*u(t); H2(W)=fourier(h2); Y(W)=simplify(X(W)*H2(W)) y(t)=ifourier(Y,t); disp( '4.31 Part a,ii' ) y=rewrite(y(t), 'sincos' ) h3(t)=2*t*exp(-t)*u(t); H3(W)=fourier(h3); Y(W)=X(W)*H3(W); y(t)=ifourier(Y,t); disp( '4.31 Part a, iii' ) y=rewrite(y, 'sincos' ) h4(t)=h1(t)+h2(t)-h3(t); %combining h1 and h2 and h3 to get a suitable LTI system H4(W)=fourier(h4); Y(W)=X(W)*H4(W); y(t)=ifourier(Y,t); disp( '4.31 Part b' ) y=rewrite(y, 'sincos' ) copy/paste result from the Command window: Solve 4.31 (b) MATLAB only below: %Homework 2, Problem 3 clc; clear all ; close all ; syms t W ; %symbolic variables t:time, W:frequency x(t)=cos(t);

UNT FALL 2021 R. Aouf X(W)=fourier(x); u(t)=heaviside(t); h1(t)=u(t); d(t)=Dirac(t); h2(t)=-2*dirac(t)+5*exp(-2*t)*u(t); h3(t)=2*t*exp(-t)*u(t); h4(t)=h1(t)+h2(t)-h3(t); %combining h1 and h2 and h3 to get a suitable LTI system H4(W)=fourier(h4); Y(W)=X(W)*H4(W); y(t)=ifourier(Y,t); disp( '4.31 Part b' ) y=rewrite(y, 'sincos' ) copy/paste result from the Command window:

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