Exercises in Signals

1 Exercises in Signals , Systems, and TransformsIvan W. SelesnickLast edit: October 27, 2014 Contents1 discrete - time Signals and System More Z Inverse Di erence Complex Frequency Summary Simple System More Continuous- time Signals and System Laplace Di erential Complex Frequency Simple System Fourier Fourier Fourier The Sampling Theorem15011 discrete - time Signals and an accurate sketch of each of the discrete - time Signals (a)x(n)=u(n+ 3) + (n 1)(b)x(n)= (n+ 3) + (n 1)(c)x(n)=2n (n 4)(d)x(n)=2n u( n 2)(e)x(n)=( 1)nu( n 4).(f)x(n)=2 (n+ 4) (n 2) +u(n 3)(g)x(n)=1Xk=04 (n 3k 1)(h)x(n)=1Xk= 1( 1)k (n 3k) a sketch of each of the following Signals (a)x(n)=1Xk= 1( )|k| (n k)(b)x(n) = cos( n)u(n)(c)x(n)=u(n) 2u(n 4) +u(n 8) (n),x1(n),x2(n), andx3(n)wherex(n)=u(n+ 4) u(n),x1(n)=x(n 3),x2(n)=x(5 n),x3(n)=nXk= 1x(k) (n) andx1(n)wherex(n)=( )nu(n),x1(n)=nXk= 1x(k) (n) andx1(n)wherex(n)=n[ (n 5) + (n 3)],x1(n)=nXk= 1x(k) a sketch of each of the following Signals (a)f(n)=1Xk=0( )k (n 3k)(b)g(n)=1Xk= 1( )|k| (n 3k)(c)x(n) = cos( n)u(n)(d)x(n) = cos( n)u(n) discrete - time Signals in plot the discrete - time impulse function:n = -10:10;f = (n == 0);stem(n,f)Usestemto plot the discrete - time step function:f = (n >= 0);stem(n,f)Make stem plots of the following Signals .

2 Decide for yourself what the range ofnshould (n)=u(n) u(n 4)(1)g(n)=r(n) 2r(n 5) +r(n 10) wherer(n):=nu(n)(2)x(n)= (n) 2 (n 4)(3)y(n)=( )n(u(n) u(n 20))(4)v(n) = cos( n)u(n)(5) System discrete - time system may be classified as follows: memoryless/with memory causal/noncausal linear/nonlinear time -invariant/ time -varying BIBO stable/unstableClassify each of the following discrete -times systems.(a)y(n) = cos(x(n)).(b)y(n)=2n2x(n)+nx(n+ 1).(c)y(n) = max{x(n),x(n+ 1)}Note: the notation max{a,b}means for example; max{4,6}= 6.(d)y(n)= x(n)whennis evenx(n 1) whennis odd(e)y(n)=x(n)+2x(n 1) 3x(n 2).(f)y(n)=1Xk=0(1/2)kx(n k).That is,y(n)=x(n)+(1/2)x(n 1) + (1/4)x(n 2) + (g)y(n)=x(2n) discrete - time system is described by the following ruley(n)= (2n)+ (2n 1)wherexis the input signal, andythe output signal.(a)Sketch the output signal,y(n), produced by the 4-point input signal,x(n) illustrated (n)5(b)Sketch the output signal,y(n), produced by the 4-point input signal,x(n) illustrated (n)(c)Classify the system discrete - time system is described by the following ruley(n)=(x(n),whennis an even integer x(n),whennis an odd integerwherexis the input signal, andythe output signal.)

3 (a)Sketch the output signal,y(n), produced by the 5-point input signal,x(n) illustrated (n)(b)Classify the system discrete - time system is described by the following ruley(n)=( 1)nx(n)+2x(n 1)wherexis the input signal, andythe output signal.(a)Accurately sketch the output signal,y(n), produced by the input signalx(n) illustrated (n)(b)Classify the system the output of an LTI impulse response of a discrete - time LTI system ish(n)=2 (n)+3 (n 1) + (n 2).Find and sketch the output of this system when the input is the signalx(n)= (n)+3 (n 1) + 2 (n 2). a discrete - time LTI system described by the ruley(n)=x(n 5) +12x(n 7).What is the impulse responseh(n) of this system? impulse response of a discrete - time LTI system ish(n)= (n)+2 (n 1) + (n 2).Sketch the output of this system when the input isx(n)=1Xk=0 (n 4k). impulse response of a discrete - time LTI system ish(n)=2 (n) (n 4).Find and sketch the output of this system when the input is the step functionx(n)=u(n). the discrete - time LTI system with impulse responseh(n)=nu(n).

4 (a)Find and sketch the outputy(n)whentheinputx(n)isx(n)= (n) 2 (n 5) + (n 10).(b)Classify the system as BIBO the output of an LTI impulse responseh(n) of an LTI system is given byh(n)= 23 nu(n).Find and sketch the outputy(n) when the input is given by(a)x(n)= (n)(b)x(n)= (n 2) the LTI system with impulse responseh(t) = cos ( t)u(n),find and sketch the step responses(t) and classify the system as BIBO the LTI system with impulse responseh(n)= (n 1).(a)Find and sketch the outputy(n)whentheinputx(n) is the impulse train with period 6,x(n)=1Xk= 1 (n 6k).(b)Classify the system as BIBO LTI system is described by the following equationy(n)=1Xk=0 13 kx(n k).Sketch the impulse responseh(n) of this the parallel combination of two LTI (n)-h1(n)x(n)?6l+-y(n)You are told thath1(n)=u(n) 2u(n 1) +u(n 2).You observe that the step response of the total system iss(n)=2r(n) 3r(n 1) +r(n 2)wherer(n)=nu(n). Find and sketchh2(n). impulse response of a discrete - time LTI system is given byh(n)= 1ifnis a positive prime number0 otherwise (a)Is the system causal?

5 (b)Is the system BIBO stable? observe an unknown LTI system and notice thatu(n) u(n 2)-S- (n 1) 14 (n 4)Sketch the step responses(n). The step response is the system output when the input is the step functionu(n). an LTI system it is known that input signalx(n)= (n)+3 (n 1)produces the following output signal:y(n)= 12 nu(n).What is the output signal when the following input signal is applied to the system?x2(n)=2 (n 2) + 6 (n 3) More and sketch the convolutionx(n)=(f g)(n)where(a)f(n)=2 (n+ 10) + 2 (n 10)g(n)=3 (n+ 5) + 3 (n 5)10(b)f(n)= (n 4) (n 1)g(n)=2 (n 4) (n 1)(c)f(n)= (n+ 2) (n+ 1) (n)g(n)= (n)+ (n+ 1) + (n+ 2)(d)f(n)=4g(n)= (n)+2 (n 1) + (n 2).(e)f(n)= (n)+ (n 1) + 2 (n 2)g(n)= (n 2) (n 3).(f)f(n)=( 1)ng(n)= (n)+ (n 1). impulse response of a discrete - time LTI system ish(n)=u(n) u(n 5).Sketch the output of this system when the input isx(n)=1Xk=0 (n 5k). signalfis given byf(n) = cos 2n .The signalgis (n)Sketch the signal,x(n), obtained by convolvingf(n) andg(n),x(n)=(f g)(n).

6 Signalsfandgare given byf(n)=2,g(n)= 12 nu(n).Sketch the signal,x(n), obtained by convolvingf(n) andg(n),x(n)=(f g)(n). signalsf(n) andg(n) are shown: 4 3 2 101234012312321f(n)n 4 3 2 101234 2 10123 12 1g(n)nSketch the convolutionx(n)=f(n) g(n). the convolution of the discrete - time signalx(n)2321-2-10123456nx(n)with each of the following Signals .(a)f(n)=2 (n) (n 1)(b)f(n)=u(n)(c)f(n)= (d)f(n)=1Xk= 1 (n 5k) signalsfandgare defined as:f(n)=anu(n)g(n)=f( n)=a nu( n)Find the convolution:x(n)=(f g)(n)Plotf,g, andxwhena= You may use a computer for average filterhas the impulse responseh(n)= 1/N0 n N 10otherwiseUse the Matlabconvcommand to computey(n)=h(n) h(n)forN=5,10,20, and in each case make a stem plot ofh(n) andy(n).What is the general expression fory(n)? convolution of two finite length Signals can be written as a matrix vector product. Look at the documentationfor the Matlabconvmtxcommand and the following Matlab code that shows the convolution of two Signals by(1) a matrix vector product and (2) theconvcommand.

7 Describe the form of the convolution matrix and whyit works.>> x = [1 4 2 5]; h = [1 3 -1 2];>> convmtx(h ,4)*x ans =1713921-110>> conv(h,x) ans = convolutiony=h g,wherehandgare finite-length Signals , can be represented as a matrix-vector product,y=HgwhereHis a convolution matrix. In MATLAB, a convolution matrixHcan be obtained with thecommandconvmtx(h(:), K).Given finite-length sequenceshandx,defineH = convmtx(h(:), M)whereMis such that the matrix-vector productHTxis defined, whereHTdenotes the transpose terms of convolution, what does the matrix-vector productHTxrepresent?Write a MATLAB function to computeHTxusing the functionconvand without creating the to your function should be vectors, (n)=u(n) u(n 5)g(n)=r(n) 2r(n 5) +r(n 10).wherer(n):=nu(n).In MATLAB, use theconvfunction to compute the following convolutions. Use thestemfunction to plot theresults. Be aware about the lengths of the Signals . Make sure the horizontal axes in your plots are correct.(a)f(n) f(n)(b)f(n) f(n) f(n)(c)f(n) g(n)(d)g(n) (n)(e)g(n) g(n)Comment on your observations: Do you see any relationship betweenf(n) f(n) andg(n) ?

8 Comparef(n)withf(n) f(n) and withf(n) f(n) f(n). What happens as you repeatedly convolve this signal with itself?Use the commandstitle,xlabel,ylabelto label the axes of your of non-causal Signals in MATLABNote that both of these Signals start to the left ofn= (n)=3 (n+ 2) (n 1) + 2 (n 3)(6)g(n)=u(n+ 4) u(n 3)(7)First, plot the signalsf,g, andf gby hand, without using MATLAB. Note the start and end , use MATLAB to make plots off,g, andf g. Be aware that theconvfunction increases the length turn in: The plots off(n),g(n),x(n), and your Matlab commands to create the data byN-point the data the course website. Load the data into Matlab using the see your variables. One of the variables will beDataEOG. For convenience, renameit toxby typing:x = DataEOG;This signal comes from measuring electrical Signals from the brain of a a stem plot of the signalx(n). You will see it doesn t look good because there are so many points. Makea plot ofx(n)usingtheplotcommand. As you can see, for long Signals we get a better plot using theplotcommand.

9 Although discrete - time Signals are most appropriately displayed with thestemcommand, forlongdiscrete- time Signals (like this one) we use the plot command for better a simple impulse response for an LTI system:h = ones(1,11)/11;Compute the convolution ofhandx:y = conv(x, h);Make a MATLAB plot of the outputy.(a)How does convolution changex? (Comparexandy.)(b)How is the length ofyrelated to the length ofxandh?(c)Plotxandyon the same graph. What problem do you see? Can you getyto line up withx?(d)Use the following commands:y2 = y;y2(1:5) = [];y2(end-4:end) = [];What is the e ect of these commands? What is the length ofy2? Plotxandy2on the same graph. Whatdo you notice now?(e)Repeat the problem, but use a di erent impulse response:h = ones(1,31)/31;What should the parameters in part (d) be now?(f)Repeat the problem, but useh = ones(1,67)/67;What should the parameters in part (d) be now?Comment on your turn in: The plots, your Matlab commands to create the Signals and plots, and Z of the discrete - time signalx(n)isX(z)= 3z2+2z 3 Accurately sketch the signalx(n).

10 The discrete - time signalx(n) asx(n)= (n+ 2) + (n)+ (n 3) (n 5)(a)Sketchx(n).(b)Write theZ-transformX(z).(c)DefineG(z)=z 2X(z). Sketchg(n). signalg(n) (n) be the length-5 signalx(n)={1,2,3,2,1}wherex(0) is underlined. Sketch the signal corresponding to each of the followingZ-transforms.(a)X(2z)(b)X(z2)(c )X(z)+X( z)(d)X(1/z) the discrete - time signalx(n)withtheZ-transformX(z)=(1+2z)( 1+3z 1)(1 z 1). three discrete - time Signals :a(n)=u(n) u(n 4)b(n)= (n)+2 (n 3)c(n)= (n) (n 1)Define three new Z-transforms:D(z)=A( z),E(z)=A(1/z),F(z)=A( 1/z)(a)Sketcha(n),b(n),c(n)(b)Write theZ-transformsA(z),B(z),C(z)(c)Write theZ-transformsD(z),E(z),F(z)(d)Sketchd( n),e(n),f(n) the Z-transformX(z) of the signalx(n)=4 13 nu(n) 23 nu(n). signalxis defined asx(n)=a|n|FindX(z) and the ROC. Consider separately the cases:|a|<1 and|a| the right-sided signalx(n) from the Z-transformX(z)=2z+1z2 56z+ the LTI system with impulse responseh(n)=3 23 nu(n)Find the outputy(n)whentheinputx(n)isx(n)= 12 nu(n). discrete - time LTI system has impulse responseh(n)= 2 15 nu(n)Find the output signal produced by the system when the input signal isx(n)=3 12 nu(n) the transfer functions of two discrete - time LTI systems,H1(z)=1+2z 1+z 2,H2(z)=1+z 1+z 2.